A Method for the Solution of Certain Non-Linear Problems of Combined Seagoing Main Engine Performance and Fixed-Pitch Propeller Hydrodynamics with Imperative Assignment Statements and Streamlined Computational Sequences

Abstract

Seagoing marine propulsion analysis in terms of main engine performance and fixed-pitch propeller hydrodynamics is an engineering problem that has not been exactly defined to date. This study utilizes an original and comprehensive mathematical approach—involving the approximate representation of one function by another—to define this problem in mathematical terms and solve it. This is achieved by imperatively applying an original and sophisticated hybrid combination of an existing, formidable and ingenious, mathematical methodology with different original comprehensive functional systems. These original functional systems approximately represent the operations of vessels under seagoing conditions, including the thermo-fluid and frictional processes of vessels’ main engines in terms of fuel oil consumption, as well as the hydrodynamic performance of the respective vessels in terms of the shaft propulsion power and the rotational speed of the fixed-pitch propellers driven by these engines. Based on the least-squares criterion, this original and sophisticated hybrid combination systematically attains remarkably close approximate representations under seagoing conditions. Apart from this novel exact definition in mathematical terms and the significance of the above original representations, this combination is also applicable for the approximation of the baselines demarcating the standard engineering context representing the ideal reference (sea trials) conditions, from the seagoing conditions.

1. Introduction

This article introduces an original, hybrid, tripartite, methodology based on the multi-dimensional, non-linear, comprehensive and imperative, approximate representation of one function by another [1], applied to facilitate the combined analysis of marine diesel main engine performance and marine fixed-pitch (screw) propulsion hydrodynamics under seagoing conditions. The novelty, contribution, significance and merit of this study go well beyond addressing the lack of standard, generally applicable and widely known solutions for such an analysis, as the seagoing marine propulsion analysis in the above context is an engineering problem that has not been exactly defined in mathematical terms to date.

In this regard, the essential and crystal-clear objectives of this original and independent research were to develop an unambiguous definition of the a/m composite problem that is outlined in as-exact-as-possible mathematical terms in Section 1.1, and to determine the solution to it. Such a solution to the a/m problem may be in the form, or in lieu, of an approximate representation of one function by another in the context of Reference [1], and/or in the form, or in lieu, of “nomographs” or “any other means”, in the standard engineering context discussed later in this section and in Appendix Z of this article. In fact, “any other means”, effectively stands in lieu of, “any other” possible, effectively equivalent, formulations applicable for the as-exact-as-possible solution to this problem.

Apart from the formal expression of this article’s attained objectives in the terms applied above, this study actually fills a critical theoretical gap by addressing the long-standing challenge of fixed-pitch propulsion analysis under seagoing conditions. In this regard, and besides the strong originality inherent in the applied hybrid tripartite methodology, a novel mathematical framework with unique academic value was developed with the following key contributions: novel formulations based on least-squares approximation; the integration of real-world environmental conditions; a dynamic recalibration methodology; actual seagoing marine propulsion modeling; and an analysis in terms of the joint thermo-fluid/frictional/hydrodynamics solutions to the respective problems inherent in the combined seagoing main engines and fixed-pitch propellers performance.

The mathematical methodology for resolving over-determined non-linear mathematical problems [1], incorporated in this analysis, was publicly presented for the first time in 1943 [1], before its original publication in 1944 [1,2], and was also republished in 1963 [3] and 1971 [4] in the form of a numerical algorithm. In such problems, the number, n [1], of instances of the equations (conditions) to be met is higher than the number of parameters, α, β, γ, … [1] (of the mathematical formulations of the above conditions pursuant to Equation (1) of Reference [1]) which before the application of this mathematical methodology [1] are of unknown values. More specifically, this methodology [1] was developed and first applied for resolving problems such as those referred to in the opening paragraph and in the closing sentence of Reference [1], as these problems are to be further considered in conjunction with the affiliation at the time of Reference [1]’s author with the Frankford Arsenal [1]. In any case, the development and first successful application of this mathematical methodology were completed earlier than 26 November 1943, when Reference [1] was read and publicly presented for the first time before the Annual Meeting of the American Mathematical Society in Chicago, Illinois [1].

Regardless of the mathematical context, depth and ingenuity of Reference [1], in its opening paragraph (page 164) and just before the author of it explained his incentive, purpose and drive, he observed publicly and defined in very precautious and inconspicuous (almost cryptographic) language, a problem in terms of applied mathematics and numerical analysis of “… failure … encountered rather frequently in connection with certain engineering applications involving the approximate representation of one function by another. The purpose of this article is to show how the problem may be solved …” [1], which he accomplished right after this opening paragraph, publicly, openly, freely, enthusiastically, and youthfully, by means of full, un-truncated, and exact terms of applied mathematics and numerical analysis. The wording above is not quoted trivially; rather, one reason for quoting this excerpt is because it underlines that the subject of Reference [1] is, since the birth of it, a mathematical method of approximate representation of one function by another [1] originally intended for engineering applications [1].

A second, among yet others, reason for the above quoting is to underline that the most probable and obvious purpose for applying such an approximate representation of one function by another [1], is that the function to be approximated, h(x, y, z,…) [1], would not be mathematically formulated in its entirety, in the first place. Instead, pursuant to Equation (1) of Reference [1], likely only certain, n [1], values of this function to be approximated, h(x_i, y_i, z_i,…) [1], at certain points, x_i, y_i, z_i,… [1], in the multi-dimensional domain of the sequence of variables, x, y, z,… [1], of this function to be approximated [1], would be evaluated [3,4,5,6].

However, this most probable and obvious reason, obscures certain other aspects inherent in this mathematical method [1]. In fact, ever since the republication of this mathematical method [1] in 1963 [3] and 1971 [4] in the form of a numerical algorithm, this reason is the root cause that this method [1] has been effectively downgraded, misconceived and merely referenced to, as a numerical algorithm, and occasionally as only a fitting one in particular.

These obscured “… other aspects inherent in this mathematical method [1].” comprise some of the remaining reasons for the above quoting of the opening paragraph (page 164) of Reference [1]; these aspects include cases where the, n [1], values of the function to be approximated, h(x_i, y_i, z_i,…) [1], at certain points, x_i, y_i, z_i,… [1], effectively represent an almost continuous distribution of the function to be approximated [1]. Such almost continuous distributions may or may not be the analogue or digital outputs of measuring, monitoring, or identification devices or systems, in the form of effectively continuous curves [7]. This is particularly discussed within the context of this article.

Furthermore, the a/m obscured other aspects, include cases where the function to be approximated, h(x, y, z,…) [1], is not evaluated at certain only, continuously distributed or not, discrete points, x_i, y_i, z_i,… [1], in the multi-dimensional domain of the sequence of variables, x, y, z,… [1]; in such cases, the function to be approximated, h(x, y, z,…) [1], is instead mathematically formulated and consequently evaluated, in the entirety of one or more specific partitions of the multi-dimensional domain of sequence of variables, x, y, z,… [1]. In these cases, the mathematically formulated function to be approximated, h(x, y, z,…) [1], may represent design, operational, and/or other engineering criteria, to be complied with in terms of the, optimized, respectively applicable design parameters and/or characteristics. Examples include the design of passenger jet aircrafts [2]; the feasibility assessment of geothermal or oil wells [8]; the optimization of thermodynamic cycles [7]; or other cases that are discussed within the context of this article.

The solution attained by this mathematical method [1] is not an exact one as this would not be possible under the above circumstances; for an exact solution, each residual value, f_i (α, β, γ, …) [1], of each of the instances, i = 1, 2, …, n [1], of the equations (conditions) to be met, would equal zero. Instead, the solution attained [1] represents a certain point in the multi-dimensional domain of all unknown parameters, α, β, γ, … [1]. At this point, all the partial derivatives of the sum, s(α, β, γ, …) [1], of squares of the above residual values, f_i (α, β, γ, …) [1], summed for all above instances, i = 1, 2, …, n [1], against all above unknown parameters, α, β, γ, … [1], equal zero pursuant to Equation Set (5) of Reference [1]. Consequently, any incremental change in the value of any one of the unknown parameters, α, β, γ, … [1], from the attained solution (set of values of the unknown parameters, α, β, γ, … [1]), would result in increasing the sum, s(α, β, γ, …) [1], of squares of the residuals, f_i (α, β, γ, …) [1], applicable to the attained solution, which is to be minimized pursuant to the set of Equations (2)–(5) of Reference [1]. To this end, and on the basis of the least-squares criterion, the attained solution is considered to be a best-fit (most probable/least uncertain) “local minimum”.

This “local” attainable solution may prove to rely (effectively, in a stochastic manner) on the set of initial values of the unknown parameters, α, β, γ, … [1], while not all possible solutions are expected to be physically significant for a specific problem and the circumstances thereof. The above dependence is related to how well the physical, technical or other problem is defined in the first place, and on how solid the transformation of the above definition is in terms of the analytical and comprehensive mathematical formulations inherent in the approximating function, H(x_i, y_i, z_i,…; α, β, γ, …) [1], pursuant to Equation (1) of Reference [1].

Judgment is required for making sure that the attained solution is the most probable and least uncertain, best-fit, solution for a particular problem and the circumstances thereof [4,5,6,7,8]. Such judgment is based mainly on knowledge and experience pertaining to solving the same or similar types of problems in the same or a similar or another way; the bounds (threshold values) of the unknown parameters, α, β, γ, … [1], which are fencing physical significance areas; and testing (sensitivity analyses).

To date, the direct use of the above mathematical methodology in marine applications [7,9,10,11,12,13,14] has been mainly focused on maneuverability and artificial neural networks (ANN) for vessel steering and control, and on the respective predictions of vessel responses. This mathematical methodology and approximate representation analysis [1] have also been very effectively applied for at least 40 years to marine diesel engines in the form of analytical, comprehensive and imperative applications of the above [7,14].

Furthermore, during recent years, a trend has become evident regarding the application of the above mathematical methodology for the implementation of artificial neural networks (ANNs)—considered by many as “pure black” or “grey box” approaches—for the fault diagnostics of marine diesel engines under seagoing conditions, as well as to predict the performance of marine diesel engines under laboratory or other shore conditions [13].

However, there are no evident trends, or even remotely similar or relevant trails, in the literature for resolving the combined problem of determining the main engine performance and fixed-pitch propeller hydrodynamics under seagoing conditions. This also holds true for the application of the above mathematical methodology in the form of an analytical, comprehensive, and imperative approximate representation of one function by another [1] for resolving the a/m combined problem as is the case in the present work. Obviously, this lack of relevant or similar references underlines the significance, merit, originality and novelty of the research work behind this article.

In the research presented in this article, the above mathematical methodology was applied to the fixed-pitch (screw) marine propulsion problem and the analytical, comprehensive and imperative approximate representation [1] of the fixed-pitch (screw) ship propulsion in particular, as well as for the dynamic recalibration of existing and new thermo-fluid and frictional analytical, comprehensive and imperative models of diesel engines. This dynamic recalibration was based on research work [7,14,15,16,17,18,19,20,21,22,23,24,25] conducted by the author between 1986 and 1995, and between 2005 and 2022 at the NTUA Internal Combustion Engines Laboratory, as well as independently, and is also based on other relevant literature referenced in this article [26,27,28,29,30,31,32,33,34,35]. In fact, this work includes research that the author conducted in collaboration with NTUA faculty and other researchers, and also other relevant research of the author that has not been reported to date.

As a member of different research teams, the author of the present article published research results between 1992 and 1993 that were attained by applying the above analytical, comprehensive and imperative approximate representation [1] on marine diesel engines. In fact, these results include solutions of in-cylinder heat transfer problems on the basis of the availability of the indicated pressure diagrams [7,14,21,22]. Such solutions are paramount for quantifying the fuel injection and combustion characteristics and scale, after the end of the compression phase of the indicator diagram [7,14,21,22,23]; for implementing analyses on the first and second laws of thermodynamics [7,14,15,19,21,22,23]; and for developing and applying original CFD models for the analysis of the gas dynamics of marine diesel engines [18,20,21,22,23].

In fact, the set of Equations (1)–(5) of Reference [1] is effectively identical with the set of Equations (44)–(48) of Reference [7], as well as the set of Equations (3.1.8)–(3.1.17) of Reference [22]. Consequently, the applicable, verbal, analytical, mathematical, numerical and computational, correlations and steps presented in detail after the paragraph (three lines and two sentences long) just before Equation (1) amid page 164 of Reference [1], should be further considered as “not falling within the scope of the present article”. The same will also apply to all similar content of References [3,4,5,6,7,8,9,10,11,12,13,14,22].

A detailed account of the hardware and software used for the analytical, comprehensive and imperative approximate representation of one function by another [1] introduced in this work is available in the third paragraph of Appendix H of this article. Furthermore, the a/m paragraph (three lines and two sentences long) just before Equation (1) amid page 164 of Reference [1], effectively lays out the context for an imperative assignment statement ready format. This context is clearly defined in the first two paragraphs of Appendix H, whereas the above should be further considered in conjunction with the fact that the author of Reference [1] “… was a major contributor for the development of the computer during the inception stage of development; …” [2]. The factual nature of the above quoting, as well as of all the other content of Reference [2], ensues from the fact that Reference [2] is a certified (regulated) attestation issued by a contemporary (1973), governmental, regional competent authority.

Such an extraordinary attestation [2] and the circumstances thereof, can be attributed to the affiliation of Reference [1]’s author with the Frankford Arsenal [1] at the time that he publicly presented and published Reference [1] (1943–1944), and to the (concealed [1]) exact nature and particular significance of those “… certain engineering applications involving … types of problems which are of much greater complexity …” [1] referred to in the opening paragraph and in the closing sentence of Reference [1]. Pursuant to the above, the following additional facts ensue from Reference [2], among others: that the author of Reference [1] “… was a mathematician of illustrious standing …” and “… a professor of such a noted fame …”; that he “…is well noted for the invention of the method of Damped Least Squares; …” (an alias for part of the contributions of Reference [1]) as early as 1943–1944 (definitely earlier than the republication of Reference [1] in 1963 [3] and 1971 [4] in the form of a numerical algorithm); and that he “… was listed in the American Men of Science in 1970 because of his accomplishments in the field of mathematics, …”.

To this end, a demarcation point is reached as soon as a set of equations originally introduced in this article is brought to the imperative assignment statement ready format, meeting the conditions set in the a/m paragraph (three lines and two sentences long) amid page 164 of Reference [1]. This paragraph reads as follows: “Let the function to be approximated be h(x, y, z, …), and let the approximating function be H(x, y, z,…; α, β, γ, …), where α, β, γ, … are the unknown parameters. Then the residuals at the points, (x_i, y_i, z_i,…), i = 1, 2, …, n, are …” (This paragraph is followed immediately by Equation (1) of Reference [1], which calculates the exact values of the residuals, f_i (α, β, γ, …) [1]).

In this regard, any further “downstream” verbal, analytical, mathematical, numerical and computational, detailed correlations and steps beyond this demarcation point, in accordance or not with the References provided herewith, were considered as “not falling within the scope of the present article”. All required background information regarding the application of the subject mathematical methodology [1] in the context of the present article, has been sufficiently provided, on the basis of the above introductory information and the relevant reasoning included or referred to in Section 2.4 and Appendix H, Appendix L, Appendix M, Appendix N, Appendix O, Appendix P, Appendix Q, Appendix R, Appendix S, Appendix T and Appendix U.

The relevant and extensive introductory and background information mentioned above includes References [7,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35] on the comprehensive functional systems approximately representing [1] the thermo-fluid and frictional processes of vessels’ main engines in terms of fuel oil consumption under actual seagoing conditions. These systems comprise existing and new thermo-fluid and frictional analytical, comprehensive, and imperative models of diesel engines. A further detailed relevant analysis and information are also presented in Section 1.1.1, Section 2.2, Section 2.4, Section 3, Section 3.1 and Section 4; in Appendix F, Appendix I, Appendix V and Appendix Z; and in the relevant international standards [36,37,38,39,40,41] and literature [42,43,44].

Section 1, Section 1.1.2, Section 2.1, Section 2.3, Section 2.4, Section 3, Section 3.2, Section 3.3 and Section 4, as well as Appendix A, Appendix B, Appendix C, Appendix D, Appendix E, Appendix F, Appendix G, Appendix I, Appendix J, Appendix K, Appendix Y and Appendix Z, present a detailed relevant analysis and additional information on the comprehensive functional systems approximately representing [1] the hydrodynamic performance of vessels under actual seagoing conditions in terms of the shaft propulsion power and the rotational speed of these vessels’ fixed-pitch propellers. Furthermore, the presented application of the approximate representation of one function by another methodology [1] to fixed-pitch propeller (FPP) hydrodynamics under actual seagoing conditions was based on the following additional considerations, including the existing engineering context relevant to this article. This context is mainly applicable to vessel design, newbuilding, classification and regulatory compliance, and comprises towing tank activities under scale, propellers testing and development, sea trials, and academic, industry standard, and research activities such as CFD analyses. Nevertheless, this engineering context is neither easily transferable nor easily applicable to actual seagoing conditions. A condensed and concise outline of this engineering context [43,44,45,46,47,48,49,50,51,52,53,54,55]—also explaining in some detail the difficulties inherent in the transfer and applicability of it to actual seagoing conditions [56,57,58,59,60,61,62,63,64,65]—is presented in Appendix Z of this article.

As discussed above, seagoing marine propulsion analysis in terms of main engine performance and fixed-pitch propeller hydrodynamics is an engineering problem that has not been exactly defined to date in mathematical terms. In fact, the analytical and other research work reported in this article (both original and independent), has been concentrated on an as-exact-as-possible definition and solution to this engineering problem, which is presented in further detail in Section 1.1. The solution to this problem is particularly important under seagoing conditions (“at sea”); in fact, these conditions are the reason that any attempted solutions of this problem to date have been extremely receptive of essential contributions with regard to their clarity, general applicability, endurance, robustness, physical/technical significance, insight on the basis of fundamental principles, and versatility.

In this regard, reference is also made to Appendix Z in its entirety and point (24) of it in particular. Attention should also be paid to References [36,66,67,68,69,70,71,72,73,74,75,76,77,78] which, directly or indirectly, acknowledge that significant opportunities for improvement exist in terms of the essential contributions mentioned above. In fact, such opportunities are specifically underlined in Reference [36] which is in perfect alignment with the context of this article. The discussion of this problem by Reference [36] (which has not been revised for the last 17 years) is also quoted in points (29) to (35) in Appendix Z.

This discussion should be considered in conjunction with the fact that neither a formulated definition of this problem, nor a solution to it, could be regulated by 2008 in a manner other than through the abstract, vague and generic hints in Sections 6.3.1.3, 6.4.3.4 and 6.4.3.4.2 of Reference [36] (which are also quoted below and in points (31) and (35) of Appendix Z). This highlights the particular importance of this problem under actual seagoing conditions (“at sea”). In fact, Reference [36] tacitly confirms that an unambiguous definition of this problem and a solution to it were not widely known by 2008 (see points (29) to (35) of Appendix Z [36]).

As already discussed in Section 1, the essential objectives of this original and independent research were to develop an unambiguous definition of the composite problem that is outlined in as-exact-as-possible mathematical terms in Section 1.1, and to determine the solution to it. The resulting practical implications for seagoing marine propulsion modeling and analyses in terms of combined seagoing main engine and fixed-pitch propeller performance, include joint thermo-fluid/frictional/hydrodynamics solutions to the respective problems, coupled by enhanced visibility with regard to the energy flow from main engines’ fuel oil consumption to the vessels’ hydrodynamic output; see also in this regard Appendix E, Appendix F and Appendix Z. Such solutions to the a/m problems may be in the form, or stand in lieu, of an approximate representation of one function by another in the context of Reference [1]; in the form, or in lieu, of “nomographs” in the context of Section 6.4.3.4.2 of Reference [36]; and/or in the form of, “any other means” in the context of Sections 6.3.1.3 and 6.4.3.4 of Reference [36] (“any other means” in the above context, means, “any other” possible, effectively equivalent, formulations applicable for the as-exact-as-possible definition of and solution to this problem).

The a/m determination is also directly relevant to, associated with, and cross correlated with different integrated regulatory frameworks (stochastic or not), and other approaches and attempts, applicable for GHG emission control and decarbonisation pursuant to the Reference groups [79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114]. Considering these regulatory frameworks, approaches and attempts from a sustainability standpoint—and particularly the stochastic ones among them [79,81]—one can easily acknowledge the possibilities offered in terms of the a/m combined main engines/fixed-pitch propeller seagoing performance and joint thermo-fluid/frictional/hydrodynamics solutions to the respective problems facilitated by the integration of real-world environmental conditions. At the same time, the application of the subject dynamic recalibration methodology based on least-squares approximation, directly answers the challenges in terms of the stochastic approaches introduced by some sustainability-oriented regulatory frameworks [79,81].

On top of the exact definition in mathematical terms and the solution to the engineering problem discussed above and presented in further detail in Section 1.1, there is at least one more equally essential contribution indicative of the significance, novelty, and originality of this work:

The analytical, comprehensive and imperative approximate representation of one function by another [1], however not only meant in the applied mathematics and numerical analysis terms already discussed in this section. Additionally, the approximate representation of one function by another [1], refers to the fact that the a/m, relevant to this article, engineering context discussed in this section and in Appendix Z, is neither easily transferable, nor easily applicable, to actual seagoing conditions.

Therefore, the approximate representation of one function by another [1] is also related to the closest possible approximate representation [1] of the function to be approximated [1] by the approximating function [1], considering as the latter the overall methodology presented in this article, along with the results of it under actual seagoing conditions. In this case, however, the function to be approximated [1] is the engineering context discussed in this section and in Appendix Z, which is neither easily transferable nor easily applicable to uncontrolled seagoing conditions, unlike the controlled conditions for the propellers’ and towing tanks under scale or sea/full-scale trials.

In fact, this additional essential novelty of this work is drawn from the obscured “… other aspects inherent in this mathematical method [1].” which “… comprise some of the remaining reasons for the above quoting of the opening paragraph (page 164) of Reference [1]; …” earlier in this section. In this case, the function not mathematically formulated in its entirety, to be approximated, h(x, y, z,…) [1], is not evaluated only at certain discrete points, x_i, y_i, z_i,… [1], in the multi-dimensional domain of its sequence of variables, x, y, z,… [1]. Instead, this function to be approximated, h(x, y, z,…) [1], is mathematically formulated and consequently evaluated in the entirety of a specific partition of this multi-dimensional domain of the sequence of variables, x, y, z,… [1].

This specific partition is defined as such, pursuant to points (1) to (16) of Appendix Z. Furthermore, in this case, Reference [1] in its entirety and Equation (1) of it in particular are applicable at points, x_i, y_i, z_i,… [1]. These points are continuously distributed along the multi-dimensional demarcation curves (effectively, surfaces) between this specific partition and the multi-dimensional domain of the sequence of variables, x, y, z,… [1] for the remaining, actual seagoing conditions. These multi-dimensional demarcation curves are referred to and defined in points (15), (16) and (26) of Appendix Z.

In this regard, the a/m correlation and the as-effective-as-possible alignment of the research presented in this article to the engineering context under discussion is dialectically considered in points (1) to (25) of Appendix Z, whereas in points (15), (16) and (26), the two counterparts eventually meet, exchanging their advantages. Consequently, these “meeting” points, (15), (16) and (26) of Appendix Z, in conjunction with Section 2.1.3 and Section 2.1.4 as well as Appendix C, Appendix D and Appendix E—which are also directly relevant in this regard—comprise an essential contribution indicative of the significance, novelty and originality of this work. The same holds true for the fact that the application of this research does not require the design of experiments, as also discussed in point (24) of Appendix Z and earlier in this section.

In fact, the different approaches outlined in Appendix Z are summarized and compared in terms of their key elements and features, including, but not limited to, the design of experiments inherent in semi-empirical phenomenological approaches [56,57,58,59,60,61,62,63,64,65] and the CFD analyses, in

Table A1. Comparison of the different approaches outlined in Appendix Z.

Key Elements & Features of Different Approaches Outlined in Appendix Z	Existing Engineering Context	Semi-Empirical Phenomenological Approach	Approximate Representation of One Function by Another
Vessels design	X
Vessels newbuilding	X
Vessels classification	X
Regulatory compliance	X
Towing tank model trials	X
Vessel models under scale	X
Propellers testing apparatus	X
Data acquisition apparatus	X	X	Table 2 and Table 3
CFD analysis	Optional	Optional
Towing theoretical concept	X
Added resistance theory	X
Full scale calculations	X
Similarity	X		Section 2.1 and Section 2.1.5 (m)
Dimensional Analysis	X		Section 2.1 and Section 2.1.5 (m)
Thrust force loss	X		Point Z(26)
Effective wake velocity loss	X		Point Z(26)
Factors/Coefficients	Point Z(8)
Shaft power determination	Point Z(9)	X	Section 1.1, Section 2.1.5 and Section 2.2
Full scale sea trials	X
Sea trials reports	Point Z(15)		Point Z(26)
Sea trials curves	Point Z(15)		Point Z(26)
Design of experiments		Points Z(23) & Z(24)
Additional data sensors	Point Z(10)	Points Z(22) & Z(23)
Met/ocean data availability	X	X	Table 2 and Table 3
Vessel data availability	X	X	Table 2 and Table 3
Vessel-tracking/monitoring			Section 2.3, Section 2.3.1 and Section 2.3.2
Data sets processing	X	X	Table 1 and Table 3, Appendix H
Actual seagoing conditions			Section 1.1, Section 2.1.5 and Section 2.2
Section 6.3.1.3 Reference [36]	Point Z(30)		Section 1, Point Z(31)
Section 6.4.3.4 Reference [36]	Point Z(30)		Section 1, Point Z(35)
Section 6.4.3.4.2 Reference [36]	Point Z(30)		Section 1, Point Z(35)

An “X” entry in Table A1 above denotes that as far as this particular key element/feature (line) of Table A1 is concerned, this element/feature is comprised by the respective approach after which the respective column of this entry is titled. Same also applies in those cases where, instead of an “X” entry, another entry (also, other than “Optional”) is provided; in those cases, such an entry denotes an additional reference to specific content of this article which is particularly applicable in this regard. Naturally, an “Optional” entry denotes that the particular key element/feature (line) of such an entry may or may not be comprised by the respective approach after which the respective column of this entry is titled.

between points (23) and (24) of Appendix Z.

1.1. Approximate Representation [1] of the Thermo-Fluid and Frictional Processes of the Main Engines in Terms of FOC and of the Hydrodynamic Performance in Terms of FPP’s Shaft Propulsive Power and Rotational Speed, Applicable to Standard Vessels Under Actual Seagoing Conditions

1.1.1. The Thermo-Fluid and Frictional Processes of the Main Engines of Standard Vessels Under Actual Seagoing Conditions (Also See Section 2.2)

Any given main engine’s specific fuel oil consumption, SFOC, shall be equal to an applicable occurrence, H′(x_i, y_i, z_i,…; α, β, γ, …; J), of the approximating function, H(x_i, y_i, z_i,…; α, β, γ, …) [1], in the context of Reference [1], where the sequence of variables, x, y, z,… [1], in the context of Equation (1) of Reference [1], will be as follows:

(1.1.1) 

The engine power, P;

(1.1.2) 

The engine rotational speed, r;

(1.1.3) 

The engine rotational acceleration, if any, dr/dt;

(1.1.4) 

The applicable correction parameters set, cor (see Section 2.2 and Section 2.4).

1.1.2. The Hydrodynamic Performance of Standard Vessels Under Actual Seagoing Conditions (See Also Section 2.1)

Any given vessel’s fixed-pitch propeller’s (FPP’s) shaft propulsive power, P, shall be equal to an applicable occurrence, H′(x_i, y_i, z_i,…; α, β, γ,…; J), of the approximating function, H(x_i, y_i, z_i,…; α, β, γ, …) [1], in the context of Reference [1], where the sequence of variables, x, y, z,… [1], in the context of Equation (1) of Reference [1], will be as follows:

(1.2.1) 

The applicable reference (correction) parameters, specifically referring to the ambient, meteorological and oceanographic, environmental, reference, ideal, still air and water, conditions and ship-tracking data pursuant to Section 2.1.2, Correlation Scheme #2, and Section 2.3, “Big data” set, and the specific vessel’s hydrostatic conditions pursuant to Section 2.1.4, Correlation Scheme #4;

(1.2.2) 

The vessel’s TTW (through-the-water) speed (or simply, the log-speed) in the vessel’s forward direction, TTWSFD;

(1.2.3) 

Either of the following:

(1.2.3.1) 

The FPP shaft’s rotational speed, r;

(1.2.3.2) 

The actual ambient, meteorological, and oceanographic environmental conditions and ship-tracking data pursuant to Section 2.1.1, Correlation Scheme #1, and Section 2.3, “Big data” set, and the specific vessel’s service margin pursuant to Section 2.1.3, Correlation Scheme #3.

In addition to the combination of (1.2.2) and (1.2.3) above, other effectively equivalent combinations of the content of (1.2.2), (1.2.3.1), and (1.2.3.2) above may be applicable. Furthermore, the FPP’s shaft power, P, and shaft rotational speed, r, in the context of Section 1.1.2, may be interchangeable (exchanged for being mutually substituted by each other). With regard to Section 1.1.2, also see Section 2.1.5 and Section 2.4.

2. Materials and Methods

2.1. The Fixed-Pitch (Screw) Marine Propulsion Approximating Functional [1] System

The fixed-pitch (screw) marine propulsion approximating functional [1] system used in this study is based on the fundamental principle of the Law of Similarity and Dimensional Analysis, as applied in ship propulsion [51,52,53]:

“For fixed hydrostatic conditions (displacement, draft/trim), fixed water temperature and salinity, fixed air barometric pressure and any given (“quasi”-) steady vessel, water and air state and conditions, the (“quasi”-) steady-state fixed-pitch screw propeller‘s (FPP) shaft power, when such a ship is making (“quasi”-) steady course way “by engines” along a fixed latitude circle, at (“quasi”-) fixed rudder angle and through sufficiently deep and otherwise unconstrained waters, depends only on the (“quasi”-) steady through-the-water (TTW) speed of it and the (“quasi”-) steady rotational speed of the FPP propeller” (see Figure 1, Figure 2, Figure 3 and Figure 4 below in conjunction with Appendix G of this article).

The above fixed-pitch (screw) marine propulsion approximating functional [1] system is applied by means of the multi (4)—step Correlation Schemes below, #1, #2, #3, and #4, referring to Section 2.1.1, Section 2.1.2, Section 2.1.3 and Section 2.1.4, respectively, keeping in mind that the content of Section 2.1 is subject and conditional to the relevant limitations laid out in Appendix Y.

2.1.1. Correlation Scheme #1

This correlation is between actual seagoing conditions, mean effective over time or instantaneous, and perfectly still (calm), water and air, conditions that are otherwise identical to the actual mean effective or instantaneous seagoing air and water conditions. It also applies to the same (actual) draft and trim and is based on the considerations described in detail in Appendix A. These considerations are to be accounted for in conjunction with Figure 1 and Figure 2 and Appendix G.

2.1.2. Correlation Scheme #2

This correlation between perfectly still (calm) water and air conditions, which are otherwise identical to the actual mean effective or instantaneous seagoing air and water conditions, and the “ideal” conditions of ship/voyage-specific “virtual” sea (power and speed) trials, is based on the considerations laid out in further detail in Appendix B and in References [115,116,117,118,119,120]. These considerations include the conditions of the same (real) draft and trim as in Section 2.1.1, Correlation Scheme #1, above.

2.1.3. Correlation Scheme #3

This correlation between ship/voyage-specific “virtual” sea trials under “ideal” conditions, and the same ship’s “virtual” sea trials under “ideal/newbuilding” conditions (conducted virtually) upon the most recent delivery of the vessel by a shipyard after her newbuilding or her major modification, is based on the considerations laid out in further detail in Appendix C. With regard to this Correlation Scheme #3, reference is also made to Figure 2, Figure 3 and Figure 4 below in conjunction with Section 1 and Section 1.1, Appendix G and Appendix Z, as well as [121,122,123]. These considerations include the condition of same (actual) draft and trim as in Section 2.1.1 and Section 2.1.2, Correlation Schemes #1 and #2, above.

2.1.4. Correlation Scheme #4

This correlation between the corrected results of the ship’s official sea trials, calculated for the ship’s actual draft and trim, and the results of the actual sea trials in laden or ballast conditions, is based on the considerations laid out in further detail in Appendix D. See also Figure 3 and Figure 4 below, Section 1 and Section 1.1, as well as Appendix G and Appendix Z.

2.1.5. Sea Margin, Speed Loss, Light Running Margin, Sea Running Margin, Apparent TTW Slip

In this article, the ratio between the FPP’s shaft power under actual seagoing conditions (“at sea”) and the FPP’s shaft power under “ideal” conditions, assuming that the draft and trim remain unchanged, is referred to as the shaft power ratio. Further details regarding this ratio and alternative means of expressing it, are provided in Appendix E of this article. The above ratio, in any one of its alternative forms/expressions (sea margin, speed loss, light running margin, and sea running margin) effectively compares and quantifies any given actual mean effective or instantaneous seagoing conditions against a ship/voyage-specific “virtual” sea (power and speed) trial in “ideal” conditions:

(a) 

The above shaft power ratio when considered to sustain the same “ideal” conditions, TTWSFD, decreased by one (or by 100% if calculated as a percentage), is defined as the sea margin.

(b) 

The above shaft power ratio, when considered to sustain the same “ideal” conditions shaft rotational speed, r, decreased by one (or by 100% if calculated as a percentage), is defined as the sea running margin.

(c) 

The light running margin is defined as the reduction percentage (%) of the shaft rotational speed, r, under “ideal conditions”, required to sustain the same “ideal” FPP’s shaft power; the light running margin is common for, and representative of, steady (fixed) sea margin and/or sea running margin values, for any given value of the FPP’s shaft power.

(d) 

The speed loss is defined as the reduction (%) in the “ideal” conditions, TTWSFD, necessary to sustain the same “ideal” FPP’s shaft power. It is representative of steady (fixed) sea margin and/or sea running margin values for any given value of the FPP’s shaft power. Regarding the different meanings of the term speed loss in the marine industry [56,57,58,59,60,61,62,63,64,65,124,125], in conjunction with the relevant discussion on the steady rotational speed, r, of the main engine, the shaft and the FPP, more details are available in Appendix F.

(e) 

The above dimensionless indexes (sea margin, sea running margin, light running margin, and speed loss) are all interrelated, meaning that when one of them is determined, the other three are determined as well. Furthermore, they may be, either individually or together, comprehensively defined based on the above actual, non-ideal, conditions. The above are not referred to in a trivial manner: instead, they are of particular value and significance as different researchers and different stakeholders report their relevant observations, conclusions, and data in terms of different indexes among those discussed in (a)–(d) above, (f) below, and in other contexts.

(f) 

One (1, unity) minus the dimensionless apparent TTW slip is a ship specific dimensionless ratio of, TTWSFD to the FPP rotational speed, r; as such, it is heavily dependent on any or all of the above dimensionless indexes (sea margin, sea running margin, light running margin, and speed loss), with a slight only additional dependence on a proper dimensionless form of the FPP rotational speed, r. The above “apparent” designation is not a trivial one. It clarifies that the apparent TTW slip is calculated based on the vessel’s TTWSFD which as far as the FPP is concerned, is the “apparent” and not the “evident” velocity at which the water enters the FPP. More relevant information and technical background regarding the above “apparent” designation is available in Appendix E, in conjunction with the relevant points of Appendix Z. Figure 1 depicts the system of axes of the FPP’s rotational speed, r vs. the FPP’s shaft power, P, and three sets of curves:

Figure 1. The (“quasi”-) steady FPP’s shaft power as a function of (“quasi”-) steady TTWSFD and the (“quasi”-) steady rotational speed of the FPP. Fixed hydrostatic conditions (displacement, draft/trim), fixed water temperature and salinity, fixed air barometric pressure, any given (“quasi”-) steady vessel, water and air state and conditions, (“quasi”-) steady course way “by engines” along a fixed latitude circle, (“quasi”-) fixed rudder angle, and in sufficiently deep and otherwise unconstrained waters.

Figure 1. The (“quasi”-) steady FPP’s shaft power as a function of (“quasi”-) steady TTWSFD and the (“quasi”-) steady rotational speed of the FPP. Fixed hydrostatic conditions (displacement, draft/trim), fixed water temperature and salinity, fixed air barometric pressure, any given (“quasi”-) steady vessel, water and air state and conditions, (“quasi”-) steady course way “by engines” along a fixed latitude circle, (“quasi”-) fixed rudder angle, and in sufficiently deep and otherwise unconstrained waters.

(g) 

The set of curves representing the steady (fixed) TTWSFD are depicted in Figure 1 (e.g., 13, 16, 19 and 22 knots).

(h) 

The set of curves representing the steady (fixed) apparent TTW slip are shown in Figure 1 (e.g., −2%, 2%, 6% and 10%).

(i) 

The, effectively, steady (fixed) weather conditions are also shown in Figure 1 (e.g., A: extremely good weather; B: average weather; and C: extremely bad weather; these are depicted, in green, blue and red, respectively).

For more details and information on Figure 1, Figure 2, Figure 3 and Figure 4, and formulations (1) to (7) below which effectively express these figures in mathematical terms, see Appendix G. As also discussed above, the dimensionless indexes of the service conditions (sea margin, sea running margin, light running margin, speed loss, and apparent TTW slip), are interrelated. This means that when one of them is determined, the other four are determined as well. In fact, they are comprehensively defined in the form of different occurrences of the approximating function, H(x_i, y_i, z_i,…; α, β, γ, …) [1], in the context of Equation (1) of Reference [1]. The sequence of variables, x_i, y_i, z_i,… [1], of the above approximating function, H(x_i, y_i, z_i,…; α, β, γ, …) [1], includes all the a/m actual, non-ideal, conditions in terms of the continuous availability of the “big data” set discussed in Section 2.3.

Figure 2. The (“quasi”-) steady-state FPP’s shaft power, as a function of the (“quasi”-) steady rotational speed of the FPP or of the (“quasi”-) steady TTWSFD. (1) During a sea trial under “ideal” conditions, upon the latest delivery of a vessel by a shipyard after her newbuilding or her major modification (see green curve). (2) During the same vessel’s “virtual sea trials” (“ideal”), voyage-specific vessel conditions (service margin) “++”, and under the same hydrostatic conditions (see red curve).

Figure 2. The (“quasi”-) steady-state FPP’s shaft power, as a function of the (“quasi”-) steady rotational speed of the FPP or of the (“quasi”-) steady TTWSFD. (1) During a sea trial under “ideal” conditions, upon the latest delivery of a vessel by a shipyard after her newbuilding or her major modification (see green curve). (2) During the same vessel’s “virtual sea trials” (“ideal”), voyage-specific vessel conditions (service margin) “++”, and under the same hydrostatic conditions (see red curve).

Pursuant to the four correlation steps above, the different occurrences of the approximating function, H(x_i, y_i, z_i,…; α, β, γ, …) [1], comprise two sets of components and elements:

(j) 

The set of purely deterministic ones;

(k) 

The set of hybrid, stochastic and deterministic, ones.

All these different occurrences, components and elements comprise a joint and common approximating functional [1] system. Furthermore, and pursuant to the four correlation steps above, the mathematical formulations relevant to the hybrid, stochastic and deterministic, components and elements of the above different occurrences of the approximating function, H(x_i, y_i, z_i,…; α, β, γ, …) [1], pursuant to Equation (1) of Reference [1], need to attain the closest possible approximate representation of the following conditions:

First, (l), for the greatest part of any voyage, the rotational acceleration, dr/dt, of the main engine, shaft and propeller of a standard vessel should be zero (steady rotational speed), while for the remaining, significantly shorter, time intervals of any voyage, it should remain as steady or as smooth as possible.

Second, (m), the a/m fundamental principle of the Law of Similarity and Dimensional Analysis, as specifically applied to ship propulsion.

The above condition, (l), is a “fact of life” in the maritime industry; more relevant information and technical background on this topic are available in Appendix F. The above two conditions, (l), and, (m), are to be approximately represented, simultaneously and jointly, in their composite form of a function to be approximated, h(x_i, y_i, z_i,…) [1].

Considering the a/m conditions, (l), and, (m), and pursuant to the four correlation steps above, a different occurrence of the approximating function, H(x_i, y_i, z_i,…; α, β, γ, …) [1] may be defined for each of the service conditions’ dimensionless indexes.

Figure 3. The (“quasi”-) steady-state FPP’s rotational speed, as a function of the (“quasi”-) steady TTWSFD: (1) in the sea trials, under the “ideal”, ballast condition, upon the latest delivery of a vessel by a shipyard after her newbuilding or her major modification (see dashed curve). (2) The same as (1), but under fully laden conditions (see green curve). (3) The same as (2), but under the same vessel’s “virtual sea trials” (“ideal”) voyage-specific vessel conditions (service margin) “+” (see blue curve). (4) The same as (2), but under the same vessel’s “virtual sea trials” (“ideal”) voyage-specific conditions (service margin) “++” (see red curve).

Figure 3. The (“quasi”-) steady-state FPP’s rotational speed, as a function of the (“quasi”-) steady TTWSFD: (1) in the sea trials, under the “ideal”, ballast condition, upon the latest delivery of a vessel by a shipyard after her newbuilding or her major modification (see dashed curve). (2) The same as (1), but under fully laden conditions (see green curve). (3) The same as (2), but under the same vessel’s “virtual sea trials” (“ideal”) voyage-specific vessel conditions (service margin) “+” (see blue curve). (4) The same as (2), but under the same vessel’s “virtual sea trials” (“ideal”) voyage-specific conditions (service margin) “++” (see red curve).

On the basis of the above definitions, and of the purely deterministic components and elements of the different occurrences of the approximating function, H(x_i, y_i, z_i,…; α, β, γ, …) [1], as well as of the a/m four correlation steps, two more occurrences of the approximating function, H(x_i, y_i, z_i,…; α, β, γ, …) [1], may be defined. These are the engine and shaft rotational speed, r, and power, P, and they may be defined as instantaneous and/or average values.

In any case, all definitions discussed above are effectively based on the following two additional conditions: (n) and (o).

Condition (n): The vessel’s TTWSFD is calculated and/or measured using the a/m big data set (see also Appendix E, Appendix G, Appendix J and Appendix K of this article).

Condition (o): The a/m service conditions’ dimensionless indexes (sea margin, sea running margin, light running margin, speed loss and apparent TTW slip) are defined in terms of different occurrences of the approximating function, H(x_i, y_i, z_i,…; α, β, γ, …) [1], whereas the sequence of variables, x_i, y_i, z_i,… [1], will include the a/m timestamped “big data” set.

The unknown parameters, α, β, γ,… [1], in the context of Equation (1) of Reference [1] are common to all different occurrences of the approximating function, H(x_i, y_i, z_i,…; α, β, γ, …) [1], for the same vessel, and they are replaced in the context of Section 2.1.5 by the calibration constants, Cj, j = 1, K. In fact, these calibration constants, Cj, j = 1, K, stand for, and are representative of, the a/m hybrid, stochastic and deterministic, nature of the respective set of components and elements of the approximating functional [1] system discussed above; this is addressed in further detail in Section 1 and Section 2.4.

Figure 4. The (“quasi”-) steady-state FPP’s shaft power, as a function of the (“quasi”-) steady TTWSFD, in the same conditions, (1), (2), (3) and (4), listed in the legend of Figure 3, and indicated by means of the same colour-code in both Figure 3 and Figure 4.

Figure 4. The (“quasi”-) steady-state FPP’s shaft power, as a function of the (“quasi”-) steady TTWSFD, in the same conditions, (1), (2), (3) and (4), listed in the legend of Figure 3, and indicated by means of the same colour-code in both Figure 3 and Figure 4.

For the different occurrences of the approximating function, H(x_i, y_i, z_i,…; α, β, γ, …) [1], in the exact context of Equation (1) of Reference [1], as applicable for the a/m service conditions’ dimensionless indexes (sea margin, sea running margin, light running margin, speed loss and apparent TTW slip), the following formulations apply, for each point (e.g., each position and UTC timestamp pairing), i, i = 1, I_obs + L_obs (see also Appendix Q):

(sea margin)_i = f1_i = f1(x_i, y_i, z_i,…; Cj, j = 1, K) = H′(x_i, y_i, z_i,…; α, β, γ,…; 1)

(1)

(sea running margin)_i = f2_i = f2(x_i, y_i, z_i,…; Cj, j = 1, K) = H′(x_i, y_i, z_i,…; α, β, γ,…; 2)

(2)

(light running margin)_i = f3_i = f3(x_i, y_i, z_i,…; Cj, j = 1, K) = H′(x_i, y_i, z_i,…; α, β, γ,…; 3)

(3)

(speed loss)_i = f4_i = f4(x_i, y_i, z_i,…; Cj, j = 1, K) = H′(x_i, y_i, z_i,…; α, β, γ,…; 4)

(4)

(apparent TTW slip)_i = f5_i = f5(x_i, y_i, z_i,…; Cj, j = 1, K) = H′(x_i, y_i, z_i,…; α, β, γ,…; 5)

(5)

By combining the two considerations, (n), and, (o), the instantaneous (and average) rotational speed, r, and power, P, of the FPP and the engine may be defined all along the vessel’s course on the basis of the a/m unknown hybrid, stochastic and deterministic, K, calibration constants, Cj, j = 1, K, and of the observed, I_obs + L_obs, pairs of positions and UTC timestamps, i = 1, I_obs + L_obs, accompanied by their matching a/m timestamped “big data” set:

P_i = P(x_i, y_i, z_i,…; Cj, j = 1, K) = H′(x_i, y_i, z_i,…; α, β, γ,…; 6), i = 1, I_obs + L_obs

(6)

r_i = r(x_i, y_i, z_i,…; Cj, j = 1, K) = H′(x_i, y_i, z_i,…; α, β, γ,…; 7), i = 1, I_obs + L_obs

(7)

Considering condition (l) above, the following conditions should be always met for the following two different sets of pairs of positions and UTC timestamps:

(p), I_obs, where the rotational acceleration, dr/dt, of the main engine and FPP is expected to equal zero, or, r_i = r(x_i, y_i, z_i,…), i = 1, I_obs, is steady; or, (dr/dt)_i = dr(x_i, y_i, z_i, …)/dt = 0, i =1, I_obs. However, for the sake of good order and as far as the perfect alignment with Reference [1] is concerned, from a numerical analysis perspective, the above are not applicable to the occurrence of the approximating function, H(x_i, y_i, z_i,…; α, β, γ, …) [1], for, dr/dt. Instead, the above are applicable to the occurrence of the function to be approximated, h(x_i, y_i, z_i,…) [1], for, dr/dt. So, pursuant to condition (l) above, h(x_i, y_i, z_i,…) [1] = 0, for dr/dt, and for, i = 1, I_obs. Consequently, and in the context of Equations (1)–(5) of Reference [1], and for each and every point (pair of position and UTC timestamp), i, i = 1, I_obs:

h′(x_i, y_i, z_i,…; 8) = (dr/dt)_i = dr(x_i, y_i, z_i, …)/dt = 0

(8h)

H′(x_i, y_i, z_i,…; α, β, γ, …; 8) = dr(x_i, y_i, z_i,…; Cj, j = 1, K)/dt = dH′(x_i, y_i, z_i, …; α, β, γ, …; 7)/dt

(8H)

n′(8) = I_obs

(8n)

and, (q), L_obs, where the rotational acceleration, dr/dt, of the main engine and FPP, (dr/dt)_i = dr(x_i, y_i, z_i, …)/dt, i = 1, L_obs is steady, or as smooth, and as near to steady, as possible; or, (d²r/dt²)_i = d²r(x_i, y_i, z_i, …)/dt² = ~0, i = 1, L_obs. However, for the sake of good order and as far as the perfect alignment with Reference [1] is concerned, from a numerical analysis perspective, the above are not applicable to the occurrence of the approximating function, H(x_i, y_i, z_i,…; α, β, γ, …) [1], for d²r/dt². Instead, the above are applicable to the occurrence of the function to be approximated, h(x_i, y_i, z_i,…) [1], for, d²r/dt². So, pursuant to condition (l) above, h(x_i, y_i, z_i,…) [1] = 0, for d²r/dt², and for, i = 1, L_obs. Consequently, and in the context of Equations (1)–(5) of Reference [1], and for each and every point (pair of position and UTC timestamp), i, i = 1, L_obs:

h′(x_i, y_i, z_i,…; 9) = (d²r/dt²)_i = d²r(x_i, y_i, z_i, …)/dt² = ~0

(9h)

H′(x_i, y_i, z_i,…; α, β, γ,…; 9) = d²r(x_i, y_i, z_i,…; Cj, j = 1, K)/dt² = d²H′(x_i, y_i, z_i,…; α, β, γ, …; 7)/dt²

(9H)

n′(9) =L_obs

(9n)

Considering the a/m perfect alignment with Reference [1] in conjunction with the content of the paragraph just before Equation (6) above, the following also apply:

(α, β, γ, …)′ = (Cj, j = 1, K)

(10a)

(Cj, j = 1, K) = (α, β, γ, …)″

(10b)

(x_i, y_i, z_i,…) = [data set (1.2.1)]_i U [data set (1.2.2)]_i U [data set (1.2.3.2)]_i

(11)

Equation (11) is applicable for each and every point (position and UTC timestamp pairing), i, i = 1, I_obs + L_obs. Regarding the solution of the Equation Sets, (7) and (8), or, (7)–(9), in conjunction with Equations (10) and (11), we refer to Table 1 and Table 2 in conjunction with Appendix H and Appendix Q. All the above also apply for the solution of the Equation Sets below (12)–(14), in conjunction with Equations (6), (7), (10) and (11), or (6), (7), (15)–(17), with or without their combination with Equation Sets, (7) and (8), or, (7), (8) and (9), respectively and pursuant to Table 1 and Table 2 as well as Appendix H and Appendix Q.

h′(x_i, y_i, z_i,…; 12) = Nrev_i

(12h)

$$\begin{gathered} H^{\prime }(x_i, y_i, z_i,\dots ;\alpha , \beta , \gamma , \dots ;12)= \\ ={\int }_{\mathit{s}\mathit{t}\mathit{a}\mathit{r}\mathit{t}}^{\mathit{e}\mathit{n}\mathit{d}}r(x_i, y_i, z_i, \dots ;Cj, j=1, K)dt={\int }_{\mathit{s}\mathit{t}\mathit{a}\mathit{r}\mathit{t}}^{\mathit{e}\mathit{n}\mathit{d}}{H}^{\prime }(x_i, y_i, z_i, \dots ;\alpha , \beta , \gamma , \dots ;7)dt \end{gathered}$$

(12H)

n′(12) = I_-Nrev

(12n)

h′(x_i, y_i, z_i,…; 13) =W_i

(13h)

$$\begin{gathered} H^{\prime }(x_i, y_i, z_i,\dots ;\alpha , \beta , \gamma , \dots ;13)= \\ ={\int }_{\mathit{s}\mathit{t}\mathit{a}\mathit{r}\mathit{t}}^{\mathit{e}\mathit{n}\mathit{d}}P(x_i, y_i, z_i, \dots ;Cj, j=1, K)dt={\int }_{\mathit{s}\mathit{t}\mathit{a}\mathit{r}\mathit{t}}^{\mathit{e}\mathit{n}\mathit{d}}{H}^{\prime }(x_i, y_i, z_i, \dots ;\alpha , \beta , \gamma , \dots ;6)dt \end{gathered}$$

(13H)

n′(13) = I_-W

(13n)

h′(x_i, y_i, z_i,…; 14) =FOC_i

(14h)

$$\begin{gathered} H^{\prime }(x_i, y_i, z_i,\dots ;\alpha , \beta , \gamma , \dots ;14)={\int }_{\mathit{s}\mathit{t}\mathit{a}\mathit{r}\mathit{t}}^{\mathit{e}\mathit{n}\mathit{d}}P(x_i, y_i, z_i, \dots ;Cj, j=1, K) SFOC(P, r, dr/dt, cor)dt= \\ ={\int }_{\mathit{s}\mathit{t}\mathit{a}\mathit{r}\mathit{t}}^{\mathit{e}\mathit{n}\mathit{d}}{H}^{\prime }(x_i, y_i, z_i, \dots ;\alpha , \beta , \gamma , \dots ;6) SFOC(P, r, dr/dt, cor)dt= \\ ={\int }_{\mathit{s}\mathit{t}\mathit{a}\mathit{r}\mathit{t}}^{\mathit{e}\mathit{n}\mathit{d}}{H}^{\prime }(x_i, y_i, z_i, \dots ;\alpha , \beta , \gamma , \dots ;6) SFOC[H^{\prime }(x_i, y_i, z_i, \dots ;\alpha , \beta , \gamma , \dots ;6), r, dr/dt, cor]dt \end{gathered}$$

(14Ha)

$$\begin{gathered} H^{\prime }(x_i, y_i, z_i,\dots ;\alpha , \beta , \gamma , \dots ;14)={\int }_{\mathit{s}\mathit{t}\mathit{a}\mathit{r}\mathit{t}}^{\mathit{e}\mathit{n}\mathit{d}}P(x_i, y_i, z_i, \dots ;Cj, j=1, K) SFOC(P, r, dr/dt, cor)dt= \\ ={\int }_{\mathit{s}\mathit{t}\mathit{a}\mathit{r}\mathit{t}}^{\mathit{e}\mathit{n}\mathit{d}}{H}^{\prime }(x_i, y_i, z_i, \dots ;\alpha , \beta , \gamma , \dots ;6) SFOC(P, r, dr/dt, cor)dt= \\ ={\int }_{\mathit{s}\mathit{t}\mathit{a}\mathit{r}\mathit{t}}^{\mathit{e}\mathit{n}\mathit{d}}{H}^{\prime }(x_i, y_i, z_i, \dots ;\alpha , \beta , \gamma , \dots ;6) SFOC[H^{\prime }(x_i, y_i, z_i, \dots ;\alpha , \beta , \gamma , \dots ;6), r, dr/dt, cor]dt= \\ ={\int }_{\mathit{s}\mathit{t}\mathit{a}\mathit{r}\mathit{t}}^{\mathit{e}\mathit{n}\mathit{d}}{H}^{\prime }(x_i, y_i, z_i, \dots ;\alpha , \beta , \gamma , \dots ;6) SFOC[H^{\prime }(x_i, y_i, z_i, \dots ;\alpha , \beta , \gamma , \dots ;6), \\ {H}^{\prime }(x_i, y_i, z_i, \dots ;\alpha , \beta , \gamma , \dots ;7), {dH}^{\prime }(x_i, y_i, z_i, \dots ;\alpha , \beta , \gamma , \dots ;7)/dt, cor]dt \end{gathered}$$

(14Hb)

n′(14) = I_-FOC

(14n)

n′(15) =I_-intervals

(15)

(α, β, γ, …)′= (Cj, j = 1, K)′

(16a)

(Cj, j = 1, K)′ = (α, β, γ, …)″

(16b)

(x_i, y_i, z_i,…) = [data set (1.2.1)]_i U [data set (1.2.2)]_i U [data set (1.2.3.1)]_i

(17)

With regard to the solution of the set of Equations (7), (12h), (12H), (15)–(17), and/or (6), (13h), (13H), (15)–(17), and/or, (6), (14h), (14Ha), (15)–(17), see Table 1 and Table 2 and Appendix H and Appendix Q.

(r) Thirty (30) different possible cases of simultaneous (combined) or singular application and solution of Equations (6)–(17) are presented in terms of different respective lines (cases) in Table 1 and Table 2, whereas the respective conditions regarding the requisite data availability and other details relevant to the a/m simultaneous (combined) application and solution, are discussed between Table 1 and Table 2. The columns of both tables are titled after the indexes of the respective equations above. In this regard and in the context of this article, the formulations above, (6), (7), (8h), (8H), (8n), (9h), (9H), (9n), (10a)/(10b), (11), (12h), (12H), (12n), (13h), (13H), (13n), (14h), (14Ha)/(14Hb), (14n), (15), (16a)/(16b), and (17), are to be considered not only as equations but also as imperative assignments statements that are readily applicable in the context of any imperative numerical analysis paradigm or any imperative programming language and/or compiler (in this regard, see also Appendix H). See also the Nomenclature, Steps (lines) 9 and 11 in Table 3 in Section 2.2, and Appendix H, for Equations (10a)/(10b) and (16a)/(16b).

An “X” entry in Table 1 denotes that as far as the particular case (line) of Table 1 is concerned, the equation after which the respective column of this entry is titled shall be respectively applicable for this particular case (line). An “X^(a)” entry in column 14H denotes that as far as the particular case (line) is concerned, equation (14Ha) is applicable, whereas an “X^(b)” entry in column 14H denotes that equation (14Hb) is applicable.

(s) For each one of these thirty (30) different possible cases pursuant to Table 1 and Table 2, two possibilities are offered in terms of whether to apply Correlation Scheme #4 pursuant to Section 2.1.4 and Appendix D. If Correlation Scheme #4 is applied, many different, ideally consecutive, voyages (legs) of the same vessel, each representative of a subset of effectively steady hydrostatic conditions and relevant data, may comprise a reporting period of the specific vessel, and any one of the thirty (30) different possible cases pursuant to Table 1 and Table 2 may be applied in this entire reporting period. On the contrary, if Correlation Scheme #4 is not applied, any one of the thirty (30) different possible cases pursuant to Table 1 and Table 2 may be applied on a per-voyage/leg basis, to be defined as such (among other commercial and/or vessel management conditions) also on the basis of effectively steady hydrostatic conditions and relevant data.

If Equations (6), (14h), (14Ha), (15), (16) and (17) are applied pursuant to line (case) 26 of Table 1 and Table 2 of Section 2.1.5, for the legs and the reporting intervals of an entire reporting period in conjunction with Correlation Scheme #4 and pursuant to Section 2.1.4 and Appendix D; then, such an application is defined as Analysis Type 2, which is also discussed in Section 2.4, Section 3, Section 3.2, Section 3.3 and Section 4 as well as in Appendix X of this article.

If Equations (6), (14h), (14Ha), (15), (16) and (17) are applied pursuant to line (case) 26 of Table 1 and Table 2 of Section 2.1.5, on a per-voyage/leg basis, for the reporting intervals of a single voyage/leg at a time, and without applying Correlation Scheme #4, then such an application is defined as Analysis Type 3, which is also discussed in Section 2.4, Section 3, Section 3.3 and Section 4 as well as in Appendix X of this article.

(t) Regarding the streamlined computational sequence for applying and solving Equations (6)–(17) above, reference is made to the six (6) Steps (lines) 7 to 12 of Table 3 in Section 2.2. As mentioned in Section 2.2, the six (6) Steps (lines) 1 to 6 of Table 3 always run before the six (6) Steps (lines) 7 to 12. The streamlined sequential nature of the computational Steps pursuant to Table 3 is directly and essentially interrelated to the nature, character, and context of any imperative numerical analysis paradigm and/or any imperative programming language and/or compiler (in this regard, see also Appendix H).

Table 1. Mapping of different applications and solutions of combinations of Equations (6)–(17).

	6	7	8h	8H	8n	9h	9H	9n	10	11	12h	12H	12n	13h	13H	13n	14h	14H	14n	15	16	17
1		X	X	X	X				X	X
2		X	X	X	X	X	X	X	X	X
3		X	X	X	X				X	X	X	X	X
4		X	X	X	X	X	X	X	X	X	X	X	X
5	X	X	X	X	X				X	X				X	X	X
6	X	X	X	X	X	X	X	X	X	X				X	X	X
7	X	X	X	X	X				X	X							X	X(b)	X
8	X	X	X	X	X	X	X	X	X	X							X	X(b)	X
9	X	X	X	X	X				X	X	X	X	X	X	X	X
10	X	X	X	X	X	X	X	X	X	X	X	X	X	X	X	X
11	X	X	X	X	X				X	X	X	X	X				X	X(b)	X
12	X	X	X	X	X	X	X	X	X	X	X	X	X				X	X(b)	X
13	X	X	X	X	X				X	X				X	X	X	X	X(b)	X
14	X	X	X	X	X	X	X	X	X	X				X	X	X	X	X(b)	X
15	X	X	X	X	X				X	X	X	X	X	X	X	X	X	X(b)	X
16	X	X	X	X	X	X	X	X	X	X	X	X	X	X	X	X	X	X(b)	X
17		X							X	X	X	X	X
18	X								X	X				X	X	X
19	X								X	X							X	X(b)	X
20	X	X							X	X	X	X	X	X	X	X
21	X								X	X				X	X	X	X	X(b)	X
22	X	X							X	X	X	X	X				X	X(b)	X
23	X	X							X	X	X	X	X	X	X	X	X	X(b)	X
24		X									X	X								X	X	X
25	X													X	X					X	X	X
26	X																X	X(a)		X	X	X
27	X	X									X	X		X	X					X	X	X
28	X													X	X		X	X(a)		X	X	X
29	X	X									X	X					X	X(a)		X	X	X
30	X	X									X	X		X	X		X	X(a)		X	X	X

Table 1. (Continued in terms of below annotations on the combined application of Table 1 and Table 2): In cases (lines) 1 to 16 of Table 1 above, the availability of the respective data sets indicated in the last column of Table 2 below is required at the frequency indicated in column 8n of Table 2. Additionally, in cases (lines) 3 to 16 of Table 1 above, the respective data sets mentioned above should also include subsets of them at the frequencies indicated in columns 12n, 13n and 14n of Table 2 below, as well as the respective values (attained at the same frequencies mentioned directly above) of the functions to be approximated [1] pursuant to columns 12h, 13h and 14h of Table 2 below respectively. Each one of the above subset should contain data in internal sub-frequencies, high enough to enable the accurate calculation of the integrals included in Equations (12H), (13H) and (14Hb) above. In cases (lines) 17 to 23 of Table 1 above, the above additional requirements on data subsets applicable for cases (lines) 3 to 16 of the same table also apply; however, the respective data sets indicated in the last column of Table 2 below are not required at a frequency additional (higher) to the sum of the internal sub-frequencies of the data subsets indicated in columns 12n, 13n and 14n of Table 2 below. In these cases (lines) 17 to 23 of Table 1 above, the sum of the frequencies indicated in columns 12n, 13n and 14n of Table 2 below is expected to be on the order of magnitude of the frequency indicated in column 8n of Table 2, to attain results of the same accuracy as the ones of the cases (lines) 1 to 16 of Table 1 above. In cases (lines) 24 to 30 of Table 1 above, where the data sets indicated in the last column of Table 2 below for cases (lines) 1 to 23 of Table 1 above are not required, Equations (12H), (13H), and/or (14Ha), above are still applicable on the basis of the values of the functions to be approximated [1] and the available data pursuant to columns 12h, 13h, 14h, as well as the last column of Table 2 below at the frequency indicated in column 15 of the same table. In this regard, the integrals included in Equations (12H), (13H) and (14Ha) above are calculated on the basis of the respective data mean effective values. In even cases (lines), 2 to 16, of Table 1 and Table 2, where timestamp and locations pairs, L_obs, are positively identified, the engine and FPP rotational acceleration, dr/dt, calculated at, L_obs, is accounted for as far as the above even cases (lines) are concerned.

Table 1. Mapping of different applications and solutions of combinations of Equations (6)–(17).

	6	7	8h	8H	8n	9h	9H	9n	10	11	12h	12H	12n	13h	13H	13n	14h	14H	14n	15	16	17
1		X	X	X	X				X	X
2		X	X	X	X	X	X	X	X	X
3		X	X	X	X				X	X	X	X	X
4		X	X	X	X	X	X	X	X	X	X	X	X
5	X	X	X	X	X				X	X				X	X	X
6	X	X	X	X	X	X	X	X	X	X				X	X	X
7	X	X	X	X	X				X	X							X	X(b)	X
8	X	X	X	X	X	X	X	X	X	X							X	X(b)	X
9	X	X	X	X	X				X	X	X	X	X	X	X	X
10	X	X	X	X	X	X	X	X	X	X	X	X	X	X	X	X
11	X	X	X	X	X				X	X	X	X	X				X	X(b)	X
12	X	X	X	X	X	X	X	X	X	X	X	X	X				X	X(b)	X
13	X	X	X	X	X				X	X				X	X	X	X	X(b)	X
14	X	X	X	X	X	X	X	X	X	X				X	X	X	X	X(b)	X
15	X	X	X	X	X				X	X	X	X	X	X	X	X	X	X(b)	X
16	X	X	X	X	X	X	X	X	X	X	X	X	X	X	X	X	X	X(b)	X
17		X							X	X	X	X	X
18	X								X	X				X	X	X
19	X								X	X							X	X(b)	X
20	X	X							X	X	X	X	X	X	X	X
21	X								X	X				X	X	X	X	X(b)	X
22	X	X							X	X	X	X	X				X	X(b)	X
23	X	X							X	X	X	X	X	X	X	X	X	X(b)	X
24		X									X	X								X	X	X
25	X													X	X					X	X	X
26	X																X	X(a)		X	X	X
27	X	X									X	X		X	X					X	X	X
28	X													X	X		X	X(a)		X	X	X
29	X	X									X	X					X	X(a)		X	X	X
30	X	X									X	X		X	X		X	X(a)		X	X	X

Table 1. (Continued in terms of below annotations on the combined application of Table 1 and Table 2): In cases (lines) 1 to 16 of Table 1 above, the availability of the respective data sets indicated in the last column of Table 2 below is required at the frequency indicated in column 8n of Table 2. Additionally, in cases (lines) 3 to 16 of Table 1 above, the respective data sets mentioned above should also include subsets of them at the frequencies indicated in columns 12n, 13n and 14n of Table 2 below, as well as the respective values (attained at the same frequencies mentioned directly above) of the functions to be approximated [1] pursuant to columns 12h, 13h and 14h of Table 2 below respectively. Each one of the above subset should contain data in internal sub-frequencies, high enough to enable the accurate calculation of the integrals included in Equations (12H), (13H) and (14Hb) above. In cases (lines) 17 to 23 of Table 1 above, the above additional requirements on data subsets applicable for cases (lines) 3 to 16 of the same table also apply; however, the respective data sets indicated in the last column of Table 2 below are not required at a frequency additional (higher) to the sum of the internal sub-frequencies of the data subsets indicated in columns 12n, 13n and 14n of Table 2 below. In these cases (lines) 17 to 23 of Table 1 above, the sum of the frequencies indicated in columns 12n, 13n and 14n of Table 2 below is expected to be on the order of magnitude of the frequency indicated in column 8n of Table 2, to attain results of the same accuracy as the ones of the cases (lines) 1 to 16 of Table 1 above. In cases (lines) 24 to 30 of Table 1 above, where the data sets indicated in the last column of Table 2 below for cases (lines) 1 to 23 of Table 1 above are not required, Equations (12H), (13H), and/or (14Ha), above are still applicable on the basis of the values of the functions to be approximated [1] and the available data pursuant to columns 12h, 13h, 14h, as well as the last column of Table 2 below at the frequency indicated in column 15 of the same table. In this regard, the integrals included in Equations (12H), (13H) and (14Ha) above are calculated on the basis of the respective data mean effective values. In even cases (lines), 2 to 16, of Table 1 and Table 2, where timestamp and locations pairs, L_obs, are positively identified, the engine and FPP rotational acceleration, dr/dt, calculated at, L_obs, is accounted for as far as the above even cases (lines) are concerned.

Table 2. Mapping of requisite data availability for applying and solving Equations (6)–(17).

	8n	9n	12h	12n	13h	13n	14h	14n	15	x, y, z, … iaw eq. 11, 17
1	Iobs + Lobs									(1.2.1); TTWSFD; (1.2.3.2)
2	Iobs + Lobs	Iobs + Lobs								(1.2.1); TTWSFD; (1.2.3.2)
3	Iobs + Lobs		Nrev	I-Nrev						(1.2.1); TTWSFD; (1.2.3.2)
4	Iobs + Lobs	Iobs + Lobs	Nrev	I-Nrev						(1.2.1); TTWSFD; (1.2.3.2)
5	Iobs + Lobs				W	I-W				(1.2.1); TTWSFD; (1.2.3.2)
6	Iobs + Lobs	Iobs + Lobs			W	I-W				(1.2.1); TTWSFD; (1.2.3.2)
7	Iobs + Lobs						FOC	I-FOC		(1.2.1); TTWSFD; (1.2.3.2)
8	Iobs + Lobs	Iobs + Lobs					FOC	I-FOC		(1.2.1); TTWSFD; (1.2.3.2)
9	Iobs + Lobs		Nrev	I-Nrev	W	I-W				(1.2.1); TTWSFD; (1.2.3.2)
10	Iobs + Lobs	Iobs + Lobs	Nrev	I-Nrev	W	I-W				(1.2.1); TTWSFD; (1.2.3.2)
11	Iobs + Lobs		Nrev	I-Nrev			FOC	I-FOC		(1.2.1); TTWSFD; (1.2.3.2)
12	Iobs + Lobs	Iobs + Lobs	Nrev	I-Nrev			FOC	I-FOC		(1.2.1); TTWSFD; (1.2.3.2)
13	Iobs + Lobs				W	I-W	FOC	I-FOC		(1.2.1); TTWSFD; (1.2.3.2)
14	Iobs + Lobs	Iobs + Lobs			W	I-W	FOC	I-FOC		(1.2.1); TTWSFD; (1.2.3.2)
15	Iobs + Lobs		Nrev	I-Nrev	W	I-W	FOC	I-FOC		(1.2.1); TTWSFD; (1.2.3.2)
16	Iobs + Lobs	Iobs + Lobs	Nrev	I-Nrev	W	I-W	FOC	I-FOC		(1.2.1); TTWSFD; (1.2.3.2)
17			Nrev	I-Nrev						(1.2.1); TTWSFD; (1.2.3.2)
18					W	I-W				(1.2.1); TTWSFD; (1.2.3.2)
19							FOC	I-FOC		(1.2.1); TTWSFD; (1.2.3.2)
20			Nrev	I-Nrev	W	I-W				(1.2.1); TTWSFD; (1.2.3.2)
21					W	I-W	FOC	I-FOC		(1.2.1); TTWSFD; (1.2.3.2)
22			Nrev	I-Nrev			FOC	I-FOC		(1.2.1); TTWSFD; (1.2.3.2)
23			Nrev	I-Nrev	W	I-W	FOC	I-FOC		(1.2.1); TTWSFD; (1.2.3.2)
24			Nrev						I-intervals	(1.2.1); TTWDFD;W; t
25					W				I-intervals	(1.2.1); TTWDFD;Nrev; t
26							FOC		I-intervals	(1.2.1); TTWDFD;Nrev; t
27			Nrev		W				I-intervals	(1.2.1); TTWDFD;Nrev; W; t
28					W		FOC		I-intervals	(1.2.1); TTWDFD;Nrev; t
29			Nrev				FOC		I-intervals	(1.2.1); TTWDFD;Nrev; W; t
30			Nrev		W		FOC		I-intervals	(1.2.1); TTWDFD;Nrev; W; t

Table 2. Mapping of requisite data availability for applying and solving Equations (6)–(17).

	8n	9n	12h	12n	13h	13n	14h	14n	15	x, y, z, … iaw eq. 11, 17
1	Iobs + Lobs									(1.2.1); TTWSFD; (1.2.3.2)
2	Iobs + Lobs	Iobs + Lobs								(1.2.1); TTWSFD; (1.2.3.2)
3	Iobs + Lobs		Nrev	I-Nrev						(1.2.1); TTWSFD; (1.2.3.2)
4	Iobs + Lobs	Iobs + Lobs	Nrev	I-Nrev						(1.2.1); TTWSFD; (1.2.3.2)
5	Iobs + Lobs				W	I-W				(1.2.1); TTWSFD; (1.2.3.2)
6	Iobs + Lobs	Iobs + Lobs			W	I-W				(1.2.1); TTWSFD; (1.2.3.2)
7	Iobs + Lobs						FOC	I-FOC		(1.2.1); TTWSFD; (1.2.3.2)
8	Iobs + Lobs	Iobs + Lobs					FOC	I-FOC		(1.2.1); TTWSFD; (1.2.3.2)
9	Iobs + Lobs		Nrev	I-Nrev	W	I-W				(1.2.1); TTWSFD; (1.2.3.2)
10	Iobs + Lobs	Iobs + Lobs	Nrev	I-Nrev	W	I-W				(1.2.1); TTWSFD; (1.2.3.2)
11	Iobs + Lobs		Nrev	I-Nrev			FOC	I-FOC		(1.2.1); TTWSFD; (1.2.3.2)
12	Iobs + Lobs	Iobs + Lobs	Nrev	I-Nrev			FOC	I-FOC		(1.2.1); TTWSFD; (1.2.3.2)
13	Iobs + Lobs				W	I-W	FOC	I-FOC		(1.2.1); TTWSFD; (1.2.3.2)
14	Iobs + Lobs	Iobs + Lobs			W	I-W	FOC	I-FOC		(1.2.1); TTWSFD; (1.2.3.2)
15	Iobs + Lobs		Nrev	I-Nrev	W	I-W	FOC	I-FOC		(1.2.1); TTWSFD; (1.2.3.2)
16	Iobs + Lobs	Iobs + Lobs	Nrev	I-Nrev	W	I-W	FOC	I-FOC		(1.2.1); TTWSFD; (1.2.3.2)
17			Nrev	I-Nrev						(1.2.1); TTWSFD; (1.2.3.2)
18					W	I-W				(1.2.1); TTWSFD; (1.2.3.2)
19							FOC	I-FOC		(1.2.1); TTWSFD; (1.2.3.2)
20			Nrev	I-Nrev	W	I-W				(1.2.1); TTWSFD; (1.2.3.2)
21					W	I-W	FOC	I-FOC		(1.2.1); TTWSFD; (1.2.3.2)
22			Nrev	I-Nrev			FOC	I-FOC		(1.2.1); TTWSFD; (1.2.3.2)
23			Nrev	I-Nrev	W	I-W	FOC	I-FOC		(1.2.1); TTWSFD; (1.2.3.2)
24			Nrev						I-intervals	(1.2.1); TTWDFD;W; t
25					W				I-intervals	(1.2.1); TTWDFD;Nrev; t
26							FOC		I-intervals	(1.2.1); TTWDFD;Nrev; t
27			Nrev		W				I-intervals	(1.2.1); TTWDFD;Nrev; W; t
28					W		FOC		I-intervals	(1.2.1); TTWDFD;Nrev; t
29			Nrev				FOC		I-intervals	(1.2.1); TTWDFD;Nrev; W; t
30			Nrev		W		FOC		I-intervals	(1.2.1); TTWDFD;Nrev; W; t

(u) Step (line) 7 of Table 3 is about determining the single applicable line of Table 1 and Table 2, among the 30 lines of each one of these two tables. This determination is based on the availability and quality of the applicable values of the occurrences of the functions to be approximated, h′(x_i, y_i, z_i,…; J), J = 8, 9, 12, 13, 14, on one hand, as well as of the values of the data sets pursuant to Equations (11) or (17) on the other. Regarding the requisite availability and quality of these applicable values pursuant to Table 2, reference is made to [126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143] in terms of TTWDFD and TTWSFD, to [36,79,80,81,82,83,84,85,86,87,97,100,110,144] in terms of FOC, and to [36,144] in terms of r, Nrev, P and W.

(v) As far as the acquisition of these functions’ and data values is concerned, reference is made to Steps (lines) A and B of Table 3 which are naturally always taken before the six (6) Steps (lines) 1 to 6 of Table 3; reference is also made to Appendix O and Appendix Q of this article. Appendix O includes information on the shipboard acquisition apparatus and systems relevant to this article. Step (line) A of Table 3 pertains to the acquisition of the a/m values by means of the respective shipboard systems, which is followed by the transmission of these values ashore and “to the cloud”, pursuant to a/m Appendix O and Appendix Q.

(w) Step (line) B of Table 3 is always taken after the a/m Step (line) A. As part of Step (line) B and pursuant to Appendix Q of this article, these functions’ and data values are screened; pre-processed; organized for input; synchronized; controlled for obvious errors and gaps; and validated, verified, supplemented and amended as far as any data gaps and obvious errors are concerned. This is achieved by means of respective data sets acquired via universal systems other than the shipboard systems, such as the ones discussed under Section 2.3, Section 2.3.1 and Section 2.3.2 of this article as well as Appendix J and Appendix K to it.

(x) Step (line) 8 of Table 3 comprises the imperative assignment statements pursuant to the outcome of Step 7 and Appendix Q. Step (line) 9 of Table 3 is about imperatively assigning a set of initially estimated values to the unknown parameters, α, β, γ, … [1], pursuant to Appendix H and Equation (10a) or (16a) in particular (depending on the outcome of Step 7).

(y) Step (line) 10 of Table 3 comprises the procedures for solving pursuant to Appendix H, those equations out of Equations (6)–(17) above, which are applicable in this regard pursuant to Step 7. After Step (line) 10 is completed, Step (line) 11 follows to finally return the set of the attained values of the unknown parameters, α, β, γ, … [1], yielding the minimum sum, s(α, β, γ, …) [1], of the squares of the residuals, f_i (α, β, γ, …) [1], pursuant to Appendix H and the applicable Equation (10b) or (16b) in particular (depending on the outcome of Step 7). Based on the assignment of Step (line) 11 discussed above, Step (line) 12 of Table 3 comprises the final definition of those of the occurrences, H′(x_i, y_i, z_i,…; α, β, γ,…; J), J = 8, 9, 12, 13, 14, of the approximating function, H(x_i, y_i, z_i,…; α, β, γ,…), in Equation (1) of Reference [1], which are applicable following Step (line) 7. These applicable occurrences yield the closest possible approximate representation [1] of those of the occurrences, h′(x_i, y_i, z_i,…; J), J = 8, 9, 12, 13, 14, of the function to be approximated, h(x_i, y_i, z_i,…) [1], which are applicable pursuant to Step 7 (line).

(z) After Step (line) 12 of Table 3 is concluded, Step (line) D of Table 3 follows. Step (line) D comprises the final assessment of whether the “local minimum” solution attained on the basis of the least-squares criterion by following Steps (lines) 10, 11 and 12 of Table 3 is the best-fit (most probable/least uncertain) one, or not. As already discussed in Section 1, this “local” attainable solution may prove to rely (effectively, in a stochastic manner), or not, on the set of the initial values of the unknown parameters, α, β, γ, … [1], assigned in Step (line) 9 of Table 3, whereas not all possible solutions are expected to be physically significant for a specific problem and circumstances thereof.

In fact, judgment is required to make sure that the attained solution is the most probable and least uncertain, best-fit, one for the particular problem and circumstances. Such judgment is based mainly on knowledge and experience pertaining to solving the same or similar types of problems in the same or a similar or another way, as well as on bounds (threshold values) of the unknown parameters, α, β, γ, … [1], fencing physical significance areas, and also on testing (sensitivity analysis). As a general rule and based on the a/m criteria, if the above assessment Step (line) D of Table 3 yields a negative or a questionable outcome, a rerun/rework loop is applicable to return the computation back to Step (line) 9 of Table 3 and assign a different set of initial values of the unknown parameters, α, β, γ, …, [1]. This is to reassess whether this new set will yield, or not, a best-fit (most probable/least uncertain) solution, and so on.

However, in practice, and as far as the particular problem and circumstances thereof (as these are examined in the context of the subject research work) are concerned, a negative or a questionable outcome is not the general case and instead is an exceptional one, even though such an exceptional case cannot be entirely ruled out at all times. There are many reasons (root causes) supporting the above observation, including the following:

One is obviously related to how well the physical and technical problem under scope is defined in the first place, and on how solid the transformation of the above definition is in terms of the, necessarily analytical and comprehensive mathematical formulations inherent in the respective applicable occurrence of the approximating function, H′(x_i, y_i, z_i,…; α, β, γ, …; J) [1], pursuant to Equation (1) of Reference [1].

Another is related to the extreme non-linearity of the a/m physical and technical problem, which significantly narrows down the physically significant areas of values of unknown parameters, α, β, γ, …, [1], especially after considering that these unknown parameters, α, β, γ, …, [1], effectively represent in a systematic (even though inconspicuous) manner, fundamental correlations inherent in the different conditions pertaining to the specific problem.

Yet another is related to the intense experience already gained in solving the specific problem, not only in other ways, but in the way presented in this article as well. In other words, after already attaining a best-fit (most probable/least uncertain) solution for a variety and a significant population of different conditions pertaining to the specific problem, it is only natural, reasonable and rewarding to proceed as follows:

Every time a set of new conditions appears, the set of initial values of the unknown parameters, α, β, γ, … [1], to be assigned in Step (line) 9 of Table 3, will rely, “mutatis—mutandis”, on those sets of “correct”, best-fit (most probable/least uncertain) values already attained in Step (line) 11 of Table 3 for other conditions, being as near as possible to the set of new conditions. In this regard, an additional, underlying data-basic aspect of the integrated framework, applicable for systematically tackling the subject problem, is highlighted.

Last but not least, and among yet other reasons (root causes) explaining the above observation, this observation generally applies in cases other than the ones of ill-conditioned appearances of the specific problem. In this regard, particular reference is made to Section 2.4, as well as to Appendix L, Appendix M, Appendix N, Appendix O, Appendix P, Appendix Q, Appendix R, Appendix S and Appendix T, of this article.

2.2. The Main Engine, Thermo-Fluid and Frictional, SFOC Approximating Functional [1] System

With regard to the following unknown, vessel/main engine’s specific, expression referred to in Section 2.1.5:

SFOC_i = SFOC(P_i, r_i, dr_i/dt, cor, Xj, j = 1, M)

(18)

SFOC_i stands for the applicable occurrence, H′(x_i, y_i, z_i,…; α,β,γ,…; J), of the approximating function, H(x_i, y_i, z_i,…; α, β, γ, …) [1], of any main engine’s specific fuel oil consumption, SFOC, in the context of Reference [1] and Equation (1) of it. The unknown parameters, α, β, γ, … [1], in the context of Equation (1) of Reference [1], equal, Xj, j = 1, M, while n, the number of points, i, in Equations (2) and (4) of Reference [1], equals, I_sfoc-obs, or, (I_sfoc-obs)′, in the context of Section 2.2.

The above function is determined by evaluating the a/m unknown parameters, α, β, γ, … [1], in the context of Equation (1) of Reference [1], which are equal to the calibration constants of the applicable composite thermo-fluid and frictional, SFOC, functional system, Xj, j = 1, M. This allows us to obtain the closest possible approximate representation [1] of SFOC for all applicable (reported) combinations of the above variables, x, y, z,… [1], pursuant to Appendix Q. The a/m unknown parameters, α, β, γ, … [1], are equal to the calibration constants of the applicable thermo-fluid and frictional, SFOC, functional system, Xj, j = 1, M, and are evaluated pursuant to the above, on the basis of the thermo-fluid and frictional components of the a/m functional system for different diesel engines.

These components have been developed and successfully applied by the author of this article on the basis of his expertise which is related to his research conducted between 1986 and 1995 and between 2005 and 2022 [7,14,15,16,17,18,19,20,21,22,23,24,25]. This includes research conducted in collaboration with other researchers and faculties at the NTUA Internal Combustion Engines Laboratory, as well as other relevant research work by the author that has not been reported to date and has been specifically extended to address the design and operation of two-stroke marine diesel main engines. These components also consider research findings reported by other relevant literature [26,27,28,29,30,31,32,33,34,35]. The above-described composite SFOC functional system comprising the a/m different thermo-fluid and frictional components for any given main engine, can be evaluated in further detail by approximating any available applicable shop tests (factory acceptance tests, or FAT), bollard tests and/or sea trials, as well as observed results data points and/or actual operational observed data points, i = 1, I_sfoc-obs, or, (I_sfoc-obs)′, of SFOC values, by satisfying the following requisite conditions:

h′(x_i, y_i, z_i,…; 19) = (SFOC)_i

(19h)

H′(x_i, y_i, z_i,…; α, β, γ, …; 19) = SFOC(P_i, r_i, dr_i/dt, cor, Xj, j = 1, M)

(19H)

n′(19) =I_sfoc-obs

(19n)

and/or:

h′(x_i, y_i, z_i,…; 20) = FOC_i/W_i

(20h)

H′(x_i, y_i, z_i,…; α, β, γ, …; 20) = SFOC(P_i, r_i, dr_i/dt, cor, Xj, j = 1, M)

(20H)

n′(20) = (I_sfoc-obs)′

(20n)

(α, β, γ, …)′ = (Xj, j = 1, M)

(21a)

(Xj, j = 1, M) = (α, β, γ, …)″

(21b)

(x_i, y_i, z_i,…) =(P_i, r_i, dr_i/dt, cor) =

= [data set (1.1.1)]_i U [data set (1.1.2)]_i U [data set (1.1.3)]_i U [data set (1.1.4)]_i

(22)

Equation Sets (19) or (20), or their combination, are solved in conjunction with Equations (18), (21) and (22) to determine the a/m calibration constants of the applicable composite thermo-fluid and frictional SFOC approximating functional [1] system, Xj, j = 1, M. If Equation Set (20) is solved in conjunction with Equations (18), (21) and (22), without however being combined with Equation Set (19), then such a solution is defined as the Analysis Type 1 discussed in Section 2.4, Section 3, Section 3.1, Section 3.2 and Section 4 and Appendix X of this article. With regard to Equation Sets (18)–(22), reference is made to Appendix H, while, n, the number of points, i, in Equations (2) and (4) of Reference [1] equals I_sfoc-obs or (I_sfoc-obs)′, pursuant to Equations (19n) or (20n). In cases where Equation Sets (18)–(22) are resolved simultaneously, reference is also made to Appendix H. In this regard, and in the context of this article, the formulations above, (19h), (19H), (19n), (20h), (20H), (20n), (21a)/(21b) and (22), are not to be considered as equations only, but also as imperative assignments statements, readily applicable in the context of any imperative numerical analysis paradigm or any imperative programming language and/or compiler. With regard to Equations (21a)/(21b), see the Nomenclature, Steps (lines) 3 and 5 in Table 3 below, and Appendix H.

Particular attention should be paid to cor, the set of parameters utilized for the correction, alignment and benchmarking of the SFOC values of the main engines with regard to fuel type, lower calorific value and other fuel quality characteristics, as well as to SFOC-related environmental, installation and other conditions considering in this regard relevant industry standards and experience, as well as Section 2.4. The above set of parameters should be predetermined, at least for the most part of them, meaning that the fuel quality data and matching environmental conditions are, ideally, expected to be known (see Section 2.3, “Big data” set).

Section 2.2 draws upon Reference [17] as well as Appendix I of this article. The solutions of the above Equation Sets (18)–(22) are equivalent to determining the most probable SFOC occurrence of the approximating function, H(x_i, y_i, z_i,…; α, β, γ, …) [1], which will produce a physically/technically significant and consistent closest possible approximate representation [1] of SFOC for all applicable (reported) combinations of variables, x, y, z,… [1], in the context of Equation (1) of Reference [1] and pursuant to points (1.1.1), (1.1.2), (1.1.3) and (1.1.4) in Section 1.1.1 (see also Appendix Q).

The streamlined computational sequence for applying and solving Equations (18)–(22) above, is depicted in the six (6) Steps (lines) 1 to 6 of Table 3 below. As also mentioned in Section 2.1.5, these six (6) Steps (lines) 1 to 6 of Table 3 below always run before the six (6) Steps (lines) 7 to 12 of Table 3. The streamlined sequential nature of the computational Steps (lines) in Table 3 is directly and essentially interrelated to the character and context of any imperative numerical analysis paradigm and/or any imperative programming language and/or compiler (see also Appendix H).

Step (line) 1 of Table 3 below is about determining if either one of Equation Sets (19) or (20) can be applied and resolved without the other, however together with Equations (18), (21) and (22), or instead if Equation Sets (19) and (20) can be applied and resolved together, always together with Equations (18), (21) and (22). This determination is mainly related to the availability and quality of the applicable values of the occurrences of the functions to be approximated, h′(x_i, y_i, z_i,…; J), J = 19, 20, on one hand, as well as of the values of the data sets pursuant to Equation (22) on the other. Regarding the requisite availability and quality of these applicable values, reference is made to [36,79,80,81,82,83,84,85,86,87,97,100,110,144] in terms of FOC, and to [36,144] in terms of r, Nrev, P and W.

As far as the acquisition of these functions’ and data values is concerned, reference is made to Steps (lines) A and B of Table 3 below (which naturally always run before the six (6) Steps (lines) 1 to 6 of Table 3), as well as to Appendix O and Appendix Q of this article. Steps (lines) A and B and Appendix O and Appendix Q are applicable in the exact manner laid out in perfect detail in the respective part of Section 2.1.5, just after the respective discussion on Step (line) 7 of Table 3 below.

Step (line) 2 of Table 3 below comprises the imperative assignments pursuant to the respective outcome of the previous Step 1 (see also Appendix Q). Step (line) 3 of Table 3 below concerns imperatively assigning a set of initially estimated values to the unknown parameters, α, β, γ, … [1], pursuant to Appendix H and Equation (21a) in particular.

Step (line) 4 of Table 3 below comprises the procedures for solving pursuant to Appendix H, the equations in Equations superset (18) to (22) above, which are applicable in this regard pursuant to Step 1 above. After Step 4 is completed, Step 5 (line) of Table 3 below follows to finally return the set of the attained values of the unknown parameters, α, β, γ, … [1], yielding the minimum sum, s(α, β, γ, …) [1], of squares of the residuals, f_i (α, β, γ, …) [1], pursuant to Appendix H and Equation (21b) in particular. Based on Step 5’s assignment discussed above, Step (line) 6 of Table 3 below comprises the final definition of those of the occurrences, H′(x_i, y_i, z_i,…; α, β, γ,…; J), J = 19, 20, of the approximating function, H(x_i, y_i, z_i,…; α, β, γ,…), in Equation (1) of Reference [1], which are applicable pursuant to Step 1 above. These applicable occurrences incorporate the final SFOC definition pursuant to Equation (18) above and yield the closest possible approximate representation [1] of those of the occurrences, h′(x_i, y_i, z_i,…; J), J = 19, 20, of the function to be approximated, h(x_i, y_i, z_i,…) [1], which are applicable pursuant to Step 1 above.

After Step (line) 6 of Table 3 below is concluded Step (line) C of Table 3 below follows. With regard to Steps (lines) 3, 4, 5, 6 and C of Table 3 below, exactly the same discussion is applicable as in the respective part of Section 2.1.5 (from point (z) until the end of Section 2.1.5) regarding Step (line) D of Table 3 below, provided however that Steps (lines) 9, 10, 11, 12 and D of Table 3 are replaced “mutatis—mutandis” by Steps (lines) 3, 4, 5, 6 and C of the same table.

Table 3. Streamlined computational sequences for applying and solving Equations (6)–(22).

Step	Applicable Equations Superset	Computational Content
A	Pursuant to Steps 1, 2, 7 & 8 below	Data acquisition pursuant to Appendix O and Appendix Q
B	Pursuant to Steps 1, 2, 7 & 8 below	Data pre-processing pursuant to Appendix Q
1	{(19h)/(19n) and/or (20h)/(20n),(22)}	Determination of applicable Equations
2	{(19h)/(19n) and/or (20h)/(20n),(22)}	Assignments as per applicable Equations
3	(21a)	Assignments as per Equation (21a)
4	{(18),(19H) and/or (20H)}	As per Appendix H & applicable equations
5	(21b)	Assignments as per Equation (21b)
6	{(18),(19H) and/or (20H)}	Finalization as per applicable equations
C	Pursuant to Steps 4, 5 & 6 above	Assessment of initial values assigned in Step 3
7	(*) {(8n),(9n),(11),(12h),(12n),(13h), (13n),(14h),(14n),(15),(17)} (*)	Determination of Tables’ 1 and 2 applicable line
8		Assignments as per Table’s 2 applicable line
9	(10a) or (16a)	Assignments as per Table’s 1 applicable line
10	{(6),(7),(8H),(9H),(12H),(13H),(14H)}	As per Appendix H & Table’s 1 applicable line
11	(10b) or (16b)	Assignments as per Table’s 1 applicable line
12	{(6),(7),(8H),(9H),(12H),(13H),(14H)}	Finalization as per Table’s 1 applicable line
D	Pursuant to Steps 10, 11 & 12 above	Assessment of initial values assigned in Step 9

Note (*): The combined (merged) content of the second column of Table 3 for Steps (lines) 7 and 8, is applicable with regard to the content of the third column of Table 3, for either Step (line) 7, or 8.

Table 3. Streamlined computational sequences for applying and solving Equations (6)–(22).

Step	Applicable Equations Superset	Computational Content
A	Pursuant to Steps 1, 2, 7 & 8 below	Data acquisition pursuant to Appendix O and Appendix Q
B	Pursuant to Steps 1, 2, 7 & 8 below	Data pre-processing pursuant to Appendix Q
1	{(19h)/(19n) and/or (20h)/(20n),(22)}	Determination of applicable Equations
2	{(19h)/(19n) and/or (20h)/(20n),(22)}	Assignments as per applicable Equations
3	(21a)	Assignments as per Equation (21a)
4	{(18),(19H) and/or (20H)}	As per Appendix H & applicable equations
5	(21b)	Assignments as per Equation (21b)
6	{(18),(19H) and/or (20H)}	Finalization as per applicable equations
C	Pursuant to Steps 4, 5 & 6 above	Assessment of initial values assigned in Step 3
7	(*) {(8n),(9n),(11),(12h),(12n),(13h), (13n),(14h),(14n),(15),(17)} (*)	Determination of Tables’ 1 and 2 applicable line
8		Assignments as per Table’s 2 applicable line
9	(10a) or (16a)	Assignments as per Table’s 1 applicable line
10	{(6),(7),(8H),(9H),(12H),(13H),(14H)}	As per Appendix H & Table’s 1 applicable line
11	(10b) or (16b)	Assignments as per Table’s 1 applicable line
12	{(6),(7),(8H),(9H),(12H),(13H),(14H)}	Finalization as per Table’s 1 applicable line
D	Pursuant to Steps 10, 11 & 12 above	Assessment of initial values assigned in Step 9

Note (*): The combined (merged) content of the second column of Table 3 for Steps (lines) 7 and 8, is applicable with regard to the content of the third column of Table 3, for either Step (line) 7, or 8.

2.3. “Big Data” Set

The “big data” set required for implementing the above-described fixed-pitch (screw) marine propulsion approximating functional [1] system, as per Equations (1)–(22) as/if applicable, comprises two distinct types of data: (1) ship-tracking data (AIS, LRIT, other) and (2) environmental, “met-ocean” (meteorological and oceanographic), “hind-cast” or actual (historical) data; see Section 2.3.1, Section 2.3.2 and Section 2.4.

2.3.1. Ship-Tracking Data (AIS, LRIT, and Other)

The AIS tracking data within the relevant “big data” set include a number of relevant data points which are listed and discussed in further details in the relevant References [17,133,134,135,136,137,138,139,140,141,142,143] and in Appendix J of this article.

AIS data are transmitted by ships at predefined frequencies related to the navigational status, speed and Rate Of Turn (ROT) of the ships transmitting them. However, they are received and made available in subsets of lesser, varying, frequencies depending on the actual circumstances and capabilities of the receiving ships, satellite stations and terrestrial stations. Although it would be really “nice”, there is not one universal system to receive and store all AIS data from all ships at all frequencies; instead, there is an aftermarket trading AIS data at different levels of wholesale and/or retail.

The above information is relevant to universal ship-tracking data. However, such data are also (not universally) available from individual vessels’ shipboard systems.

2.3.2. Environmental, “Met-Ocean” (Meteorological and Oceanographic) Data

The environmental “met-ocean” (meteorological and oceanographic), “hind-cast” or actual (historical), data in the relevant “big data” set referred to in this article, include a number of relevant data points which are listed in detail in Reference [17], and in Appendix K of this article.

The above information is relevant to universal environmental “met-ocean” data. However, such data are also (not universally) available from individual vessels’ shipboard systems.

2.4. Reduced Uncertainty, Reasonable Degree of Certainty, Reasonable Assurance and Materiality

The methodological concept, context and background information of the closest possible approximate representation of one function by another [1] applicable to “… certain engineering applications involving … types of problems which are of much greater complexity …” [1] in the context of this article, is available in Appendix L.

The approximation of residuals as such is applied in the context of Reference [1] and discussed in Appendix M. Appendix N introduces two requisite conditions for systematically attaining remarkably close approximate representations [1].

With regard to the different components of the functional systems to be approximated, h(x_i, y_i, z_i,…) [1], reference is made to Appendix O. The error component of the functional system to be approximated, h(x_i, y_i, z_i,…) [1], is discussed in Appendix P. Appendix Q describes the contemporary, shipboard and universal, “big data” acquisition systems falling within the scope of this article.

For details on the, stochastic nature of the functional system to be approximated, h(x_i, y_i, z_i,…) [1] see Appendix R. The hybrid, potentially stochastic definition of the deterministic approximating function, H(x_i, y_i, z_i,…; α, β, γ, …) [1], is discussed in Appendix S; this discussion also explains why the original tripartite, combined methodology presented in this article is referred to as a hybrid, deterministic and stochastic, methodology. The original methodology presented in this article is designated as hybrid based on the above grounds; the same designation as hybrid is true for this combined methodology based on the seamless integration, articulation and amalgamation of three distinct disciplines: mechanical engineering, and the thermo-fluid and frictional analysis of diesel engines under seagoing conditions; marine engineering, and the hydrodynamics of fixed-pitch propellers under seagoing conditions; and applied mathematics, numerical analysis, algorithmic computing and statistics.

Each one of the above disciplines is considered in the necessary depth to justify the originality and the significance of the present article, of the research behind it, and of the designation of the combination of all three of them as hybrid.

Appendix T provides a detailed discussion on the regulated [79,81] definition of uncertainty in the context and within the scope of this article and Reference [1]. Summarizing Appendix L, Appendix M, Appendix N, Appendix O, Appendix P, Appendix Q, Appendix R, Appendix S and Appendix T of this article, it is crystal-clear that 72 years after the first public presentation and publication of Reference [1], the publications from 2015 to 2016 effecting the stochastic, and sustainability-related regulatory framework [79,80,81,82,83] referring to and/or regulating the reduced uncertainty, reasonable degree of certainty, reasonable assurance and materiality [79,81] discussed above are in perfect and remarkable alignment with Reference [1]. In fact, connecting Reference [1] to References [79,81] as far as their regulatory stochastic context [79,80,81,82,83] is concerned, as well as to other relevant references such as [54,145,146], is also a latent contribution of the present article.

2.4.1. FOC Uncertainty Type 1.1 (Analysis Type 1, per Voyage)

FOC Uncertainty Type 1.1, related to the results of Analysis Type 1 (see Table 4, Table 5 and Table 6 in Section 3; Section 3.1; correlated to the above Table 4, Table 5 and Table 6, Table 7 in Section 3.1; Column 2 of Table 7), is calculated on a per-voyage basis as follows (particular reference is made to the absolute value function including the summation of the numerator of this fraction):

$$\mathit{Uncertainty}\mathit{Type}1.1={\displaystyle \frac{| {\sum }_{i=1}^{Isfoc-voy}\{ Wi SFOC\left(Pi, ri,\frac{dri}{dt},cor,Xj, j = 1,M\right) - FOCi \} | }{{\sum }_{i=1}^{Isfoc-voy}Wi SFOC\left(Pi, ri,\frac{dri}{dt},cor,Xj, j = 1,M\right)}}$$

(23)

Column 2 of Table 7 in Section 3.1 of this article lists the total FOC percentage over the whole reporting period of vessel voyages with FOC Uncertainty Type 1.1 > 10% calculated on a per-voyage basis pursuant to Equation (23), whereas pursuant to the regulated uncertainty definition [79] (see Appendix T), the percentage in Column 2 of Table 7 in Section 3.1 of this article should be lower than 5%.

2.4.2. FOC Uncertainty Type 1.2 (i) (Analysis Type 1, for a Specific Reporting Interval, i)

FOC Uncertainty Type 1.2 (i), related to the results of Analysis Type 1 (see Section 3.1; Column 3 of Table 7), for each specific reporting interval i, i = 1, (I_sfoc-obs)′, is calculated as follows (particular attention should be given to the absolute value function including the entire right side of the equation below):

$$\mathit{Uncertainty}\mathit{Type}1.2 (i)=|1- FOCi/\left\{Wi SFOC\left(Pi, ri,{\displaystyle \frac{dri}{dt}},cor,Xj, j = 1,M\right)\right\}|$$

(24)

Column 3 of Table 7 in Section 3.1 of this article lists the total FOC percentage over the whole reporting period, of vessels’ reporting intervals with FOC Uncertainty Type 1.2 (i) > 10% calculated for each one specific reporting interval, i, pursuant to Equation (24).

2.4.3. FOC Uncertainty Type 2.1 (Analysis Type 2, per Voyage)

FOC Uncertainty Type 2.1, related to the results of Analysis Type 2 (see Section 3.2 and Table 8 in it; Column 2 of Table 8), is calculated on a per-voyage basis as follows (particular reference is made to the absolute value function including the summation of the numerator of this fraction):

$$\mathit{Uncertainty}\mathit{Type}2.1={\displaystyle \frac{|{\sum }_{i=1}^{I-voy}\{ FOCi- {\int }_{start}^{end} Pi\left(Cj, j = 1, K\right) SFOC\left(Pi, ri,\frac{dri}{dt},cor\right) dt \} |}{{\sum }_{i=1}^{I-voy} {\int }_{start}^{end} Pi\left(Cj, j = 1, K\right) SFOC\left(Pi, ri,\frac{dri}{dt},cor\right) dt }}$$

(25)

Column 2 of Table 8 in Section 3.2 of this article lists the total FOC percentage over the whole reporting period of vessel voyages with FOC Uncertainty Type 2.1 > 10% calculated on a per-voyage basis pursuant to Equation (25), whereas pursuant to the regulated uncertainty definition [79] (see Appendix T), the percentage in Column 2 of Table 8 in Section 3.2 of this article should be lower than 5%.

2.4.4. FOC Uncertainty Type 2.2 (i) (Analysis Type 2, for a Specific Reporting Interval, i)

FOC Uncertainty Type 2.2 (i), related to the results of Analysis Type 2 (see Section 3.2; Column 3 of Table 8), for each specific reporting interval i, i = 1, I_-intervals, is calculated as follows (particular attention should be given to the absolute value function including the entire right side of the equation below):

$$\mathit{Uncertainty}\mathit{Type}2.2 (i)=|1-FOCi/{\int }_{start}^{end}Pi\left(Cj, j = 1, K\right) SFOC\left(Pi, ri,{\displaystyle \frac{dri}{dt}},cor\right)dt|$$

(26)

Column 3 of Table 8 in Section 3.2 of this article lists the total FOC percentage over the whole reporting period, of vessels’ reporting intervals with FOC Uncertainty Type 2.2 (i) > 10% calculated for each one specific reporting interval, i, pursuant to Equation (26).

2.4.5. FOC Uncertainty Type 3.1 (Analysis Type 3, per Voyage)

FOC Uncertainty Type 3.1, related to the results of Analysis Type 3 of (see Section 3.2 and Section 3.3 of this article, and Column 6 of Table 8 in Section 3.2 in particular), is calculated on a per-voyage basis pursuant to Equation (25) in Section 2.4.3. However, the values of shaft/engine power, P, and, SFOC, calculated using Equations (6) and (18) pursuant to Section 3.3 of this article (Analysis Type 3, per voyage), are different to those calculated using the same equations pursuant to Section 3.2 of this article (Analysis Type 2, on a per-voyage basis).

Column 6 of Table 8 in Section 3.2 of this article lists the total FOC percentage over the whole reporting period of vessel voyages with FOC Uncertainty Type 3.1 > 10% calculated on a per-voyage basis pursuant to Equation (25) in Section 2.4.3 and to this Section 2.4.5, whereas pursuant to the regulated uncertainty definition [79] (see Appendix T), the percentage in Column 6 of Table 8 in Section 3.2 of this article should be lower than 5%.

2.4.6. FOC Uncertainty Type 3.2 (i) (Analysis Type 3, for a Specific Reporting Interval, i)

FOC Uncertainty Type 3.2(i), related to the results of Analysis Type 3 (see Section 3.2 and Section 3.3 of this article, and Column 7 of Table 8 in Section 3.2 in particular), is calculated for each specific reporting interval i, i = 1, I_-intervals, pursuant to Equation (26) in Section 2.4.4. However, the values of shaft/engine power, P, and, SFOC, calculated using Equations (6) and (18) with reference to Section 3.3 of this article (Analysis Type 3, for a specific reporting interval, i), are different to the ones calculated using the same equations with reference however to Section 3.2 of this article (Analysis Type 2, for a specific reporting interval, i).

Column 7 of Table 8 in Section 3.2 of this article lists the total FOC percentage over the whole reporting period, of vessels’ reporting intervals with FOC Uncertainty Type 3.2 (i) > 10% calculated for each one specific reporting interval, i, pursuant to Equation (26) in Section 2.4.4 and to this Section 2.4.6.

2.4.7. FOC Materiality Type 1 (Analysis Type 1)

The materiality level is defined and actually regulated [81] as the quantitative threshold or cut-off point above which any erroneous entries inherent in the acquired data, individually or taken together, are considered to be material [81]. The above materiality level for FOC, on a ship-specific, whole reporting period basis, is regulated to 5% [81]. For more details on the regulatory stochastic context first published on 2015 [79], pertaining to uncertainty, certainty, materiality and assurance, see Appendix T of this article.

FOC Materiality Type 1, related to Analysis Type 1 (see Section 3.1; Column 1 of Table 7), is calculated on a ship-specific whole-reporting-period basis as follows (particular reference is made to the absolute value function including the summation of the numerator of this fraction):

$$\mathit{Materiality}\mathit{Type}1={\displaystyle \frac{|{\sum }_{i=1}^{{(Isfoc-obs)}^{\prime }}\{ Wi SFOC\left(Pi, ri,\frac{dri}{dt},cor,Xj, j =1,M\right) - FOCi\}| }{{\sum }_{i=1}^{{(Isfoc-obs)}^{\prime }} FOCi }}$$

(27)

Column 1 of Table 7 in Section 3.1 lists the FOC Materiality Type 1 (Analysis Type 1) values calculated over the whole reporting period of vessel voyages pursuant to Equation (27), whereas pursuant to the a/m regulated materiality definition [81] the percentages in Column 1 of Table 7 should be lower than 5%.

2.4.8. FOC Materiality Type 2 (Analysis Type 2)

FOC Materiality Type 2, related to Analysis Type 2 (see Section 3.2; Column 1 of Table 8), is calculated on a ship-specific whole-reporting-period basis as follows (particular reference is made to the absolute value function including the summation of the numerator of this fraction):

$$\mathit{Materiality}\mathit{Type}2={\displaystyle \frac{|{\sum }_{i=1}^{I-intervals}\{ FOCi-{\int }_{start}^{end}Pi\left(Cj, j=1, K\right) SFOC\left(Pi, ri,\frac{dri}{dt},cor\right)dt\}| }{{\sum }_{i=1}^{I-intervals} FOCi }}$$

(28)

Column 1 of Table 8 in Section 3.2 lists the FOC Materiality Type 2 (Analysis Type 2) values calculated over the whole reporting period of vessel voyages pursuant to Equation (28), whereas pursuant to the regulated materiality definition [81] (see Section 2.4.7) the percentages in Column 1 of Table 8 should be lower than 5%.

2.4.9. FOC Materiality Type 3 (Analysis Type 3)

FOC Materiality Type 3, related to Analysis Type 3 (see Section 3.2 and Section 3.3, and Column 5 of Table 8 in Section 3.2 in particular), is calculated on a ship-specific whole-reporting-period basis pursuant to Equation (28) in Section 2.4.8. However, the values of shaft/engine power, P, and, SFOC, calculated using Equations (6) and (18) and pursuant to Section 3.3 of this article (Analysis Type 3), are different to those calculated using the same equations with reference to Section 3.2 of this article (Analysis Type 2).

Column 5 of Table 8 in Section 3.2 lists the FOC Materiality Type 3 (Analysis Type 3) values calculated over the whole reporting period of vessel voyages pursuant to Equation (28) in Section 2.4.8 and to this Section 2.4.9, whereas pursuant to the regulated materiality definition [81] (see Section 2.4.7) the percentages in Column 5 of Table 8 should be lower than 5%.

2.4.10. FOC Standard Deviation Type 1 (Analysis Type 1)

FOC standard deviation is calculated on a ship specific, whole-reporting-period basis for the Analysis Types 1 and 2 (see Section 3.1 and Section 3.2 of this article), pursuant to the equations in Section 2.4.10 and Section 2.4.11. Relevant context and background information are available in Appendix U.

FOC Standard Deviation Type 1, related to Analysis Type 1 (see Section 3.1; Column 4 of Table 7 in particular), on a ship-specific whole-reporting-period basis, is equal to:

$$\begin{gathered} \mathit{Standard}\mathit{Deviation}\mathit{Type}1= \\ =\sqrt{{\sum }_{i=1}^{{(Isfoc-obs)}^{\prime }}\{FOCi/{\sum }_{i=1}^{{(Isfoc-obs)}^{\prime }} FOCi\} \{{ 1-Wi SFOC\left(Pi, ri,{\displaystyle \frac{dri}{dt}},cor,Xj, j =1,M\right)/FOCi \}}^2} \end{gathered}$$

(29)

Column 4 of Table 7 in Section 3.1 lists the FOC Standard Deviation Type 1 (Analysis Type 1) values calculated over the whole reporting period of vessel voyages pursuant to Equation (29).

2.4.11. FOC Standard Deviation Type 2 (Analysis Type 2)

FOC Standard Deviation Type 2, related to Analysis Type 2 (see Section 3.2; Column 4 of Table 8 in particular), on a ship-specific whole-reporting-period basis, is equal to:

$$\begin{gathered} \mathit{Standard}\mathit{Deviation}\mathit{Type}2= \\ =\sqrt{{\sum }_{i=1}^{I-intervals}\{FOCi/{\sum }_{i=1}^{I-intervals} FOCi\} \{{ 1 - {\int }_{start}^{end}Pi\left(Cj, j =1, K\right) SFOC\left(Pi, ri,{\displaystyle \frac{dri}{dt}},cor\right)dt/FOCi \}}^2} \end{gathered}$$

(30)

Column 4 of Table 8 in Section 3.2 lists the FOC Standard Deviation Type 2 (Analysis Type 2) values calculated over the whole reporting period of vessel voyages pursuant to Equation (30).

3. Results

The results of the original analysis introduced in the previous sections of this article, are based on shipboard and universal “big data” sets from/for 10 vessels. These data were acquired via the process described in Appendix O and Appendix Q of this article, as well as in those parts of Section 2.1.5 referring to Steps (lines) A and B of Table 3 in Section 2.2.

Some specific details of these vessels’ main engines are presented in Table 4 and more information about these engines may be found in Appendix V. Reference is also made to the last three columns of Table 7 of Section 3.1 which are considered in conjunction with the information presented in Table 4 with particular focus on the geometrical data. Section 4 and Appendix V also feature this information.

Table 4. Main engines specifics for the fleet of 10 vessels.

Vessel Number	Power (MCR, KW)	RPM (MCR)	Number of Cylinders	Cylinder Bore (m)	Stroke (m)
1	13,560	105	6	0.6	2.4
2	13,560	105	6	0.6	2.4
3	13,560	105	6	0.6	2.4
4	11,080	83	6	0.62	2.658
5	11,080	83	6	0.62	2.658
6	11,080	83	6	0.62	2.658
7	11,080	83	6	0.62	2.658
8	15,088	71.8	6	0.72	3.086
9	15,088	71.8	6	0.72	3.086
10	15,200	90	7	0.65	2.73

Table 4. Main engines specifics for the fleet of 10 vessels.

Vessel Number	Power (MCR, KW)	RPM (MCR)	Number of Cylinders	Cylinder Bore (m)	Stroke (m)
1	13,560	105	6	0.6	2.4
2	13,560	105	6	0.6	2.4
3	13,560	105	6	0.6	2.4
4	11,080	83	6	0.62	2.658
5	11,080	83	6	0.62	2.658
6	11,080	83	6	0.62	2.658
7	11,080	83	6	0.62	2.658
8	15,088	71.8	6	0.72	3.086
9	15,088	71.8	6	0.72	3.086
10	15,200	90	7	0.65	2.73

Some approximate hull data for the 10 vessels are presented in Table 5. For more information on the types of data presented in the different columns of Table 5, see Appendix W. All the values in Table 5 are nominal (reference) approximate values.

During the data acquisition period each of the above vessels was monitored during a number of legs (voyages) and the respective leg-specific (voyage-specific) hydrostatic conditions (draft, trim, displacement) were maintained, “effectively”, steady (during each one of these legs—voyages). The number of legs (voyages) per vessel and other relevant information are presented in Table 6.

Table 5. Approximate hull data for the fleet 10 vessels.

Vessel Number	Type	DWT (MT)	GT (RT)	Draft (m)	LOA (m)	LBP (m)	Beam (m)	Speed (knots)
1	Tanker	115,700	61,200	15	250	240	44	15
2	Tanker	115,700	61,300	15	250	240	44	15
3	Tanker	115,700	61,300	15	250	240	44	15
4	Tanker	114,300	62,900	15	250	240	45	14
5	Tanker	114,300	62,900	15	250	240	45	14
6	Tanker	114,300	62,900	15	250	240	45	14
7	Tanker	114,400	62,900	15	250	240	45	14
8	Tanker	157,500	81,600	17	275	265	48	14
9	Tanker	157,500	81,600	17	275	265	48	14
10	Tanker	159,200	82,600	17	275	265	48	14

Table 5. Approximate hull data for the fleet 10 vessels.

Vessel Number	Type	DWT (MT)	GT (RT)	Draft (m)	LOA (m)	LBP (m)	Beam (m)	Speed (knots)
1	Tanker	115,700	61,200	15	250	240	44	15
2	Tanker	115,700	61,300	15	250	240	44	15
3	Tanker	115,700	61,300	15	250	240	44	15
4	Tanker	114,300	62,900	15	250	240	45	14
5	Tanker	114,300	62,900	15	250	240	45	14
6	Tanker	114,300	62,900	15	250	240	45	14
7	Tanker	114,400	62,900	15	250	240	45	14
8	Tanker	157,500	81,600	17	275	265	48	14
9	Tanker	157,500	81,600	17	275	265	48	14
10	Tanker	159,200	82,600	17	275	265	48	14

For each of the 10 vessels, three different types of analysis were conducted using the data acquired during each reporting period.

In Analysis Type 1, Equation Set (18), (20)–(22) was applied for all legs and all reporting intervals thereof.

Table 6. Number of legs (voyages) and reporting intervals per vessel.

Vessel Number	Number of Legs	Analysis Type 1	Analysis Type 2	Analysis Type 3	Reporting Intervals
1	38	38	38	38	306
2	31	31	31	31	333
3	35	35	35	35	297
4	29	29	29	29	284
5	5	5	5	5	87
6	4	4	4	4	81
7	32	32	32	32	298
8	13	13	13	13	172
9	30	30	30	27	165
10	5	5	5	5	49

Table 6. Number of legs (voyages) and reporting intervals per vessel.

Vessel Number	Number of Legs	Analysis Type 1	Analysis Type 2	Analysis Type 3	Reporting Intervals
1	38	38	38	38	306
2	31	31	31	31	333
3	35	35	35	35	297
4	29	29	29	29	284
5	5	5	5	5	87
6	4	4	4	4	81
7	32	32	32	32	298
8	13	13	13	13	172
9	30	30	30	27	165
10	5	5	5	5	49

Next, in Analysis Type 2, Equations (6), (14h), (14Ha) and (15)–(17) were applied for all legs and all reporting intervals pursuant to line (case) 26 of Table 1 and Table 2 in Section 2.1.5.

As an alternative to Analysis 2, in Analysis Type 3, Equations (6), (14h), (14Ha) and (15)–(17) were applied for all voyages (legs), pursuant to line (case) 26 of Table 1 and Table 2 of Section 2.1.5; however, the above equations were applied independently “per leg” and for the reporting intervals of each leg.

As far as vessel 9 is concerned, Analysis Type 3 could not be applied to three discrete legs, since the number of reporting intervals for each one of these legs was lower than the minimum required. Thus, the number of legs indicated in Table 6 for vessel 9 and Analysis Type 3 is 27 instead of 30.

Table 6 also indicates the total number of reporting intervals, n′(15) = I_-intervals, pursuant to Equation (15), which is also equal to, n′(20) = (I_sfoc-obs)′, pursuant to Equation (20). This was applicable for Equations (6), (14h), (14Ha), (15)–(17) pursuant to line (case) 26 of Table 1 and Table 2 in Section 2.1.5, as well as for the Equation Set (18) and (20)–(22).

3.1. Analysis Type 1

For Analysis Type 1, the Equation Set (18) and (20)–(22) was applied for all 10 vessels, and for all legs during the reporting period for each vessel (see Table 6), as well as for all reporting intervals (see Table 6). The primary (essential) data for applying Equation Set (18) and (20)–(22) were the main engine running hours, t; the average shaft/main engine power (effectively, mechanical work, W);the average shaft/main engine rotational speed, r (effectively, revolutions, Nrev); and FOC (with regard to the above, reference is also made to Section 2.2).

The same composite thermo-fluid and frictional, SFOC, approximating functional [1] system based on relevant existing diesel engines models [7,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,42] was specifically extended to also cover also two-stroke main engines layout and their operation and was applied for all 10 vessels/main engines, while for each vessel/man engine, the applicable geometrical data in Table 4 were used. The same also applied for each engine’s other technical characteristics, and for cor (standing for the set of parameters to be utilized for the correction, alignment and benchmarking of the main engines’ SFOC values with regard to fuel type, the fuel’s lower calorific value, other fuel quality characteristics, and SFOC related environmental and other conditions pursuant to relevant industry standards and experience).

M, the number of unknown calibration constants of Xj, j = 1, M, was equal to 7 (M = 7), meaning that for each of the 10 vessels, an over-determined mathematical problem was resolved to determine the values of a set of seven unknown calibration constants on the basis of a number of observations for each vessel equal to the respective value of the last column of Table 6. This was achieved by applying the Equation Set (18) and (20)–(22) instead of attempting to obtain an exact solution to the above problem (which in any case, would be impossible). Each set of the seven unknown calibration constants determined for each vessel was applicable for each vessel/main engine throughout the whole reporting period and all their legs and reporting intervals, meaning that as far as Analysis Type 1 is concerned, no voyage/leg specific distinction was applicable, or necessary, throughout the whole reporting period.

Table 7. Analysis Type 1: Some key/critical results.

Vessel Number	Column 1 (%)	Column 2 (%)	Column 3 (%)	Column 4 (%)	Column 5 (%)	Column 6 (%)	Column 7 (%)
1	1.38	0.00	4.33	5.79	50.1	92.1	46.2
2	0.89	0.00	9.40	6.99	50.1	91.8	46.0
3	0.01	0.00	4.76	4.92	47.6	89.5	42.6
4	0.06	0.00	1.71	3.74	52.8	91.6	48.4
5	0.05	3.74	2.62	3.82	50.9	92.1	46.9
6	0.10	0.00	5.21	6.65	48.3	92.4	44.6
7	0.02	0.00	0.88	2.62	53.2	91.6	48.7
8	4.17	0.00	0.98	4.38	56.2	91.7	51.5
9	0.06	4.30	5.70	5.68	55.8	90.2	50.4
10	4.89	0.00	0.06	5.20	57.8	91.5	52.8

Table 7. Analysis Type 1: Some key/critical results.

Vessel Number	Column 1 (%)	Column 2 (%)	Column 3 (%)	Column 4 (%)	Column 5 (%)	Column 6 (%)	Column 7 (%)
1	1.38	0.00	4.33	5.79	50.1	92.1	46.2
2	0.89	0.00	9.40	6.99	50.1	91.8	46.0
3	0.01	0.00	4.76	4.92	47.6	89.5	42.6
4	0.06	0.00	1.71	3.74	52.8	91.6	48.4
5	0.05	3.74	2.62	3.82	50.9	92.1	46.9
6	0.10	0.00	5.21	6.65	48.3	92.4	44.6
7	0.02	0.00	0.88	2.62	53.2	91.6	48.7
8	4.17	0.00	0.98	4.38	56.2	91.7	51.5
9	0.06	4.30	5.70	5.68	55.8	90.2	50.4
10	4.89	0.00	0.06	5.20	57.8	91.5	52.8

On the basis of the above, a different occurrence of the SFOC function defined by means of Equation (18) was effectively determined for each of the 10 vessels/main engines throughout the whole reporting period of each vessel, whereas each one of the determined occurrences of the SFOC function, was also comparable in a technically and physically meaningful manner to the respective shop test SFOC values (curve) of each vessel/specific main engine, and to similar main engines in general.

Given the type of data available, the transient operation effect, dr/dt, in Equation Set (18) and (20) was not considered at all, meaning that in all cases (quasi-)steady-state performance was only considered on the basis of average data values, while transient main engine operation intervals (such as maneuvering in, maneuvering out, pilotage, etch) were not considered at all. Some key/critical results of Analysis Type 1 are presented in Table 7. With regard to the different results (percentages) presented in the different columns of Table 7, each one of these columns is explained in Appendix X and Appendix V of this article in the context of the respective formulations in Section 2.4.

3.2. Analysis Type 2

For Analysis Type 2, pursuant to line (case) 26 of Table 1 and Table 2 of Section 2.1.5, Equations (6), (14h), (14Ha) and (15)–(17) were applied for all legs during the reporting periods of all 10 vessels (see Table 6), as well as for all of their reporting intervals (see Table 6). The primary (essential) data for applying Equations (6), (14h), (14Ha) and (15)–(17) pursuant to line (case) 26 of Table 1 and Table 2 of Section 2.1.5 were the main engine running hours, t; the average shaft/main engine rotational speed, r (effectively, revolutions: Nrev); FOC; the average TTW (log) speed in the forward direction values (effectively, TTWDFD, values); and hydrostatic data (draft, trim and displacement, which in any case are voyage/leg-specific) pursuant to Correlation Scheme #4 in Section 2.1.4.

Regarding the different occurrences of the SFOC function defined by means of Equation (18) for each of the 10 vessels/main engines throughout the entire reporting period required for each vessel before applying Equations (6), (14h), (14Ha) and (15)–(17) pursuant to line (case) 26 of Table 1 and Table 2 of Section 2.1.5, these occurrences are available as a result of Analysis Type 1 (see Section 3.1). In fact, Analysis 2 commences only after Analysis Type 1 is completed.

As far as the function for defining and effectively determining shaft power, P, is concerned, applying and resolving Equations (6), (14h), (14Ha) and (15)–(17) pursuant to Reference [1] as per line (case) 26 of Table 1 and Table 2 of Section 2.1.5, is equivalent to determining the most probable occurrence of the approximating function, H(x_i, y_i, z_i,…; α, β, γ, …) [1], for Equation (6), H′(x_i, y_i, z_i,…; α, β, γ,…; 6), which would equal, P(x_i, y_i, z_i,…; Cj, j = 1, K).

Furthermore, this determination of the most probable occurrence for Equation (6), H′(x_i, y_i, z_i,…; α, β, γ,…; 6), is equivalent to determining the, physically/technically significant and consistent, closest possible approximate representation [1] of shaft power, P, for all applicable (reported) combinations of variables, x, y, z,… [1], in the context of Equation (1) in Reference [1] and pursuant to points, (1.2.1), (1.2.2), and, (1.2.3.1), in Section 1.1.2. In this regard, reference is also made to Correlation Schemes #1, #2, #3 and #4, as these are discussed in Section 2.1.1, Section 2.1.2, Section 2.1.3 and Section 2.1.4, as well as to the “big data” set availability and utilization discussed in Section 2.3 and in other parts of this article.

K, the number of unknown, reporting period-specific calibration constants, (Cj, j = 1, K)′, pursuant to Equation (16), is equal to 6 (K = 6), meaning that for each of the 10 vessels an over-determined mathematical problem was resolved to determine the values of a set of six unknown calibration constants on the basis of a number of observations for each vessel (equal to the value of the last column of Table 6). This solution was achieved by applying Equations (6), (14h), (14Ha) and (15)–(17) pursuant to line (case) 26 of Table 1 and Table 2 of Section 2.1.5, instead of attempting to obtain an exact solution to the problem (which in any case, would be impossible). Each set of unknown calibration constants for each vessel was applicable to each vessel/main engine throughout the reporting period and all the relevant legs and reporting intervals, meaning that as far as Analysis Type 2 is concerned, no voyage/leg-specific distinction was applicable, or necessary, throughout the entire reporting period. In this regard, particular reference is also made to Correlation Scheme 4 in Section 2.1.4.

Table 8. Analysis Types 2 and 3: Some key/critical results.

Vessel Number	Column 1 (%)	Column 2 (%)	Column 3 (%)	Column 4 (%)	Column 5 (%)	Column 6 (%)	Column 7 (%)
1	2.72	10.63	20.29	15.11	0.73	0.00	7.27
2	4.13	9.43	20.47	12.67	0.86	0.00	10.58
3	2.02	4.64	12.38	8.73	0.52	0.00	4.29
4	1.39	0.00	12.36	13.64	1.04	0.00	9.79
5	0.85	3.73	7.05	8.92	0.16	0.00	1.87
6	0.36	0.00	10.10	8.11	0.63	0.00	11.21
7	2.02	1.76	11.50	16.19	0.81	0.00	8.19
8	1.13	5.85	9.12	12.27	1.76	2.22	6.33
9	2.96	7.24	13.70	10.79	1.97	1.96	11.65
10	1.18	0.00	7.95	6.20	1.42	0.00	10.50

Table 8. Analysis Types 2 and 3: Some key/critical results.

Vessel Number	Column 1 (%)	Column 2 (%)	Column 3 (%)	Column 4 (%)	Column 5 (%)	Column 6 (%)	Column 7 (%)
1	2.72	10.63	20.29	15.11	0.73	0.00	7.27
2	4.13	9.43	20.47	12.67	0.86	0.00	10.58
3	2.02	4.64	12.38	8.73	0.52	0.00	4.29
4	1.39	0.00	12.36	13.64	1.04	0.00	9.79
5	0.85	3.73	7.05	8.92	0.16	0.00	1.87
6	0.36	0.00	10.10	8.11	0.63	0.00	11.21
7	2.02	1.76	11.50	16.19	0.81	0.00	8.19
8	1.13	5.85	9.12	12.27	1.76	2.22	6.33
9	2.96	7.24	13.70	10.79	1.97	1.96	11.65
10	1.18	0.00	7.95	6.20	1.42	0.00	10.50

A different occurrence of the shaft power, P, function defined in Equation (6), was effectively determined for the hull, appendages, propeller, propeller shaft line and stern tube of each of the 10 vessels, throughout the whole reporting period of each vessel. These determined occurrences of the shaft power, P, function are also technically and physically comparable to the respective main engine and propeller data, as well as to similar main engines and propellers in general.

Given the type of data available, any transient operation effect in Equations (6), (14h), (14Ha) and (15)–(17) pursuant to line (case) 26 of Table 1 and Table 2 of Section 2.1.5 was not considered, meaning that in all cases (quasi-)steady-state performance was only considered on the basis of data average values, while transient operation intervals, including but not limited to maneuvering in, maneuvering out, pilotage, and similar, were not considered. Some of the key results obtained via Analysis Type 2 are presented in Table 8.

The different percentages presented in Columns 1, 2, 3 and 4 of Table 8 are the results of Analysis Type 2, and are explained in Appendix X in the context of the respective formulations in Section 2.4.

3.3. Analysis Type 3

Analysis Type 3 is quite similar to Analysis Type 2 (see Section 3.2). The only difference is that as far as the function for defining and effectively determining shaft power, P, is concerned, Equations (6), (14h), (14Ha) and (15)–(17) were applied pursuant to line (case) 26 of Table 1 and Table 2 of Section 2.1.5 for each of the 10 vessels/main engines, but not throughout the whole reporting period for each vessel. Instead, the above equations were applied on per voyage/per leg basis.

K, the number of unknown, voyage-specific, calibration constants (Cj, j = 1, K)′, pursuant to Equation (16), is equal to 3 (K = 3), meaning that for each of the voyages/legs of each of the 10 vessels, an over-determined mathematical problem was resolved to determine the values of a set of three unknown calibration constants on the basis of the number of observations for each voyage (each leg) of each vessel equal to the respective number of reporting intervals per voyage (per leg), instead of attempting to obtain an exact solution to the above problem (which in any case, would be impossible).

Each set of three unknown calibration constants for each of the voyages/legs of each of the 10 vessels was applicable for each individual voyage/leg (but not throughout the entire reporting period) of each vessel, meaning that as far as Analysis Type 3 is concerned, a voyage/leg-specific distinction was applicable during the entire reporting period of each vessel.

On this basis, different occurrences of the shaft power, P, function defined using Equation (6) were effectively determined for the hull, appendages, propeller, propeller shaft line and stern tube for each of the voyages/legs of each of the 10 vessels, during the same reporting period applicable for each vessel. The determined occurrences of the shaft power, P, function were also technically and physically comparable to the respective main engine and propeller data, as well as to similar main engines and propellers in general.

Some key results of Analysis Type 3 are presented in Columns 5, 6, and 7 of Table 8. The results (in percentages) of Analysis Type 3 are presented in the last 3 columns of Table 8 and are explained in Appendix X of this article in the context of the respective formulations in Section 2.4 of this article.

4. Discussion

The results presented in Section 3, comprise only a very small portion of the data, results, information and experience available with regard to the original research presented in this article. The above results have been carefully selected to underline the nature of this work and to balance the structure of it with regard to the obvious and self-explanatory space-related and other limitations in this regard as well as in the application of the methods presented above (See also relevant content in and in-between Table 1 and Table 2 in Section 2.1.5 and in Appendix Y).

As far as the purely technical aspects and insight capabilities of Analysis Type 1 are concerned, reference is made to the last three columns of Table 7, which are considered in conjunction with the information and data in Table 4, with particular emphasis on the geometrical data. The different types of control and operation of the 10 main engines in Table 4 (for which mainly mechanical control is used for the first three engines, mainly electronic is used for the next six engines (4 to 9), and a control system comprising electronic and hybrid, electronic-and-hydraulic, elements is used for the 10th engine), also provide valuable insights.

These considerations are of particular relevance to the indicated efficiency and the mechanical (frictional losses) efficiency of the main engines (see Columns 5 and 6 of Table 7), which are critical for the technical management of vessels and their main engines, and can be evaluated via Analysis Type 1. Further detailed technical discussion with regard to the indicated efficiency and the mechanical (frictional losses) efficiency of the main engines (see Columns 5 and 6 of Table 7) is included in Section 1, Section 2.2 and Section 3.1; in Appendix V; and in References [36,37,38,39,40,41,42,43,44,144,147].

When comparing Table 7 (Analysis Type 1 results) and Table 8 (Analysis Types 2 and 3 results), Analysis Type 1 (seagoing main engine thermo-fluid/gas-dynamics and frictional analysis) appears to be far more successful than Types 2 and 3 (seagoing hydrodynamics analysis). An initial attempt to explain the above comparison would consider the fact that the approximating functional [1] systems applied in the thermo-fluid/gas-dynamics and frictional analysis of the main engines of the relevant vessels and reporting periods are more deterministic and less stochastic than those applied in the respective hydrodynamics analysis (Analysis Types 2 and 3). Additionally, the former use seven calibration “degrees of freedom” compared to the six and three calibration “degrees of freedom” used by the latter Analysis Types respectively. In fact, resolving such an over-determined, extremely non-linear, system of seven unknown “degrees of freedom” and of (up) to 333 (or even 49) observations/“equations” (reporting intervals), with the quality characteristics of the solution depicted in Columns 1, 2, 3 and 4 of Table 7, would only underline the deterministic strength, significance and merit of the approximating functional [1] systems applied in Analysis Type 1 above. Such a conclusion may also be derived based on this research work in its entirety and on the retrospective analysis of the existing literature duly considered in this research in particular.

Notwithstanding any of the above, and revisiting the a/m comparison between Table 7 (Analysis Type 1), and Table 8 (Analysis Types 2 and 3), one can only agree that a closer look at the above would reveal some interesting aspects of Analysis Types 2 and 3, particularly considering that the general context of seagoing hydrodynamics is “by definition”/“by birth” more stochastic and less deterministic than the main engine seagoing thermo-fluid/gas-dynamics and mechanical (frictional) efficiency analysis. Particular reference is made to Section 2.4, and its content and context with regard to the stochastic nature of the approximate representation [1] methodology presented in this article and to the relevant systematic and random factors.

In fact, a closer comparison of the results of Analysis Type 2 (Columns 1, 2, 3 and 4 of Table 8) and Analysis Type 3 (Columns 5, 6 and 7 of Table 8), easily reveals that the latter results are, from a qualitative standpoint and for all considered 10 vessels, almost as successful as those of Analysis Type 1. Analysis Type 3, as far as the data and results presented in this work are concerned, is based on the solution of an over-determined, extremely non-linear system of three unknown “degrees of freedom” and of (up) to 33 observations/“equations” (maximum number of reporting intervals per voyage/leg and per vessel). In this regard, the quality characteristics of the solutions depicted in the last three columns of Table 8 only underlines the deterministic strength, significance and merit of the hydrodynamics approximating functional [1] systems applied in Analysis Type 3.

Considering all of the above in conjunction with the fact that the results of Analysis Type 2 in Columns 1, 2, 3, and 4 of Table 8 are not distinctively worse than the results of Analysis Types 1 and 3 in their entirety—but only as far as the data analysis of only 4 or 5 out of the 10 vessels under consideration is concerned—and furthermore, that the data analysis of the remaining five or six vessels is much stronger (albeit not as strong as the analyses for Types 1 and 3 above), one can safely conclude that the comparative lack of success of Analysis Type 2, which encompassed almost half the vessels’ data under consideration, has to do with the differences between the Analyses of Types 2 and 3. These differences are described in detail in Section 3.3. In summary, the key feature that differentiates (“for all the good reasons”) Analysis Type 2 from Analysis Type 3 is the application of Correlation Scheme #4, as laid out in detail in Section 2.1.4. This scheme requires solid hydrostatic data (draft, trim, displacement) to be available, practically at all times, assuming the acquisition of such data is possible and meaningful.

Considering the multiple, digital, analogue or manual, shipboard systems from which these data may be acquired—including but not limited to a “draft survey” during which the seaside drafts are read from the (still or not) water level at the different hull markings “for” (“forward” or “ahead”, near to the bow), “aft” (“astern”, near to the stern) and amidships, by means of a launch boat or tender—and the commercial sensitivity of these data as far as any vessel’s loading conditions are concerned: it is possible that an erroneous data flow, or even data entry or data control, in conjunction with any permanent or temporary effect of loading distribution on the hull geometry irrespective of any given waterline position relative to the hull, may represent significant limitations. As far as the particular data acquired from the specific vessels are concerned, the above factors may serve as a solid example of the validity, significance and versatility of the integrated methodology presented in this work.

5. Conclusions

After the original formation of the hybrid tripartite combined application of:

(a) 

the existing, formidable and ingenious, “… method for the solution of certain non-linear problems in least squares …” [1,2] publicly presented for the first time in 1943 [1];

(b) 

the thermo-fluid and frictional functional system originally presented in this article that mathematically approximates the fuel oil consumption of vessels’ main engines under actual seagoing conditions;

(c) 

the functional system originally presented in this article that mathematically approximates the hydrodynamic performance of the respective vessels in terms of the shaft propulsion power and the shaft rotational speed of the fixed-pitch propellers driven by the a/m engines, also under actual seagoing conditions.

A very successful application of the above hybrid trinity was experienced by systematically attaining remarkably close approximate representations [1] of functions to be approximated [1], evaluated under actual seagoing conditions, on the basis of the least-squares criterion. The comprehensive, verbal, analytical, mathematical and imperative assignment statement ready correlations and reasoning of this original, hybrid tripartite, combined application also reveal the novelty and the significance of this research work. In fact, the original, comprehensive and imperative combination of (a) the methodology of the approximate representation of one function by another [1] with the original approximating functions [1] of (b) main engines’ thermo-fluid and frictional performance and of (c) standard/FPP seagoing vessels’ propulsion hydrodynamic performance, was applied to develop an integrated solution to the combined marine propulsion problem for standard ships under seagoing conditions, as specified in detail in Section 1.1 of this article. This original analysis was applied on the basis of the requisite availability of shipboard and/or universal “big data” sets and enabled the closest possible approximate representation [1] of the main engine’s fuel oil consumption, yielding formidable results in terms of the uncertainty, materiality, and standard deviation.

The specific and detailed conditions and limitations regarding the a/m requisite availability of shipboard and/or universal “big data” sets discussed above are laid out in and between Table 1 and Table 2 in Section 2.1.5. This also applies to Equation Sets (18)–(22) in Section 2.2. Apart from the a/m requisite availability of shipboard and/or universal “big data” sets, additional limitations and conditions are applicable pursuant to Appendix Y of this article, before the methodology of the approximate representation of one function by another [1] is applied.

One key finding of technical significance behind this research, “fit for purpose” of an “elevator pitch” speech, is Equation (8h), which may or may not be complemented by Equation (9h) (Equation (9h) is of lesser importance). In fact, the technical significance behind Equation (8h) lies in condition (l) in Section 2.1.5 and in Appendix F and Appendix G, which are directly relevant to the above condition. There are many verbal, analytical, mathematical and computational ways to express and apply this condition, resulting in alternative solutions to essentially the same problem. However, Equation (8h) is the most significant, precise, exact and concise formulation among them, for the reason that, among all the other observed occurrences of functions to be approximated [1], h′(x_i, y_i, z_i,…; 8) = (dr/dt)i = dr(x_i, y_i, z_i, …)/dt = 0 is the only one to always be exactly evaluated, regardless of any other conditions. Furthermore, Equation (8h) is the equation that facilitates other mathematical formulations that are applicable in the context of this article, which may or may not directly involve Equation (8h). In this way, this equation is one of the verbal, analytical and mathematical cornerstones, foundations and assets of this article and of the research behind it. In any case, Equation (8h) is nearly as important as the perfect alignment and adherence of this research to the form, content, context, significance, merit, effectiveness, and ingenuity of Reference [1]. Nevertheless, the above form, content, context, significance, merit, effectiveness, and ingenuity comprise the primary verbal, analytical, and mathematical cornerstones, foundations and assets of this research.

An additional essential contribution indicative of the significance, novelty and originality of this work, is related to the obscured “… other aspects inherent in this mathematical method [1].” which “… comprise some of the remaining reasons for the above quoting of the opening paragraph (page 164) of Reference [1]; …” in Section 1 of this article. In this regard, a reference is made to Section 1 and to all parts of it related to Appendix Z in its entirety; to points (1) to (16) of Appendix Z; to the multi-dimensional demarcation curves referred to and defined in points (15), (16) and (26) of Appendix Z; to the as-effective-as-possible alignment of the research presented in detail in this article to the engineering context of Appendix Z, dialectically considered in points (1) to (25) of it; to points (15), (16), and (26) of Appendix Z, where the two counterparts eventually meet, exchanging their advantages; to Section 2.1.3 and Section 2.1.4 as well as Appendix C, Appendix D and Appendix E in this article; to the fact that the application of the subject research does not require the design of experiments, as also discussed in point (24) of Appendix Z and in Section 1; and to the different approaches outlined, summarized and compared in Table A1 of Appendix Z as well as in Appendix Z in its entirety, in terms of their key elements and features.

Some specific benefits originating from all of the above are discussed in Section 1; the additional potential benefits include, but are not limited to, the following: comprehensive prediction, verification, certification of the shaft power and/or mechanical work, the RPM and/or revolution values, the fuel oil and/or other (forms of) fuels’ consumption, and CO2 and/or other emissions, subject to any applicable regulatory or other limitations and accompanied by the respectively applicable uncertainty and materiality levels; the support of relevant business/financing plans and/or “newbuilding” projects; the resolution and/or support of relevant charter-party-related claims and/or disputes; the optimization of the relevant voyage performance parameters (i.e., the vessel’s speed, RPM, weather-routing, fuel oil consumption, and others); the comprehensive determination of the vessel’s main engine condition and deterioration level, in terms of the indicated and mechanical/frictional efficiency, as well as hull and propeller fouling in terms of the service margin and in conjunction with the hull and propeller inspection and/or polishing interval frequency (underwater or not); comprehensive performance insight into hydro and thermal analyses of vessels under sea trial reference conditions and service conditions; and the integration of any of the above to AIS or other vessel-tracking data and/or weather-routing services.

As already discussed in Section 4, the results presented in this article represent a very small portion of the data, results, information, experience and original research work behind this article; in fact, these results have been carefully selected to highlight the nature of this research and to balance the structure of this article with regard to the obvious and self-explanatory space-related and other limitations in this regard.

Therefore, the simple plan for further research work in this area, is the dissemination of all the above in the form of further publications and/or other academic or research activities relevant to the research methods presented in this article, including, but not limited to, the broader application of these methods in the marine industry (also see the last paragraph of Appendix G).

On top of the above, this article indicates in full detail how “… certain engineering …” [1] or not “… applications involving … types of problems which are of much greater complexity …” [1], other than the ones presented in detail in this article along with the solutions to their respective “… types of problems which are of much greater complexity …” [1], may be implemented by means of the analytical, comprehensive and imperative approximate representation of one function by another [1], reconnected and aligned to by the present work.

Considering this article as a whole, as well as the original and independent research work behind it, the author fully and unconditionally confirms the closing sentence of Reference [1]. This closing sentence indicates that “… the method of damped least squares …” [1,2] “… for the solution of certain non-linear problems in least squares …” [1] introduced then (1943–1944) “… has solved, with a comparatively rapid rate of convergence, types of problems which are of much greater complexity than those to which the principle of least squares is ordinarily applied.” For confirming this closing sentence however, the author of this article does not have in his mind the same “… certain engineering applications involving … types of problems which are of much greater complexity …” [1] that the author of Reference [1] had in his mind when he drew his above conclusion (before 1943–1944, while affiliated with the Frankford Arsenal [1]). Instead, the author of this article has, since 1986, been studying, practicing, and researching on the “… engineering applications involving … types of problems which are of much greater complexity …” [1], laid out in perfect detail and resolved herewith.