Experimental and Numerical Studies of the Heat Transfer in Thin-Walled Rectangular Tubes under Fire

: The paper studies, experimentally and numerically, the ﬁre behavior of some structural elements of symmetric-tubular (rectangular in this study) shapes using modern dimensional analysis (MDA). A model at a certain scale of the real prototype is analyzed in order to obtain its response in case of ﬁres. Experimental measurements are performed on a 1:10 scale model of a real support pillar and compared with the results of the numerical simulation. The obtained results can have useful applications in engineering practice, allowing fast obtaining of results with minimal costs.


Introduction
In the analysis of large structures, modern dimensional analysis (MDA) has lately become frequently used [1][2][3]. Historically, the correlation between the behavior of the model and the prototype has been gradually studied, using different mathematical methods, such as geometric analogy (GA), theory of similarity (TS), and classical dimensional analysis (CDA) [4][5][6]. Although the CDA allows a comprehensive approach, due to its rather cumbersome methodology, it has not become sufficiently widespread among researchers. Properties of symmetry, as it is presented in this study, make it possible to reduce the number of the significant parameters that are necessary to take into account. Here are some of the shortcomings of the method, presented in [7][8][9]: • Model Law (ML), which includes the correlations of model behavior and prototype through dimensionless variables π j , j = 1, 2, . . . , n, and requires a skill obtained from previous applications; • This protocol is non-unitary, requiring both deep knowledge in the field from the one who wants to apply it, and a certain ingenuity in separating and grouping the variables involved in describing the phenomenon in order to obtain the desired dimensionless variables; • The protocol is not only difficult, but also precarious, because only in relatively simple cases does it allow us to obtain all the desired dimensionless variables (which would serve us to the complete description of the model-prototype correlation); • The protocol also involves significant mathematical knowledge, which ensures the correct formulation of laws that describe as accurately as possible the pursued phenomenon (for example, heat transfer).
As a result, the CDA had and has limited applicability. With the development of MDA, these shortcomings were solved, and the method became unitary, efficient, and easy, and, last but not least, it allowed the complete obtaining of the Model Law (ML), including all dimensionless variables, which describe the correlation between model and prototype [10,11]. Applications of the method are found in numerous works, i.e., [12][13][14][15][16]. The obtaining of the ML elements will be briefly illustrated in Appendix A based on the results presented in the works [10,11,17].
In their contribution, the authors of this paper briefly presented the method called MDA in order to study and to show its net advantages compared to other methods that use the behavior of the model in viewing the predicted behavior of the prototype. Next, reference is made only to the particularities of applying this MDA method to bars with tubular-rectangular sections. New results concerning the MDA are presented in [18][19][20][21][22]. The results of the experiments obtained in this paper come to confirm the validity of the applied method.
In [9], the authors presented the applicability of MDA to heat transfer in bars and bar structures of circular sections. An analysis of the straight bars of rectangular-tubular sections is made in [23], where the ML is deduced for this type of heat transfer. A comparison made in [24][25][26] proves that for similar results, the time and efforts made using MDA are significantly reduced compared to where experimental determinations would be made. Other new results concerning the method and new application are presented in [27][28][29][30].
In the paper, experimentally and numerically, the fire behavior of some structural elements of a symmetric-tubular shape using MDA is presented. A model at a certain scale of the real prototype is analyzed in order to obtain its response in case of fires. There are experimental measurements performed on a 1:10 scale model of a real support pillar, and these are compared with the results of the numerical simulation. This allows the designers to obtain results in a short time and with minimal costs.

Materials and Methods
In a normal case, the approach to the problem of fire resistance of structural elements is conducted either in specially designed furnaces or in enclosures of considerable size, and therefore at a particularly high cost.
In order to substantially reduce material costs and working time, the authors proceeded from the results of thorough investigations carried out by [31] on the size of the heat transfer coefficient (thermal conductivity) to a thermal insulation material often used in the protection of fire (intumescent solvent-based paint type: Interchar 404, from International Marine and Protective Coatings).
Regarding this, one paper [31] examines two ways of heating the resized structural element (a steel bar), namely: • Direct heating of the structural element covered with intumescent paint, i.e., when the thermal flux was applied from the outside to the protected element, resulting in λ = 0.1089 W m·K = 0.1089 W m· • C , a value specified in the technical data sheet of the paint; • Reverse heating, when the thermal flux was applied (through an electric current) from inside the structural element to the outside, thus obtaining λ = 0.1102 W m·K = 0.1102 W m· • C , a difference of approximately 1%, which is highly acceptable considering the fact that these investigations did not have metrological conditions. Consequently, even if at first glance the result seems to be a little curious, the thermal flux direction does not have a special significance in the behavior of the thermal insulation layer, respectively on the parameters of the thermal flux propagation, obtaining practically identical effects.
However, from the point of view of the necessary equipment, the effects are significant. Compared to direct heating, where the equipment has the disadvantages mentioned before, in the case of reverse heating, the equipment becomes very simple and cheap, with a practically 100 % reproducibility of the effects of fires or simply the pursuit of the heat flux propagation in a structural element.
Heating (reverse) with the help of electric power sources, if the control system is efficient, will lead to significant savings in the involved equipment and material costs, staff, This simple and safe way of testing will open new perspectives, both in the field of testing intumescent paints and in the propagation of heat flux along structural elements made of steel.
In this paper, the authors opted for the reverse heating method where both the heating source and the investigation protocol were based on this method. In the present paper, the authors started from the experimental study of the 1:10 scale model of a real 5 m high support pillar, provided with four beams with a length of 4 m, from a domestic industrial hall, with the dimensions of 20 × 25 m 2 .
The column, being of a tubular-square shape, and the adapted model also presented these characteristics, shown in Figure 1. before, in the case of reverse heating, the equipment becomes very simple and cheap, with a practically 100 % reproducibility of the effects of fires or simply the pursuit of the hea flux propagation in a structural element.
Heating (reverse) with the help of electric power sources, if the control system is ef ficient, will lead to significant savings in the involved equipment and material costs, staff and, not least, fire protection conditions, which must be fulfilled in the case of open fir sources, applied to direct heating.
This simple and safe way of testing will open new perspectives, both in the field o testing intumescent paints and in the propagation of heat flux along structural element made of steel.
In this paper, the authors opted for the reverse heating method where both the heat ing source and the investigation protocol were based on this method. In the present paper the authors started from the experimental study of the 1:10 scale model of a real 5 m high support pillar, provided with four beams with a length of 4 m, from a domestic industria hall, with the dimensions of 20 × 25 m 2 .
The column, being of a tubular-square shape, and the adapted model also presented these characteristics, shown in Figure 1. . In order to eliminate the undesirable influence of some accidental current of air from our lab and taking the reduced thermal inertia of this column into consideration, the col umn was foreseen with a m 025 0.
height (see Figure 2). The radius of this heat-insulation cylinder i comparable (at the reduced scale of 1:10) with the half-distance between the columns. A steel cylinder with a diameter of m 105 0. and a height of m 015 0.
is fixed at the bottom of the column, which provides the electric heating. In advance, on the bottom of the col umn, a 0.080 0.080 0.003m  steel plate was welded in order to make the mounting o the above-mentioned steel cylinder (Figure 1) easier. The temperature at the bottom of th support column is known from the heating performance and measurements, so the circu lar heating was not modelled. At the bottom of the column, a constant nominal tempera ture was prescribed (T = 100; 200; 300; 350; 400; 500 °C).
To monitor the thermal regimes inside the column, three PT 100-402 type therma resistors were fixed on a thin steel rod, with 0.150 m-long terminals and a measuring rang of ( )  In order to eliminate the undesirable influence of some accidental current of air from our lab and taking the reduced thermal inertia of this column into consideration, the column was foreseen with a 0.025 m-thick heat-insulation open cylinder, with a 0.45 m diameter and 0.65 m height (see Figure 2). The radius of this heat-insulation cylinder is comparable (at the reduced scale of 1:10) with the half-distance between the columns. A steel cylinder with a diameter of 0.105 m and a height of 0.015 m is fixed at the bottom of the column, which provides the electric heating. In advance, on the bottom of the column, a 0.080 × 0.080 × 0.003 m steel plate was welded in order to make the mounting of the above-mentioned steel cylinder (Figure 1) easier. The temperature at the bottom of the support column is known from the heating performance and measurements, so the circular heating was not modelled. At the bottom of the column, a constant nominal temperature was prescribed (T = 100; 200; 300; 350; 400; 500 • C).
To monitor the thermal regimes inside the column, three PT 100-402 type thermal resistors were fixed on a thin steel rod, with 0.150 m-long terminals and a measuring range of (−70 . . . + 500 • C), at well-defined distances from the base of the column (at 0.165 m, 0.245 m and 0.305 m). It can be observed that the subassembly was mounted inside the column (Figures 2 and 3).
Additionally, to monitor the ambient temperature thus delimited by the thermal protection cylinder, another PT was fixed on a thin steel support at a distance of 0.060 m from the top of the column (Figures 2 and 3a).  (a) Experimental setup. One column (b) One column with one bar    Additionally, to monitor the ambient temperature thus delimited by the thermal protection cylinder, another PT was fixed on a thin steel support at a distance of 0.060m from the top of the column (Figures 2 and 3a). Figure 3 shows the single pole with the PTs mounted, as well as the set of PTs intended for monitoring the temperature inside the pole in three places (on its height), and the ambient environment, with the PT mounted on the crossbar of wire located above the pole; in Figure 3b, it is the first beam installed, in Figure 3c it is all the beams installed, Figure 3. Stages of mounting the PT heat resistors on the pillar, indoors and environment, respectively, on the four beams [12].  Figure 3 shows the single pole with the PTs mounted, as well as the set of PTs intended for monitoring the temperature inside the pole in three places (on its height), and the ambient environment, with the PT mounted on the crossbar of wire located above the pole; in Figure 3b, it is the first beam installed, in Figure 3c it is all the beams installed, and in Figure 3d, it is shown as a detail with the post and two beams.
The trusses were made of flat steel with dimensions of 0.010 × 0.0015 m and a length of 0.200 m, in compliance with the law of the model, deduced and validated in [13], especially with reference to the Shape Factor ς 1 m related to the real beams and the model [10,11]. The arrangement of the locations of the PTs fixed on the beams is shown in Figure 4. Here, the distances are presented in mm. Figure 3. Stages of mounting the PT heat resistors on the pillar, indoors and environment, re tively, on the four beams [12].
Additionally, to monitor the ambient temperature thus delimited by the thermal tection cylinder, another PT was fixed on a thin steel support at a distance of 0.06 from the top of the column (Figures 2 and 3a). Figure 3 shows the single pole with the PTs mounted, as well as the set of PT tended for monitoring the temperature inside the pole in three places (on its height), the ambient environment, with the PT mounted on the crossbar of wire located abov pole; in Figure 3b, it is the first beam installed, in Figure 3c it is all the beams insta and in Figure 3d, it is shown as a detail with the post and two beams.
The   In addition, in order to be able to properly place this column (together with its cylindrical support) on the laying area of the support in the form of a pyramid triangle on the stand, an intermediate support (a stainless steel plate with dimensions 0.450 × 0.450 × 0.003 m) has been provided, on which the pole could be placed in good conditions ( Figure 5). A layer of paste with high thermal conduction properties (heat-sink compounds Hy410, with thermal conductivity λ > 1.42 W/m·K) was applied between the cylindrical support of the mast and the intermediate plate, identical to that used to fix the PTs on structure. ) has been provided, on which the pole could be placed in good conditions ( Figure 5). A layer of paste with high thermal conduction properties (heat-sink compounds Hy410, with thermal conductivity λ > 1.42 W/m.K) was applied between the cylindrical support of the mast and the intermediate plate, identical to that used to fix the PTs on structure. In the present paper, the effects of heating the column at the base on the propagation of the thermal field along it are analyzed. According to Figure 5, the structural element 1, together with the stainless-steel plate fixed to its base, after a translation along the arrow, will be placed on the upper area of the dome in the form of a pyramid trunk 2, which in Figure 5. Heating stand, together with the tested pillar model [12]. In the present paper, the effects of heating the column at the base on the propagation of the thermal field along it are analyzed. According to Figure 5, the structural element 1, together with the stainless-steel plate fixed to its base, after a translation along the arrow, will be placed on the upper area of the dome in the form of a pyramid trunk 2, which in turn rests on rigid frame 3 and support legs 4.
During the experimental investigations, the free surface of the laying plate is covered with such a thermal insulation blanket ( Figure 6). The image with the set of 12 Silite bars during operation is shown in Figure 7.  Silite bars during operation [12].
As an illustration of the efficiency of thermal insulation, it can be mentioned that at the nominal heating temperature . The authors designed an original electronic control and adjustment system, presented in the paper [12], which ensured optimal operation in a wide range of dissipated power, as well as accurate monitoring of electricity consumed for heating. Thus, at the nominal value of the supply voltage, the heating system ensures a power of approx. 25 kW, as well as reaching temperatures The protocol of the experimental investigations was as follows:  Silite bars during operation [12].
As an illustration of the efficiency of thermal insulation, it can be mentioned that at the nominal heating temperature . The authors designed an original electronic control and adjustment system, presented in the paper [12], which ensured optimal operation in a wide range of dissipated power, as well as accurate monitoring of electricity consumed for heating. Thus, at the nominal value of the supply voltage, the heating system ensures a power of approx. 25 kW, as well as reaching temperatures The protocol of the experimental investigations was as follows: • Placing the structural element 1 on the assembled stand, by interposing between Figure 7. Silite bars during operation [12].
As an illustration of the efficiency of thermal insulation, it can be mentioned that at the nominal heating temperature t 0, nom = 500 • C of the tested structural elements, the temperature around the support frame 3 and the pyramid trunk 2 did not exceed (45 . . . 50) • C.
The authors designed an original electronic control and adjustment system, presented in the paper [12], which ensured optimal operation in a wide range of dissipated power, as well as accurate monitoring of electricity consumed for heating. Thus, at the nominal value of the supply voltage, the heating system ensures a power of approx. 25 kW, as well The protocol of the experimental investigations was as follows: • Placing the structural element 1 on the assembled stand, by interposing between them a segment of thermal insulating mattress on the effective placement area (effective contact area), in order to ensure a perfect contact without thermal losses ( Figure 6); • Mounting, on the lower plate of the tested structural element as close as possible to their junction area, a thermocouple type K, which will be connected to the temperature controller ATR121B, in order to monitor the desired nominal temperature at the base of the structural element; •  The ATR121B temperature controller also has the self-learning function; practically, after the first cycle of reaching the nominal temperature t 0, nom , it will ensure the temperature regulation within very determined limits. Thus, for example, based on the measurements performed at one t 0, nom = 500 • C, the thermal oscillations related to the adjustment were of maximum (4 . . . 5) • C; • A stabilized temperature regime was considered to be achieved when at the level of the last thermoresistance PT 100-402 (near the upper part of the tested structural element) and the maximum temperature oscillations (0.2 . . . 0.3) • C were observed for a minimum period of (120 . . . 180) s .

Experimental Results
In the following section, the results obtained on the column provided with four beams in the non-thermally protected version (unpainted with intumescent paint) will be presented. In Figures 8-10, the thermal fields at the level of the column in the stabilized operating regime, the evolution in time of the thermal regimes, and the changes in the ambient temperature and inside the column temperature are shown, in the cases of the elements not covered with thermal paint.
In Figure 8, the horizontal axis refers to the length of the pole, and the vertical one to the imposed temperatures and their evolution along the pole at the level of the nine PTs (thermo-resistances), each marked by the characteristic points of the diagrams; the seven curves T1 . . . T7 refer to the imposed temperatures. Figure 9 shows the variation curves of the temperatures at the level of the nine PTs in relation to the duration of the tests; in addition to the color codes, the coordinates of these thermoresistances are also mentioned. Figure 10 shows the variation curves of the temperatures: First at the level of the three PTs arranged inside the pillar (Tst.int1 . . . 3), second at the level of the one that monitored ambient temperatures (Tamb) and third at three other thermoresistances: at the base of the pillar z(0), the next z(1), and respectively from the top of the pillar z(9), in relation to the time of the tests.          The exponential law of temperature propagation along a straight, full-section bar is well-known [1][2][3]: T r (z) = T(z) − T a = c 1 ·e m 0 ·z + c 2 ·e −m 0 ·z (1) where: T r [ • C] represents the relative temperature, i.e., the difference between the temperature of the bar T(z) and that of the environment T a ; c 1 , c 2 [ • C]-integration constants; The generally accepted parameter m 0 is also used in establishing by experiment the size of the heat transfer coefficient α n , for which it becomes important to establish it as simple, accurate and as safe as possible. In this sense, the methodology developed in [12], based on a polynomial approach instead of the classical, exponential one, is particularly efficient.
The hypothesis m 0 = const. along the entire bar is valid only in the case of bars with the full section. Taking the previous results obtained in [12,27,32,33] into account by which, on the basis of rigorous experimental investigations, it was established that the exponential law of well-known distribution (5) of temperature from solid bars cannot be applied to tubular ones, it results that the condition m 0 = const. is valid only at short intervals.
In the latter case, instead of a single exponential law of the type (5), and therefore of a single condition m 0 = const., at least three intervals must be delimited on the total length of the bar L  , thus making it possible to accurately approximate these curves by polynomial functions with a relatively small degree.
These new intervals are shown in Figure 11a-c. At first glance, the curves in Figure  11c seem to show strong gradients of temperature variation, but if we look closely at the z(m) scale, it is found that, in fact, these lengths are much larger than in the first two sets of diagrams, resulting in very smooth changes in temperatures per portion (see Figure 8). Figure 12 shows the variation, with a small gradient, of the thermal fields along a beam.  (see Figure 8). Figure 12 shows the variation, with a small gradient, of the thermal fields along a beam.

CFD Simulation/Numerical Solution
A three-dimensional numerical model was built to investigate the temperature distribution on the above-described frame. The numerical approach with the commercial software package ANSYS Fluent was applied, based on the Finite Volume Method (FVM).
The temperature at the bottom of the support column is known from the heating Figure 12. Temperature variation along a thermally unprotected beam.

CFD Simulation/Numerical Solution
A three-dimensional numerical model was built to investigate the temperature distribution on the above-described frame. The numerical approach with the commercial software package ANSYS Fluent was applied, based on the Finite Volume Method (FVM).
The temperature at the bottom of the support column is known from the heating performance and measurements, so the circular heating was not modelled.
On the surface of the frame, the standard no-slip condition was used for the velocity. Natural convection develops in the closed domain around the column; a flow can be induced due to the force of gravity acting on the density variations [34]. The ambient air temperature is prescribed based on the experimental data from Figure 10, while the maximum heating temperature of the frame is 500 • C. The Boussinesq model should not be used if the temperature differences in the domain are large [32], so fluid density must be specified as a function of temperature. The fluid properties such as viscosity, density and thermal conductivity were treated as temperature-dependent using polynomial functions. The coefficients of the functions were obtained by fitting the data of VDI-Wärmeatlas [31] in the range 0 • C ≤ T ≤ 600 • C: • Density: • Thermal conductivity: The dependence of the viscosity on temperature is given by Sutherland's formula [31]. The external insulation is 0.025 m-thick Fiberfrax Durablanket, the density of which is 128 kg/m 3 , while its thermal conductivity depends on the temperature: In the computational domain, hexagonal elements were used. It can be seen in Figure 13 that the cells are logarithmically spaced, providing a fine grid scale near the solid bodies (frame and insulations) and relatively coarse grid in the far field. temperature is prescribed based on the experimental data from Figure 10, while the maximum heating temperature of the frame is 500 °C. The Boussinesq model should not be used if the temperature differences in the domain are large [32], so fluid density must be specified as a function of temperature. The fluid properties such as viscosity, density and thermal conductivity were treated as temperature-dependent using polynomial functions. The coefficients of the functions were obtained by fitting the data of VDI-Wärmeatlas [31] in the range 0 °C ≤ T ≤ 600 °C: • Density: (3) • Thermal conductivity: (4) The dependence of the viscosity on temperature is given by Sutherland's formula [31]. The external insulation is 0.025m -thick Fiberfrax Durablanket, the density of which is 128 kg/m 3 , while its thermal conductivity depends on the temperature: In the computational domain, hexagonal elements were used. It can be seen in Figure  13 that the cells are logarithmically spaced, providing a fine grid scale near the solid bodies (frame and insulations) and relatively coarse grid in the far field. Two types of mesh were created, i.e., a coarser mesh with approximately 5.66 million cells and a finer mesh with roughly 11.6 million cells. For the two different mesh designs, the temperature values were compared on the column at a height of z = 0.495 m at heating temperatures of 500 • C. There was a relative error of about 6% between the two mesh designs.
The obtained values were compared to the measured values, and it is found that the coarser mesh is not suitable for determining the exact values, as there was a difference of almost 9%, while the difference for the finer mesh was 3%.
Therefore, further calculations were performed on the finer mesh, although the calculation time thus increased by several hours.
The governing equations with the boundary conditions are solved using Finite Volume Method. The SIMPLE (Semi-Implicit Method for Pressure-Linked Equation) algorithm was used for pressure-velocity coupling. The second order upwind scheme was used in momentum and energy equations. The calculi included the effect of the internal heat-flux during the heating of the column.
The measurement and calculation results are compared for different heating conditions. Figure 14 shows the temperature distribution along the column, where the measurement results are denoted by EM = experimental measurement, while the numerical values are denoted by CFD = computational fluid dynamics. The governing equations with the boundary conditions are solved using Finite Volume Method. The SIMPLE (Semi-Implicit Method for Pressure-Linked Equation) algorithm was used for pressure-velocity coupling. The second order upwind scheme was used in momentum and energy equations. The calculi included the effect of the internal heat-flux during the heating of the column.
The measurement and calculation results are compared for different heating conditions. Figure 14 shows the temperature distribution along the column, where the measurement results are denoted by EM = experimental measurement, while the numerical values are denoted by CFD = computational fluid dynamics. At the top of the column, the measurement temperature values agree well with numerical data, but the distribution in the column, i.e., the slope of the curve fitted to the measured and calculated values, differs from each other.
In order to evaluate the obtained differences between experiments and calculi, one can put in evidence the following: • The thermal radiation was neglected in the calculation; • It was quite difficult to reproduce the exact environmental conditions from experiments during the simulation; • The specific case of the hole beams from the . 0 const m = condition (see Equation (5)) must be studied, where, instead of a single domain, the length of the beam has to be divided into three subdomains; • This last demand was not realized due to the software's particularities and consequently, in the authors' opinion, this can be the main influence factor on the curves' differences.
Distributions of temperature and velocity in the middle plane of the column at 100, 300 and 500 °C are shown in Figure 15. It can be seen that the flow velocity increases along the hot column and flows upwards, and then it hits the insulation on top and the flow continues to spread downwards along the side insulation. At the top of the column, the measurement temperature values agree well with numerical data, but the distribution in the column, i.e., the slope of the curve fitted to the measured and calculated values, differs from each other.
In order to evaluate the obtained differences between experiments and calculi, one can put in evidence the following:

•
The thermal radiation was neglected in the calculation; • It was quite difficult to reproduce the exact environmental conditions from experiments during the simulation; • The specific case of the hole beams from the m 0 = const. condition (see Equation (5)) must be studied, where, instead of a single domain, the length of the beam has to be divided into three subdomains; • This last demand was not realized due to the software's particularities and consequently, in the authors' opinion, this can be the main influence factor on the curves' differences.
Distributions of temperature and velocity in the middle plane of the column at 100, 300 and 500 • C are shown in Figure 15. It can be seen that the flow velocity increases along the hot column and flows upwards, and then it hits the insulation on top and the flow continues to spread downwards along the side insulation.

Conclusions
• In the paper, experimental investigations were performed on a reduced model at a scale of 1:10 of a pillar supporting an industrial hall, applying the method of reverse heating; • A high-power electrical stand was designed, built and tested, with verified performances on a prototype (pillar segment on a natural scale), detailed in the paper [13];

Conclusions
• In the paper, experimental investigations were performed on a reduced model at a scale of 1:10 of a pillar supporting an industrial hall, applying the method of reverse heating; • A high-power electrical stand was designed, built and tested, with verified performances on a prototype (pillar segment on a natural scale), detailed in the paper [13]; • The numerical simulation of the heating phenomenon of the model was also performed, considering a thermal insulation cylinder, which delimits the space of the model from the environment; • The obtained differences between the measured values and those from the numerical analysis may be mainly due to the fact that the software could not directly take into account the anomaly valid only on the sub-intervals of the tubular bar compared to the generally valid one (along the entire bar) from those of full sections. • Among the goals for the near future of the authors are, among others: • Improvement of the numerical method, in order to be able to recognize these intervals (   The lines related to the D-C matrices offer in a unique, simple and safe way the elements of the Model Law. Another great advantage of MDA is that if a variable is either physically or dimensionally insignificant [10,11], it can be removed automatically without the ML to undergo significant changes (in essence) regarding the characterization of the studied phenomenon. MDA allows the obtaining of the complete set of dimensionless variables, which will constitute the ML. It is a very flexible method, allowing disregarding some variables, compared to the general case previously developed; thus, for a particular set of measurements, where the operating conditions of the prototype and the model coincide, certain variables can be excluded from the model law, without affecting the rest of the elements already deduced, because their scale factors (see below) are equal to unity.
The set of relations contained in the ML does not represent computational relations in the usual sense, but only firm correlations established between the scale factors S E j = E j, m E j, p of the variables involved, E j, m , E j, p , in describing the analyzed phenomenon, where the m index always refers to the model and the p to the prototype.
Thus, each element of the DS (see Equation (A1)) represents an exponent, namely: in cases A and B there are the exponents of the basic dimensions involved, while in D and C they are of the variables involved.
Consider, for example, DS, with 6 basic dimensions (r 1 , r 2 , . . . , r 6 ) and 14 variables involved (E 1 , E 2 , E 3 , . . . , E 14 ), given below in the following image. The calculations for matrices C and D showed the form below DS.
How to obtain the fifth element of the ML is presented below. This element is related to the connection of the variables E 5 from the prototype and model through the line related to the dimensionless variable π 5 , marked in green, i.e., the line π 5 elements, in correlation with the column related to the variable E 5 . The exponents of the independent variables involved ( )