Multi-Objective Constructal Optimization for Marine Condensers

: A marine condenser with exhausted steam as the working fluid is researched in this paper. Constructal designs of the condenser are numerically conducted based on single and multi-objective optimizations, respectively. In the single objective optimization, there is an optimal dimensionless tube diameter leading to the minimum total pumping power required by the condenser. After constructal optimization, the total pumping power is decreased by 42.3%. In addition, with the increase in mass flow rate of the steam and heat transfer area and the decrease in total heat transfer rate, the minimum total pumping power required by the condenser decreases. In the multi-objective optimization, the Pareto optimal set of the entropy generation rate and total pumping power is gained. The optimal results gained by three decision methods in the Pareto optimal set and single objective optimizations are compared by the deviation index. The optimal construct gained by the TOPSIS decision method corresponding to the smallest deviation index is recommended in the optimal design of the condenser. These research ideas can also be used to design other heat transfer devices.


Introduction
A shell-and-tube heat exchanger (STHE) has the advantages of low cost, easy cleaning, large processing capacity, and reliable operation [1,2]. STHE is commonly used in marine condensers. Some researchers have conducted in-depth research on the STHE. Johnson et al. [3] studied the marine condenser with the goal of minimum total pumping power (TPP), and reduced the TPP by 35% after optimizing the external diameter and length of heat transfer tube (HTT). Patankar and Spalding [4], as well as Prithiviraj and Andrews [5], established the STHE models with a porous medium, and introduced the distributed resistance method to analyze the flow performances at shell side. Guo et al. [6] studied an STHE with different parameters and gained its optimal entropy generation performances under two different conditions. Mirzabeygi and Zhang [7] conducted multiobjective optimization of a surface condenser and obtained its optimal tube diameter, tube thickness, and tube spacing, respectively. Rodrigues et al. [8] proposed a new ecological function objective for evaluating the performance of an STHE and obtained the Pareto optimal solution set considering the new ecological function and total cost objective. Xiao et al. [9] minimized the total annual cost of an STHE network with phase change and reduced the cost of the network by up to 23.7%. Yu et al. [10] proposed a compound STHE with longitudinal vortex generator and pointed out that the rise in height and the attack angle of the generator could improve its overall performance. Sridhar et al. [11] investigated the performance of an STHE with SnO2-water and Ag-water nanofluids based on experimentation and found that thermal conductivity increased by 29% and 39% after adding the two nanoparticles, respectively. Li et al. [12] discussed the STHE arrangement problem for an organic Rankine cycle, and pointed out that the largest difference of the thermo-economic performance was 14.7% for the five considered STHE arrangements. Miansari et al. [13] studied the performance of an STHE with circular fins and pointed out that the circular fin had an obvious effect on the thermal efficiency of the STHE.
The heat exchanger (HE) is also the research object of constructal theory. On the basis of this theory, a group of scholars have carried out performance optimization research on HEs. Varga and Bejan [52] studied the performance of the HEs with fins and smooth surfaces and obtained the same optimal results for the two structures. Bejan [53] optimized the structure of a dendritic HE and gained the maximum heat transfer rate (HTR) density corresponding to the optimal construct of the HE. Da Silva et al. [54,55] and Zimparov et al. [56] further conducted constructal designs of various two-dimensional tree-shaped HEs and compared their performance under different flow patterns and shapes of HEs. Azad and Amipour [57] optimized the structure parameters of the HE with the goal of minimum total cost. They reduced the total cost of the HE by 50% after constructal optimization. Yang et al. [58,59] built a Y-shaped STHE model and evidently reduced the cost of the STHE after constructal optimization compared to that of the initial design. Mirzaei et al. [60] further conducted multi-objective optimization of the Y-shaped STHE model, and increased its thermal efficiency by more than 28%. Manjunath and Kaushik [61] further explored the heat transfer performance of an H-shaped HE and found that the comprehensive performance of the H-shaped HE was superior to that of the traditional HE. Bejan et al. [62] further optimized the construct of a cross-flow HE, obtained the optimal construct with the maximum HTR, and analyzed the influences of the total volume of the HE and the total number of flow channels on the constructal optimization results. Hajabdollahi [63] optimized a plate fin HE with multi-objective and obtained the optimal parameters about the fin and size of the HE. Ariyo and Bello-Ochende [64] optimized a subcooled microchannel HE and obtained the optimal performance of the microchannel better than that with single phase fluid. In addition, by applying constructal theory, the constructs of the regenerators [65,66], underground HEs [67,68], low temperature evaporators [69,70], steam evaporator [71], superheater [72] and economizer [73] of the boiler, biomass boilers [74,75], and steam generators, [76][77][78][79] were optimized, respectively.
Condenser is one of the usual HE types. It has also been optimized by few researchers using constructal design. Bejan et al. [80] optimized the arrangement of the tube bundles with the goal of maximum condensation rate of a condenser and obtained the optimal tube bundle arrangement and optimal condensation performance. Li et al. [81] performed constructal design of grooved condenser wick structures and formulated it as a general "area-to-point" heat conduction problem with disk-shaped structure.
The condenser is an important component of the marine steam power plant. The structure of the marine condenser has an important effect on its performance, which has not been optimized based on constructal theory in open published literatures. Therefore, a marine condenser will be researched in this paper. According to constructal law, in the conditions of fixed total HTR and heat transfer area (HTA), constructal design of the condenser will firstly be conducted with the goal of minimum TPP. The optimal outer diameter (OD) of the HTT will be obtained. The effects of cooling water inlet temperature (CWIT), steam mass flow rate (MFR), total HTA and HTR on constructal optimization results will be analyzed. Then, the multi-objective optimization considering the performances of entropy generation rate (EGR) and TPP will be further conducted, and the Pareto optimal set of the two indexes will be gained. The first novelty of this paper is the adoption of constructal theory in the performance of the marine condenser, which is expected to significantly improve its performance. Another novelty of this paper is the adoption of three decision methods to evaluate the Pareto optimal set gained by NSGA-II, which will choose a reasonable optimal design scheme for the marine condenser to satisfy different design requirements.

Model of the Marine Condenser
The simple model of a marine shell-and-tube condenser is shown in Figure 1. The exhausted steam enters from inlet 1 and flows through the outside of the HTT. The steam releases heat to the cooling water (CW), and finally flows out from exit 4. The CW (seawater) enters the HTTs of the condenser from inlet 5, absorbs heat from the steam, and then flows out from outlet 6. Since the heat transfer rates of the steam cooling and supercooling stages are small in the actual condenser, only the isothermal condensation process of the steam is considered in the simplified model. Thus, the working fluid is approximately viewed as the saturated states at inlet 1 and outlet 4. The corresponding T-s diagram is shown in Figure 2

Total HTR and Heat Balance Equation
The HTR (heat load) of the condenser is the heat transferred through the HTTs per unit time. The HTR of the condenser is calculated as where K is the total heat transfer coefficient (HTC), c A is the total HTA, and Δ m T is the logarithmic mean temperature difference (MTD). The total HTA of the condenser is where e n is the number of HTTs.
In the STHE, the temperature difference of the two fluids is not constant along the HTS. Therefore, the logarithmic MTD is introduced [82] The HTR on the steam side is where LH r is the latent heat of the steam.
The HTR on the CW side is where pc c is the specific heat capacity.
According to the energy conservation, the following equation should be satisfied

Total HTC
According to the heat transfer principle based on the multilayer wall, the total HTC is [83] ( ) where c d is the diameter of the HTT, e r is the fouling resistance of the HTT, , c wall λ is the thermal conductivity (TC) of the HTT, and c α is the convective HTC, respectively.
Generally, the CW flowing in the tube is in a fully developed or turbulent state, therefore, the convective HTC on the inner surface of the HTT can be calculated by the Gnielinski formula [83] ( ) ( )  The condensation HTC at the exhausted steam side is formulated as [83] ( ) where g , LH r , l λ , l ρ , ν l , s T and are the gravity acceleration, latent heat of vaporization, liquid film thermal conductivity, liquid film density, kinematic viscosity, steam temperature, and cooling wall temperature, respectively.

Total EGR
Ignoring the EGRs caused by the fluid flow and heat loss, the EGR at the exhausted steam side is wf,in wf,out g, wf s where wf m  is the MFR of the exhausted steam, s T is the condensation temperature, and The EGR for the heat absorbing process in HTTs is Combining Equations (11) and (12), the total EGR of the condenser is ( ) wf,in wf,out g g ,wf g, c c c c, out c, in s

Calculations of Pressure Drop (PD) and Required TPP
For the HE, the TPP consumption is related with the PDs of the fluids in the tube and shell sides. Therefore, the PD is one of the important indicators to measure the HE performance.
The PD inside the tube is expressed as [82] ( ) where c l is the length of tube, , c water u is the velocity, and b is the diameter correction factor. The PD outside the tube is expressed as [82] 2.5 3 , where wf v is the specific volume of the steam at the working fluid inlet of the condenser.
Because the effect of the tube arrangement is not considered, the PD calculated in Equation (15) is an approximate value. Combing Equations (11) and (12) where p η is the pump efficiency.

Constructal Design of the Marine Condenser
To study different performances of the marine condenser, constructal designs with single and multi-objective optimizations will be conducted in this section.

Constructal Design with Single Objective Optimization
The idea of the constructal design method is to find an optimal structure channel for a "flow" under the condition of a fixed structural constraint. Thus, the constructal design of the marine condenser in this paper is conducted by taking the TPP as the optimization objective and the tube diameter as the optimization variable with the constraints of constant heat load (HL) c Q and total HTA c A . The HTA constraint that reflects the investment cost of the condenser can be achieved by adjusting the length of the HTT when the tube diameter is varied. The TPP should be reduced as far as possible in the constructal design, which provides easier channels for the flows and reduces the operating cost of the condenser. To launch the constructal design of the marine condenser, the initial design parameters are given as:  , the TPP of the condenser is decreased by 42.3% after constructal optimization, which shows that the flow performance of the condenser can be greatly improved by selecting a suitable OD of the HTT. Moreover, the condensation temperatures (CTs) derived by the theoretical calculation (this paper) and real device (reference [82]) are 60.8 °C and 63.5 °C, respectively. This shows that the difference between them is small, which proves the correctness of the theoretical calculation results to some extent.  reduces the MTD of the condenser. To achieve the design HL of the condenser, the CW flow rate must be increased, and the increase in the flow rate causes the increases of flow resistance and TPP. In addition, the CWIT has little effect on the optimum OD corresponding to the minimum TPP, thus the optimal structure of the condenser can be stably obtained according to its design standards.   . This is because under certain HL, increasing the total HTA means that the MTD between the steam and CW will diminish. When the CT changes little, the outlet temperature of the CW will increase, which ultimately leads to the decreases of the MFR and corresponding TPP. When the steam MFR is unchanged, increasing the HL means that the condensation latent heat of the steam needs to be increased and the CT is diminished correspondingly. Meanwhile, when the HTC and HTA are constants, the MTD will increase. This reduces both the average temperature and outlet temperature of the CW, and ultimately leads to the increases in the MFR and corresponding TPP.

Constructal Design with Multi-Objective Optimization
In the previous section, the TPP of the condenser is minimized. In effect, the EGR performance is another important index for the condenser. The performances of the EGR and TPP cannot reach the minimum at the same time, thus the multi-objective optimization is conducted in this section.
The flow chart of the multi-objective optimization is shown in Figure 11. From this figure, the dimensionless EGR ( g be gained, and three decision methods can be used to evaluate the Pareto optimal set with the smallest deviation index. Figure 12 further shows the flow chart of the NSGA-II. In the NSGA-II, the population number, mutation probability, and generation number are set as 100, 0.9, and 20, respectively. When the maximum generation number is reached, the Pareto front can be exported.   Figure 13 shows the Pareto optimal set of g  S and  c W gained by multi-objective optimization. There are 100 points to describe the Pareto optimal set, which is represented by the blue symbol of "*" in Figure 13. From Figure 13, the Pareto optimal set locates between the ideal solution (point C) and nadir solution (point D), which is the compromise between the EGR and TPP under different design requirements. To compare the optimal results of the Pareto optimal set, the decision methods of LINMAP, TOPSIS, and Shannon entropy  are introduced. Table 1 lists the optimization results of the condenser gained by different decision methods and single objective optimizations. Obviously, the EGRs (or TPPs) of the condenser gained by the three decision methods are not smaller than those gained by EGR (or TPP) minimization at point A (or point B). One can further adopt the deviation index i D to evaluate the optimization results. It shows that the deviation index of the TOPSIS decision method is the smallest one in Table 1. Therefore, the optimal construct gained by the TOPSIS decision method is recommended in the optimal design of the condenser.

Conclusions
Constructal design of a marine condenser is conducted in this paper. The TPP required by the condenser is minimized with fixed total HTR and HTA, and the optimal OD of the HTT is obtained. The multi-objective optimization considering the performances of EGR and TPP is further conducted, and the Pareto optimal set of the two indexes is gained. The results reveal that: (1) There is an optimal dimensionless tube diameter , , =1.49  c out opt d leading to the minimum TPP required by the condenser. Compared with , =1.0 c out d  , the TPP of the condenser is reduced by 42.3% after constructal optimization, which greatly improves the fluid flow performance and reduces the operation cost of the condenser. As the steam MFR and HTA increase and the total HTR decreases, the minimum TPP required by the condenser decreases.
(2) The Pareto optimal set of the g  S and  c W gained by multi-objective optimization is the compromise between the EGR and TPP under different design requirements. The EGRs (or TPPs) of the condenser gained by the three decision methods are not smaller than that gained by EGR (or TPP) minimization at point A (or point B). The deviation index of the TOPSIS decision method is the smallest one. Therefore, the optimal construct gained by the TOPSIS decision method is recommended in the optimal design of the condenser.
The constructal design method, which is adopted in this paper, can be used as a theoretical instruction for optimal designs of various condensers. The next step is to consider more structure variables, such as the number, spacing, and arrangement of the HTTs, to further improve the performance of the condenser. Furthermore, a more practical nonisothermal condensation model will be established with CFD software. All of these works can make the optimization research of the condenser more meaningful. The research model and optimized results can also be used to guide the modelling and optimization of the whole steam power plant.

Acknowledgments:
The authors wish to thank the reviewers for their careful, unbiased, and constructive suggestions, which led to this revised manuscript.

Conflicts of Interest:
The authors declare no conflicts of interest.