A novel method based on the calculus of variations to optimize the cooling passage configuration in thermal protection structure

A novel method based on the calculus of variations to obtain the optimization of cooling structure was developed in this work. The optimization of heat sink designs for better heat-dissipating effects has been researched substantially. However, the optimization of cooling passage configurations in thermal protection structures to improve the comprehensive heat transfer performance has remained a long-term interest and unsolved problem for researchers. Due to the intrinsic complexity, the laminar flow is usually considered before while the turbulent flow is seldom treated. It is valuable to provide timely and effective solutions to optimize the cooling structure, where the turbulent flow is involved, for practical use. A novel method to optimize cooling structure based on the calculus of variations has been developed to meet this need. The average temperature, temperature inhomogeneity, and coolant flow pressure drop were chosen as objective functions. The cooling channels can be established depending on the geometric and thermal boundary conditions of the structure, and then the quasi-three-dimensional simulation of the fluid/solid coupling field was calculated to get the temperature distribution of the cooling structure, the pressure drop, etc. Hence, this work explores the feasibility of combining coupled heat transfer and the calculus of variations to realize optimal cooling channels. By the novel method, the minimum objective function can be obtained.


Introduction
Thanks to the emergence of new and advancement of existing manufacturing technologies, the designers have more freedom to optimize the thermal protection structure for high-speed engines.The optimization method itself becomes more vital due to a trend toward higher heat flux in the engine [1].As the flight speed increases, the total temperature of free stream in the rocket engines or the scramjet/ramjet engines will grow dramatically.The combustor wall is exposed to a high heat flux due to high-temperature gas in the combustor.The engine design under high Mach number should withstand high pressure and temperature, which will bring thermo-temperature stresses.Regenerative cooling has been proposed as an effective thermal management approach to handle high heat loads for combustor walls [2], where the coolant fuel flows through the channels embedded in the wall to partially cool down the wall before being injected into the combustor.
In the active cooling system, coolant fuel circulates in the channels to help maintain the wall temperature below the structure's safe temperature limit.The fuel bringing the energy absorbed back to the injector improves the combustion efficiency.Challenges in the optimal design of the cooling 2 structure are the temperature limit of the combustor material and the restriction that fuel should not coke when flowing [3].Thermal stresses occurring due to temperature inhomogeneity and pressure drop of the fuel along the channel need to be considered as well.Therefore, the objective function should take average temperature, temperature inhomogeneity, and pressure drop into account.
The efficient cooling technique to bring the heat load away is a long-time pursuit for researchers [4].Convective heat transfer and forced fluid flow in cooling channels and their application in the cooling field have drawn the great attention of researchers in recent years.A typical optimization approach for cooling structure design by parametric optimization of channels has developed.Zhang et al. [5] used global and local methods to study the thermal management of fuel in cooling channels.Pizzarelli et al. [6] identified that the optimum aspect ratio does exist that minimizes the fuel losses in the cooling channel.Wang et al. [7] aimed to seek the optimal number of channels in the engine.
However, only the parametric optimization is not sufficient for the cooling channel optimization.Different optimization methods emerge topology optimization, genetic algorithm, bifurcation, particle swarm optimization, and so on.
Dede et al. [8] proposed a multipass branching microchannel for cooling applications.Dilgen et al. [9] presented a fast and practical approach to topology to optimize complex fluid flow systems.Mital et al. [10] showed that narrow channels are desired for efficient heat extraction.Xie et al. [11] applied the microchannel with bifurcation flow to improve the thermal performance.The laminar flow of bifurcation ratio and length ratio is researched.Zhao et al. [12] used a topology optimization methodology with a low-cost model to optimize cooling channels.However, there are some limitations to the optimization methods above.Although the optimization of cooling structure can be obtained by utilizing those methods, the cost of the calculation is often high and the distribution of cooling channels is too complicated.In this paper, the optimization of the cooling structure is based on the calculus of variations, where cooling channel optimizations conform to the geometric characteristics of the combustor chamber while automatically adapting to the realistic heat flux or temperature distribution.Here the flow can be turbulent flow, which is thought to be steady and incompressible.
This work aims to design a cooling channel having high heat transfer performance while the pressure drop ranges in an acceptable scope.Numerical simulation for its heat transfer and flow have been developed.In the following, the modeling and optimization methodology will be introduced and followed by its physical and mathematical models.Then, the numerical results will be demonstrated along with the performance comparisons for different cases.In consequence, some conclusions can be drawn.

Optimization method based on the calculus of variations
In order to exhibit the generality of the methodology, the cooling structure is arbitrary, as shown in Figure 1, in which the cooling channel is designed, with coolant flowing in from one side and out from the other side of the cooling structure.The top of the cooling structure is subjected to heat flux and the bottom is an adiabatic boundary.
The geometry and the thermal boundary condition of the structure should be determined.The heat flux imposed on the top of the structure is q and the remaining boundaries are adiabatic.The independent parameters of the cooling inlet are determined with inlet mass flowrate ms, inlet temperature T and coolant thermal property parameters which are thermal conductivity K, constant pressure specific heat capacity Cp and density ρ.According to the target, the shape and the thermal boundary conditions of the cooling structure, the independent parameters of the cooling inlet should be combined to establish an adaptive optimization design model based on the variational method.To implement this model, the adaptive design cooling channel of the structure can be obtained.To illustrate the generality of the methodology, in Figure 2, here is an example of arbitrary physical space.To quantify the mapping in Figure 2 between the physical space (x, y) and computational space (ξ, η) by relations between (x, y) and (ξ, η).The inlets of all the cooling channels start from the left side, with outlets on the right side.The η line at a specific value is the center of the cooling channel and the width of cooling channel d is proportional to the grid density.The physical grid density varies with the temperature or q distribution.The grids become denser where the temperature or q is larger, then the d tends to be narrow.Step 1.A conformal grid is generated based on the geometry of the cooling structure.In this paper, the Laplace equation is used to generate a conformal grid [13].
The component functions ξ and η are harmonic, satisfying Laplace's equation: Whereby computation, here is 2 0 2 0 With coefficients: x y (3) In this way, the initial mapping is established.
Step 2. The developed calculus of variations method is one of the optimizations, where the objective function combining the given function, local orthogonality, and grid smoothness is minimized based on the initial mapping [14].
The weighting function is measured by: Where = ( , ) x y

 
is the given function, namely heat flux distribution in the following section.The global smoothness of the domain is measured by the integral.
The orthogonality of the domain is measured by the integral.(8) A system of the second-order PDE can be solved by the Gauss-Seidel iteration in the computational space.
Step 3. The mapping can be established by using the calculus of variations method, where the grid can be denser when ω is bigger.Based on this principle, the width of cooling channel d should be proportional to the and .The η line at the constant is the center of the cooling channel and the d is vertical to the η line.In the mapping where the ω increases, the grids become denser, then the hydraulic diameter d will decrease.In this principle, ( ( , ), ( , )) Where n is a variable number.In addition, the pressure drop is sensitive to the d.When the d becomes narrow, the pressure drop will increase dramatically.The number of cooling channels should be determined by coupling with the thermal analysis method to derive the ideal cooling channel distribution.

Mathematical formulation and numerical methods
Due to the complexity of flow in the cooling channel, to analyze the cooling structure established via the methodology mentioned above and to calculate the flow, and the heat transfer in the cooling channel should be simplified.The following assumptions need to be made.(2) 1Cr18Ni9Ti is employed everywhere in the wall.
(3) The fuel in the cooling channel has no change in the physical properties of the radial direction.(4) The influence of the thermal boundary layer is not considered.Since the channel hydrodynamic diameter is much smaller than the length, the cooling channel is divided into a few units, as shown in Figure 3. Before the calculation of heat transfer, the hydraulic diameter needs to be defined: 4 4 In Figure 4, according to the governing equations: continuity equation, momentum equation and energy equation, flow and physical parameters can be achieved by iterations in the calculation process.
Considering that the temperature gradient in the flow direction is smaller than that in the crosssection, the heat conduction between adjacent sections can be neglected when enough cross sections (Figure 5) are divided along the flow direction.In this way, the problem can be transformed into twodimensional heat conduction, which can reduce the cost of calculation.In this way, the twodimensional structure temperature distribution of the cooling structure is achieved via a conservative method [14].

Objective functions
The purpose of the study is to obtain the optimal distribution of the cooling channels that minimizes average temperature, temperature inhomogeneity, and pressure drop simultaneously.The objective functions can be listed as follows: Where A represents the average temperature, B is the temperature inhomogeneity, C means the pressure drop, Ω s is the solid domain, V s is the solid structure volume, and N s is the number of the solid domain nodes.
Combine the objective functions above, the cooling structure optimization function can be demonstrated as follows:   , , min + . .( ) 0, ( ) 0 The objective function F0 can be chosen in many ways.Here, the average temperature of the cooling structure should not be too high in the practical application, otherwise, the temperature will exceed the tolerance limit of the material.The temperature inhomogeneity should not be too large, or the temperature stress will affect the lifetime of the structure.The loss of flow pressure should not be too large, or an additional burden will be added to the system.Hence, the minimum of average temperature, temperature inhomogeneity, and loss of flow pressure are treated as the objective functions.Among these, the weight coefficients 1  and 3  are selected according to different conditions. ( ) ( ) are the constraint functions.In this study, the shape and the distribution of the cooling channel both need to be determined.The constraint can be depicted as: The objective function normalized can be described as: (16) Among them, Amax, Bmax, and Cmax are the maximum values of each factor in the range, the weight coefficients 1  and 3  are selected according to different conditions.Here, there are four cases to be optimized.The specific weight coefficients are set in Table 1.Case 1 considers the influence of various factors on average.Case 2 focuses on the decrease in average temperature.Case 3 eyes on the reduction of temperature inhomogeneity.Case 4 emphasizes the reduction of pressure drop.The optimization results can be achieved when: The procedure for optimization in this research is demonstrated in the flow chart (Figure 6).

Problem Description
To illustrate the advantages of the optimization method, three examples are presented here.
An initial area was determined with the inlet and outlet as well as a heat resource.The method was employed to generate the cooling channels under the geometric and thermal boundary conditions between the inlet and outlet (Figure 1).
The first example demonstrates that the cooling channels can be optimized according to the geometry of the structure.The second example shows that cooling channels are optimized according to the thermal boundary condition.The third example reveals that the cooling channels can be optimized not only under the geometrical boundary condition but also dependent on the thermal boundary condition.The optimization cases studied and baseline analysis were evaluated.Under the uniform heat flux, the shape and distribution of cooling channels depend on the geometry of the cooling structure (Figure 7).It can be concluded that the width of the cooling channel tends to be wide in the narrow zone under the uniform heat flux (Figure 8).Since the total heat of the narrow zone is less than the wide zone, the cooling channel in the narrow zone does not need to take as much heat as the wide zone.Under the uniform geometry of the cooling structure, here is the cylindrical section in Figure 9.The heat flux reaches the peak value of 6 MW/m 2 at 0.1 m to 0.15 m.As seen in Figure 10, the width of the cooling channel becomes narrow where heat flux climbs high.The reason why the width of the cooling channel is narrow is due to the total heat is large.In the example here (Figure 11), the shape and distribution of the cooling channels depend not only on the geometric boundary condition but also on the thermal boundary condition.It is a compromise between the influence of geometry and heat flux.It demonstrates that the channel width varies based on the geometry and thermal boundary condition, hence the channel width is not a constant.The optimal channel is presented in Figure 12.Due to the combustor wall being thin, the three-dimensional wall can be projected into twodimensional.The heat flux can be treated as one-dimensional, where a specified heat flux is used on the inner wall for the solid domain.

Optimized according to the thermal boundary condition (example 2)
For Case 3, the details are discussed here.In Figure 13, the configuration of the cooling structure including the cooling inlet, cooling outlet, and geometry structure with convergence and divergence.The target is to obtain optimal cooling channel distribution between the inlet and outlet under the constraints.The schematic drawing of the cross-section of the cooling structure for analysis is given in Figure 13, among which, δ is the rib thickness, W is the width of the cooling channel, H is the depth of the cooling channel, Hc is the cold side wall thickness, H h is the hot side wall thickness, Tw 1 represents inner wall temperature, Tw 2 represents cooling channel wall temperature, Tw 3 represents the outside wall temperature.The inner wall is subjected to the heat flux of high-temperature gas and the outside wall is considered to be adiabatic.Due to the same hot environment of every channel, a single cooling channel can be taken as the research object to be optimized.Since the cooling structure is fixed, the parameters should satisfy: H s is the total thickness of the cooling structure wall, W s is the total width of the cooling structure wall, m s is the total mass flow rate, and N is the number of cooling channels.
As known, the heat flux in the combustor usually reaches a few megawatts.Hence, the heat flux imposed on the cooling structure is depicted in Figure 11.All available details are demonstrated in Table 2.

Inlet mass flowrate Inlet Pressure
Inlet Temperature Length 1 kg/s 60 bar 300 K 270 mm The shape and distribution of cooling channels are formulated via the method described above.Whereas cooling channels can be generated in this way, the number of cooling channels still needs to be chosen.
The cooling structure is made of 1Cr18Ni9Ti with a temperature limit of 1473 K.The density and specific heat capacity can be treated as constants, and thermal conductivity is treated as a linear function of temperature.
(W/m•K) (21) The RP-1 kerosene serves as the coolant and the five-component model is adopted, then the thermal properties of kerosene at the required pressure and temperature can be calculated according to the NIST Supertrapp thermal properties calculation program.Figure 15.The variation of different cases with the number of cooling channels.Channel number N, aspect ratio α, rib to channel width ratio β are of great importance in the thermal design of cooling channels.First, the influence of channel number N is investigated.Based on the calculus of variations, the objective functions vary along with the N. Keeping the channel height H = 2 mm, H h = 0.6 mm, Hc = 1.2 mm, the inlet mass flow rate is set to be a constant of ms = 1 kg/s.

Channel number and width ratio
In the view of Figure 14, with the growth of the cooling channel number, the average temperature of the cooling structure decreases.Until the number of cooling channels reaches a certain number, the IOP Publishing doi:10.1088/1742-6596/2764/1/01203810 average temperature tends to be stable.Furthermore, when the number of cooling channels increases, the temperature inhomogeneity drops rapidly and then drops slowly.The loss of flow pressure reduces with the increase of cooling channel number.In addition, all the factors decrease dramatically with the rise of cooling channel number when the number is in less quantity, then decrease slowly with that in high quantity.When the channel number changes from 70 to 91, the average temperature decreases by 8%, the temperature inhomogeneity reduces by 8.5%, and the pressure drop drops by 40%.Besides, the higher the channel number is, the thinner the rib thickness is, and the machining process becomes challenging.Hence, the rib thickness should not be below 1 mm.
The four cases of different weight coefficients mentioned above show that the objective function becomes smaller when the number of cooling passages grows (Figure 15).It can be found that for all the cases, the minimum objective function is located in cooling channel number 91.

Effect of channel height (channel aspect ratio α)
The influence of α should be investigated as well.The conditions are as same as those mentioned above except for the channel height.The different α means different channel height, although the channel height changes under channel number 91, the channel width keeps still and the aspect ratio is proportional to the channel height (Figure 16).
The channel height can be adjusted as H = 1 mm, 1.5 mm, 2 mm, 2.5 mm.When the channel number equals 91, the parameters can vary along the channel height.Figure 17 shows the variation of the average temperature of the structure, temperature inhomogeneity, and pressure drop under different channel heights H.With the channel height increase, the average temperature decreases slowly from 603 K to 574 K, the temperature inhomogeneity keeps nearly still, around 115 K, while the pressure drop decreases from 20.1 bar to 2.4 bar.In addition, the higher H is, the lower the pressure drop can reach.The pressure drop decrease is significant at a relatively lower value of H, when H is larger than 2 mm, the variation tendency of pressure drop becomes slow.As channel height changes from 2 mm to 2.5 mm, the pressure drop decreases by 40%.Besides, for higher fin, the strength of the structure should be taken into consideration.Therefore, H = 2 mm is appropriate in this case.
It is clear that the highest temperature of x = 0.1 m in the cross-section in the largest channel height is higher than that in other channel heights (Figure 18).

Effect of mass flowrate
The optimal cooling structure parameters are found to be H = 2 mm, and N = 91.The influence of the mass flow rate ms is investigated, as shown in Figure 19.For different mass flow rates, pressure drop has a quadratic relationship with mass flow rate.In the different mass flow rates, the average temperature drops from 859 K to 508 K, the temperature inhomogeneity changes from 140 K to 105 K, and the pressure drop increases from 0.45 bar to 16 bar, when the flux increases from 300 g/s to 2000 g/s.Compromise of the thermal performance and pressure drop demonstrates that ms = 1 kg/s is appropriate.It shows that the highest temperature of x = 0.1 m in the cross-section in the lowest mass flow rate is greater than that in other mass flow rates (Figure 20).

Details of Case 3
To discuss the details of Case 3, it is concluded that when the channel number arrives at N = 91, the minimum objective function can be achieved at n = 0.05.After the calculation, the width of the cooling channel not only depends on the heat flux but also on the geometric boundary condition based on the principle.As a consequence of a combination of heat flux and geometric boundary conditions, the width of the channel varies in Figure 12.In this case, the minimum rib thickness is 1.03 mm, which nearly reaches the limitation of 1 mm.Table 3.The parameters of the optimization channel.21.It is found that the width of the optimization channel varies in different places.Taking the place where the highest heat flux is located (X = 0.1 m) as an example, the highest temperature stays above 800 K in the optimization channel.In addition, the Tw 1 of the optimization channel is 831 K.

Average
In specific analysis, the velocity in the optimization channel changes in different places apparently (Figure 22).The heat transfer coefficient in the optimization channel can be as high as 6400 W/m 2 K (Figure 23).Cross-section temperature in optimization channel.By virtue of the heat transfer coefficient in the optimization channel, Tw 1 of the optimization channel is reversed to the heat transfer coefficient, which can be seen in Figure 24.Tw 1 in the highest flux place is 830 K when the channel is optimized.
In addition, the average temperature of a cross-section along the flow of the optimization channel can be shown in Figure 25.
The pressure drop in the optimization channel is 4 bar (Figure 26).The cooling temperature climbs from 300 K to 458 K (Figure 27).

Conclusion
The thermal performance and pressure drop characteristics for the flow in cooling channels are analyzed in this paper.The results are shown for the cooling structure with different shapes.Based on the results and analysis, the conclusions can be obtained as follows.
(1) From the substance described above, the optimization method based on the calculus of variations has been developed and the cooling structure can be designed more effectively than regular design.The merits of the optimization developed here are: the cooling channel not only depends on the geometric shape of the structure but also the heat flux.Combining the geometric boundary condition and thermal boundary condition, the cooling channel can be generated.By the novel method, the minimum objective function can be obtained.
(2) For Case 3 illustrated here, the average temperature is 585 K, temperature inhomogeneity is 117 K, with the cost that the pressure drop is 4 bar.

Figure 1 .
Figure 1.Schematic diagram of design domain.To illustrate the generality of the methodology, in Figure2, here is an example of arbitrary physical space.To quantify the mapping in Figure2between the physical space (x, y) and computational space (ξ, η) by relations between (x, y) and (ξ, η).The inlets of all the cooling channels start from the left side, with outlets on the right side.The η line at a specific value is the center of the cooling channel and the width of cooling channel d is proportional to the grid density.The physical grid density varies with the temperature or q distribution.The grids become denser where the temperature or q is larger, then the d tends to be narrow.

Figure 2 .
Figure 2. Physical and computational space.Step 1.A conformal grid is generated based on the geometry of the cooling structure.In this paper, the Laplace equation is used to generate a conformal grid[13].The component functions ξ and η are harmonic, satisfying Laplace's equation:

Figure 3 .
Figure 3.The schematic of the cooling channel calculation unit division.(1) The cooling channel outside the wall is adiabatic.(2)1Cr18Ni9Ti is employed everywhere in the wall.(3)The fuel in the cooling channel has no change in the physical properties of the radial direction.(4) The influence of the thermal boundary layer is not considered.Since the channel hydrodynamic diameter is much smaller than the length, the cooling channel is divided into a few units, as shown in Figure3.

Figure 4 .
Figure 4.The schematic of the cooling channel element.

Figure 6 .
Figure 6.Flowchart of the optimization procedure.For the sake of the ultimate goal, the factor n and cooling channel number N can be adjusted to obtain the optimization structure.The optimization results can be achieved when:

Figure 7 .
Figure 7. Schematic diagram of the design domain.Under the uniform heat flux, the shape and distribution of cooling channels depend on the geometry of the cooling structure (Figure7).It can be concluded that the width of the cooling channel tends to be wide in the narrow zone under the uniform heat flux (Figure8).Since the total heat of the narrow zone is less than the wide zone, the cooling channel in the narrow zone does not need to take as much heat as the wide zone.

Figure 8 .
Figure 8.The schematic of a single channel for geometric boundary condition.

Figure 9 .
Figure 9.The schematic of the combustor heat flux distribution.Under the uniform geometry of the cooling structure, here is the cylindrical section in Figure9.The heat flux reaches the peak value of 6 MW/m 2 at 0.1 m to 0.15 m.As seen in Figure10, the width of the cooling channel becomes narrow where heat flux climbs high.The reason why the width of the cooling channel is narrow is due to the total heat is large.

8 Figure 10 .
Figure 10.The schematic of a single channel for thermal boundary condition.

4. 3 .Figure 11 .
Figure 11.The heat flux distribution of the combustor.In the example here (Figure11), the shape and distribution of the cooling channels depend not only on the geometric boundary condition but also on the thermal boundary condition.It is a compromise between the influence of geometry and heat flux.It demonstrates that the channel width varies based on the geometry and thermal boundary condition, hence the channel width is not a constant.The optimal channel is presented in Figure12.

Figure 12 .
Figure 12.The width of the cooling channel in the structure.Due to the combustor wall being thin, the three-dimensional wall can be projected into twodimensional.The heat flux can be treated as one-dimensional, where a specified heat flux is used on the inner wall for the solid domain.For Case 3, the details are discussed here.In Figure13, the configuration of the cooling structure including the cooling inlet, cooling outlet, and geometry structure with convergence and divergence.The target is to obtain optimal cooling channel distribution between the inlet and outlet under the constraints.The schematic drawing of the cross-section of the cooling structure for analysis is given in Figure13, among which, δ is the rib thickness, W is the width of the cooling channel, H is the depth of the cooling channel, Hc is the cold side wall thickness, H h is the hot side wall thickness, Tw 1 represents inner wall temperature, Tw 2 represents cooling channel wall temperature, Tw 3 represents the outside wall temperature.The inner wall is subjected to the heat flux of high-temperature gas and the outside wall is considered to be adiabatic.Due to the same hot environment of every channel, a single cooling channel can be taken as the research object to be optimized.Since the cooling structure is fixed, the parameters should satisfy:

9 Figure 13 .
Figure 13.The geometry of a single channel and schematic diagram of the design domain.H s is the total thickness of the cooling structure wall, W s is the total width of the cooling structure wall, m s is the total mass flow rate, and N is the number of cooling channels.As known, the heat flux in the combustor usually reaches a few megawatts.Hence, the heat flux imposed on the cooling structure is depicted in Figure11.All available details are demonstrated in Table2.Table2.Boundary conditions.

Figure 14 .
Figure 14.The variation of variables with the number of cooling channels.

Figure 16 .
Figure 16.The aspect ratio for different channel height.

Figure 17 .Figure 18 .
Figure 17.The variation of variables with different channel height.

Figure 19 .Figure 20 .
Figure 19.The variation of variables with the number of cooling channels for different mass flow rates.

Figure 21 .
Figure 21.The cross-section temperature diagram of the optimization channel.The temperature of cross sections is analyzed here.The temperature of cross sections is shown in Figure21.It is found that the width of the optimization channel varies in different places.Taking the place where the highest heat flux is located (X = 0.1 m) as an example, the highest temperature stays above 800 K in the optimization channel.In addition, the Tw 1 of the optimization channel is 831 K.In specific analysis, the velocity in the optimization channel changes in different places apparently (Figure22).The heat transfer coefficient in the optimization channel can be as high as 6400 W/m 2 K (Figure23).

Figure 22 .
Figure 22.The flow velocity in the optimization channel.

Figure 23 . 13 Figure 24 .
Figure 23.The heat transfer coefficient in the optimization channel.

Figure 25 .
Figure25.Cross-section temperature in optimization channel.By virtue of the heat transfer coefficient in the optimization channel, Tw 1 of the optimization channel is reversed to the heat transfer coefficient, which can be seen in Figure24.Tw 1 in the highest flux place is 830 K when the channel is optimized.In addition, the average temperature of a cross-section along the flow of the optimization channel can be shown in Figure25.The pressure drop in the optimization channel is 4 bar (Figure26).The cooling temperature climbs from 300 K to 458 K (Figure27).

Figure 26 .
Figure 26.The flow pressure drop in the optimization channel.

Figure 27 .
Figure 27.The coolant temperature in the optimization channel.

Table 1 .
Different weight coefficients for different cases.