Regenerative cooling multi-channel layout design method based on fuel reaction progress uniformity optimization

Abstract

The application discloses a kind of based on fuel reaction progress uniformity optimization's regenerative cooling multi-channel layout design method, comprising: setting initial channel number and determining the design fuel flow of each channel, initial state is equidistantly distributed and section geometric parameter is set;Cooling structure is divided into multiple sites perpendicular to fuel flow direction;According to the non-uniform thermal field of engine, load thermal boundary condition;Through calculating heat conduction process and fuel cracking reaction, obtain the fuel temperature and cracking reaction data of each site;The position of channel is adjusted between adjacent sites by optimization algorithm, and the value of objective function is minimized;Through repeated optimization calculation, obtain the channel layout of all sites, and connect each site by cubic spline curve to obtain the final multi-channel layout optimization result.The application solves the layout adaptation problem of regenerative cooling channel in complex thermal environment, improves the cooling efficiency, reduces fuel excessive cracking and cooling capacity waste.

Description

Technical Field

[0001] This invention relates to the field of thermal protection technology for scramjet engines, and in particular to a regenerative cooling multi-channel layout design method based on fuel reaction progress uniformity optimization. Background Technology

[0002] Scramjet engines, as a key power system for future hypersonic vehicles, must meet the demands for high specific impulse and high thrust. With increasing flight speeds, especially at Mach 5 and above, the temperature of the combustion gases within the engine rapidly rises, exceeding 2500K, surpassing the temperature resistance limits of most materials. Therefore, effectively cooling the combustion chamber is a critical issue that must be addressed in the design of scramjet engines.

[0003] To address this challenge, regenerative cooling technology using hydrocarbon fuels as coolants has become a common thermal protection method for scramjet engines. Its basic principle is to utilize the coolant (usually the engine's load fuel) flowing through cooling channels on the combustion chamber walls. The combustion chamber walls are cooled by the physical heating and heat absorption of the fuel through its cracking reaction. During this process, the cooled fuel, after absorbing heat, enters the combustion chamber, mixes with air, and burns, recovering and reusing the absorbed heat to form a closed-loop cooling system.

[0004] However, the layout and design of cooling channels present significant challenges due to the typically significant non-uniformity of heat load distribution within the combustion chamber. Particularly in high heat load regions, the fuel within the cooling channels may undergo cracking reactions due to excessive heating. The unsaturated hydrocarbons in the cracking products can then form coke at high temperatures, causing channel blockage and leading to cooling failure. Conversely, in low heat load regions, the fuel temperature within the cooling channels is relatively low, resulting in underutilization of cooling capacity and wasted cooling resources.

[0005] Therefore, in the design of the cooling system of scramjet engines, how to rationally arrange the cooling channels under uneven thermal loads, coordinate the fuel cracking reaction progress in different regions, and avoid excessive cracking and coking, as well as wasted cooling capacity, has become a key issue in research and design. This not only relates to the engine's thermal protection performance but also directly affects its operating efficiency and safety. Summary of the Invention

[0006] To address the shortcomings of existing technologies, this invention provides a regenerative cooling multi-channel layout design method based on fuel reaction progress uniformity optimization.

[0007] To achieve the above-mentioned objectives, the technical solution adopted by the present invention is as follows:

[0008] A regenerative cooling multi-channel layout design method based on fuel reaction progress uniformity optimization includes the following steps:

[0009] Step 1: Set the initial number of channels. Divide the total available cooling fuel flow rate of the design area by the number of channels to determine the design fuel flow rate of each channel. Set the initial geometric parameters of the channel cross-section with the channels evenly spaced as the initial state.

[0010] Step 2: Divide the cooling structure into several stations perpendicular to the overall fuel flow direction and number them. Each station is a two-dimensional geometric region containing the flow cross-section of all channels and the cross-section of solid material.

[0011] Step 3: Transform the non-uniform thermal field of the engine into a second or third type of thermal boundary condition and apply it to the boundary of the solid material geometric region at each site;

[0012] Step 4: Take two adjacent stations and calculate the two-dimensional heat conduction process in the solid material region. The calculation method is to solve the heat conduction equation using finite volume and finite difference methods, and calculate the endothermic heating and cracking reaction of the fuel through the quasi-one-dimensional reactive flow control equation. Couple the heat transfer calculation to obtain the temperature distribution of the solid material cross section, and obtain the fuel temperature and cracking reaction data information at the station channel outlet.

[0013] Step 5: In two adjacent stations, the center coordinates of the channel flow section of the downstream station are used as design variables. The optimal design variables are obtained by calculating the minimum objective function value through the optimization algorithm.

[0014] Step 6: Starting from the first station at the inlet of the engine regenerative cooling channel, proceed downstream by repeating the optimization calculations of Step 4 and Step 5 until the outlet of the regenerative cooling channel.

[0015] Step 7: Export the center coordinates of the cooling channels of all stations obtained from the optimization calculation, sort them along the flow direction, and connect the sorted points using the piecewise cubic spline curve method to obtain the optimized design results of the regenerative cooling multi-channel layout.

[0016] Furthermore, the quasi-one-dimensional reactive flow governing equation in step 4 is as follows:

[0017]

[0018] In the formula, ρ, u, P, and e represent the fluid's density, velocity, pressure, and internal energy, respectively. i ω i Let A and Q represent the molar molecular weight, net reaction rate, and mass fraction of species i, respectively. e and a e These represent the cross-sectional area, heat flux density, and surface area per unit length of the channel, respectively.

[0019] Furthermore, in step 5, the objective function is determined by the temperature deviation of each channel when the pyrolysis conversion rate is zero; when there is a conversion rate greater than zero, the objective function is determined by the degree of deviation of variables such as the pyrolysis rate or the content of a certain component in the pyrolysis products from the mean.

[0020] Furthermore, the optimization algorithm used in step 5 is a Bayesian algorithm, a genetic algorithm, or a machine learning optimization algorithm.

[0021] Furthermore, the Bayesian algorithm selects Gaussian process regression as an alternative model, generates initial samples through Latin hypercube sampling, and selects the expected boosting function as the acquisition function.

[0022] Furthermore, step 5 includes the following:

[0023] Between two adjacent stations, the center coordinates of the downstream station's channel flow section were selected as the design variable. The fuel temperature and pyrolysis conversion rate of each channel at the downstream station were statistically analyzed. When the pyrolysis conversion rate was 0, the temperature deviation of each channel was used as the objective function. When a conversion rate greater than 0 existed, the kerosene pyrolysis rate Z of each channel was used as the objective function. i The degree of deviation relative to the mean is used as the optimization objective function;

[0024] The following system of nonlinear equations is constructed using constraints:

[0025]

[0026] The objective function f(X) is expressed as:

[0027]

[0028] Among them, T total The sum of the temperatures of all six channels is calculated using the following formula: The formula for calculating the sum of the fragmentation rates of all six channels is as follows: X is the set of optimization variables; T i Let be the temperature of the i-th channel.

[0029] The Bayesian optimization algorithm is used to adjust the design variables through a Gaussian process regression model. The expected improvement function is selected as the acquisition function, and step 4 calculations are performed multiple times to optimize the design variables by minimizing the objective function value.

[0030] The Bayesian optimization update formula is as follows:

[0031] a(x)=σ(x|D)[γ(x)Φ(γ(x))+φ(γ(x))],x∈X

[0032] in, x∈X, Φ(·) and φ(·) are the cumulative distribution function (CDF) and probability density function under the standard Gaussian distribution, respectively, D is the current dataset, μ and σ represent the mean and standard deviation of the Gaussian regression process, respectively, f best σ(x|D) represents the optimal target value, σ(x|D) represents the standard deviation of the Gaussian process regression model, and γ(x) represents the standardized predicted value.

[0033] Through optimized iterative calculations, when the wall temperature difference change is less than 0.01, the objective function value is output, and the current optimal design variable x is used as the basis for the calculation. new Update the dataset until the convergence condition is met:

[0034] ||X * -X‖ <E

[0035] Where E is the allowable error of the design variable, X* is the design variable of the current iteration, and X is the design variable value of the previous iteration.

[0036] Furthermore, the axial advance distance S in step 6 n-1,n The calculation formula is:

[0037]

[0038] Where, x n,i y n,i z n,i Let be the center coordinates of the cooling channel, n and n-1 be the labels of the divided stations, and i be the label of the cooling channel.

[0039] Furthermore, in step 7, the center coordinates of all the optimized cooling channels at each site are sorted along the flow direction according to their respective cooling channel numbers. The sorted points are then connected using a piecewise cubic spline curve method to finally obtain the optimized regenerative cooling multi-channel layout design.

[0040] Compared with the prior art, the advantages of the present invention are as follows:

[0041] 1. This invention optimizes the layout of the cooling channels, enabling the fuel pyrolysis reaction to proceed uniformly, effectively preventing local overheating and excessive pyrolysis and coking, significantly improving the cooling efficiency of the combustion chamber walls, and avoiding thermal runaway problems caused by uneven cooling.

[0042] 2. By optimizing the channel distribution, the risk of coking of unsaturated hydrocarbons caused by excessive fuel cracking is reduced, thereby improving the reliability and long-term operational stability of the cooling system.

[0043] 3. For low heat load areas, by adjusting the design of the cooling channels, the waste of fuel cooling capacity in low temperature areas is avoided, thereby maximizing the cooling effect and improving the overall thermal management capability of the system.

[0044] 4. Compared with the traditional uniformly distributed channel layout, the channel layout optimized in this invention reduces the outlet pyrolysis conversion rate non-uniformity index by 90.27% and the outlet temperature non-uniformity index by 83.97% under the same operating conditions, which greatly improves the stability and efficiency of engine operation.

[0045] 5. This invention effectively solves the problem of traditional cooling channel design being unable to adapt to complex thermal environments, especially in dealing with the challenges brought by circumferential non-uniform thermal loads, and has important application value and practical significance. Attached Figure Description

[0046] Figure 1 This is a flowchart of the regenerative cooling multi-channel layout design method based on fuel reaction progress uniformity optimization in an embodiment of the present invention;

[0047] Figure 2 This is a schematic diagram of a cooling plate with six cooling channels according to an embodiment of the present invention; (a) is a geometric model diagram, and (b) is a two-dimensional cross-sectional view.

[0048] Figure 3 This is a schematic diagram of site division according to an embodiment of the present invention;

[0049] Figure 4 This is a diagram showing the distribution of cooling channels before and after optimization in an embodiment of the present invention;

[0050] Figure 5 This is a comparison of coolant outlet oil temperature and conversion rate before and after optimization in the embodiments of the present invention. Detailed Implementation

[0051] To make the objectives, technical solutions, and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the accompanying drawings and examples.

[0052] This invention provides a regenerative cooling multi-channel layout design method based on fuel reaction progress uniformity optimization, the process of which is shown in the attached figure. Figure 1 As shown, the specific steps include the following:

[0053] Step 1: Set the initial number of channels. Divide the total available cooling fuel flow rate of the design area by the number of channels to determine the design fuel flow rate of each channel. Assuming the channels are evenly spaced, set the initial geometric parameters of the channel cross-sections.

[0054] Step 2: Divide the cooling structure into several stations perpendicular to the overall fuel flow direction and number them. Each station is a two-dimensional geometric region containing the cross-sections of all channels and the solid material.

[0055] Step 3: Transform the non-uniform thermal field of the engine into a second or third type of thermal boundary condition and apply it to the boundary of the solid material geometric region of each site.

[0056] Step 4: Select two adjacent stations and calculate the two-dimensional heat conduction process in the solid material region. The calculation method uses finite volume and finite difference methods to solve the heat conduction equation. Taking the corresponding channel flow sections at the two adjacent stations as inlets and outlets, the endothermic heating and decomposition reaction of the fuel between the inlets and outlets are calculated using the quasi-one-dimensional reactive flow control equation. Coupled heat transfer from the solid material to the cooling channel is calculated through convective boundary conditions. Data such as the temperature distribution of the solid material cross-section, fuel temperature at the station channel outlet, and decomposition reaction are obtained.

[0057] Step 5: For adjacent stations, use the center coordinates of the flow cross-section of the downstream station as the design variable. Statistically analyze the fuel temperature and pyrolysis conversion rate of each channel at the downstream station. When the pyrolysis conversion rate is 0, use the temperature deviation of each channel as the objective function; when there is a conversion rate greater than 0, use the deviation of variables such as the pyrolysis rate or the content of a certain component in the pyrolysis products relative to the mean as the optimization objective function. The optimization variables must conform to the geometric structure as a constraint. Using optimization algorithms such as Bayesian algorithm, genetic algorithm, and machine learning optimization, adjust the design variables under the constraints, performing several calculations in Step 4 to obtain a set of optimal design variables that minimizes the objective function value.

[0058] Step 6: Starting from the first station at the inlet of the engine regenerative cooling channel, take two adjacent stations each time and repeat the optimization calculations of Step 4 and Step 5 to advance the operation downstream until the outlet of the regenerative cooling channel.

[0059] Step 7: Export the center coordinates of the cooling channels of all stations obtained from the optimization calculation, sort them along the flow direction according to their corresponding cooling channel numbers, and connect the sorted points using methods such as piecewise cubic spline curves to obtain the optimized design results of the regenerative cooling multi-channel layout.

[0060] The quasi-one-dimensional reactive flow control equation in step 4 should also include necessary mathematical and physical models such as the kinetic mechanism of the cracking reaction, the actual gas equation of state, and the thermodynamic equation.

[0061] The following specific examples illustrate the present invention. A cooling plate containing six cooling channels is selected as an optimized example, as shown in the attached figure. Figure 2 As shown.

[0062] Step 1: Set the initial number of channels to 6, and the total available cooling fuel flow rate in the design area to 33.36 g / s, evenly distributed across the 6 cooling channels. Assuming the channels are evenly spaced, set the initial geometric parameters of the channel cross-sections. Set the width and height of the cooling channels to 2 mm, the distance between the channels and the upper and lower walls to 1.5 mm, and the coolant to be kerosene with an initial temperature of 300 K and a pressure of 5 MPa.

[0063] Step 2: Divide the cooling structure into 51 stations perpendicular to the overall fuel flow direction and number the cooling channels sequentially from 1 to 6. Each station is a two-dimensional geometric region containing the flow cross-section of all channels and the cross-section of the solid material, as shown in the attached diagram. Figure 3 As shown.

[0064] Step 3: Transform the non-uniform thermal field of the engine into the third type of thermal boundary conditions, namely heat transfer coefficient and adiabatic temperature, and apply them to the boundary of the solid material geometric region of each site.

[0065] Step 4: Taking two adjacent sites, calculate the two-dimensional heat conduction process in the solid material region. The calculation method uses a finite volume method based on triangular meshes to solve the variable-property two-dimensional heat conduction equation. Using the corresponding channel flow sections at the two adjacent sites as inlets and outlets, the endothermic heating and pyrolysis reaction of the fuel between the inlets and outlets are calculated using the quasi-one-dimensional reactive flow governing equation. The quasi-one-dimensional governing equation is as follows:

[0066]

[0067] In equations (1)-(4), the variables are defined as follows: ρ, u, P, and e represent the fluid's density, velocity, pressure, and internal energy, respectively. i ω i and Y i A and Q represent the molar molecular weight, net reaction rate, and mass fraction of species i, respectively. e and a e These represent the cross-sectional area, heat flux density, and surface area per unit length of the channel, respectively. The third term on the left side of equation (2) represents the effect of fluid viscous forces, which can be expressed by Darcy's law. The change in mass fraction in component equation (4) is mainly due to the net reaction rate w of each component. i This is caused by [the mechanism of reaction]. The net reaction rate needs to be calculated based on the pyrolysis reaction mechanism.

[0068] Secondly, the Penn-Robinson (PR) real gas equation of state and its corresponding thermodynamic model are used to calculate parameters such as density and internal energy in equations (1)-(4). Finally, the heat transfer in the cooling channel is transformed into convective boundary conditions to realize the coupled heat transfer calculation from the solid material to the cooling channel. Data such as the temperature distribution of the solid material cross section and the fuel thermometer cracking reaction at the station channel outlet are obtained.

[0069] Step 5: In the case of two adjacent stations, the center coordinates of the flow cross section of the downstream station are used as design variables. The fuel temperature and pyrolysis conversion rate of each channel at the downstream station are statistically analyzed. When the pyrolysis conversion rate is 0, the temperature deviation of each channel is used as the objective function; when there is a conversion rate greater than 0, the deviation of the kerosene pyrolysis rate Z of each channel from the mean is used as the optimization objective function. With the constraint that the optimization variables must conform to the geometric structure, the following constrained nonlinear equation system is constructed:

[0070]

[0071] in,

[0072]

[0073] Using a Bayesian algorithm, the design variables are adjusted under constraints, and step 4 is performed several times to obtain a set of optimal design variables that minimize the objective function. Gaussian process regression is chosen as the alternative model for the original optimization problem in the Bayesian algorithm, and initial samples are generated through Latin hypercube sampling. The expected improvement function is selected as the data collection function in the optimization policy.

[0074] a(x)=σ(x|D)[γ(x)Φ(γ(x))+φ(γ(x))],x∈X (8)

[0075]

[0076] Φ(·) and φ(·) are the cumulative distribution function (CDF) and probability density function under the standard Gaussian distribution, respectively. D is the current dataset, and μ and σ represent the mean and standard deviation of the Gaussian regression process, respectively.

[0077] Based on the current optimal design variable x new Parametric modeling and generation of a two-dimensional mesh are performed. Through several calculations in step 4, the convergence condition is that the change in wall temperature difference between the previous and subsequent iterations is less than 0.01. The objective function f(x) is then output.

[0078] Additionally, the Bayesian algorithm involves updating the dataset, specifically x. new Add the corresponding function value f(x) to the dataset X, and repeat step 4. The convergence condition for the optimization iteration is:

[0079] ||X * -X‖ <E(10)

[0080] E represents the allowable error of the design variables, and X represents... * X is the design variable for the current iteration, and X is the design variable value for the previous iteration. In this example, E takes the value 1e. -2 .

[0081] Step 6: Starting from the first station at the inlet of the engine regenerative cooling channel, repeat the optimization calculations of Steps 4 and 5 for each adjacent station, advancing downstream until the outlet of the regenerative cooling channel. The axial advancement distance between adjacent stations is calculated as follows:

[0082]

[0083] The subscript i represents the cooling channel number, and n and n-1 represent the station numbers. x, y, and z are the center coordinates of each cooling channel.

[0084] Step 7: Export the center coordinates of the cooling channels of all stations obtained from the optimization calculation, sort them along the flow direction according to their corresponding cooling channel numbers, and connect the sorted points using methods such as piecewise cubic spline curves to obtain the optimized design results of the regenerative cooling multi-channel layout.

[0085] The traditional uniform cooling channel layout and the channel layout obtained by the above optimization are shown in the attached figure. Figure 4 As shown. Figure 4 As shown, the optimized channels are densely distributed in the high-temperature region. Secondly, comparing the outlet temperature and conversion rate of the two channel layouts, the optimized channel layout exhibits more uniform outlet temperature and conversion rate. Figure 5 As shown, using standard deviation as a measure of the non-uniformity of outlet temperature and conversion rate, the optimized outlet temperature and conversion rate decreased by 83.97% and 90.27%, respectively, compared to the traditional uniform channel arrangement.

[0086] The methods described above according to the invention can be implemented in hardware, firmware, or as software or computer code that can be stored in a recording medium (such as CD-ROM, RAM, floppy disk, hard disk, or magneto-optical disk), or as computer code originally stored on a remote recording medium or a non-transitory machine-readable medium and subsequently stored on a local recording medium, downloaded via a network. Thus, the methods described herein can be stored as software processing on a recording medium using a general-purpose computer, a dedicated processor, or programmable or dedicated hardware (such as an ASIC or FPGA). It is understood that the computer, processor, microprocessor controller, or programmable hardware includes storage components (e.g., RAM, ROM, flash memory, etc.) capable of storing or receiving software or computer code. When said software or computer code is accessed and executed by the computer, processor, or hardware, the regenerative cooling multi-channel layout design method based on fuel reaction progress uniformity optimization described herein is implemented. Furthermore, when a general-purpose computer accesses the code used to implement the processes shown herein, the execution of the code transforms the general-purpose computer into a dedicated computer for performing the processes shown herein.

[0087] Those skilled in the art will recognize that the embodiments described herein are intended to help the reader understand the implementation methods of the present invention, and should be understood that the scope of protection of the present invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific modifications and combinations based on the technical teachings disclosed in this invention without departing from the spirit of the invention, and these modifications and combinations are still within the scope of protection of the present invention.

Claims

1. A regenerative cooling multi-channel layout design method based on fuel reaction progress uniformity optimization, characterized in that, Includes the following steps: Step 1: Set the initial number of channels. Divide the total available cooling fuel flow rate of the design area by the number of channels to determine the design fuel flow rate of each channel. Set the initial geometric parameters of the channel cross-section with the channels evenly spaced as the initial state. Step 2: Divide the cooling structure into several stations perpendicular to the overall fuel flow direction and number them. Each station is a two-dimensional geometric region containing the flow cross-section of all channels and the cross-section of solid material. Step 3: Transform the non-uniform thermal field of the engine into a second or third type of thermal boundary condition and apply it to the boundary of the solid material geometric region at each site; Step 4: Take two adjacent stations and calculate the two-dimensional heat conduction process in the solid material region. The calculation method is to solve the heat conduction equation using the finite volume and finite difference method, and calculate the endothermic heating and cracking reaction of the fuel through the quasi-one-dimensional reactive flow control equation. Couple the heat transfer calculation to obtain the temperature distribution of the solid material cross section, and obtain the fuel temperature and cracking reaction data information at the station channel outlet. Step 5: In two adjacent stations, the center coordinates of the channel flow section of the downstream station are used as design variables. The optimal design variables are obtained by calculating the minimum objective function value through the optimization algorithm. Specifically, it includes the following: Between two adjacent stations, the center coordinates of the downstream station's channel flow section were selected as the design variable. The fuel temperature and pyrolysis conversion rate of each channel at the downstream station were statistically analyzed. When the pyrolysis conversion rate was 0, the temperature deviation of each channel was used as the objective function. When a conversion rate greater than 0 existed, the kerosene pyrolysis rate Z of each channel was used as the objective function. i The degree of deviation relative to the mean is used as the optimization objective function; The following system of nonlinear equations is constructed using constraints: ； Wherein, objective function Represented as: ； in, The sum of the temperatures of all six channels is calculated using the following formula: ； X is the set of optimization variables; Let i be the temperature of the i-th channel; The Bayesian optimization algorithm is used to adjust the design variables through a Gaussian process regression model. The expected improvement function is selected as the acquisition function, and step 4 is calculated multiple times to optimize the design variables by minimizing the objective function value. The Bayesian optimization update formula is as follows: ； in, , (⋅) and (⋅) represent the cumulative distribution function (CDF) and probability density function under the standard Gaussian distribution, respectively, and D is the current dataset. and and represent the mean and standard deviation of the Gaussian regression process, respectively. To achieve the optimal target value, The standard deviation of the Gaussian process regression model is given. These are standardized predicted values; Through optimized iterative calculations, when the wall temperature difference change is less than 0.01, the objective function value is output, and the current optimal design variable x is used as the basis for the calculation. new Update the dataset until the convergence condition is met: ； Where E is the allowable error of the design variable, X* is the design variable of the current iteration, and X is the design variable value of the previous iteration; Step 6: Starting from the first station at the inlet of the engine regenerative cooling channel, proceed downstream by repeating the optimization calculations of Step 4 and Step 5 until the outlet of the regenerative cooling channel. Step 7: Export the center coordinates of the cooling channels of all stations obtained from the optimization calculation, sort them along the flow direction, and connect the sorted points using the piecewise cubic spline curve method to obtain the optimized design results of the regenerative cooling multi-channel layout.

2. The regenerative cooling multi-channel layout design method according to claim 1, characterized in that: The governing equations for the quasi-one-dimensional reactive flow in step 4 are as follows: ； ； ； ； In the formula, ρ, u, P, and e represent the fluid's density, velocity, pressure, and internal energy, respectively; M i ω i A and Yi represent the molar molecular weight, net reaction rate, and mass fraction of species i, respectively; A and Q e and a e These represent the cross-sectional area, heat flux density, and surface area per unit length of the channel, respectively.

3. The regenerative cooling multi-channel layout design method according to claim 1, characterized in that: Axial advance distance in step 6 The calculation formula is: ； in, , , Let be the center coordinates of the cooling channel, n and n−1 be the labels of the divided stations, and i be the label of the cooling channel.

4. The regenerative cooling multi-channel layout design method according to claim 1, characterized in that: In step 7, the center coordinates of all the optimized cooling channels at each site are sorted along the flow direction according to their respective cooling channel numbers. The sorted points are then connected using a piecewise cubic spline curve method to finally obtain the optimized regenerative cooling multi-channel layout design.

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