Method for rapidly acquiring transient response characteristics of supercritical fluid under small pressure disturbance

Abstract

The invention discloses a method for quickly acquiring transient response characteristics of supercritical fluid under small pressure disturbance, and belongs to the field of energy power systems. According to the reasonable hypothesis, a control equation is simplified, and the control equation comprises a continuity equation, a momentum equation, an energy equation, a state equation and a speed correction equation; for a given working condition, a steady-state result is calculated based on the simplified control equation; and performing transient calculation by taking the calculated steady-state result as an initial field to obtain the transient response characteristic of the supercritical fluid. Under the condition that working condition parameters are known, the flow heat exchange transient response characteristics of the working condition under inlet pressure disturbance can be rapidly calculated, and the calculation magnitude and the calculation efficiency are greatly improved. The method is suitable for various supercritical fluids such as supercritical hydrocarbon fuel and supercritical water, the suitable structure is a single-tube cooling channel, the transient calculation magnitude can reach the nanosecond level, and the method has wide engineering applicability.

Description

Technical Field

[0001] This invention belongs to the field of energy and power system technology, specifically relating to a method for rapidly obtaining the transient response characteristics of supercritical fluids under small pressure disturbances. Background Technology

[0002] With the ever-increasing cooling demands of energy and power systems, supercritical fluids, possessing both the strong heat absorption capacity of liquids and the low viscosity of gases, have become ideal cooling media in key technology fields such as next-generation nuclear energy systems, solar thermal power generation, aerospace propulsion, and high-power electronic equipment cooling. In practical engineering cooling systems, the system is often subjected to minute pressure disturbances induced by factors such as pump and valve regulation, load variations, pulsating heat input, or external environmental disturbances, leading to changes in the fluid velocity and temperature field distribution within the system. Supercritical fluids undergo supercritical pressure flow heat transfer in cooling systems, and their temperature distribution typically spans the quasi-critical temperature region. Within this region, the fluid properties are extremely sensitive to temperature; even a small change in temperature can cause drastic nonlinear jumps in properties, resulting in flow instability phenomena such as drift or oscillation of system parameters like flow rate, pressure, and temperature. This leads to a decrease in cooling system efficiency and may also cause localized thermal stress concentration, seriously threatening the safety and reliability of the cooling structure. Therefore, accurately and efficiently revealing the transient response characteristics of supercritical fluid flow heat transfer under small pressure disturbances is of great significance for ensuring the safe, stable, and efficient operation of advanced energy systems.

[0003] Existing research largely focuses on the heat transfer characteristics of supercritical fluid flow under steady-state conditions, with limited coverage of the transient response characteristics under small pressure disturbances. Current methods for studying transient response characteristics of flow heat transfer primarily rely on experimental and three-dimensional numerical methods. On the one hand, experimental research faces challenges due to the high cost of constructing experimental systems and the difficulty of accurately capturing millisecond-level transient physical processes using existing measurement techniques. On the other hand, while three-dimensional numerical simulations can analyze such transient responses, their computational cost is enormous, and they impose extremely stringent requirements on mesh scale, time step, and turbulence model, making implementation difficult. In conclusion, the application of experimental and three-dimensional simulation methods in analyzing transient flow responses under pressure disturbances remains constrained, hindering rapid prediction of engineering problems in this field.

[0004] Therefore, how to construct a method for rapidly obtaining the transient response characteristics of supercritical fluids under small pressure disturbances applicable to various fluids is a technical challenge that must be solved to realize the calculation of the transient response characteristics of supercritical fluid flow and heat transfer. Summary of the Invention

[0005] To address the issue of transient response characteristics of supercritical fluids under small pressure disturbances within cooling channels, this invention provides a method for rapidly obtaining the transient response characteristics of supercritical fluids under small pressure disturbances.

[0006] This invention overcomes the limitations of experimental methods, such as high cost and difficulty in capturing millisecond-level transient processes, and avoids the problems of high computational cost and strict requirements on mesh and time step in three-dimensional numerical simulation. It can quickly calculate the transient response characteristics of the system under inlet pressure disturbance (not exceeding 1% of pipeline pressure drop), such as changes in pressure, temperature and velocity fields.

[0007] This invention provides a fast and reliable method for obtaining transient response characteristics, which is especially suitable for flow heat transfer analysis under small pressure disturbances.

[0008] The technical solution adopted by this invention to solve the technical problem is as follows:

[0009] This invention provides a method for rapidly obtaining the transient response characteristics of supercritical fluids under small pressure disturbances, which specifically includes the following steps:

[0010] Step S1: Make reasonable assumptions;

[0011] Step S2: Based on the reasonable assumptions of Step S1, the governing equations are simplified. The governing equations include: continuity equation, momentum equation, energy equation, state equation, and velocity correction equation.

[0012] Step S3: For a given operating condition, calculate the steady-state result based on the simplified control equations;

[0013] Step S4: Use the steady-state result calculated in step S3 as the initial field to perform transient calculations and obtain the transient response characteristics of the supercritical fluid.

[0014] Furthermore, in step S1, the reasonable assumptions are as follows:

[0015] (1) Simplify the control equation to one dimension;

[0016] (2) Ignore unsteady pressure pulsation terms;

[0017] (3) Neglect the work done by surface forces;

[0018] (4) Ignore the viscous dissipation term;

[0019] (5) Ignore axial heat conduction terms;

[0020] (6) Ignore the effect of pressure drop on physical properties.

[0021] Furthermore, in step S2, the continuity equation is as follows:

[0022]

[0023] Where ρ is the density of the supercritical fluid, t is the flow time of the supercritical fluid, u is the flow velocity of the supercritical fluid, and x is the axial coordinate.

[0024] Furthermore, in step S2, the momentum equation is as follows:

[0025]

[0026] Where ρ is the density of the supercritical fluid, t is the flow time of the supercritical fluid, u is the flow velocity of the supercritical fluid, x is the axial coordinate, p is the pressure of the supercritical fluid, f is the friction coefficient, d is the inner diameter of the pipe, and g is the axial acceleration.

[0027] Furthermore, in step S2, the energy equation is as follows:

[0028]

[0029] Where ρ is the density of the supercritical fluid, t is the flow time of the supercritical fluid, u is the flow velocity of the supercritical fluid, x is the axial coordinate, h is the specific enthalpy of the supercritical fluid, and q... w Let S be the heat flux through the wall, S be the perimeter of the pipe cross-section, and A be the cross-sectional area of ​​the pipe.

[0030] Furthermore, in step S2, the state equation is as follows:

[0031] ρ=F(p,T) (4)

[0032] Where ρ is the density of the supercritical fluid, T is the temperature, and F(p,T) represents the state equation function.

[0033] Furthermore, in step S2, the velocity correction equation is as follows:

[0034]

[0035] Among them, u' e For the speed correction value of interface e, A e Let e ​​be the interface area, and a be the interface area. e Let P' be the coefficient of the momentum equation for interface e. P P' is the pressure correction value for the central node. E This is the pressure correction value for downstream nodes.

[0036] Furthermore, the specific implementation process of step S3 is as follows:

[0037] S3.1: Given boundary conditions: inlet flow rate m 0 Initial field velocity distribution u 0 Initial field of pressure distribution p 0 and initial field T of temperature distribution 0 ;

[0038] S3.2: Solve the state equation (4) to obtain the n-step iteration density ρ. n ;

[0039] S3.3: Solve the momentum equation (2) to obtain the predicted velocity value u. * ;

[0040] S3.4: Substitute the velocity correction equation (5) into the continuity equation (1) to obtain the pressure iteration value p at iteration step n+1. n+1 With velocity iteration value u n+1 ;

[0041] S3.5: Solve the energy equation (3) to obtain the temperature iteration value T at iteration step n+1. n+1 ;

[0042] S3.6: Solve the state equation (4) to obtain the iteration density ρ for the n+1th iteration step. n+1 ;

[0043] S3.7: If the difference between the density iteration values ​​of adjacent levels is less than the residual, then the steady-state result is obtained through convergence. If the difference between the density iteration values ​​of adjacent levels is greater than the residual, then repeat steps S3.1-S3.6 until the steady-state result is obtained through convergence: steady-state pressure field p steady Steady-state temperature field T steady and steady-state velocity field u steady .

[0044] Furthermore, in step S4, a small pressure disturbance is applied to the inlet at the initial moment of calculation. This small pressure disturbance does not exceed 1% of the pipeline pressure drop, and the changes of system physical quantities such as flow rate over time are monitored and recorded to obtain the transient response characteristics of the supercritical fluid under the small pressure disturbance.

[0045] Furthermore, the specific implementation process of step S4 is as follows:

[0046] S4.1: Given boundary conditions: initial field velocity distribution u 0 =u steady Initial field of pressure distribution p 0 =p steady and initial field T of temperature distribution 0 =T steady And give the inlet a small pressure disturbance;

[0047] S4.2: Solve the state equation (4) to obtain the iteration density ρ of time step j and iteration step n. j,n ;

[0048] S4.3: Solve the momentum equation (2) to obtain the predicted velocity value u. * ;

[0049] S4.4: Substitute the velocity correction equation (5) into the continuity equation (1) to obtain the pressure iteration value p at time step n+1. j,n+1 With velocity iteration value u j,n+1 ;

[0050] S4.5: Solve the energy equation (3) to obtain the temperature iteration value T at time step j+1. j,n+1 ;

[0051] S4.6: Solve the state equation (4) to obtain the iteration density ρ of time step n+1. j,n+1 ;

[0052] S4.7: If the difference between the density iteration values ​​of adjacent layers is less than the residual, then the steady-state result at time step j is obtained through convergence. If the difference between the density iteration values ​​of adjacent layers is greater than the residual, then steps S4.2-S4.6 are repeated until the steady-state result is obtained through convergence. After convergence, the process proceeds to time step j+1 and steps S4.2-S4.7 are repeated. Finally, the transient response characteristics of the supercritical fluid under small pressure disturbances are obtained: transient pressure field p transient The transient temperature field T changes with time transient The transient velocity field u changing with time transient .

[0053] The beneficial effects of this invention are:

[0054] Compared to experimental methods and 3D numerical simulation methods, this invention significantly improves computational efficiency. Based on reasonable assumptions, this invention simplifies the governing equations (such as one-dimensionalization and ignoring multiple minor factors), greatly reducing computational complexity. This invention features small computational load and high speed, enabling nanosecond-level transient response calculations, making it suitable for rapid prediction and analysis in engineering applications.

[0055] This invention is applicable to a variety of supercritical fluids, such as supercritical hydrocarbon fuels and supercritical water; it is suitable for single-pipe cooling channel structures, has a wide range of applications, and has good engineering applicability and versatility.

[0056] This invention is easy to implement and can be used for safety assessment, system optimization, and stability analysis of practical energy and power systems (such as nuclear energy, solar energy, and aerospace propulsion).

[0057] In summary, this invention has significant advantages in terms of computational efficiency, applicability, and engineering suitability. It provides a fast, economical, and reliable technical means for transient response analysis of supercritical fluids under pressure disturbances, which is of great significance for ensuring the safe and stable operation of advanced energy systems. Attached Figure Description

[0058] Figure 1The flowchart illustrates a method for rapidly obtaining the transient response characteristics of a supercritical fluid under small pressure disturbances, as provided by this invention.

[0059] Figure 2 Temperature field calculation verification was performed to implement the method for rapidly obtaining transient response characteristics of supercritical fluid under small pressure disturbances provided by this invention.

[0060] Figure 3 This invention provides a method for rapidly obtaining transient response characteristics of supercritical fluids under small pressure disturbances, which is used to calculate and verify the transient response characteristics of flow rate. Detailed Implementation

[0061] The present invention will be further described in detail below with reference to the accompanying drawings.

[0062] This invention provides a method for rapidly obtaining the transient response characteristics of supercritical fluids under small pressure disturbances. It is applicable to a variety of fluids, including supercritical hydrocarbon fuels and supercritical water. The applicable structure is mainly a single-tube cooling channel, and the transient calculation can reach the nanosecond level.

[0063] This invention provides a method for rapidly obtaining the transient response characteristics of supercritical fluids under small pressure disturbances. Based on reasonable assumptions, the governing equations are simplified, and by solving these equations, the transient response characteristics of the supercritical fluid under small pressure disturbances are obtained. The specific implementation process is as follows:

[0064] Step S1: Make reasonable assumptions;

[0065] (1) Simplify the control equation to one dimension;

[0066] (2) Ignore unsteady pressure pulsation terms;

[0067] (3) Neglect the work done by surface forces;

[0068] (4) Ignore the viscous dissipation term;

[0069] (5) Ignore axial heat conduction terms;

[0070] (6) Ignore the effect of pressure drop on physical properties.

[0071] Step S2: Based on the reasonable assumptions made in Step S1, simplify the governing equations;

[0072] The simplified governing equations are as follows:

[0073] Continuity equation:

[0074]

[0075] Where ρ is the density of the supercritical fluid, t is the flow time of the supercritical fluid, u is the flow velocity of the supercritical fluid, and x is the axial coordinate.

[0076] Momentum equation:

[0077]

[0078] Where p is the pressure of the supercritical fluid, f is the friction coefficient, d is the inner diameter of the pipe, and g is the axial acceleration.

[0079] Energy equation:

[0080]

[0081] Where h is the specific enthalpy of the supercritical fluid, q w Let S be the heat flux through the wall, S be the perimeter of the pipe cross-section, and A be the cross-sectional area of ​​the pipe.

[0082] Equations of state:

[0083] ρ=F(p,T) (4)

[0084] Where T is the temperature, and F(p,T) represents the state equation function.

[0085] Velocity correction equation:

[0086]

[0087] Among them, u' e For the speed correction value of interface e, A e Let e ​​be the interface area, and a be the interface area. e Let P' be the coefficient of the momentum equation for interface e. P P' is the pressure correction value for the central node. E This is the pressure correction value for downstream nodes.

[0088] Step S3: Calculate the steady-state result;

[0089] For a given operating condition, the steady-state result is calculated based on the simplified control equations above. The specific implementation process is as follows:

[0090] S3.1: Initial boundary conditions;

[0091] Given inflow m 0 Initial field velocity distribution u 0 Initial field of pressure distribution p 0 and the initial field T of temperature distribution 0 Equal boundary conditions;

[0092] S3.2: Solve the state equation (4) to obtain the n-step iteration density ρ. n This is the old density field;

[0093] S3.3: Solve the momentum equation (2) to obtain the predicted velocity value u. * This is the predicted velocity field;

[0094] S3.4: Substitute the velocity correction equation (5) into the continuity equation (1) to obtain the pressure iteration value p at iteration step n+1. n+1 With velocity iteration value u n+1 ;

[0095] S3.5: Solve the energy equation (3) to obtain the temperature iteration value T at iteration step n+1. n+1 This is the new temperature field;

[0096] S3.6: Solve the state equation (4) to obtain the iteration density ρ for the n+1th iteration step. n+1 This is the new density field;

[0097] S3.7: If abs(ρ n+1 -ρ n If the difference between adjacent density iteration values ​​is less than the residual, then the convergence yields a steady-state result. If the difference between adjacent density iteration values ​​is greater than the residual, then repeat steps S3.1-S3.6 until the convergence yields a steady-state result.

[0098] For a given operating condition, the steady-state result can be calculated by performing steps S3.1-S3.7, and then the steady-state pressure field p can be obtained. steady Steady-state temperature field T steady and steady-state velocity field u steady .

[0099] Step S4: Use the steady-state result calculated in step S3 as the initial field to perform transient calculations and obtain the transient response characteristics of the supercritical fluid.

[0100] By applying a small pressure disturbance to the inlet at the initial moment of calculation (typically not exceeding 1% of the pipeline pressure drop) and monitoring and recording the changes in system physical quantities such as flow rate over time, the transient response characteristics of the supercritical fluid under the small pressure disturbance can be obtained. The specific implementation process is as follows:

[0101] S4.1: Given the initial field u of the velocity distribution 0 =u steady Initial field of pressure distribution p 0 =p steady Initial field of temperature distribution T 0 =T steady Equal boundary conditions, and a small pressure disturbance is applied to the inlet;

[0102] S4.2: Solve the state equation (4) to obtain the iteration density ρ of time step j and iteration step n.j,n This is the old density field;

[0103] S4.3: Solve the momentum equation (2) to obtain the predicted velocity value u. * This is the predicted velocity field;

[0104] S4.4: Substitute the velocity correction equation (5) into the continuity equation (1) to obtain the pressure iteration value p at time step n+1. j,n+1 With velocity iteration value u j,n+1 ;

[0105] S4.5: Solve the energy equation (3) to obtain the temperature iteration value T at time step j+1. j,n+1 This is the new temperature field;

[0106] S4.6: Solve the state equation (4) to obtain the iteration density ρ of time step n+1. j,n+1 This is the new density field;

[0107] S4.7: If abs(ρ j,n+1 -ρ j,n If the difference between adjacent density iteration values ​​is less than the residual, then the steady-state result at time step j is obtained. If the difference between adjacent density iteration values ​​is greater than the residual, then steps S4.2-S4.6 are repeated until the steady-state result is obtained. After convergence, the process proceeds to time step j+1 and steps S4.2-S4.7 are repeated.

[0108] For a given operating condition, by executing steps S4.1-S4.7, the transient response characteristics of the supercritical fluid under small pressure disturbances can be obtained, and thus the transient pressure field p can be obtained. transient The transient temperature field T changes with time transient The transient velocity field u changing with time transient .

[0109] This invention provides a method for rapidly obtaining the transient response characteristics of supercritical fluids under small pressure disturbances. The transient response characteristics of the temperature field and flow rate of a supercritical fluid in a heating channel are calculated. The temperature field calculation results are as follows: Figure 2 As shown, the calculation results of the transient response characteristics of the flow rate are as follows: Figure 3 As shown, the temperature field and flow transient response characteristics obtained by implementing the rapid acquisition method for supercritical fluid transient response characteristics under small pressure disturbances provided by the present invention are compared with the experimental values ​​or reference values ​​given in the references, which confirms the correctness and feasibility of the rapid acquisition method for supercritical fluid transient response characteristics under small pressure disturbances provided by the present invention.

[0110] Figure 2The temperature field calculation results obtained by the method proposed in this invention are compared with the experimental results in Reference 1 (Yang Z, Shan Y, Zhang B, et al. Hydrodynamic characteristics of cyclohexane in a horizontal mini-tube at trans-and supercritical pressures[J]. Applied Thermal Engineering, 2018, 129: 62-69.). The maximum error does not exceed 8%. The experimental data in Reference 1 is Exp in the figure.

[0111] Figure 3 The transient response characteristics of the flow obtained by the method proposed in this invention are compared with the numerical calculation results in reference 2 (Du W, Pan Y, Wang N, et al. Investigation on hydrodynamic characteristic and flow instability in regenerative cooling channels[J]. Applied Thermal Engineering, 2025, 269: 125983.). The maximum error does not exceed 10%. The numerical calculation data in reference 2 is shown in Ref in the figure.

[0112] In summary, the method for rapidly obtaining the transient response characteristics of supercritical fluids under small pressure disturbances provided by this invention is easy to apply to the rapid calculation of the transient response characteristics of supercritical fluids under small pressure disturbances, and has extremely strong engineering application value.

[0113] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. However, these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for rapidly obtaining the transient response characteristics of a supercritical fluid under small pressure disturbances, characterized in that, Includes the following steps: Step S1: Make reasonable assumptions; Step S2: Based on the reasonable assumptions of Step S1, the governing equations are simplified. The governing equations include: continuity equation, momentum equation, energy equation, state equation, and velocity correction equation. Step S3: For a given operating condition, calculate the steady-state result based on the simplified control equations; Step S4: Use the steady-state result calculated in step S3 as the initial field to perform transient calculations and obtain the transient response characteristics of the supercritical fluid.

2. The method for rapidly obtaining the transient response characteristics of supercritical fluid under small pressure disturbances according to claim 1, characterized in that, In step S1, the reasonable assumptions are as follows: (1) Simplify the control equations to one dimension; (2) Ignore unsteady pressure pulsation terms; (3) Neglect the work done by surface forces; (4) Ignore the viscous dissipation term; (5) Ignore axial heat conduction terms; (6) Ignore the effect of pressure drop on physical properties.

3. The method for rapidly obtaining the transient response characteristics of a supercritical fluid under small pressure disturbances according to claim 1, characterized in that, In step S2, the continuity equation is as follows: Where ρ is the density of the supercritical fluid, t is the flow time of the supercritical fluid, u is the flow velocity of the supercritical fluid, and x is the axial coordinate.

4. The method for rapidly obtaining the transient response characteristics of supercritical fluid under small pressure disturbances according to claim 1, characterized in that, In step S2, the momentum equation is as follows: Where ρ is the density of the supercritical fluid, t is the flow time of the supercritical fluid, u is the flow velocity of the supercritical fluid, x is the axial coordinate, p is the pressure of the supercritical fluid, f is the friction coefficient, d is the inner diameter of the pipe, and g is the axial acceleration.

5. The method for rapidly obtaining the transient response characteristics of supercritical fluid under small pressure disturbances according to claim 1, characterized in that, In step S2, the energy equation is as follows: Where ρ is the density of the supercritical fluid, t is the flow time of the supercritical fluid, u is the flow velocity of the supercritical fluid, x is the axial coordinate, h is the specific enthalpy of the supercritical fluid, and q... w Let S be the heat flux through the wall, S be the perimeter of the pipe cross-section, and A be the cross-sectional area of ​​the pipe.

6. The method for rapidly obtaining the transient response characteristics of a supercritical fluid under small pressure disturbances according to claim 1, characterized in that, In step S2, the state equation is as follows: ρ=F(p,T) (4) Where ρ is the density of the supercritical fluid, T is the temperature, and F(p,T) represents the state equation function.

7. The method for rapidly obtaining the transient response characteristics of a supercritical fluid under small pressure disturbances according to claim 1, characterized in that, In step S2, the velocity correction equation is as follows: Among them, u' e For the speed correction value of interface e, A e Let e ​​be the interface area, and a be the interface area. e Let P' be the coefficient of the momentum equation for interface e. P P' is the pressure correction value for the central node. E This is the pressure correction value for downstream nodes.

8. The method for rapidly obtaining the transient response characteristics of a supercritical fluid under small pressure disturbances according to claim 1, characterized in that, The specific implementation process of step S3 is as follows: S3.1: Given boundary conditions: inlet flow rate m 0 Initial field velocity distribution u 0 Initial field of pressure distribution p 0 and initial field T of temperature distribution 0 ; S3.2: Solve the state equation (4) to obtain the n-step iteration density ρ. n ; S3.3: Solve the momentum equation (2) to obtain the predicted velocity value u. * ; S3.4: Substitute the velocity correction equation (5) into the continuity equation (1) to obtain the pressure iteration value p at iteration step n+1. n+1 With velocity iteration value u n+1 ; S3.5: Solve the energy equation (3) to obtain the temperature iteration value T at iteration step n+1. n+1 ; S3.6: Solve the state equation (4) to obtain the iteration density ρ for the n+1th iteration step. n+1 ; S3.7: If the difference between the density iteration values ​​of adjacent levels is less than the residual, then the steady-state result is obtained through convergence. If the difference between the density iteration values ​​of adjacent levels is greater than the residual, then repeat steps S3.1-S3.6 until the steady-state result is obtained through convergence: steady-state pressure field p steady Steady-state temperature field T steady and steady-state velocity field u steady .

9. The method for rapidly obtaining the transient response characteristics of a supercritical fluid under small pressure disturbances according to claim 1, characterized in that, In step S4, a small pressure disturbance is applied to the inlet at the initial moment of calculation. This small pressure disturbance does not exceed 1% of the pipeline pressure drop, and the changes of system physical quantities such as flow rate over time are monitored and recorded to obtain the transient response characteristics of the supercritical fluid under the small pressure disturbance.

10. A method for rapidly obtaining the transient response characteristics of a supercritical fluid under small pressure disturbances according to claim 1 or 9, characterized in that, The specific implementation process of step S4 is as follows: S4.1: Given boundary conditions: initial field velocity distribution u 0 =u steady Initial field of pressure distribution p 0 =p steady and initial field T of temperature distribution 0 =T steady And give the inlet a small pressure disturbance; S4.2: Solve the state equation (4) to obtain the iteration density ρ of time step j and iteration step n. j,n ; S4.3: Solve the momentum equation (2) to obtain the predicted velocity value u. * ; S4.4: Substitute the velocity correction equation (5) into the continuity equation (1) to obtain the pressure iteration value p at time step n+1. j,n+1 With velocity iteration value u j,n+1 ; S4.5: Solve the energy equation (3) to obtain the temperature iteration value T at time step j and iteration step n+1. j,n+1 ; S4.6: Solve the state equation (4) to obtain the iteration density ρ of time step n+1. j,n+1 ; S4.7: If the difference between the density iteration values ​​of adjacent layers is less than the residual, then the steady-state result at time step j is obtained through convergence. If the difference between the density iteration values ​​of adjacent layers is greater than the residual, then steps S4.2-S4.6 are repeated until the steady-state result is obtained through convergence. After convergence, the process proceeds to time step j+1 and steps S4.2-S4.7 are repeated. Finally, the transient response characteristics of the supercritical fluid under small pressure disturbances are obtained: transient pressure field p transient The transient temperature field T changes with time transient The transient velocity field u changing with time transient .

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