One-dimensional fluid pipe network simulation method suitable for secondary air system
By constructing a constrained set of control equations and a damping mechanism using FloTides software, and combining the Newton-Raphson iteration method and the fixed-point iteration method, the problems of inaccurate coupling between the fluid and solid domains and the black-box nature of the solver in the secondary air system are solved, achieving high-precision and stable simulation results and supporting the development of custom components.
Patent Information
- Application Number
- CN202511945414.6
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-12-22
- Publication Date
- 2026-02-06
AI Technical Summary
Existing one-dimensional fluid network simulation software suffers from limitations in convergence and accuracy in secondary air systems, inaccurate coupling between fluid and solid domains, and difficulties in algorithm optimization and model verification due to the black-box nature of the solver.
The constrained control equations are constructed using the domestic FloTides software. Combined with the damping mechanism, the multiphysics field is solved iteratively in groups to achieve steady-state/transient solutions. The energy balance equations of fluid and solid nodes are established through the coordinated iteration of the Newton-Raphson iteration method and the fixed-point iteration method. The modular solver architecture supports the development of custom components.
It improves the solution accuracy and convergence stability of nonlinear systems, realizes complete simulation of fluid-solid thermal coupling, is applicable to high-temperature secondary air systems, supports the development of custom components, and solves the black-box problem of the solution process in commercial software.
Smart Images
Figure CN121479938A_ABST
Abstract
Description
TECHNICAL FIELD
[0001] The present application relates to the technical field of fluid mechanics simulation, in particular to a one-dimensional fluid pipe network simulation method suitable for a secondary air system. BACKGROUND
[0002] A secondary air system (SAS) is commonly used in gas turbines, aero-engines and large-scale turbomachinery. The system is responsible for cooling, sealing, purging and driving auxiliary components, and has an important influence on the efficiency, stability and service life of the whole machine. Due to the existence of multi-branch flow, complex geometric constraints, and thermal-flow-rotation multi-physical coupling characteristics in the system, its analysis usually relies on one-dimensional fluid network simulation methods. With the increasing performance requirements of modern gas turbines, the simulation of the secondary air system gradually needs to consider the swirl transfer effect, solid heat capacity and heat conduction process, and the coupling of complex transient state equations. Although existing one-dimensional fluid pipe network simulation software (such as Flomaster, Flownex, etc.) can realize the steady-state and transient calculation of complex fluid networks, its numerical calculation framework and model mechanism still have the following shortcomings: first, the approximate linearization leads to limited convergence and precision. When the system contains strong nonlinear elements, the linearization error accumulates significantly, and the calculation result deviates from the true physical law. When in non-design point or strong coupling working condition, the convergence is significantly reduced, and the iteration may oscillate or diverge; second, the fluid domain and solid domain are usually weakly coupled through the boundary heat transfer coefficient or the empirical heat transfer model, which cannot accurately reflect the strong interaction between the fluid temperature, wall temperature and heat conduction structure in the secondary air system. Especially in non-steady state conditions (such as cold start, transition heat exchange, wall temperature rise response), the calculation error is significant. In addition, the multi-physical interaction effects of swirl, leakage flow and solid heat conduction are often simplified or ignored in the existing system, resulting in a large deviation between the simulation result and the actual operation; third, the solver part of the existing commercial software is mostly a closed black box algorithm, and users cannot customize or intervene in the solving strategy (such as damping, linearization update step, convergence control criterion), which limits the embedding of algorithm optimization, model verification and self-research control strategy, lacks bottom interface support, and it is difficult to realize the expandable implementation of algorithm-level innovation or cross-domain coupled solution. SUMMARY
[0003] The main purpose of the present application is to provide a one-dimensional fluid pipe network simulation method suitable for a secondary air system, which is used to solve the problems in the background art.
[0004] To achieve the above-mentioned purpose, the technical scheme adopted by the present application is as follows: The present application provides a one-dimensional fluid pipe network simulation method suitable for a secondary air system, and the steps are as follows: S1: Using the domestic one-dimensional fluid pipeline simulation software FloTides, drag and drop components to build the pipeline system of the secondary air system; S2: Using the nodal dwell mass as the core state variable, a set of constrained control equations is constructed; the equations are coupled and constrained to form a unified steady-state / transient solution model. S3: The governing equations are grouped and iteratively solved based on the coupling strength and solution characteristics between the equations, and combined with the damping mechanism to achieve stable and coordinated convergence of the multi-physics field; S4: The steady-state / transient solution model performs steady-state and transient calculations respectively under a unified governing equation system; S5: Decouple the solution and result output, perform data iteration, until the system converges.
[0005] Furthermore, in S1, the FloTides software automatically identifies the topology of the pipe network based on the connection relationships between the components in the secondary air system's pipe network. Locations with fluid storage capacity in the pipe network are defined as nodes, and one-dimensional flow channels between nodes are defined as branches. Nodes are used to carry the state variables of fluid pressure and temperature, while branches are used to describe the mass flow and momentum transfer relationships between nodes and the physical characteristics of solid heat transfer units. For solid regions in the system, the FloTides software discretizes them into solid nodes and solid conductor components to characterize the heat conduction process inside the solid and its coupling heat transfer with the fluid.
[0006] Furthermore, in S2, the set of governing equations includes the mass conservation equation, momentum conservation equation, energy balance equation, state equation, and vortex transfer equation. The mass conservation equation, momentum conservation equation, and vortex transfer equation all explicitly consider the dynamic changes of the resident mass at the nodes and are combined with the state equations of the nodes to achieve a complete description and accurate simulation of the steady-state / transient behavior of the pipeline system.
[0007] Furthermore, a mass conservation equation is established at the fluid node to constrain the balance between the change in the resident mass at the node and the mass flow rate of the connected branch. The mass conservation equation is expressed as follows: , in, Indicates the first Fluid retention mass at each fluid node; , These represent the branch mass flow rates flowing into and out of the node, respectively. , Representing the nodes respectively A set of connected entry branches and a set of connected exit branches; Indicates time; A momentum conservation equation driven by the nodal pressure difference is established for each branch. The momentum conservation equation is expressed as follows: , in, and These represent the pressure at upstream and downstream nodes of the branch, respectively. This is represented as a pressure term, which is the main driving force of the branched flow; For resistance, This is the equivalent loss coefficient, used to uniformly characterize frictional resistance and local resistance; and These represent the fluid density and temperature within the branch, respectively. For branch quality flow; This is the inertial term (unsteady momentum term); The equivalent inertia coefficient for the branch.
[0008] Furthermore, energy balance equations are established for fluid and solid nodes respectively; The energy balance equation for the fluid node is expressed as follows: , in, For nodes enthalpy value, For heat source items, This represents the convective heat transfer between the current fluid node and the solid. The energy balance equation for solid nodes is expressed as follows: , in, and Solid nodes Specific heat capacity and temperature; and These represent the convective heat transfer between the solid node and the fluid, and the heat transfer between adjacent nodes within the solid, respectively. In transient simulations, the coupling relationship between the change in nodal resident mass and pressure and temperature is described using state equations, the expression of which is: , in, and They are nodes Pressure and temperature; Equivalent control volume for nodes; is the fluid gas constant.
[0009] Furthermore, to establish a swirling flow momentum balance within a chamber or gap exhibiting swirling flow characteristics, and considering the swirling flow attenuation factor, the following swirling flow transfer equation is adopted: , in, , expressed as vortex intensity per unit mass, and These are the tangential velocities of the fluid and the rotating wall, respectively. and They are respectively with nodes A set of connected entry branches and a set of connected exit branches; For nodes Entry Branch mass flow rate; For branches The swirl intensity carried by medium flow rates; For nodes Export Branch mass flow rate; For branches The swirl intensity carried by medium flow rates; For nodes vortex source term; is the vortex attenuation coefficient.
[0010] Furthermore, in S3, the mass conservation equation, momentum conservation equation, and state equation in the governing equations form a strongly coupled set. This set of equations includes dynamic changes in nodal mass, branch momentum balance, and closed-loop pressure-temperature-volume relationships at the nodes, and they are strongly coupled with each other through nodal mass, branch flow rate, and pressure variables. The Newton-Raphson iterative method is employed, and the Jacobian matrix is constructed... Iterative update of unknown vectors , , in, The nonlinear residual vector is composed of the mass conservation equation, momentum conservation equation, and state equation. Iteration is performed until the residual converges to the preset accuracy. The Jacobian matrix is constructed; It is the inverse of the Jacobian matrix; For the first The vector value calculated in the next iteration; This refers to the damping factor during the variable update process. Relaxation control is applied to the correction amount to limit the variation range of a single iteration; The energy balance equation and the swirl transfer equation are constructed as relatively independent sets. These equations mainly rely on nodal mass, flow rate, and swirl variables, but their coupling with momentum constraints is relatively weak, allowing for separate solutions. The solution method employs a fixed-point iteration method: using the mass and flow rate calculated in the previous step as known inputs, the nodal enthalpy is updated sequentially. With swirling variables The iteration continues until the nodal enthalpy and swirling variable converge; the equations for the fixed-point iteration method are as follows: , in, For the enthalpy update function, and the current th The nodes calculated in the next iteration enthalpy value Quality of stay and upstream and downstream branches mass flow rate Related, For the vortex update function, and the current _th The nodes calculated in the next iteration vortex Quality of stay and upstream and downstream branches mass flow rate Related; and are the fixed-point iterative relaxation factors for nodal enthalpy and swirling variable, respectively.
[0011] Furthermore, in each Newton-Raphson iteration, after updating the node mass, node pressure, and branch mass flow rate, convective heat transfer between the fluid and solid is introduced into the node energy calculation to update the enthalpy and temperature of the fluid node. Simultaneously, based on the updated fluid node enthalpy, the temperature change of the solid node is calculated. In the next iteration, the heat transfer is recalculated using the new solid node temperature and fed back into the fluid energy balance equation, thereby achieving coordinated iteration and overall convergence among multiple physical field variables such as fluid momentum, heat, and swirling flow. When the residuals of all equation sets meet the convergence conditions, the entire pipeline system converges.
[0012] Furthermore, in S4, when performing steady-state calculations, it is assumed that the resident mass of each node in the pipeline system does not change with time. The time-varying terms in the mass conservation equation, momentum conservation equation, and fluid energy balance equation are set to zero, forming a static nonlinear algebraic equation system. The mass flow rate, node pressure, and temperature distribution of the pipeline system under steady-state conditions are obtained by iterative solution. In transient calculations, the governing equations are solved using a time-stepping method. Within each time step, the time-varying terms in the mass conservation equation, momentum conservation equation, and energy balance equation are discretized using the finite difference method. The changes in the nodal dwell mass and related state variables are calculated. By progressively advancing the time steps, the dynamic response process of the pipeline system is simulated.
[0013] Furthermore, in S5, the model is built, the control equations are established and solved, and the results are output functionally decoupled. The model building and results output are implemented based on the user graphical interface of the FloTides software. The solver is embedded in the system as an independent module to complete the automatic construction and numerical solution of the control equations, realizing a modular solution architecture. After constructing and solving the global control equations, the solver calculates the pressure loss coefficient and heat transfer of each component according to the latest distribution and feeds it back to the next iteration, iterating until the system converges.
[0014] Compared with existing technologies, this invention uses Newton's method to accurately solve the mass conservation equation, momentum conservation equation, and state equation, overcoming the accuracy loss of traditional approximate linearization algorithms in strongly nonlinear systems, thus improving the solution accuracy and convergence stability of nonlinear equation sets. Simultaneously, it establishes fluid-solid node energy balance equations, which can simulate complex heat transfer paths and wall thermal inertia, applicable to high-temperature secondary air systems, ensuring complete fluid-solid thermal coupling. The swirling flow transfer equation and energy balance equation characterize swirling flow loss, swirling flow ratio, and swirling flow attenuation characteristics within the cavity, providing efficient and accurate simulation for secondary air systems under complex multi-condition scenarios. By introducing a damping mechanism, it effectively suppresses iterative oscillations and convergence difficulties in strongly nonlinear systems. Finally, through an independent modular solver architecture, it supports custom component development, and the source of the control equations is traceable, solving the problems of black-box solving processes and difficulties in model expansion in commercial software. Attached Figure Description
[0015] Figure 1 This is a schematic diagram of the piping system construction for the secondary air system of the present invention.
[0016] Figure 2 This is a flowchart of the coupled solution process of Newton's method and fixed-point iteration method of the present invention.
[0017] Figure 3 This is a flowchart of the steady-state solution process for the secondary air system of the present invention.
[0018] Figure 4 This is a flowchart of the transient solution process for the secondary air system of the present invention. Detailed Implementation
[0019] To make the objectives, technical means and advantages of the present invention readily understood, the present invention will be further described below in conjunction with specific embodiments.
[0020] like Figures 1 to 4 As shown, this invention provides a one-dimensional fluid pipeline network simulation method suitable for secondary air systems, with the following steps: S1: Using the domestic one-dimensional fluid pipeline simulation software FloTides, drag and drop components to build the pipeline system of the secondary air system; S2: Using the nodal dwell mass as the core state variable, a set of constrained control equations is constructed; the equations are coupled and constrained to form a unified steady-state / transient solution model. S3: The governing equations are grouped and iteratively solved based on the coupling strength and solution characteristics between the equations, and combined with the damping mechanism to achieve stable and coordinated convergence of the multi-physics field; S4: The steady-state / transient solution model performs steady-state and transient calculations respectively under a unified governing equation system; S5: Decouple the solution and result output, perform data iteration, until the system converges.
[0021] As a preferred embodiment, such as Figure 1 As shown, in S1, the FloTides software automatically identifies the topology of the secondary air system's piping network based on the connection relationships between components. Locations with fluid storage capacity are defined as nodes, and one-dimensional flow channels (i.e., components) between nodes are defined as branches. Nodes carry the state variables of fluid pressure and temperature, while branches (such as orifice plates, throttle valves, and pipe sections) describe the mass flow and momentum transfer relationships between nodes, as well as the physical characteristics of solid heat transfer units. For solid regions in the system, the FloTides software discretizes them into solid nodes and solid conductor components to characterize the internal heat conduction process of the solid and its coupled heat transfer with the fluid. After completing the automatic node-branch discretization of the piping network system, this invention does not directly apply traditional fluid control equations. Instead, it constructs a set of constrained control equations for calculation, using the resident mass of nodes as the core state variable, to achieve a unified description of the dynamic behavior of complex piping network systems.
[0022] In a preferred embodiment, in S2, the governing equations include mass conservation equations, momentum conservation equations, energy balance equations, state equations, and vortex transfer equations. The mass conservation equations, momentum conservation equations, and vortex transfer equations all explicitly consider the dynamic changes in the resident mass at nodes and are combined with the state equations of the nodes to achieve a complete description and accurate simulation of the steady-state / transient behavior of the pipeline system. The equation sets are coupled and mutually constrained at the node and branch levels, jointly forming a unified steady-state / transient solution model.
[0023] In a preferred embodiment, a mass conservation equation is established at the fluid node to constrain the balance between the change in the resident mass at the node and the mass flow rate of the connected branch. The mass conservation equation is expressed as follows: , in, Indicates the first Fluid retention mass at each fluid node; , These represent the branch mass flow rates flowing into and out of the node, respectively. , Representing the nodes respectively A set of connected entry branches and a set of connected exit branches; The mass conservation equation is used to describe the dynamic change of the internal mass of a node over time under transient conditions, and constitutes an important constraint condition for the state equation and the energy conservation equation. A momentum conservation equation driven by the nodal pressure difference is established for each branch. The momentum conservation equation is expressed as follows: , in, and These represent the pressure at upstream and downstream nodes of the branch, respectively. This is represented as a pressure term, which is the main driving force of the branched flow; For resistance, This is the equivalent loss coefficient, used to uniformly characterize frictional resistance and local resistance; and These represent the fluid density and temperature within the branch, respectively. For branch quality flow; This is the inertial term (unsteady momentum term); The equivalent inertia coefficient for the branch.
[0024] As a preferred embodiment, energy balance equations are established for fluid and solid nodes respectively; The energy balance equation for the fluid node is expressed as follows: , in, For nodes enthalpy value, For heat source items, This represents the convective heat transfer between the current fluid node and the solid. The energy balance equation for solid nodes is expressed as follows: , in, and Solid nodes Specific heat capacity and temperature; and These represent the convective heat transfer between the solid node and the fluid, and the heat transfer between adjacent nodes within the solid, respectively. In transient simulations, the coupling relationship between the change in nodal resident mass and pressure and temperature is described using state equations, the expression of which is: , in, and They are nodes Pressure and temperature; Equivalent control volume for nodes; As the fluid gas constant, high-precision two-way thermal coupling simulation between fluid and solid was achieved by constructing energy balance equations based on fluid nodes and solid nodes.
[0025] As a preferred embodiment, a swirling flow momentum balance is established in a chamber or gap with swirling flow characteristics. Considering the swirling flow attenuation factor, the following swirling flow transfer equation expression is adopted: , in, , expressed as vortex intensity per unit mass, and These are the tangential velocities of the fluid and the rotating wall, respectively. and They are respectively with nodes A set of connected entry branches and a set of connected exit branches; For nodes Entry Branch mass flow rate; For branches The swirl intensity carried by medium flow rates; For nodes Export Branch mass flow rate; For branches The swirl intensity carried by medium flow rates; For nodes vortex source term; is the vortex attenuation coefficient.
[0026] In a preferred embodiment, in S3, the mass conservation equation, momentum conservation equation, and state equation in the governing equations form a strongly coupled set. This set of equations includes dynamic changes in nodal mass, branch momentum balance, and closed-loop pressure-temperature-volume relationships at the nodes, and they are strongly coupled with each other through nodal mass, branch flow rate, and pressure variables. The Newton-Raphson iterative method is used to construct the Jacobian matrix. Iterative update of unknown vectors , , in, The nonlinear residual vector is composed of the mass conservation equation, momentum conservation equation, and state equation. Iteration is performed until the residual converges to the preset accuracy. The Jacobian matrix is constructed; It is the inverse of the Jacobian matrix; For the first The vector value calculated in the next iteration; This refers to the damping factor during the variable update process. The correction amount is relaxed to limit the change range of a single iteration, thus preventing numerical oscillation or divergence caused by excessive nonlinearity.
[0027] The energy balance equation and the swirl transfer equation are constructed as relatively independent sets. These equations mainly rely on nodal mass, flow rate, and swirl variables, but their coupling with momentum constraints is relatively weak, allowing for separate solutions. The solution method employs a fixed-point iteration method: using the mass and flow rate calculated in the previous step as known inputs, the nodal enthalpy is updated sequentially. With swirling variables The iteration continues until the nodal enthalpy and swirling variable converge; the equations for the fixed-point iteration method are as follows: , in, For the enthalpy update function, and the current th The nodes calculated in the next iteration enthalpy value Quality of stay and upstream and downstream branches mass flow rate Related, For the vortex update function, and the current _th The nodes calculated in the next iteration vortex Quality of stay and upstream and downstream branches mass flow rate Related; and These are the fixed-point iterative relaxation factors for nodal enthalpy and swirling variables, respectively, used to slow down the variable update rate and avoid divergence in energy or swirling calculations under strongly coupled conditions.
[0028] In a preferred embodiment, after updating the node mass, node pressure, and branch mass flow rate during each Newton-Raphson iteration, convective heat transfer between the fluid and solid is introduced into the node energy calculation to update the enthalpy and temperature of the fluid node. Simultaneously, the temperature change of the solid node is calculated based on the updated fluid node enthalpy. In the next iteration, the heat transfer is recalculated using the new solid node temperature and fed back into the fluid energy balance equation, thereby achieving coordinated iteration and overall convergence among multiple physical field variables such as fluid momentum, heat, and swirling flow. When the residuals of all equation sets meet the convergence conditions, the entire pipeline system converges.
[0029] As a preferred embodiment, such as Figures 2 to 4 As shown, in S4, when performing steady-state calculations, it is assumed that the resident mass of each node in the pipeline system does not change with time. The time-varying terms in the mass conservation equation, momentum conservation equation, and fluid energy balance equation are set to zero, forming a set of static nonlinear algebraic equations. The mass flow rate, node pressure, and temperature distribution of the pipeline system under steady-state conditions are obtained by iterative solution. In transient calculations, a time-stepping approach is used to solve the governing equations. Within each time step, the time-varying terms in the mass conservation equation, momentum conservation equation, and energy balance equation are discretized using the finite difference method. The changes in nodal resident mass and related state variables are calculated. By progressively advancing the time steps, the dynamic response process of the pipeline system is simulated. This approach enables steady-state and transient calculations to be performed within a unified governing equation framework, balancing computational efficiency and dynamic simulation accuracy.
[0030] In a preferred embodiment, in S5, model building, the establishment of the control equations, and the decoupling of the output functions are performed. Model building and output are implemented based on the user graphical interface of the FloTides software. The solver is embedded in the system as an independent module to automatically build and numerically solve the control equations, realizing a modular solution architecture. After building and solving the global control equations, the solver calculates the pressure loss coefficient and heat transfer of each element according to the latest distribution and feeds it back to the next iteration, iterating until the system converges. Under this architecture, local pressure loss equations are defined at the element level to calculate the drag or loss coefficients in the momentum equation, and local heat transfer equations are defined to characterize the heat source term in the energy equation. The solver, as an independent module embedded in the system, can expand the pipeline element model without modifying the core algorithm of the solver, improving the scalability and maintainability of the system.
[0031] This invention uses Newton's method to accurately solve the mass conservation equation, momentum conservation equation, and state equation, overcoming the accuracy loss of traditional approximate linearization algorithms in strongly nonlinear systems. It also establishes energy balance equations for fluid and solid nodes, enabling the simulation of complex heat transfer paths and wall thermal inertia, making it suitable for high-temperature secondary air systems and ensuring complete fluid-solid thermal coupling. The invention characterizes swirling losses, swirling ratios, and swirling attenuation characteristics within the cavity using swirling transfer equations and energy balance equations, providing efficient and accurate simulation for secondary air systems under complex multi-condition scenarios. By introducing a damping mechanism, it effectively suppresses iterative oscillations and convergence difficulties in strongly nonlinear systems. Finally, through an independent modular solver architecture, it supports the development of custom components, and the source of the control equations is traceable, solving the problems of black-box solving processes and difficulties in model expansion in commercial software.
[0032] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely illustrative of the principles of the invention. Various changes, equivalent substitutions, and improvements can be made without departing from the spirit and scope of the invention, and all such changes should fall within the scope of the present invention as claimed. The scope of protection of this invention is defined by the appended claims and their equivalents.
Claims
1. A one-dimensional fluid network simulation method suitable for secondary air systems, characterized in that, Includes the following steps: S1: Using the domestic one-dimensional fluid pipeline simulation software FloTides, drag and drop components to build the pipeline system of the secondary air system; S2: Using the nodal dwell mass as the core state variable, a set of constrained control equations is constructed; the equations are coupled and constrained to form a unified steady-state / transient solution model. S3: The governing equations are grouped and iteratively solved based on the coupling strength and solution characteristics between the equations, and combined with the damping mechanism to achieve stable and coordinated convergence of the multi-physics field; S4: The steady-state / transient solution model performs steady-state and transient calculations respectively under a unified governing equation system; S5: Decouple the solution and result output, perform data iteration until the system converges.
2. The one-dimensional fluid pipeline simulation method for secondary air systems according to claim 1, characterized in that, In S1, the FloTides software automatically identifies the topology of the pipeline network based on the connection relationship between the components in the secondary air system's pipeline network, defining the locations in the pipeline network with fluid storage capacity as nodes, and the one-dimensional flow channels between nodes as branches. Nodes are used to carry the state variables of fluid pressure and temperature, while branches are used to describe the mass flow and momentum transfer relationships between nodes and the physical properties of solid heat transfer units. For the solid region in the system, FloTides software discretizes it into solid nodes and solid conductor components to characterize the heat conduction process inside the solid and its coupling heat transfer with the fluid.
3. The one-dimensional fluid pipeline simulation method for secondary air systems according to claim 1, characterized in that... In S2, the set of governing equations includes the mass conservation equation, momentum conservation equation, energy balance equation, state equation, and vortex transfer equation. The mass conservation equation, momentum conservation equation, and vortex transfer equation all explicitly consider the dynamic changes of the resident mass at the nodes and are combined with the state equation of the nodes to achieve a complete description and accurate simulation of the steady-state / transient behavior of the pipeline system.
4. The one-dimensional fluid network simulation method for secondary air systems according to claim 3, characterized in that, A mass conservation equation is established at the fluid node to constrain the balance between the change in the resident mass at the node and the mass flow rate of the connected branches. The mass conservation equation is expressed as follows: ， in, Indicates the first Fluid retention mass at each fluid node; , These represent the branch mass flow rates flowing into and out of the node, respectively. , Representing the nodes respectively A set of connected entry branches and a set of connected exit branches; Indicates time; A momentum conservation equation driven by the nodal pressure difference is established for each branch. The momentum conservation equation is expressed as follows: ， in, and These represent the pressure at upstream and downstream nodes of the branch, respectively. This is represented as a pressure term, which is the main driving force of the branched flow; For resistance, This is the equivalent loss coefficient, used to uniformly characterize frictional resistance and local resistance; and These represent the fluid density and temperature within the branch, respectively. For branch quality flow; It is the inertial term; The equivalent inertia coefficient for the branch.
5. The one-dimensional fluid network simulation method for secondary air systems according to claim 3, characterized in that, Energy balance equations were established for fluid and solid nodes respectively; The energy balance equation for the fluid node is expressed as follows: ， in, For nodes enthalpy value, For heat source items, This represents the convective heat transfer between the current fluid node and the solid. The energy balance equation for solid nodes is expressed as follows: ， in, and Solid nodes Specific heat capacity and temperature; and These represent the convective heat transfer between the solid node and the fluid, and the heat transfer between adjacent nodes within the solid, respectively. In transient simulations, the coupling relationship between the change in nodal resident mass and pressure and temperature is described using state equations, the expression of which is: ， in, and They are nodes Pressure and temperature; Equivalent control volume for nodes; is the fluid gas constant.
6. The one-dimensional fluid network simulation method for secondary air systems according to claim 3, characterized in that, To establish swirling flow momentum equilibrium in a chamber or gap exhibiting swirling flow characteristics, and considering the swirling flow attenuation factor, the following swirling flow transfer equation is used: ， in, , expressed as vortex intensity per unit mass, and These are the tangential velocities of the fluid and the rotating wall, respectively. and They are respectively with nodes A set of connected entry branches and a set of connected exit branches; For nodes Entry Branch mass flow rate; For branches The swirl intensity carried by medium flow rates; For nodes Export Branch mass flow rate; For branches The swirl intensity carried by medium flow rates; For nodes vortex source term; is the vortex attenuation coefficient.
7. The one-dimensional fluid pipeline simulation method for secondary air systems according to claim 3, characterized in that, In S3, the mass conservation equation, momentum conservation equation, and state equation in the governing equations form a strongly coupled set; This set of equations includes dynamic changes in nodal mass, branch momentum balance, and closed-loop pressure-temperature-volume relationships at the nodes, which are strongly coupled through nodal mass, branch flow rate, and pressure variables. The Newton-Raphson iterative method is employed, and the Jacobian matrix is constructed as a method. Iterative update of unknown vectors , ， in, The nonlinear residual vector is composed of the mass conservation equation, momentum conservation equation, and state equation. Iteration is performed until the residual converges to the preset accuracy. The Jacobian matrix is constructed; It is the inverse of the Jacobian matrix; For the first The vector value calculated in the next iteration; This refers to the damping factor during the variable update process. Relaxation control is applied to the correction amount to limit the variation range of a single iteration; The energy balance equation and the swirl transfer equation are constructed as relatively independent sets. These equations mainly rely on nodal mass, flow rate, and swirl variables, but their coupling with momentum constraints is relatively weak, allowing for separate solutions. The solution method employs a fixed-point iteration method: using the mass and flow rate calculated in the previous step as known inputs, the nodal enthalpy is updated sequentially. With swirling variables The iteration continues until the nodal enthalpy and swirling variable converge; the equations for the fixed-point iteration method are as follows: ， in, For the enthalpy update function, and the current th The nodes calculated in the next iteration enthalpy value Quality of stay and upstream and downstream branches mass flow rate Related, For the vortex update function, and the current _th The nodes calculated in the next iteration vortex Quality of stay and upstream and downstream branches mass flow rate Related; and are the fixed-point iterative relaxation factors for nodal enthalpy and swirling variable, respectively.
8. The one-dimensional fluid network simulation method for secondary air systems according to claim 7, characterized in that, In each Newton-Raphson iteration, after updating the nodal mass, nodal pressure, and branch mass flow rate, convective heat transfer between the fluid and solid is introduced into the nodal energy calculation to update the enthalpy and temperature of the fluid nodes. Simultaneously, based on the updated fluid node enthalpy, the temperature change of the solid nodes is calculated. In the next iteration, the heat transfer is recalculated using the new solid node temperature and fed back into the fluid energy balance equation, thereby achieving coordinated iteration and overall convergence among multiple physical field variables such as fluid momentum, heat, and swirling flow. When the residuals of all equation sets meet the convergence conditions, the entire pipeline system converges.
9. A one-dimensional fluid network simulation method for secondary air systems according to claim 3, characterized in that, In S4, when performing steady-state calculations, it is assumed that the resident mass of each node in the pipeline system does not change with time. The time-varying terms in the mass conservation equation, momentum conservation equation, and fluid energy balance equation are set to zero to form a static nonlinear algebraic equation system. The mass flow rate, node pressure, and temperature distribution of the pipeline system under steady-state conditions are obtained by iterative solution. In transient calculations, the governing equations are solved using a time-stepping method. Within each time step, the time-varying terms in the mass conservation equation, momentum conservation equation, and energy balance equation are discretized using the finite difference method. The changes in the nodal dwell mass and related state variables are calculated. By progressively advancing the time steps, the dynamic response process of the pipeline system is simulated.
10. A one-dimensional fluid network simulation method for secondary air systems according to claim 1, characterized in that, In S5, the model is built, the control equations are established and solved, and the results are output functionally decoupled. The model building and results output are implemented based on the user graphical interface of the FloTides software. The solver is embedded in the system as an independent module to complete the automatic construction and numerical solution of the control equations, realizing a modular solution architecture. After building and solving the global control equations, the solver calculates the pressure loss coefficient and heat transfer of each component according to the latest distribution and feeds it back to the next iteration, iterating until the system converges.