CN110968967A - Heat transfer coupling simulation order reduction method for underground pipe heat exchanger - Google Patents

Heat transfer coupling simulation order reduction method for underground pipe heat exchanger Download PDF

Info

Publication number
CN110968967A
CN110968967A CN201911215035.6A CN201911215035A CN110968967A CN 110968967 A CN110968967 A CN 110968967A CN 201911215035 A CN201911215035 A CN 201911215035A CN 110968967 A CN110968967 A CN 110968967A
Authority
CN
China
Prior art keywords
heat exchanger
temperature
heat transfer
reduced
soil
Prior art date
Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
Granted
Application number
CN201911215035.6A
Other languages
Chinese (zh)
Other versions
CN110968967B (en
Inventor
孔琼香
蒋耀林
马真迪
Current Assignee (The listed assignees may be inaccurate. Google has not performed a legal analysis and makes no representation or warranty as to the accuracy of the list.)
Xian Jiaotong University
Original Assignee
Xian Jiaotong University
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Application filed by Xian Jiaotong University filed Critical Xian Jiaotong University
Priority to CN201911215035.6A priority Critical patent/CN110968967B/en
Publication of CN110968967A publication Critical patent/CN110968967A/en
Application granted granted Critical
Publication of CN110968967B publication Critical patent/CN110968967B/en
Active legal-status Critical Current
Anticipated expiration legal-status Critical

Links

Images

Classifications

    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02E—REDUCTION OF GREENHOUSE GAS [GHG] EMISSIONS, RELATED TO ENERGY GENERATION, TRANSMISSION OR DISTRIBUTION
    • Y02E10/00—Energy generation through renewable energy sources
    • Y02E10/10—Geothermal energy

Abstract

A heat transfer coupling simulation order reduction method for a ground heat exchanger comprises the following steps: step 1, establishing a control equation; step 2, determining a boundary; step 3, obtaining an original system of a state space form of a mathematical model of the heat transfer problem of the ground heat exchanger; step 4, reducing the original system obtained in the step 3 by adopting a model reduction method to obtain a reduced price system formula; step 5, carrying out numerical solution on the reduced system formula to obtain a solution of the reduced system; and 6, calculating to obtain an output variable through the reduced system coefficient matrix obtained in the step 4 and the reduced system state variable obtained in the step 5, wherein the output variable is an approximate solution of the original system output variable, namely a numerical solution of the output variable of the control equation under corresponding boundary conditions. The scale of the order-reducing system is far smaller than that of the original system, and only one system with smaller scale needs to be solved, so that the computing resources are saved, and the running time of a computing program is greatly shortened.

Description

Heat transfer coupling simulation order reduction method for underground pipe heat exchanger
Technical Field
The invention belongs to the field of heat transfer numerical value calculation methods, and particularly relates to a heat transfer coupling simulation order reduction method for a ground heat exchanger.
Background
The geothermal energy has the characteristics of large reserve capacity, wide distribution, cleanness, environmental protection, stability, reliability and the like. The geothermal energy heating (refrigeration) is mainly realized by a ground source heat pump system, and the key for improving the efficiency of the system is to improve the heat exchange performance of the ground heat exchanger. However, heat exchange of the ground heat exchanger in rock and soil is a complex and variable dynamic heat transfer process, and the related factors are many, such as rock and soil thermophysical parameters, structure setting of the heat exchanger, fluid operation parameters and the like. The buried depth of the buried pipe heat exchanger utilizing shallow geothermal heat is 100-300m, the buried depth of the buried pipe heat exchanger in the middle-deep layer can reach 2000m, the calculation of not only a single buried pipe heat exchanger but also the whole well group is required, and the calculated space area is large. In addition, because the heat transfer of the fluid and the soil in the heat exchanger are mutually influenced and change along with the time, the simulation of the heat transfer process of the whole life cycle of the ground heat exchanger needs to be carried out, and the calculated time domain is long. Therefore, for the heat exchange problems, the numerical simulation is greatly challenged, and an accurate and efficient numerical calculation method is urgently needed to be found to guide the design of the ground heat exchanger so as to improve the efficiency of the ground heat exchanger.
The Model Order Reduction (MOR) method is to convert a large complex system into an approximate small system, thereby reducing the difficulty of theoretical analysis of the large complex system, reducing the data storage capacity and the calculation amount, and accelerating the simulation calculation of the system, so that the method is widely applied to the engineering fields of large-scale integrated circuits, automatic control, mechanical and building performance simulation and the like. However, the application of the model order reduction method in the field of simulation of the heat transfer performance of the ground heat exchanger does not exist.
Disclosure of Invention
The invention aims to provide a heat transfer coupling simulation order reduction method for a ground heat exchanger, which aims to solve the problems.
In order to achieve the purpose, the invention adopts the following technical scheme:
a heat transfer coupling simulation order reduction method for a ground heat exchanger comprises the following steps:
step 1, analyzing the characteristics of heat transfer between the ground heat exchanger and rock soil, and establishing a control equation: a two-dimensional calculation model of the heat exchanger of the ground heat exchanger in the radial direction and the burial depth direction;
step 2, determining the temperature of a fluid inlet in the buried pipe, the temperature of a far boundary of rock and soil, the temperature of an upper boundary and a lower boundary and the initial temperature of the rock and soil;
step 3, performing space dispersion on the control equation established in the step 1 by adopting a finite volume method, and combining the boundary conditions determined in the step 2 to obtain an original system of a state space form of a mathematical model of the heat transfer problem of the ground heat exchanger;
step 4, reducing the original system obtained in the step 3 by adopting a model reduction method to obtain a reduced price system formula;
step 5, carrying out numerical solution on the reduced system formula to obtain a solution of the reduced system;
and 6, calculating to obtain an output variable through the reduced system coefficient matrix obtained in the step 4 and the reduced system state variable obtained in the step 5, wherein the output variable is an approximate solution of the original system output variable, namely a numerical solution of the output variable of the control equation under corresponding boundary conditions.
Further, the step 1 specifically comprises: simplifying the U-shaped buried heat exchanger into a straight pipe section by using an equivalent diameter method, and expressing a control equation into a two-dimensional calculation model in the radial r direction and the burial depth direction z direction by using radial symmetry;
based on different heat transfer mechanisms of fluid and rock-soil solid in the heat exchanger, the control equation comprises two parts of a control equation of convective heat transfer of the fluid in the ground heat exchanger and a control equation of heat conduction of rock-soil outside the ground heat exchanger:
the control equation of the convection heat transfer of the fluid in the buried pipe is as follows:
Figure BDA0002299271660000021
the rock-soil heat conduction control equation is as follows:
Figure BDA0002299271660000022
ρ: density, kg/m3；cp: specific heat capacity, J/(kg. DEG C.); w: axial flow velocity, m/s; t: temperature, deg.C; λ: thermal conductivity, W/(m.DEG C); lambda [ alpha ]t: turbulent heat conductivity, W/(m.deg.C); t: time, s.
Further, in a laminar flow state, a turbulent heat conductivity coefficient lambdatIs 0, the axial flow rate w is based on the fluidCross sectional average flow velocity wmObtaining the analytic solution;
in a turbulent flow state, solving a momentum control equation and solving a turbulent flow viscosity coefficient by applying a mixing length theory to obtain velocity distribution; firstly, a velocity field is assumed, and the mixing length l is calculated according to an empirical formulamAnd then calculating a turbulent viscosity coefficient ηtThen η according to the turbulent viscosity coefficienttSolving the momentum equation with the dynamic viscosity coefficient η to obtain a new velocity field, repeating the above processes until convergence to obtain a velocity field and ηt(ii) a Then solving the temperature control equation by solving the turbulent heat conductivity coefficient lambdat.
Further, the step 2 specifically comprises: determining the temperature of a fluid inlet in the buried pipe according to the simulated working condition; the temperature of the far boundary of the rock soil is the same as the original temperature of the rock soil; the temperature of the upper and lower boundaries is the original temperature of the rock and soil at the depth, and is determined according to the earth temperature data obtained by geological survey; the initial temperature of the rock soil is determined according to the temperature of the rock soil at the beginning of the simulation working condition.
Further, in step 3, the original system formula is as follows:
Figure BDA0002299271660000031
in the formula (I), the compound is shown in the specification,
Figure BDA0002299271660000032
the state variable of the system is the temperature of each space discrete node, wherein n is the number of the space discrete nodes;
Figure BDA0002299271660000033
inputting variables for the system, namely boundary temperature of rock soil and inlet fluid temperature of a buried pipe in the model, wherein p is the number of the input variables;
Figure BDA0002299271660000034
the system output variable is the outlet fluid temperature of the buried pipe, and m is the output variable number;
Figure BDA0002299271660000035
is a coefficient matrix, wherein A and B are mainly related to physical parameters of fluid and rock soil and the positions of nodes, and C and D are determined by required output variables.
Further, the step 4 specifically comprises:
reducing the n-order original system formula (3) obtained in the step (3) by adopting a model order reduction method to obtain an r-order reduced system as follows
Figure BDA0002299271660000036
Figure BDA0002299271660000037
In the formula (I), the compound is shown in the specification,
Figure BDA0002299271660000038
for the corresponding coefficient matrix of the reduced order system,
Figure BDA0002299271660000039
v is a projection matrix obtained by a model order reduction method;
Figure BDA00022992716600000310
in order to order the state variables of the reduced order system,
Figure BDA00022992716600000311
Figure BDA00022992716600000312
is the output variable of the reduced order system.
Further, step 6 specifically includes: reduced system coefficient matrix obtained by step 4
Figure BDA00022992716600000313
And the reduced order system state variable obtained in step 5
Figure BDA00022992716600000314
Calculated from the formula (3b)Output variable
Figure BDA00022992716600000315
The approximate solution of the output variable y (t) of the original system formula (3) is the numerical solution of the output variables of the control equations (1) and (2) under the corresponding boundary conditions.
Compared with the prior art, the invention has the following technical effects:
the model order reduction method comprises the steps of carrying out series expansion on state variables in a frequency domain, constructing an approximate system, and calculating a coefficient matrix of the system so as to obtain an order reduction system. The scale of the reduced order system is much smaller than the original system, we need only solve a system of smaller scale (r order,
Figure BDA0002299271660000041
) Therefore, the computing resources are saved, and the running time of the computing program is greatly shortened.
The invention reduces the time consumption of simulation calculation under the condition of keeping the required precision, provides a rapid calculation method for the design of the ground heat exchanger and the simulation of the whole life cycle, thereby saving the cost and time, and the larger the calculated space and time domain is, the more the calculation method has high efficiency. The Krylov subspace method can be expanded at different points to obtain reduced-order systems with different calculation accuracies, has good adaptability and flexibility to calculation of different models, and is convenient to apply to engineering design.
Drawings
FIG. 1 is a schematic diagram of heat transfer between a ground heat exchanger and rock soil
FIG. 2 is a computational model diagram of heat transfer between a heat exchanger of a ground heat exchanger and rock soil
FIG. 3 is a technical roadmap for a step-down method for a heat transfer model of a borehole heat exchanger;
FIG. 4 is a schematic diagram of computational domain node partitioning;
FIG. 5 is a graph of inlet and outlet water temperature directly solved and tested;
FIG. 6 is a graph of the relative error of outlet temperature versus time for the direct solution and the KS-MOR solution
FIG. 7 is a graph of the average relative error of the outlet temperature for the direct solution and the KS-MOR solution;
FIG. 8 is a graph of the time taken for the calculation of the KS-MOR method at different orders;
FIG. 9 is a graph of the elapsed time of calculation of KS-MOR in the time domain for different calculations;
FIG. 10 is a scale graph of the total time consumed for calculation of KS-MOR/direct solution in the time domain for different calculations;
Detailed Description
The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.
A method for solving the reduced order of heat transfer of the ground heat exchanger takes a Krylov subspace model reduced order method as an example, the state variable of a system is expanded on a frequency domain, and then a required reduced order matrix is constructed, wherein the technical route is shown in figure 3, and the detailed steps are as follows.
The method comprises the following steps: and analyzing the heat transfer process between the ground heat exchanger and rock and soil, and establishing a physical mathematical model of the heat transfer of the ground heat exchanger.
Based on different heat transfer mechanisms of fluid and rock-soil solids in the heat exchanger, the control equation comprises two parts, namely a control equation of convective heat transfer of the fluid in the ground heat exchanger and a control equation of heat conduction of rock-soil outside the ground heat exchanger.
The control equation of the convection heat transfer of the fluid in the buried pipe is as follows:
Figure BDA0002299271660000051
the rock-soil heat conduction control equation is as follows:
Figure BDA0002299271660000052
ρ: density, kg/m3；
cp: specific heat capacity, J/(kg. DEG C.);
w: axial flow velocity, m/s;
t: temperature, deg.C;
λ: thermal conductivity, W/(m.DEG C);
λt: turbulent heat conductivity, W/(m.deg.C);
t: time, s.
Under laminar flow conditions, the turbulent heat conductivity coefficient lambdatAt 0, the axial flow rate w is determined by:
w/wm＝2[1-(r/R1)2](3)
R1: pipe radius, m.
wm: cross-sectional average flow velocity, m/s.
In a turbulent state, the velocity distribution needs to be obtained by solving a momentum control equation and solving a turbulent viscosity coefficient by applying a mixing length theory.
The momentum control equation:
Figure BDA0002299271660000061
η dynamic viscosity coefficient, kg/(m.s);
ηt: the turbulent viscosity coefficient, kg/(m · s);
p: pressure, Pa.
Carrying out dimensionless processing on the momentum equation (3):
Figure BDA0002299271660000062
wherein, R/R1，
Figure BDA0002299271660000063
Solving the turbulent viscosity coefficient by applying a mixing length theory:
Figure BDA0002299271660000064
the mixing length is calculated as follows:
lm＝(DF)V·(lm)N(7)
wherein (l)m)NCalculated according to the Nikuards formula, and the Damping Factor (DF)VDetermined as follows:
(lm)N＝[0.14-0.08(r/R1)2-0.06(r/R1)4]R1(8)
Figure BDA0002299271660000065
wherein, tauwFor wall shear stress, a is 26.
Turbulent heat conductivity coefficient
Figure BDA0002299271660000066
σTThe number of Prandtl is 0.9-1.
Because ηtAssuming a velocity field, η is determined according to equations (7), (8) and (9)tThen solving the momentum equation, repeating the process until convergence to obtain a velocity field and ηt. Then, obtain λtThe solution of the temperature control equation can be performed.
Step two: and determining and establishing initial conditions and boundary conditions.
In order to numerically solve the control equations (1) and (2), initial conditions and boundary conditions thereof need to be determined from the problem. The temperature of the fluid inlet in the buried pipe is determined according to the simulated working condition. The temperature at the far boundary of the rock soil is not influenced by the heat exchanger of the buried pipe any more, so the temperature is the same as the original temperature of the rock soil, and the temperature of the upper boundary and the lower boundary are also regarded as the original temperature of the rock soil at the depth and can be determined according to the earth temperature data obtained by geological survey. The initial temperature of the rock soil is determined according to the temperature of the rock soil at the beginning of the simulated working condition and determined according to different working conditions. In this example, the specific boundary conditions are as follows:
Figure BDA0002299271660000071
Tin: inlet temperature, deg.C;
TW: wall temperature in deg.C.
Step three: and (3) obtaining an original system for solving the problem by the spatial dispersion of the heat transfer control equation of the ground heat exchanger.
As shown above, the control equations (1) and (2) of the heat transfer process of the ground heat exchanger are partial differential equations with respect to time and space, and when numerical solution is adopted, spatial dispersion is firstly required to be carried out, and the partial differential equations are converted into ordinary differential equations with respect to time. Nodal division of the calculation region as shown in FIG. 3, radial k1A node, axial k2A node, n ═ k1·k2The inner nodes, i.e. the n control volumes. Wherein (Delta r)iIndicating the radial length of the radial ith control volume, (δ r)iRepresents the distance from the ith node to the (i +1) th node; (Δ z)jAxial length of axial jth control volume (δ z)jRepresents the distance from the jth node to the (j +1) th node. The heat transfer partial differential equations (1) and (2) of the ground heat exchanger are spatially dispersed by adopting a finite volume method, and an original system of a heat transfer mathematical model of the ground heat exchanger described in a state space form is obtained by combining a boundary condition (10), wherein the original system is as shown in the following formula (11).
Figure BDA0002299271660000072
In the formula (I), the compound is shown in the specification,
Figure BDA0002299271660000073
the state variable of the system is the temperature of each space discrete node, wherein n is the number of the space discrete nodes;
Figure BDA0002299271660000074
the method comprises the following steps of (1) inputting variables of a system, namely the temperature of a rock-soil boundary and the temperature of an inlet fluid of a ground heat exchanger in a model, wherein p is the number of the input variables, and p is 4;
Figure BDA0002299271660000075
for system output variables, it may be the level of the outlet fluid of the borehole heat exchangerMean temperature, m is the number of output variables, and m is 1.
Figure BDA0002299271660000081
Is a coefficient matrix, wherein A and B are mainly related to physical parameters of fluid and rock soil and the positions of nodes, C and D are determined by required output variables,
Figure BDA0002299271660000082
when the temperature of the far boundary, the upper boundary and the lower boundary of rock and the temperature of the inlet fluid of the ground heat exchanger are used as input variables and the average temperature of the outlet fluid of the ground heat exchanger is used as an output variable, the expressions of the variables T (t), u (t) and y (t) and the coefficient matrix A, B, C, D are calculated as follows:
Figure BDA0002299271660000083
Figure BDA0002299271660000084
Figure BDA0002299271660000085
Figure BDA0002299271660000086
Figure BDA0002299271660000091
Figure BDA0002299271660000092
ai,j,P＝ai,j,E+ai,j,W+ai,j,N+ai,j,S，i＝1,2,…,k1；j＝1,2,…,k2
step four: and (4) reducing the original system by adopting a Krylov subspace order reduction method to obtain the order reduction system. The order r of the reduced system is much smaller than the order n of the original system, as shown in equation (12). Generally, r is reduced, the calculation error is increased, and the calculation amount is reduced, so for each specific problem, a reasonable order of the order-reduced system needs to be found out by experiments.
Figure BDA0002299271660000093
Coefficient matrices in equation (12)
Figure BDA0002299271660000094
The solution process of (2) is as follows:
1) inputting the coefficient matrix A, B, C, D and the order r of the reduction order;
2) constructing a calculation space: the original system is transformed by Laplace to obtain a transfer function H(s) ═ C (sI-A)-1B, if there is a point
Figure BDA0002299271660000095
So that A-s0I is not singular, then the Krylov subspace of the original system can be constructed
Figure BDA0002299271660000096
Wherein
Figure BDA0002299271660000097
3) Applying the Arnoldi procedure to the two resulting computation spaces results in a set of orthonormal bases of the computation spaces:
(1) initializing a vector:
Figure BDA0002299271660000098
(2) calculate a second orthonormal vector:
Figure BDA0002299271660000099
(3) computing the r-th orthonormal vector:
Figure BDA00022992716600000910
(4) constructing an orthogonal projection matrix V ═ V1,v2,…,vr]；
(5) Respectively through a relational expression
Figure BDA00022992716600000911
Solving the coefficient matrix of the reduced system, and outputting the coefficient matrix of the reduced system:
Figure BDA00022992716600000912
step five: the numerical solution is carried out on the reduced system formula (12a) to obtain the solution of the reduced system
Figure BDA00022992716600000913
Step six: according to equation (12b), the output variables in the reduced order system
Figure BDA0002299271660000101
The reduced system coefficient matrix obtained by the fourth step
Figure BDA0002299271660000102
And step five, obtaining the state variable of the reduced order system
Figure BDA0002299271660000103
Calculating to obtain an approximate solution of the output variable y (t) of the original system formula (11); the state variable T (t) of the original system expression (11) can be obtained according to the relational expression
Figure BDA0002299271660000104
And the projection matrix V and the state variable of the reduced order system obtained in the fourth step
Figure BDA0002299271660000105
The calculation is a numerical solution of the governing equations (1) and (2) under the boundary condition (10).
And writing and running a calculation program according to the detailed steps of the technical route by combining the specific case, and analyzing the reliability and the beneficial effect of the heat transfer model order reduction method of the ground heat exchanger according to the calculation result.
Without loss of generality, the heat release process of a certain U-shaped pipe ground heat exchanger is taken as an example. The method of equivalent diameter is adopted to simplify the U-shaped pipe into a straight pipe section, and the radius of each part is R1＝0.0378m，R2＝0.0406m，R3＝0.08m，R42.5 m. The buried depth of the ground heat exchanger is L100 m. The initial temperature of the rock soil is 18 ℃, the far boundary temperature is 17 ℃, the calculated time domain is 48h, and the water temperature of the fluid inlet changes along with the time as shown in figure 5. The physical parameters of the fluid, the pipe wall, the backfilled soil and the rock soil are shown in table 1.
TABLE 1 physical Properties of fluids, pipe walls, backfilled soils and rock and soil
Figure BDA0002299271660000106
Then, the control equation is dispersed, and the total number of the inner nodes of the radial grid is k 140, where the number of internal nodes of the fluid is kpThe total number of inner nodes of the axial grid is k as 82The total number of internal nodes n is 4000, i.e. the order of the original system is 4000. The time step is taken to be 5 s. When the order reduction processing is carried out by using a Krylov subspace model order reduction method (KS-MOR), the orders of a reduction system are respectively 10,20,40,50,80,100 and 200.
For the known conditions (parameters of fluid, pipeline, backfill soil and rock soil) of the case, a calculation program is written and operated according to the detailed steps of the technical route of figure 3, and numerical simulation results are obtained by using KS-MOR and a direct solving method. The direct solving method is to directly solve the original system obtained in the step three, that is, there are no step four (construction of projection matrix and coefficient matrix of reduced order system) and step five (solving of reduced order system). The computer program is written by MatlabR2016a, and a workstation for running the program is mainly configured as follows: CPU (central processing unit)
Figure BDA0002299271660000111
E5-2630 v4@2.2GHz, memory 64G and video card NVIDIA Quadro P2000.
For the problem of heat transfer of the ground heat exchanger, the calculation result based on the Krylov subspace and the direct solving method and the comparison with the experimental result are as follows.
FIG. 5 is the inlet and outlet water temperatures for the direct solution and experiment, FIG. 6 is the relative error of the outlet average temperature for the direct solution and KS-MOR solution over time, and FIG. 7 is the average relative error of the outlet average temperature for the direct solution and KS-MOR solution.
The relative error definition and the average relative error are defined as follows: .
Figure BDA0002299271660000112
Figure BDA0002299271660000113
Wherein the content of the first and second substances,
Figure BDA0002299271660000114
the exit average temperatures of the direct solution and KS-MOR solution, respectively, and m is the number of time nodes.
It can be seen that the calculated results of the KS-MOR method and the direct solving method are high in coincidence degree, the error between the KS-MOR method and the direct solving method is small, in the calculated orders of the order-reduced system, the maximum relative error is less than 7%, the maximum average relative error is less than 0.7%, and when the orders of the order-reduced system are more than 40 orders, the average relative error gradually tends to be stable and is less than 0.1%. Therefore, the calculation accuracy of the Krylov subspace model order reduction method is high.
The program running time (namely the calculation time) of the KS-MOR method is shown in figure 7, the total calculation time, the matrix construction time and the solution time of the direct solution method are respectively 19984.45s, 0.14s and 19984.31s, and the total calculation time and the solution time of the KS-MOR method are far less than the time of the direct solution method, and the time of constructing the matrix of the KS-MOR method is slightly more than the time of the direct solution method.
By comprehensively considering the relative error and the calculation time, it can be seen that the order selection of the model order reduction system is proper, namely about 1% of the original system, from 40 to 50 orders.
And then, KS-MOR calculation is adopted to obtain reduced systems with orders of 40 and 50, and numerical simulation with calculation time domains of 6h, 12h, 24h, 36h and 48h is carried out. FIGS. 8 and 9 show the computation time of the Krylov subspace method in different computation time domains and the comparison between the computation time and the direct solution computation time, and it can be seen that the total computation time of the KS-MOR method increases with the increase of the computation time domain. And when the time domain of the calculation is increased from 6h to 48h, the total time consumption of the KS-MOR method is reduced from 0.182% (r ═ 40) and 0.178% (r ═ 50) of the total time consumption of the direct solution to 0.043% (r ═ 40) and 0.051% (r ═ 50) of the total time consumption of the direct solution. For direct solving and KS-MOR solving at a certain order of a reduced-order system, along with the increase of a calculation time domain, the time consumption for constructing a matrix is unchanged, and the time consumption for solving is in direct proportion to the calculation time domain; moreover, on a time node, the direct solution needs to solve an equation of an n order, and the KS-MOR only needs to solve an equation of an r order, wherein the time consumption of the former is far more than that of the latter, so that the time consumption of the direct solution is increased rapidly along with the increase of a time domain, and the KS-MOR is relatively slow.
Therefore, the model order reduction method is high in calculation accuracy, program operation time can be effectively shortened, and calculation efficiency is remarkably improved. Therefore, as the time domain or space of computation increases, the computation of KS-MOR is much less time consuming than the direct solution, and its efficiency is more evident.

Claims (7)

1. A heat transfer coupling simulation order reduction method for a ground heat exchanger is characterized by comprising the following steps:
step 1, analyzing the characteristics of heat transfer between the ground heat exchanger and rock soil, and establishing a control equation: a two-dimensional calculation model of the heat exchanger of the ground heat exchanger in the radial direction and the burial depth direction;
step 2, determining the temperature of a fluid inlet in the buried pipe, the temperature of a far boundary of rock and soil, the temperature of an upper boundary and a lower boundary and the initial temperature of the rock and soil;
step 3, performing space dispersion on the control equation established in the step 1 by adopting a finite volume method, and combining the boundary conditions determined in the step 2 to obtain an original system of a state space form of a mathematical model of the heat transfer problem of the ground heat exchanger;
step 4, reducing the original system obtained in the step 3 by adopting a model reduction method to obtain a reduced price system formula;
step 5, carrying out numerical solution on the reduced system formula to obtain a solution of the reduced system;
and 6, calculating to obtain an output variable through the reduced system coefficient matrix obtained in the step 4 and the reduced system state variable obtained in the step 5, wherein the output variable is an approximate solution of the original system output variable, namely a numerical solution of the output variable of the control equation under corresponding boundary conditions.
2. The simulated order reduction method for heat transfer coupling of the ground heat exchanger according to claim 1, wherein the step 1 comprises the following steps: simplifying the U-shaped buried heat exchanger into a straight pipe section by using an equivalent diameter method, and expressing a control equation into a two-dimensional calculation model in the radial r direction and the burial depth direction z direction by using radial symmetry;
based on different heat transfer mechanisms of fluid and rock-soil solid in the heat exchanger, the control equation comprises two parts of a control equation of convective heat transfer of the fluid in the ground heat exchanger and a control equation of heat conduction of rock-soil outside the ground heat exchanger:
the control equation of the convection heat transfer of the fluid in the buried pipe is as follows:
Figure FDA0002299271650000011
the rock-soil heat conduction control equation is as follows:
Figure FDA0002299271650000012
ρ: density, kg/m3；cp: specific heat capacity, J/(kg. DEG C.); w: axial flow velocity, m/s; t: temperature, deg.C; λ: thermal conductivity, W/(m.DEG C); lambda [ alpha ]t: turbulent heat conductivity, W/(m.deg.C); t: time, s.
3. A method of simulating order reduction in heat transfer coupling for a borehole heat exchanger according to claim 2 wherein the turbulent heat transfer coefficient λ is such that, in laminar flow conditions, the turbulent heat transfer coefficienttIs 0, the axial flow velocity w is determined by the average flow velocity w based on the cross section of the fluidmObtaining the analytic solution;
in a turbulent flow state, solving a momentum control equation and solving a turbulent flow viscosity coefficient by applying a mixing length theory to obtain velocity distribution; firstly, a velocity field is assumed, and the mixing length l is calculated according to an empirical formulamAnd then calculating a turbulent viscosity coefficient ηtThen η according to the turbulent viscosity coefficienttSolving the momentum equation with the dynamic viscosity coefficient η to obtain a new velocity field, repeating the above processes until convergence to obtain a velocity field and ηt(ii) a Then solving the temperature control equation by solving the turbulent heat conductivity coefficient lambdat.
4. A method for simulating reduction of the heat transfer coupling of a borehole heat exchanger according to claim 1, wherein step 2 specifically comprises: determining the temperature of a fluid inlet in the buried pipe according to the simulated working condition; the temperature of the far boundary of the rock soil is the same as the original temperature of the rock soil; the temperature of the upper and lower boundaries is the original temperature of the rock and soil at the depth, and is determined according to the earth temperature data obtained by geological survey; the initial temperature of the rock soil is determined according to the temperature of the rock soil at the beginning of the simulation working condition.
5. The simulated order reduction method for heat transfer coupling of the ground heat exchanger according to claim 1, wherein the original system formula in step 3 is as follows:
Figure FDA0002299271650000021
in the formula (I), the compound is shown in the specification,
Figure FDA0002299271650000022
the state variable of the system is the temperature of each space discrete node, wherein n is the number of the space discrete nodes;
Figure FDA0002299271650000023
inputting variables for the system, namely boundary temperature of rock soil and inlet fluid temperature of a buried pipe in the model, wherein p is the number of the input variables;
Figure FDA0002299271650000024
the system output variable is the outlet fluid temperature of the buried pipe, and m is the output variable number;
Figure FDA0002299271650000025
is a coefficient matrix, wherein A and B are mainly related to physical parameters of fluid and rock soil and the positions of nodes, and C and D are determined by required output variables.
6. A method for simulating the reduction of the heat transfer coupling of a borehole heat exchanger according to claim 1, wherein step 4 specifically comprises:
reducing the n-order original system formula (3) obtained in the step (3) by adopting a model order reduction method to obtain an r-order reduced system as follows
Figure FDA0002299271650000026
Figure FDA0002299271650000027
In the formula (I), the compound is shown in the specification,
Figure FDA0002299271650000031
for the corresponding coefficient matrix of the reduced order system,
Figure FDA0002299271650000032
v is a projection matrix obtained by a model order reduction method;
Figure FDA0002299271650000033
in order to order the state variables of the reduced order system,
Figure FDA0002299271650000034
is the output variable of the reduced order system.
7. A method for simulating reduction of the heat transfer coupling of a borehole heat exchanger according to claim 1, wherein step 6 specifically comprises: reduced system coefficient matrix obtained by step 4
Figure FDA0002299271650000035
And the reduced order system state variable obtained in step 5
Figure FDA0002299271650000036
The output variable is calculated from the formula (3b)
Figure FDA0002299271650000037
The approximate solution of the output variable y (t) of the original system formula (3) is the numerical solution of the output variables of the control equations (1) and (2) under the corresponding boundary conditions.
CN201911215035.6A 2019-12-02 2019-12-02 Heat transfer coupling simulation order reduction method for underground pipe heat exchanger Active CN110968967B (en)

Priority Applications (1)

Application Number Priority Date Filing Date Title
CN201911215035.6A CN110968967B (en) 2019-12-02 2019-12-02 Heat transfer coupling simulation order reduction method for underground pipe heat exchanger

Applications Claiming Priority (1)

Application Number Priority Date Filing Date Title
CN201911215035.6A CN110968967B (en) 2019-12-02 2019-12-02 Heat transfer coupling simulation order reduction method for underground pipe heat exchanger

Publications (2)

Publication Number Publication Date
CN110968967A true CN110968967A (en) 2020-04-07
CN110968967B CN110968967B (en) 2022-11-04

Family

ID=70032608

Family Applications (1)

Application Number Title Priority Date Filing Date
CN201911215035.6A Active CN110968967B (en) 2019-12-02 2019-12-02 Heat transfer coupling simulation order reduction method for underground pipe heat exchanger

Country Status (1)

Country Link
CN (1) CN110968967B (en)

Cited By (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN111649969A (en) * 2020-05-14 2020-09-11 西安交通大学 Semi-analytic method for predicting performance of deep butt-joint buried heat exchanger
CN112597639A (en) * 2020-12-11 2021-04-02 山东中瑞新能源科技有限公司 Heat transfer dimension reduction analysis method and system for porous ground heat exchanger
CN112906194A (en) * 2021-01-20 2021-06-04 安徽理工大学 Technology for establishing laminar rock mass ground temperature vertical conduction-convection model
CN113792462A (en) * 2021-09-16 2021-12-14 深能科技(山东)有限公司 Method for rapidly acquiring heat extraction energy of intermediate-deep geothermal buried pipe
CN117744953A (en) * 2024-02-19 2024-03-22 天津大学 Simplified analysis method for shallow soil source buried pipe group of large-scale ground source heat pump system

Citations (11)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US20160209083A1 (en) * 2013-10-08 2016-07-21 University Of Seoul Industry Cooperation Foundation Underground heat-exchange system
CN206245054U (en) * 2016-10-31 2017-06-13 四川意龙印染有限公司 A kind of energy-saving environmental-protection developing and soaping range
CN107194123A (en) * 2017-06-22 2017-09-22 北京理工大学 Retarder braking ability emulation mode and device
CN107907564A (en) * 2017-11-07 2018-04-13 山东科技大学 A kind of definite method of ground thermal property parameter and vertical ground heat exchanger thermal resistance
CN109086560A (en) * 2018-09-30 2018-12-25 安徽工业大学 The vertical single U-shaped underground pipe fluid temperature (F.T.) distribution forecasting method of earth source heat pump under variable working condition
CN109214113A (en) * 2018-09-30 2019-01-15 西安交通大学 A kind of new model depression of order method for solving of architectural exterior-protecting construction dynamic heat transfer
US20190056147A1 (en) * 2017-08-16 2019-02-21 Art Brisebois Thin multi-channel heat exchanger
CN109636034A (en) * 2018-12-11 2019-04-16 石化盈科信息技术有限责任公司 A kind of optimization method of low temperature heat system
CN109946103A (en) * 2019-04-23 2019-06-28 山东建筑大学 A kind of underground heat parameter test system and method based on mid-deep strata ground heat exchanger
CN110059389A (en) * 2019-04-10 2019-07-26 北京石油化工学院 A kind of solar cross-season soil thermal storage POD method for quick predicting
CN111539130A (en) * 2020-05-29 2020-08-14 山东建筑大学 Design and calculation method for drilling depth of geothermal heat exchanger of middle-deep buried pipe

Patent Citations (11)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US20160209083A1 (en) * 2013-10-08 2016-07-21 University Of Seoul Industry Cooperation Foundation Underground heat-exchange system
CN206245054U (en) * 2016-10-31 2017-06-13 四川意龙印染有限公司 A kind of energy-saving environmental-protection developing and soaping range
CN107194123A (en) * 2017-06-22 2017-09-22 北京理工大学 Retarder braking ability emulation mode and device
US20190056147A1 (en) * 2017-08-16 2019-02-21 Art Brisebois Thin multi-channel heat exchanger
CN107907564A (en) * 2017-11-07 2018-04-13 山东科技大学 A kind of definite method of ground thermal property parameter and vertical ground heat exchanger thermal resistance
CN109086560A (en) * 2018-09-30 2018-12-25 安徽工业大学 The vertical single U-shaped underground pipe fluid temperature (F.T.) distribution forecasting method of earth source heat pump under variable working condition
CN109214113A (en) * 2018-09-30 2019-01-15 西安交通大学 A kind of new model depression of order method for solving of architectural exterior-protecting construction dynamic heat transfer
CN109636034A (en) * 2018-12-11 2019-04-16 石化盈科信息技术有限责任公司 A kind of optimization method of low temperature heat system
CN110059389A (en) * 2019-04-10 2019-07-26 北京石油化工学院 A kind of solar cross-season soil thermal storage POD method for quick predicting
CN109946103A (en) * 2019-04-23 2019-06-28 山东建筑大学 A kind of underground heat parameter test system and method based on mid-deep strata ground heat exchanger
CN111539130A (en) * 2020-05-29 2020-08-14 山东建筑大学 Design and calculation method for drilling depth of geothermal heat exchanger of middle-deep buried pipe

Non-Patent Citations (4)

* Cited by examiner, † Cited by third party
Title
QIONGXIANG KONG等: ""Fast simulation of dynamic heat transfer through building envelope"", 《BUILDING SIMULATION》 *
ZHENDI MA等: ""NUMERICAL SIMULATION OF GROUND HEAT EXCHANGERS BASED ON MODEL ORDER REDUCTION METHOD"", 《APPLIED ENERGY SYMPOSIUM 2019: LOW CARBON CITIES AND URBAN ENERGY SYSTEMS》 *
杨卫波等: ""非连续运行工况下垂直地埋管换热器的换热特性"", 《东南大学学报 (自然科学版)》 *
魏建平: "竖直地埋管换热器短时热响应建模及应用研究", 《中国博士学位论文全文数据库 工程科技Ⅱ辑》 *

Cited By (8)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN111649969A (en) * 2020-05-14 2020-09-11 西安交通大学 Semi-analytic method for predicting performance of deep butt-joint buried heat exchanger
CN111649969B (en) * 2020-05-14 2021-03-23 西安交通大学 Semi-analytic method for predicting performance of deep butt-joint buried heat exchanger
CN112597639A (en) * 2020-12-11 2021-04-02 山东中瑞新能源科技有限公司 Heat transfer dimension reduction analysis method and system for porous ground heat exchanger
CN112597639B (en) * 2020-12-11 2023-06-09 山东中瑞新能源科技有限公司 Heat transfer dimension reduction analysis method and system for porous buried pipe heat exchanger
CN112906194A (en) * 2021-01-20 2021-06-04 安徽理工大学 Technology for establishing laminar rock mass ground temperature vertical conduction-convection model
CN113792462A (en) * 2021-09-16 2021-12-14 深能科技(山东)有限公司 Method for rapidly acquiring heat extraction energy of intermediate-deep geothermal buried pipe
CN117744953A (en) * 2024-02-19 2024-03-22 天津大学 Simplified analysis method for shallow soil source buried pipe group of large-scale ground source heat pump system
CN117744953B (en) * 2024-02-19 2024-04-19 天津大学 Simplified analysis method for shallow soil source buried pipe group of large-scale ground source heat pump system

Also Published As

Publication number Publication date
CN110968967B (en) 2022-11-04

Similar Documents

Publication Publication Date Title
CN110968967B (en) Heat transfer coupling simulation order reduction method for underground pipe heat exchanger
Diao et al. Improvement in modeling of heat transfer in vertical ground heat exchangers
Al‐Khoury et al. Efficient finite element formulation for geothermal heating systems. Part II: transient
Rees et al. A three-dimensional numerical model of borehole heat exchanger heat transfer and fluid flow
Yang et al. A two-region simulation model of vertical U-tube ground heat exchanger and its experimental verification
Wołoszyn et al. Modelling of a borehole heat exchanger using a finite element with multiple degrees of freedom
CN111125921A (en) Method for rapidly and accurately realizing dynamic simulation of performance of vertical U-shaped ground heat exchanger
CN112597639B (en) Heat transfer dimension reduction analysis method and system for porous buried pipe heat exchanger
Gadgil On convective heat transfer in building energy analysis
Wołoszyn et al. Sensitivity analysis of efficiency thermal energy storage on selected rock mass and grout parameters using design of experiment method
CN107274473A (en) Three-dimensional numerical model for heat transfer of vertical buried pipe of ground source heat pump and establishment method thereof
CN111144054A (en) Modeling method for natural circulation characteristic of villiaumite cooling high-temperature reactor passive waste heat discharge system
Hirsch et al. An efficient numerical solution method for detailed modelling of large 5th generation district heating and cooling networks
Morchio et al. Modelling and validation of a new hybrid scheme for predicting the performance of U-pipe borehole heat exchangers during distributed thermal response test experiments
Jahangir et al. A numerical study into effects of intermittent pump operation on thermal storage in unsaturated porous media
Croucher et al. Geothermal Supermodels Project: an update on flow simulator development
Belzile et al. Semi-analytical model for geothermal borefields with independent inlet conditions
Liu et al. Multilayer quasi-three-dimensional model for the heat transfer inside the borehole wall of a vertical ground heat exchanger
Nabi et al. An efficient finite volume model for shallow geothermal systems—Part II: Verification, validation and grid convergence
CN102663243A (en) Numerical simulation method of buried tube temperature field of ground source heat pump under thermoosmosis coupling
Hermanns et al. Asymptotic analysis of vertical geothermal boreholes in the limit of slowly varying heat injection rates
Xu et al. A comprehensive investigation on U-tube ground heat exchanger performance considering insulation layer in deep geothermal system
Picard et al. A new hybrid model for borefield heat exchangers performance evaluation
Vaganova et al. Optimization of location of injection wells in an open geothermal system
Hu et al. An improved decoupling algorithm for low Mach number near-critical fluids

Legal Events

Date Code Title Description
PB01 Publication
PB01 Publication
SE01 Entry into force of request for substantive examination
SE01 Entry into force of request for substantive examination
GR01 Patent grant
GR01 Patent grant