## Numerical Methods for Temperature Field about Random Heat Source Model of Ground Source Heat Pump

**Guan Changsheng**^{1,}, **Yang Shaopan**^{1}

^{1}School of civil engineering and architecture, Wuhan University of Technology, Wuhan, China

2. The Current Research Situation of GSHP Buried Pipe

3. The Randomness of Heat Transfer about GSHP Buried Pipes

4. The Model for Random Temperature Field of GSHP

5. Finite Element Method for Random Heat Transfer of GSHP

6. The Numerical Method for the Model of Random Heat Transfer

### Abstract

On basis of stochastic analysis method and finite element numerical theory, the paper which considered the conditions of random heat source did research into the temperature field of underground heat transfer. The analysis of heat transfer random factors about ground source heat pump buried pipe, establishment of general model about rock-soil random heat transfer, presentation of finite element equations about random temperature field as well as statistical regularity of space and time about temperature field are covered in this paper. The numerical simulation of given model has been realized with computer utilized. The research shows that the effect which ground source heat pump random heat source has on the temperature field of ground source heat pump is random obviously, which gives an important theoretical method to the analysis of mean value about the temperature field of heat transfer, the discrete analysis, as well as the design of ground source heat pump buried pipe. The engineering case shows that random effect of buried pipe heat transfer about ground source heat pump cannot be neglected.

### At a glance: Figures

**Keywords:** ground source heat pump, temperature field, random heat transfer, random finite element, numerical simulation

*American Journal of Industrial Engineering*, 2013 1 (2),
pp 20-27.

DOI: 10.12691/ajie-1-2-3

Received August 16, 2013; Revised September 11, 2013; Accepted September 16, 2013

**Copyright**© 2013 Science and Education Publishing. All Rights Reserved.

### Cite this article:

- Changsheng, Guan, and Yang Shaopan. "Numerical Methods for Temperature Field about Random Heat Source Model of Ground Source Heat Pump."
*American Journal of Industrial Engineering*1.2 (2013): 20-27.

- Changsheng, G. , & Shaopan, Y. (2013). Numerical Methods for Temperature Field about Random Heat Source Model of Ground Source Heat Pump.
*American Journal of Industrial Engineering*,*1*(2), 20-27.

- Changsheng, Guan, and Yang Shaopan. "Numerical Methods for Temperature Field about Random Heat Source Model of Ground Source Heat Pump."
*American Journal of Industrial Engineering*1, no. 2 (2013): 20-27.

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### 1. Introduction

Nowadays, the application about the technology of ground source heat pump (GSHP) for heating and refrigeration has entered a practical stage and brings about obvious energy-saving benefits. The randomness which is presented by the complicated structures of GSHP buried pipe and rock-soil materials in the process of heat transfer can not be ignored. Especially, for thermophysical parameters of rock-soil media, many parameters for design have inherent randomness when the porosity of materials and complex working conditions are taken into consideration. This paper which takes GSHP buried pipe and randomness of rock-soil heat transfer as research objects studied the process of random heat transfer and the distribution of random temperature field as well as put forward the random finite element method for the analysis of random heat transfer. That the uncertainty in the process of random heat transfer is taken as random parameters can constitute a random vector. In addition, a mathematical-physical model can be established according to the specific temperature boundaries. The perturbation method is utilized to achieve the equations of random finite element whose numerical solution can be obtained by the means of numerical analytical theory.

### 2. The Current Research Situation of GSHP Buried Pipe

The buried pipe whose model about heat transfer is difficult for GSHP air conditioning technology acts as the key component for GSHP system. In recent years, some foreign scholars have done some researches into experiment method, heat transfer model and the theory of analysis and calculation. For example, in 2013. LLamarche gave a study to the short-term behaviour of classical analytic solutions for the design of GSHP ^{[1]}. In 2013, a rigorous mathematical computational approach to size the GSHP with a hybrid system was presented in the study which was carried out by Masih Alavy and his coagents ^{[2]} In 2013, an experimental and modeling analysis of a GSHP system was proposed by Montagud.C ^{[3]}. In 2012, Li. Min and Lai. Alvin. C.K did a research in which the model development and validation was referred in the analytical model for short-time responses of ground heat exchangers with U-shaped tubes ^{[4]}. In 2012, a study which focused on the experimental investigation and optimization of a GSHP system under different indoor set temperature was conducted by X.Q.Zhai, etc ^{[5]}. With based on test data and analysis, a algorithm to predict the refrigerant amount direction in GSHP unit was developed, which was proposed by HangSeok Choi, Hongyun Cho and Jong Min Choi ^{[6]}. Benoit Beauchamp did research into computation theory of limited length heat source ^{[7]}. Liuxianying who utilized the systematic energy balance combined with equations of thermal conductivity has established diabatic models of heat exchangers about underground vertically buried pipes ^{[8]}.The heat transfer characteristics of geothermal heat exchanger were studied by Fangzhaohong ^{[9]}. Diaonairen proposed methods for optimization design of GSHP heat exchanger ^{[10]}. In 2004, Mustafa studied the thermal performance evaluation of the level GSHP and James studied optimization layout in depth of GSHP heat exchangers ^{[11]}. In 2005, Onder analyzed the performance of solar energy with GSHP ^{[12]}. In 2007, Guohui studied the test and simulation of the rain GSHP, Onder as well as Ozgener proposed economic evaluation parameters of thermal energy about GSHP, Louis did research into numerical solution and improvement of finite linear source about heat exchanger of ground source well hole ^{[13, 14, 15, 16, 17]}. In 2008, Hikmet used the neural network and fuzzy theory to study the prediction and evaluation of GSHP system. Obviously, there is much randomness in the process of design, installation, operation and maintenance for GSHP heat exchangers, Sarkhi studied the efficiency evaluation and the application of standard cylinder wells, Katsura studied the approaches which were used to calculate the ground source temperature of heat exchangers in groups ^{[18, 19, 20]} etc.

The porosity of geotechnical media and randomness of structure are their inherent properties. The thermophysical properties about this kind of material, such as the coefficient of thermal conductivity, thermal resistance etc, must have the property of randomness. The problems will become more complicated if the effect of groundwater, climate changes, the variation of factors about energy consumption and the uncertainty of the fact that GSHP has been used for a long time are taken into consideration. Therefore, it is imperative that the principle of random heat transfer about GSHP buried pipe and ground, random factors, distributions laws and random thermophysical parameters of geotechnical condition should be studied further.

### 3. The Randomness of Heat Transfer about GSHP Buried Pipes

**3.1. Random Thermophysical Properties for Rock and Soil Media**

The performance of GSHP design is closely relevant to the thermophysical properties for rock and soil media. That parameters about thermophysical properties for rock an soil media are confirmed is the foundation for GSHP design. It is difficult for GSHP design that the uncertainty about thermophysical properties consist of material composition and structural randomness because the thermophysical properties for rock and soil media are hard to test as well as the factors which affect heat exchangers are complicated.

**3.2. The Randomness Which Affects the Form of Buried Pipes**

The factors which have an effect on the forms of buried pipes are the following: pipe materials, pipe space length, fluid in pipes, same distance and different distance etc. The fact that randomness exists in buried pipe materials as well as parameters about heat transfer leads to a result that pipe distance, effectiveness about buried pipes and working fluid also have randomness.

**3.3. The Randomness for Initial Condition and Boundary Condition**

The initial condition as well as boundary condition usually have an impact on the underground heat transfer of GSHP and the effect caused by initial condition on the problem of unsteady heat transfer can not be ignored. The initial condition should take the initial temperature of rock-soil materials as well as climate effects into consideration and that it can act as random condition may be more reasonable because its variation coefficient can be huge. The boundary condition of temperature field can take ground-surface temperature as well as the inner boundary condition of the fact that buried pipes transfer heat to their surroundings into account. It can be taken as random boundary condition when these factors are random.

### 4. The Model for Random Temperature Field of GSHP

**4.1. The Random Temperature Field of Heat Transfer about GSHP Buried Pipes**

During the period when GSHP are under operation, because of the effects from all kinds of factors, the variation about temperature field for rock and soil media is very complicated and in some cases the temperature accumulation can give rise to failure for the function of GSHP. The temperature field for heat transfer of GSHP buried pipes are involved in the issues about rock-soil heat transfer, namely the inhomogeneity about rock-soil material and the randomness of thermophysical parameters for rock and soil.

**4.2. The Model for Random Temperature of Rock and Soil**

According to the theory about thermal conductivity, the rock-soil temperature field can be presented by the following definite solution problem:

(1) |

(2) |

(3) |

In the equations: is the coefficient of temperature conductivity, is the surface exothermic coefficient, is the coefficient of thermal conductivity, stands for environmental temperature, stands for the initial temperature, is the quantity of adiabatic temperature rise.

The random factors which affect the rock-soil temperature field include the following: initial temperature about rock and soil, environmental temperature out of ground surface, velocity and pressure of the flowing fluid in the buried pipes, backfill materials in the borehole, boundary conditions for the temperature of buried pipes and boundary temperature around the temperature field etc. The model for random heat transfer can come into existence when some random parameters appear in the equations (1)~(3).

### 5. Finite Element Method for Random Heat Transfer of GSHP

**5.1. The General Method for Unsteady Heat Transfer of GSHP Buried Pipes**

At present the U-shaped vertically buried pipes which act as heat exchangers are extensively employed in the design of soil heat pumps. The heat exchangers in the form of U-shaped buried pipes are importantly integral components for GSHP system. The heat exchange performance is relevant to the following factors: texture quality of the heat exchangers, flow rate of the medium, forms of buried pipes, layout forms, connection degree between buried pipes and soil, load variation of the heat exchangers, imbalance between heat absorption and heat release throughout the year, soil type, physical parameters, soil moisture constant, etc. The corresponding two-dimensional axisymmetric unsteady heat transfer model can be established with the heat transfer of U-shaped buried pipes considered. The solution for the problem about heat transfer in the shaped part is shown in Figure 1, namely the equations (4)~(6)

(4) |

(5) |

(6) |

In the equations, consists of line, , and . stands for the area which is surrounded by . is the coefficient of thermal conductivity. Boundary conditions are the following:

on line segment , , is the coefficient of convective heat transfer between the ground surface and air. is the air temperature around ground surface. On line , the radius of heat action is adiabatic (), namely . is the radius of heat action. It is assumed that the line in the bottom does not transfer heat downward, namely ().On line, as pipe wall, it is assumed that the heat flow is . which can also be a random quantity stands for the heat exchange amount of the drill hole per unit in depth. The equations from (4) to (6) are equations of random heat conduction. In the equations, the randomness is shown by the following: material properties for thermal conductivity, initial temperature and boundary temperature field.

**Fig**

**ure**

**1.**The model for GSHP buried pipes

**5.2. Random Finite Element Method for Unsteady Heat Transfer of GSHP**

The research gives priority to general issues. The parameters and the unknown variables in the equations are expanded into Taylor series at which is the mean value of the basic random variables . stands for the number of basic random variables that should be considered. is noted down here. and are relevant parameters and variables. For the sake of simplicity, only and are taken as random variables and they can meet the following formula:

(7) |

and are taken into the formula (7) and the value is gotten at . Then the equation series which are corresponding to , and are acquired through the equations (4)~(6) with the corresponding boundary conditions and initial conditions achieved.

**5.3. The Statistical Analysis for the Model of the Unsteady Random Heat Transfer about GSHP**

As for the steady or unsteady cases of heat transfer,

is assumed. The Taylor series of at is the following:

(8) |

The mean and variance of are the following:

(9) |

(10) |

**5.4. The Model for Random Impacts on the GSHP Boundaries**

The model for unsteady heat transfer about GSHP is considered. It is a model for the random distribution about the inner temperature field, which results from the boundary heat transfer. To meet the equations, a basic random variable is in view. Now is expanded into Taylor series at the mean , as is the following:

(11) |

In the equation, is the mean of random variable . is a minute number whose mean value is zero. According to the perturbation method, the finite element equations are formed. With put into the equation (4), the result is the following:

(12) |

According to the equation (12), the solution for can be broken down into two sets of definite solutions, which is shown as the following: the equations (13) that the mean can meet and the equations (14) which the argument of can meet

(13a) |

(13b) |

(13c) |

(14a) |

(14b) |

(14c) |

According to these above, the random temperature field for GSHP can be solved. The equations (13)~(14) can be solved by numerical method.

### 6. The Numerical Method for the Model of Random Heat Transfer

**6.1. The Approximable Difference Iteration Scheme for the Mean Temperature**

The result that the equations (13) are deduced and simplified is the following:

(15a) |

(15b) |

(15c) |

in the equation above, is defined. The partial difference formulas are introduced to find the numerical solution for the partial differential equation about thermal conductivity. To be simple, is used to substitute for. The partial derivative is indicated by the forward difference about. The partial derivative is processed by the central difference about. The partial derivative and are processed by the second-order central difference respectively. Then there are the following:

(16a) |

(16b) |

(16c) |

(16d) |

As for the partial differential equations above, whose step-length is stands for the depth of heat action, namely and . whose step-length is stands for the radius of heat action, namely and . In the process of computer simulation, the interval of time is with the time step-length being,namely and . The partial difference schemes above are put into the equation. And the process is the following:

(17) |

After the formula above is simplified, the approximable difference scheme iteration is the following:

(18) |

For the sake of simplification, it is allowable that is defined. The approximable difference iteration scheme which has been established is turned into the following:

(19) |

, and are in the scheme. There is a limited condition, namely to guarantee that the scheme is steady and convergent.

**6.2. The Approximable Difference Iteration Scheme for the Temperature**

The sign is utilized to signify and the equations (14) are simplified as the following:

(20a) |

(20b) |

(20c) |

In the equations above, and are defined. To be convenient, is used to substitute for and is still signified by.The partial derivatives and are processed by the forward difference about time.The partial derivative is indicated by the central difference about .The partial derivatives and are processed by the second-order central difference respectively. Then there are the following:

(21a) |

(21b) |

(21c) |

(21d) |

(21e) |

The partial difference schemes above are put into the equationand the process is the following:

(22) |

After the formula above is simplified, the approximable difference scheme iteration is the following:

(23) |

For the sake of simplification, it is allowable that is defined. The approximable difference iteration scheme which has been established is turned into the following:

(24) |

The initial value for the temperature can be determined by the given condition, namely. The specific meanings for and are the same as the solution for the former partial differential equation. There is a limited condition, namely to guarantee that the scheme is steady and convergent.

**6.3. The Numerical Simulation Method**

With regard to two sets of partial differential equations above, the mathematical software Matlab and the computer language are applied to the numerical simulation which needs to use the approximable difference iteration schemes, the relevant parameters corresponding to the Figure 1 and the boundary conditions of area . It not only realizes the numerical solution for two sets of partial differential equations in the form of pictures but also achieves a fact that the mean temperature and the parameter temperature field about, within the specific space and time, can be inferred. As for the minute random variable, it is assumed that complies with standard normal distribution. A sample for the random number can be achieved by the random simulation of Matlab. In the end, the real temperature for any moment at any position in the area can be inferred by the formula .

### 7. The Simulation Analysis of Computer

**7.1. The Engineering Case and the Conditions for Computation**

A certain engineering case for GSHP in city of is taken for example. The coefficient of thermal conductivity is. The rock-soil density is. The specific heat capacity is.The pipe radius, the radius of heat action and the pipe length are,and respectively. The initial value of temperature for the partial differential equation which has been simplified is. According to the real-world conditions, the temperature of boundary belonging to is taken as which stands for the soil mean temperature in area. The mean temperature on boundary is because there is no heat exchange on, namely it is adiabatic. The reason why the mean temperature on is taken as is that the heat exchange on is always a constant . As for the mean temperature on ground surface, with the equation being taken into consideration, a function is given, which can mirror the real-world mean temperature for the parts of pipes at ground surface. is taken as and (the unit is degree centigrade) is given. The time interval is and the value of is 60 (the unit is hour). According to the physical parameters and boundaries above, the random temperature field for underground heat transfer in this area can be given a numerical analysis and simulation with the numerical simulation theory in the above being utilized.

**7.2. The Distribution of Mean Temperature Field**

**Fig**

**ure**

**2**

**.**The transient temperature distribution of for area at

**Fig**

**ure**

**3**

**.**The transient temperature distribution of for area at

The moments at ,, and are taken for example to reveal the transient temperature distribution of the mean temperature as the following:

(The units for X-Axis which stands for the rock-soil depth and R-Axis which stands for the radius are meter, the unit for the random temperature is degree centigrade).

**Fig**

**ure**

**4.**The transient temperature distribution of for area at

**Fig**

**ure**

**5**

**.**The transient temperature distribution of for area at

The transient temperature distribution of random parameter is the following:

**Fig**

**ure**

**6**

**.**

**The transient temperature distribution of for area at**

**Fig**

**ure**

**7**

**.**

**The transient temperature distribution of for area at**

**Fig**

**ure**

**8**

**.**

**The transient temperature distribution of for area**

**Fig**

**ure**

**9**

**.**

**The transient temperature distribution of for area**

**7.3. The Variability Effect on Temperature Field**

As for the minute random variable whose mean is zero, it is considered that complies with the standard normal distribution. A statistical series about can be simulated by computer.

With regard to an arbitrary which belongs to the sample of random variable, the real temperature in the area can be solved by the equation if the paired temperature which stands for the values of and at any moment and position in area can be determined. The paired temperature is studied as the following. On condition that the moment is and the radius of heat action is , the position coordinate in different depth as well as the corresponding paired temperature are taken for example by the numerical solution for the partial differential equation about heat conductivity, which is shown in Table 1. The other paired temperature at a certain moment can be studied in the similar way. (the unit of temperature is degree centigrade).

As for the minute random variable, the equation can explain the property for the unsteady random heat transfer of GSHP buried pipes reasonably.

### 8. Conclusion

Currently, there are some models which can describe the principle of heat transfer for GSHP heat exchanger. However, big gaps still exist between these models and the engineering practice. To find the theoretical model and application technology for GSHP heat exchanger in complex environment is always difficult but important for the design of GSHP system.

In recent years, the GSHP researches have taken the concealment of GSHP project, systematic durability, stability, prediction for system operation and process control into consideration. The researches on rock-soil coupled heat and mass transfer can not only preferably facilitate the simulation on the situation for GSHP heat exchangers but also decrease the initial investment for rock-soil management.

This paper analyzed the random factors about heat transfer for GSHP buried pipes, established the general model for rack-soil random heat transfer and did research into the random temperature field for GSHP underground heat transfer. The finite element analysis method for random temperature field was established and the numerical calculation was given to the random temperature field by computer. The research shows that the effect which the GSHP random heat source exerts on the GSHP temperature field is obvious. The engineering case indicates that the random impacts on the heat transfer of GSHP buried pipes cannot be negligible.

Obviously, there is much randomness in the process of design, installation, operation and maintenance for GSHP heat exchangers. It is significant for the popularization of GSHP technology and the theory for underground heat transfer of GSHP is being carried out.

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