1 SIMULATION OF GROUND COUPLED VERTICAL U- TUBE HEAT EXCHANGERS. Steven P. Rottmayer. A thesis submitted in partial fulfillment of

Transcription

1 1 SIMULATION OF GROUND COUPLED VERTICAL U- TUBE HEAT EXCHANGERS by Steven P. Rottmayer A thesis submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE (MECHANICAL ENGINEERING) at the UNIVERSITY OF WISCONSIN-MADISON 1997

2 2 ABSTRACT{ TC "ABSTRACT" \l 1 } Ground oupled heat pumps are an effiient alternative to onventional methods of onditioning homes, beause instead of using the ambient air they utilize the ground as an energy soure or sink. However, ground oupled heat pumps have high installation osts that makes it ritial to design the system to maximize performane. Vertial u-tube heat exhangers are ommonly used as the ground oupled heat exhanger, but estimating their performane is diffiult beause of the unique heat transfer onditions of this onfiguration. This thesis fouses on modeling the vertial u-tube heat exhanger. Several initial attempts to model the heat exhanger were made, and finally an expliit euler finite differene numerial tehnique was employed. The ground storage volume is divided axially into setions, and eah setion is a two dimensional ylindrial mesh representing the fluid, tubes, grout, and soil at a speifi depth. The tubes are approximated by non irular setions of the mesh, and is aurate to within 8%. A loal oupling fator an inrease this auray to 3% for most systems, and omparisons with an existing model showed good agreement. The finite differene model has provided an approah that is fundamental and readily extended to more realisti onditions. It is aurate and fast enough to be useful as both a omparison to existing models and as a design tool.

3 3 ACKNOWLEDGMENTS{ TC "ACKNOWLEDGMENTS" \l 1 } I would like to express my gratitude to the Professors at the Solar Energy Lab whose expertise and work ethi enabled me to omplete my projet. Thanks to Bill Bekman for advising me through some extremely diffiult problems and allowing me the opportunity to work in the Solar Energy Lab. John Mithell's patiene, optimism, and insight enabled me to ontinue through some very overwhelming situations, and I would like to thank him for all the extra hours he spent advising me. Thanks to Sandy Klein whose ideas helped guide the projet in the right diretion. To the students and staff I also wish to thank them for their help and support. Jeff Thornton for helping me with my projet and providing insight with his knowledge and experiene with ground oupled heat pumps. Nathan Blair for always being patient and helpful with any problems I enountered. Thanks to Steve Zehr for always providing exellent advie in times of stress. Thanks to the late night rew, Pat, Olaf, Annette, and Abdulrahman who made it muh easier to work in the lab at midnight. Finally, I wish to express my thanks to everyone else with whom I had the pleasure to work with. I will definitely miss everyone.

4 4 TABLE OF CONTENTS{ TC "TABLE OF CONTENTS" \l 1 } Abstrat...i Aknowledgments...ii Table of Contents...iii List of Figures...v List of Tables...vii Nomenlature...viii ABSTRACT...2 ACKNOWLEDGMENTS...3 TABLE OF CONTENTS...4 List of Figures...6 List of Tables...7 Introdution Literature Review Projet Sope...14 Ground Soure Heat Pump Fundamentals Basi Heat Pump Operation Advantages of Ground Coupled Heat Exhangers Classifiation of Geothermal Heat Pumps Disussion of U-Tube Heat Exhangers Line Soure Theory...26 Modeling a U-Tube Heat Exhanger Thermal Properties and Behavior of Soil Thermal System of a Vertial U-Tube Heat Exhanger Finite Element Models Finite Differene Model for Two Isolated Tubes Retangular Finite Differene Grids...38 U-Tube Geothermal Heat Exhanger Model Finite Differene Method Modeling of U-Tube Heat Exhanger Testing and Validation of the Model...55

5 4.4 Comparison with the Fort Polk Installation...62 Reommendations and Conlusions Conlusions Reommendations for Future Work Appendix A Additions to TRNSYS type...59 Appendix B TRNSYS ode for the ground oupled heat exhanger model...60 BIBLIOGRAPHY...96

6 6 List of Figures{ TC "List of Figures" \l 1 } Figure 2.1 Temperature-entropy diagram of vapor ompression yle...18 Figure 2.2 Cooling and heating yles for a heat pump...19 Figure 2.3 Example of two u-tubes in series...22 Figure 2.4 Example of fluid flow through a u-tube heat exhanger during ooling...23 Figure 2.5. The effet of interferene on the heat transfer to the ground...26 Figure 2.6 Temperature distribution along the old tube as the oupling inreases...27 Figure 2.7 Fluid temperature profiles for one and two setion loop models...28 Figure 3.1 Kusuda Relationship for Temperature Distribution in Ground 34 Figure 3.2 Two dimensional finite element mesh of ground oupled heat exhanger...36 Figure 3.3 Single tube approximation using the shape fator...38 Figure 3.4 Retangular mesh for u- tube model...40 Figure 3.5 Retangular grids are ombined to make a three dimensional model...40 Figure 4.1. The ylindrial grids are ombined to make a three dimensional model...44

7 Figure D finite differene grid used to model system at one depth (m=n=3)...45 Figure 4.3 Resistane network for ylindrial mesh...46 Figure 4.4 Resistane network for the soil to grout interfae...47 Figure 4.5 Non irular tube setion Figure 4.6 Independene of R on enter-enter separation of tubes...51 Figure 4.7 Energy balane on the fluid node...52 Figure 4.8 Resistanes from the enter node to its surrounding nodes...53 Figure 4.9 Inreasing the separation distane between the nodes...54 Figure 4.10 Effet of splitting the system into t1 and t Figure 4.11 Negleting the ground apaitane near the tube walls...58 Figure 4.12 Comparison between a irular and non-irular tube...60 Figure 4.13 Comparison between one irular and one non-irular tube...61 Figure 4.14 Contours for non irular and irular tubes (realisti fluid speifi heat)

8 8 List of Tables{ TC "List of Tables" \l 1 } Table 3.1 Condutivity (Btu/hr-ft-F) of typial grouts/bakfills...32 Table 3.2 Thermal properties of soil...33 Table 4.1 Single Tube Finite Differene Comparison...59 Table 4.2. Values used to determine the largest and smallest geometry fator...62 Table 4.3 Parameters for finite element and finite differene omparison...62 Table 4.4 Parameters used to ompare with the University of Lund model...64

9 9 Nomenlature Variables: α δ R t t 1 t 2 t TRNSYS θ θmid EWT F h int 1 int 2 int 3 k L m Ý m Thermal diffusivity Speifi Heat Thikness Radial distane between nodes Time step Time step for the fluid nodes Time step for the soil nodes TRNSYS time step for the soil nodes Angular distane between nodes Angular distane between nodes Exiting water temperature Fall Fluid to tube heat transfer oeffiient Number of fluid temperature updates between ground temperature updates Fration of ground temperature updates between TRNSYS alls Fration of ground temperature updates between TRNSYS alls Thermal ondutivity of soil Length of bore hole Radial position of the tubes and edge of grout Mass flow rate

10 10 n N i N o ρ π ÝQ Q out Q oupling r r m R R fs r half s Sp Su T (UA) i (UA) o V x W Ý W Z Number of multiples of θ ontained in 90 Nondimensional ondutane between hot and old legs of u-tubes Nondimensional ondutane between hot and old legs of u-tubes Density Mathematial onstant pi Heat transfer Heat transfer from tubes Heat transfer from tubes Radius of node Radius half-way between nodes Resistane Resistane from fluid to ground Radius between the enter and r(2) Center to enter spaing between u-tubes Spring Summer Temperature Thermal ondutane between hot and old legs of u-tubes Thermal ondutane between hot/old legs and the ground Volume urrent axial position in u-tube Winter Compressor work Axial distane between nodes

11 Array Variables: 11 i j k z z' Radial position of the nodes Cirumferential position of the nodes Time step Axial position of the nodes Coupling node Subsripts: f g h s t Cold leg of u-tube Fluid Grout Hot leg of u-tube Soil Tube

12 12 Chapter 1 Introdution{ TC "Introdution" \l 1 } Residential heating and ooling aount for more than 25% of the nations eletrial energy onsumption. Sine the onsumer demand for eletriity is ontinuously rising, improvements in tehnology must be made. Air Soure heat pumps (ASHP) have proven to be an eonomial means of heating and air onditioning homes in southerly limates. However, these heat pumps employ air to air heat exhangers and their performane is highly dependent on the environment in whih they operate. In regions with low winter temperatures the effiieny of an ASHP dereases signifiantly. This had limited their market to regions with primarily mild winters. Ground soure or geothermal heat pumps (GSHPs) exhange heat with the ground, and maintain a high level of performane even in older limates. This results in more effiient use of energy. For this reason many publi utilities endorse the use of geothermal heat pumps and are ative in an effort to persuade the HVAC industry to inrease the number installed. The use of GSHPs, unfortunately, remains low. One major reason for this is the unertainty in the design of the ground oupled heat exhanger. There is a high installation ost assoiated with the heat exhanger so it is probably the most important omponent. The lak of aurate models is espeially apparent for vertial u-tube heat exhangers. These are expensive, but ommonly used beause they are easier to install in the available spae. The u-tube heat exhanger usually onsists of a plasti tube 1 inh in diameter with a bend in the middle that

13 13 reverses the diretion of fluid flow. Many models have been reated to simulate their operation, but it is diffiult to aurately represent the unique onditions of this heat transfer problem. Most of today's design tools still rely mainly on a primitive model referred to as the line soure theory (Ingersoll 1954). The potential energy and eonomial savings from the use of GSHPs is substantial, but they are also very expensive to install. Poor design an negate any savings and therefore the lak of adequate modeling undermines the onfidene of industry in the design tools and GSHPs in general. 1.1 Literature Review{ TC "1.1 Literature Review" \l 2 } There have been many models whih have been used to simulate the use of a vertial u-tube ground oupled heat exhanger. These models have handled the omplex geometry of the system by using a variety of assumptions and approximations. Although many models have been reated (twenty are disussed in greater detail in Muraya 1994), the assumptions used by eah may be ombined into three ategories. The first is a single tube approximation. This an be used with numerial methods or the analytial line soure theory. An equivalent single tube diameter has been used with either onstant temperature soures or onstant heat flux soures. The feasibility of this solution was first investigated by Bose (1984). The model developed by Bose used a onstant heat flux soure as approximated by the line soure theory. Through experimentation an empirial relationship between the single tube and the u-tubes was determined to be 2 times the u-tube diameter. This is aurate for some situations, but it is limited to a speifi range of onditions. The tubes may be unusually lose/far, or the ondutivity of the grout material unommonly high/low. Both of these onditions have been shown to make the relationship inaurate (Muraya 1994). This model was expanded by Kavanaugh (1992) whih employed the ylindrial soure solution, adjusted for the u- tube separation, and inluded the effets of thermal oupling. Kavanaugh empirially determined

14 14 another oeffiient that was used to alulate an equivalent heat transfer oeffiient for the tubes. A seond approah is to use a finite element method of analysis. This is beause it is easier to aurately model a two tube system with a finite element grid. The problem with a finite element model is the exessive omputer time required to perform three dimensional simulations. For this reason, these models have been limited to two dimensions, and the thermal interferene is estimated by using an appropriate guess value for the temperature of the fluid returning to the heat pump the amount of interferene an be estimated from the two dimensional model. These results are then extrapolated to three dimensions. The temperatures of the fluid entering and exiting the heat pump are typially used beause this results in the largest fluid temperature differene, the most interferene, and in effet a worst ase senario. This provides muh insight into the effets of interferene, but it overpredits its influene. Often, a two dimensional finite element model is used to provide aurate parameters for other three dimensional models. The third type of model used to investigate this system uses the line soure theory. The heat flux from the tubes to the ground is onsidered to our in multiple step pulses. These step pulses are used by the line soure theory to approximate the short term transient response of the heat exhanger. The temperature of the ground around the tube is updated and ontinues to hange as time passes. The step pulses derease in intensity as the temperature of the ground inreases and eventually a steady state is reahed. One model using the line soure theory was reated by the University of Lund in Sweden (Hellstrom 1990). An equivalent fluid to ground thermal resistane is alulated with a sophistiated mathematial proedure alled the multipole method (Claesson 1988). Step pulses are applied to this resistane to obtain an average temperature at the bore hole edge. One this temperature is determined a finite differene method is employed from the bore hole edge to an adiabati boundary far from the tubes. This model has proven to be quite aurate, but two dimensional testing has shown that the approximation of an average bore hole edge temperature an be inaurate. The

15 15 multipole method also requires the step pulse inputs used by the line soure theory. This model is ommonly onsidered to be the best available in the industry. Another model (Dobson 1991) eliminates the assumption of an average temperature at the bore hole edge, but still uses the ylindrial soure theory (Claesson 1988) The heat flux from the two tubes are onsidered separately and then superimposed. The ylindrial soure solution is an evolution of the line soure theory so this model still relies on the auray of the step pulses. It is also a onern that this model would not aurately aount for the thermal interferene between the tubes. Many geothermal heat pump design tools determine the size of the ground oupled heat exhanger with the models developed by the third method. In fat, all the ones researhed use the line soure or ylindrial soure solution to the problem. Some of them use the original idea of line soure in whih the entire length of the tube is onsidered to have a onstant average heat flux. The others use the idea of a line step in whih the tubes are split axially into several setions. Either way an assumed heat flux is required. These models are extremely flexible and thorough in their analysis, but they need to be ompared with alternative three dimensional models. 1.2 Projet Sope{ TC "1.2 Projet Sope" \l 2 } This projet proposes a method of simulation for a vertial u-tube ground oupled heat exhanger. The model will enhane the use of urrent models by providing a rapid alternative that an be used for design or as a omparison to existing models. The heat exhanger is modeled using an expliit euler finite differene approah. The model is a three dimensional transient heat transfer model that inludes the fluid temperature distribution through the tubes. The program allows the user to hange the following inputs: bore hole depth, flow rate, properties of the fluid, ground, and grout, and temperatures of the ground and inlet fluid. The model is ompatible with TRNSYS, a transient

16 16 simulation program, and results obtained from simulations are ompared to results from the model reated at the University of Lund. The evaluation will provide insight into the auray of urrent design tools. This projet is intended to inrease the onfidene of industry in the design of ground oupled heat pumps. Taking full advantage of the inrease in energy savings provided by ground oupled heat pumps will help ensure that their use will inrease and the eletrial demands of the nation are met in the future.

17 17 Chapter 2 Ground Soure Heat Pump Fundamentals{ TC "Ground Soure Heat Pump Fundamentals" \l 1 } This hapter desribes the fundamentals of a ground soure heat pump. The hapter begins with a desription of the basi operation of a GSHP and their advantages over traditional ASHP's. This is followed by a desription of the unique heat transfer situation GSHP's present, and onludes with a disussion of the urrent method for modeling vertial u-tube heat exhangers. 2.1 Basi Heat Pump Operation{ TC "2.1 Basi Heat Pump Operation" \l 2 } Heat pumps are used to provide heating and ooling for residenes using a vapor ompression yle for operation. Figure 2.1 shows an example of an ideal vapor ompression yle. In the ideal yle refrigerant enters the ompressor as saturated vapor (state 1) and is ompressed to the ondenser pressure. The refrigerant is now a superheated vapor (state 2). It is then ooled in the ondenser by an external fluid until it beomes a saturated liquid (state 4). The refrigerant next passes through an expansion valve and is throttled to the evaporator pressure. The temperature of the refrigerant drops below the temperature of a seond external fluid (state 5), and is heated by this fluid returning the refrigerant to its original ondition (state 1).

18 18 Temperature [F] 4 Expansion Valve Condenser 3 Compressor Evaporator Entropy [Btu/lbm - R] Figure 2.1 Temperature-entropy diagram of vapor ompression yle{ TC "Figure 2.1 Temperature-entropy diagram of vapor ompression yle" \l 8 } The heat pump is able to either heat or ool a spae by using a valve to reverse the flow diretion of the refrigerant. During ooling, the heat pump evaporator ools and removes moisture from the indoor air stream, and the ondenser, loated outdoors, rejets heat to the environment. For heating, the "evaporator" is loated outside, and absorbs heat from the environment, while the ondenser disharges heat to the indoor air stream. Both of these proesses are shown in Figure 2.2.

19 Cooling Mode Heating Mode 19 Residential Air Str eam Compr essor 1 2 Residential Air Stream 2 Compressor 1 W W Evaporator Condenser Condenser Evaporator Q Q Q Q Fluid Stream From Environment Fluid Stream From Environment 5 Expansion Valve Expansion Valve Figure 2.2 Cooling and heating yles for a heat pump{ TC "Figure 2.2 Cooling and heating yles for a heat pump" \l 8 } The outdoor ondenser/evaporator an be either an air to refrigerant heat exhanger or a water-torefrigerant heat exhanger. Air soure heat pumps (ASHP) exhange heat with the environment by irulating ambient air through an air-to-refrigerant heat exhanger. Alternatively, water soure heat pumps (WSHPs) and ground soure heat pumps (GSHPs) transfer heat to the environment with a water-to-refrigerant heat exhanger. ASHPs and GSHPs are the two types of heat pumps used in residential appliations. 2.2 Advantages of Ground Coupled Heat Exhangers{ TC "2.2 Advantages of Ground Coupled Heat Exhangers" \l 2 } There are several advantages that GSHPs have over ASHPs. The first is the replaement of an outdoor fan with a fluid irulating pump. This may redue power and it enables the heat pump, exept for the ground oupled heat exhanger, to be ompletely ontained indoors. Minimizing the

20 20 ontat of equipment with the environment an prolong its life expetany and GSHPs usually last longer than ASHPs. Another advantage is the water-to-refrigerant heat exhanger. Water has better heat transfer properties than air and this improves the performane of the heat pump. GSHPs an also utilize the benefits of a desuperheater, whih is a devie that will use the superheated refrigerant (state 2 to state 3) and produe domesti hot water. During the ooling season this heat would be wasted by an ASHP. Other advantages are the result of the GSHPs' use of the earth as a heat soure/sink. The ambient air temperature hanges signifiantly throughout the year, while the temperature of the ground is relatively onstant. Sine the ground temperature remains loser to the temperature suitable for human omfort the heat pump performane (COP) remains high throughout the year. A related positive effet is that the summer peak demand is redued ompared to traditional air onditioners due to the higher COP in the summer. ASHPs do not provide this savings beause they are operating with the same external fluid (ambient air) as traditional air onditioners. The final advantage is the elimination of the defrost yle. When the air temperature drops below freezing a defrost yle is required to prevent frost build up on the evaporator surfae. The defrost yle redues the performane of ASHPs onsiderably, and ASHPs are not usually run when the ambient air temperature drops below 20 F. 2.3 Classifiation of Geothermal Heat Pumps{ TC "2.3 Classifiation of Geothermal Heat Pumps" \l 2 } There are many types of GSHPs and they are lassified by the type of ground oupled heat exhanger they employ. The two main groups of heat exhangers are open loop and losed loop. Open loop systems onsist of open ended tubes that extrat water from the environment, use it in the water to refrigerant heat exhanger, and then disharge it bak to the environment. These heat exhangers obtain water from a well or river, and then release the water at the surfae, into a river,

21 21 or into a well. Closed loop heat exhangers irulate the heat exhange fluid through tubes buried in the ground. The heat exhange fluid is first used in the heat pump's water to refrigerant heat exhanger, it then leaves the heat pump, irulates through the tubes and finally returns to the heat pump and is reused in the water to refrigerant heat exhanger. Closed loop heat exhangers an be subdivided into pond, horizontal, and vertial loops. Pond loops are tubes in the formation of a oil or slinky that are plaed in the bottom of a nearby pond, river, or lake. Horizontal loops an be slinky, series layered, or parallel layered, and these are buried in horizontal trenhes that are typially 3-8 feet deep. Sine a large amount of spae is required several tubes are often plaed in a single trenh. The final type of heat exhanger is the vertial heat exhanger, whih is the subjet of the thesis. These an be either u-tubes, tube-in-tube, or slinky and several are plaed in parallel or series. Vertial heat exhangers are expensive, but are ommonly used instead of horizontal units beause they are easier to install in the available spae. The most ommon type of vertial ground oupled heat exhanger is the u-tube, usually onsisting of a plasti tube 1-2 inhes in diameter with a bend in the middle that reverses the diretion of fluid flow. The tube is buried vertially in the ground and the heat exhanger fluid travels through it exhanging heat with the ground. An example of a two u-tube system in series is shown in Figure 2.3.

22 22 G C H P W at er I n Water O ut Figure 2.3 Example of two u-tubes in series{ TC "Figure 2.3 Example of two u-tubes in series" \l 8 } The two omponents for a GSHP are the heat pump and ground oupled heat exhanger. Modeling the operation of heat pump is well understood, but the modeling of a u-tube ground oupled heat exhanger is not Therefore, the remainder of this thesis fouses on the operation of the ground oupled heat exhanger. 2.4 Disussion of U-Tube Heat Exhangers{ TC "2.4 Disussion of U-Tube Heat Exhangers" \l 2 } A typial vertial ground oupled heat exhanger is made by drilling a hole in the ground, inserting a u-tube, and then refilling the spae around the tubes with a new material referred to as grout. The grout an be a variety of materials ranging from onrete to sand. The heat exhanger fluid is

23 23 ommonly water or a glyol solution. The fluid exhanges heat with the refrigerant in the heat pump, then exits the heat pump and enters the u-tube. The fluid flows down to the bottom of the tube, reverses diretion, and returns to the heat pump. The fluid exhanges heat with the ground as it irulates through the u-tube, resulting in two tubes, also alled legs, that ontain fluid at different temperatures. One leg will be referred to as the "hot" tube and one will be referred to as the "old" tube as illustrated in Figure 2.4. Borehole Ground Hot Tube Cold Tube Hot Tube Cold Tube Fluid Flow Heat Flow Figure 2.4 Example of fluid flow through a u-tube heat exhanger during ooling{ TC "Figure 2.4 Example of fluid flow through a u-tube heat exhanger during ooling" \l 8 } The geometry of a u-tube heat exhanger presents a unique heat transfer situation. The two tubes transfer heat to the ground, and also exhange heat with eah other. The heat transfer between the tubes is ommonly referred to as thermal interferene or thermal oupling. Thermal oupling redues the amount of energy transferred to the ground beause the average temperature differene between the two tubes and the ground is less than if the thermal oupling did not our. As the amount of heat transfer between the tubes inreases the heat transfer to the ground dereases.

24 24 The effet of thermal oupling was investigated previously as a phenomena present in animals (Mithell and Myers 1968). This study is not diretly appliable to the u-tube heat exhanger system beause the ground ondutane (UA) is not the same as (UA) o in equation 2.1. This is due to the hange in the ground ondutane with time as the ground is either heated or ooled. The temperature of the ground is also a funtion of depth and time of year, and in this study the temperature boundary is onsidered onstant. Although this study is not diretly appliable to u- tube heat exhangers, it does provide insight into the signifiane of thermal oupling. The thermal oupling between the blood in the artery and the blood in the vein of a heron leg ats as an insulator and enables the heron to withstand the old temperatures of a pond or lake. An analytial solution to the heat transfer problem was obtained, and the effets of various parameters on the temperature distribution of the blood analyzed. The geometry of a heron leg is similar to a u- tube and equations 2.1 and 2.2 show the appropriate energy balane. m Ý dt h dx + ( UA) i ( Th T)+ ( UA) o ( Th T ) = 0 (2.1) m Ý dt dx + ( UA) i ( Th T)+ ( UA) o ( T T ) = 0 (2.2) In these equations T and T h represent the fluid temperature in the old and hot tubes respetively. T is the ground temperature at the farfield radius. For this analysis, the farfield radius is onsidered the radial distane from the tubes at whih the ground temperature is onstant. The fluid flow rate and speifi heat are given the notation Ý m. (UA) o represents both the ondutanes from eah leg of the u-tube to the ground loated at the farfield radius. Sine the tubes are small (1" diameter) relative to their distane from the onstant temperature boundary (8 ft) these are virtually idential. The amount of oupling between the tubes is defined as (UA) i. The equations are solved with the boundary onditions of T h =T o (inlet fluid temperature) at x = 0, and T h = T v at x=l. The

25 25 problem is rewritten in non dimensional form, and the solution is given in equations 2.3 through 2.8. ( T h T ) ( T o T ) = BoshA ( 1 ξ )+ sinh A ( 1 ξ ) B osha + sinh A (2.3) ( T T ) BoshA( 1 ξ) sinh A ( 1 ξ ) = ( T o T ) BoshA + sinh A (2.4) A = No 1+ 2 N i (2.5) N o B = 1+ 2 N i (2.6) N o No = ( UA) o L m Ý (2.7) Ni = ( UA) i L m Ý (2.8) ξ = x L (2.9) Figures 2.5 and 2.6 show the effets of exessive heat transfer between the two tubes. Figure 2.5 shows that as the oupling inreases the net heat transfer to the ground dereases drastially. For large ratios of (UA) i to (UA) o the heat transfer approahes zero, and the value of (UA) o beomes almost irrelevant. Figure 2.6 shows the effet of thermal oupling on the temperatures of the fluid in the tubes. The oordinate x is equal to 1 at the bottom of the u-tube and 0 at the top of the u-tube. It an be seen that the temperature of the fluid reahes the temperature of the surroundings more rapidly as (UA) i

26 26 inreases. When (UA) i beomes infinity the fluid temperature immediately reahes the surrounding temperature, and the amount of heat transferred to the ground beomes zero. Figure 2.5. The effet of interferene on the heat transfer to the ground{ TC "Figure 2.5. The effet of interferene on the heat transfer to the ground" \l 8 }

27 27 Bottom of the U-Tube Top of the U-Tube Figure 2.6 Temperature distribution along the old tube as the oupling inreases{ TC "Figure 2.6 Temperature distribution along the old tube as the oupling inreases" \l 8 } An infinite value for (UA) i may seem extreme, but it emphasizes the importane of orretly modeling a u-tube heat exhanger, espeially as it pertains to the grout material and the separation between the legs of the tubes. With the great expense of installing the ground oupled heat exhangers even slight errors in modeling ould result in signifiant eonomi losses. 2.5 Line Soure Theory{ TC "2.5 Line Soure Theory" \l 2 } Currently, the most ommon methods for modeling vertial u-tube heat exhangers employ the line soure theory. This solution assumes that a tube is buried in a ylinder of soil. The soil loated along the edge of this ylinder is not affeted by the heat exhanger's absorption or rejetion of energy. This distane was disussed previously as a onstant temperature boundary and alled the farfield radius. With the line soure theory, the farfield radius must inrease with the operation time of the heat exhanger. This ours beause the soil an not replenish/dissipate the energy as fast as

28 28 the heat exhanger removes/supplies it. The effetiveness of the heat exhanger delines beause the temperature differene between the tube and the soil will derease. Finite differene and finite element models often use a onstant temperature boundary far from the u-tube. This prevents the farfield radius from expanding beyond this distane, and the result is an artifiially high heat transfer with the ground (Giardina 1995). In its original form, the line soure theory assumes that the buried tube has a uniform heat flux along its entire surfae, thus disregarding temperature gradients in the axial diretion (Hart 1986). This spreads the load evenly over the entire tube length resulting in the inlet and outlet fluid experiening the same load. This assumption underestimates the amount of heat transfer to the ground beause in reality the load hanges with length. Figure 2.7 shows the differene between a single setion and a tube split into two setions. For a tube being heated, the soil temperature around the single setion dereases evenly. If the tube is divided into two axial setions the first setion experienes a greater load than the seond. This brings the soil around the first setion to a lower temperature than the seond setion. The seond setion of the two setion model now has higher soil temperatures ompared to the single setion tube. This results in a higher fluid temperature for the two setion model. To help orret this problem several models using the line soure theory have divided the tubes into several setions and then applied a separate solution to eah setion. Temperature Soil Temperature For One Setion Model Fluid Temperature for Two Setion Model Soil Temperature For Two Setion Model Fluid Temperature for One Setion Model Loop In Distane Along the Loop Loop Out Figure 2.7 Fluid temperature profiles for one and two setion loop models{ TC "Figure 2.7 Fluid

29 temperature profiles for one and two setion loop models" \l 8 } 29

30 30 Chapter 3 Modeling a U-Tube Heat Exhanger{ TC "Modeling a U- Tube Heat Exhanger" \l 1 } This hapter disusses onepts for modeling a ground oupled heat exhanger. First, the properties of the materials and the effets of ground heat transfer are disussed. The hapter onludes with brief summaries of the early attempts to model the heat exhangers. This is intended to assist the future researh in this field. 3.1 Thermal Properties and Behavior of Soil{ TC "3.1 Thermal Properties and Behavior of Soil" \l 2 } In order to aurately simulate the use of a ground oupled heat exhanger the phenomena assoiated with ground heat transfer must be understood, and the thermal properties of the materials known. When the ground is used as a heat soure/sink two things happen that may signifiantly affet the performane of the heat exhanger. Moisture migration ours as the temperature gradient in the ground inreases, and in older limates, the ground freezes. Although these aspets of ground heat transfer are not investigated in this thesis it is important to understand their effets. Moisture migration ours during both heating and ooling seasons, but its effets are most important when ooling. In ooling season heat is rejeted to the ground, resulting in higher temperatures around the tubes. The ground moisture moves in the diretion of hot to old temperatures, and this dries the soil around the tubes. Heat transfer to the ground dereases

31 31 beause the thermal ondutivity of dry soil is muh less than damp or saturated soil (Table 3.2), and the soil will shrink as it dries forming air gaps. The thermal ondutivity of these air gaps (0.015 Btu/hr-ft-F) is far lower than dry soil and ats as an insulator between the fluid and the ground. This adversely affets the exit temperature of the heat exhanger fluid and auses the performane of the heat pump to deline. Freezing is benefiial to ground oupled heat exhangers beause the thermal properties of frozen soil are more favorable for heat transfer than unfrozen soil. Freezing ours near the surfae and so its effets are not as important for vertial heat exhangers. The u-tube material is not hosen solely on the basis of its heat transfer properties, but also with the onern of resistane to wear, expense, and ease of installation. The International Ground Soure Heat Pump Assoiation (OSU 1988) provided standards whih desribed aeptable tube material. The two most ommon materials used for the u-tubes are made of polyethylene or polybutylene (OSU 1988). Polyethylene has a thermal ondutivity of Btu/hr-ft-F and was used for all the simulations. The grout material is very important when onsidering the performane of a geothermal heat pump. This material should have a large ondutivity to inrease the amount of heat transfer to the ground, but if the ondutivity is too high the amount of interferene between the two legs of the u-tube inreases and the ground heat transfer dereases as desribed in setion 2.4. The ondutivity of the most ommon types of grout material are listed in table 3.1 (Kavanaugh).

32 32 Table 3.1 Condutivity (Btu/hr-ft-F) of typial grouts/bakfills{ TC "Table 3.1 Condutivity (Btu/hr-ft-F) of typial grouts/bakfills" \l 9 } Grouts without Additives k Grout with Additives k (Btu/hr-ft-F) (Btu/hr-ft-F) 20 % Bentonite % Bentonite - 40% Quartzite % Bentonite % Bentonite - 40% Quartzite Cement Mortar % Bentonite - 30% Iron Ore /150 lb/ft / % Quartzite - Flowable Fill 1.07 Conrete (50% quartz sand) (Cement+Fly+Ash+Sand) The type of soil as well as its moisture ontent signifiantly affet the soil's thermal properties. There are hundreds of soil lassifiations used for different soils found in the United States. These are ombined into five ategories for modeling purposes, and are shown in Table 3.2 (OSU 1988). The property of the soil an also vary with depth, and as the model beomes more sophistiated this variation an be taken into onsideration.

33 Table 3.2 Thermal properties of soil{ TC "Table 3.2 Thermal properties of soil" \l 9 } 33 Thermal Condutivity (Btu/hr-ft-F) Density (lbm/ft 3) Speifi Heat (Btu/lbm-F) Heavy Soil Saturated Heavy Soil Damp Heavy Soil Dry Light Soil Damp Light Soil Dry The temperature of the undisturbed soil is a funtion of depth and time of year. A funtion derived by Kusuda in 1965 estimates the seasonal variation of the ground temperature with depth. This is given as equation T(Zdepth,tyear) = Tmean Tamp exp Zdepth π 365 αs os 2 π 365 t year t shift Z depth π α s 0.5 (3.1) T mean is the mean value of the ground temperature for the entire year. T amp is the amplitude of the ground temperature at the surfae over the ourse of the year. The surfae temperature drops to a value of T mean - T amp during the winter, and rises to a temperature of T mean + T amp during the summer. Both of these values are dependent on the geographial loation, and both are measured in Fahrenheit. The parameter t shift is the differene between the beginning of the alendar year and the time at whih the minimum ground temperature ours. The Kusuda equation is represented graphially in Figure 3.1.

34 34 T mean - T amp T mean T mean + T amp W F Sp Su Depth (ft) Figure 3.1 Kusuda Relationship for Temperature Distribution in Ground{ TC "Figure 3.1 Kusuda Relationship for Temperature Distribution in Ground" \l 8 } The urve W shows the ground temperatures at their minimum during the winter, and the line Su shows the ground temperatures at their maximum during the summer. The ground temperature will fall between these two urves at any times other than these extremes. The deeper the soil the slower the ground temperature hanges. This is shown by the two urves F and Sp whih represent times in the spring and summer of the year. 3.2 Thermal System of a Vertial U-Tube Heat Exhanger{ TC "3.2 Thermal System of a Vertial U-Tube Heat Exhanger" \l 2 } The thermal system of a u-tube heat exhanger onsists of: heat transfer fluid, u-tubes, bore hole grout, and the surrounding ground. The volume of the ground that is affeted by the u-tube heat

35 35 exhanger is often referred to as the ground storage volume. For modeling purposes, the ground storage volume an be onsidered a ylinder with a height equal to the depth of the bore hole. The radius of the ylinder is large enough to ensure that the soil at this distane is not signifiantly affeted by the u-tubes at the enter. As indiated, this distane is referred to as the farfield radius, and varies with onfigurations. The temperature of the ground at the farfield radius is onsidered to be a funtion of depth using the relationship of Kusuda. When modeling ground oupled heat exhangers the ground storage volume is divided axially into setions, and eah setion represents the thermal system at a speifi depth as shown in Figures 3.3, 3.5 and 4.1. The heat transfer is symmetrial about a vertial plane passing through the enter of the tubes. Consequently, it is only neessary to model one half of the ground storage volume and its assoiated tubes, fluid, and grout. A two dimensional mesh represents the fluid, tubes, grout, and soil at one axial setion as shown in Figures 3.2, 3.4 and 4.2. These an then be used at eah axial setion to reate a three dimensional model. 3.3 Finite Element Models{ TC "3.3 Finite Element Models" \l 2 } The first models reated to simulate vertial u-tube heat exhangers use finite element methods. This was done beause it is muh easier to model the two legs of the u-tube with a finite element mesh. The two tubes an be nearly irular, and onneting the nodes into a finite element mesh is straight forward. The finite element model was reated with the program FEHT, and it represents the fluid, tubes, grout material, and ground at one axial setion. The fluid is onsidered lumped and either a onstant temperature or onstant heat flux soure. The program enables the user to easily vary the temperature or heat flux of the fluid, as well as the material properties of the omponents. The geometry of the system (tube diameter, bore hole radius, et.) is less flexible, and was unhanged for eah simulation. Figure 3.2 shows a piture of the finite element mesh. Steady state and

36 36 transient simulations were used to investigate various onditions. Tubes Figure 3.2 Two dimensional finite element mesh of ground oupled heat exhanger{ TC "Figure 3.2 Two dimensional finite element mesh of ground oupled heat exhanger" \l 8 } Initially, the model was used to gain insight onerning the magnitude of the oupling between the two tubes. The heat exhanger fluid was set to a onstant temperature and an attempt was made to isolate the heat transfer due to the temperature differene between the two tubes. These attempts were unsuessful beause it was diffiult to isolate this heat transfer. Sine a three dimensional model using FEHT would be diffiult to reate, the model was only used to investigate assumptions, and provide a test for the final model desribed in hapter Finite Differene Model for Two Isolated Tubes{ TC "3.4 Finite Differene Model for Two Isolated Tubes" \l 2 } The seond model is a finite differene single tube approximation. A single tube of twie the length

37 37 of the bore hole is used to represent a u-tube as shown in Figure 3.3. The model approximates the two dimensional heat transfer in the ground as one dimensional, and did not allow axial ondution. The assumption of no axial ondution along the tube length is justified beause of the large distanes between the axial nodes relative to their temperature differene. The assumption of no axial ondution is extended to the ground below the heat exhanger and the ground and/or air above the heat exhanger. The ground temperature is alulated using a single tube approximation disussed in hapter 1, exept an equivalent radius of twie the diameter is used, and thermal oupling is determined by equation 3.2. The temperatures alulated for the ground are then used at two loations for the single tube approximation as shown in Figure 3.3. The oupling between the tubes is approximated with a shape fator relationship for two ylinders in an infinite medium (Kreith) and is given by equation 3.1. Shape = 2 π s osh 1 r 2 r ( ) 2 2 (3.1) The variable s is the enter to enter separation of the two legs of the u-tube, and the variable r is the u-tube radius. The heat transfer from the oupling is then inluded for the energy balane on the fluid nodes as expressed in equations 3.2. The apaitane of the tube and grout material was negleted. In equation 3.2a, R fs is the resistane from the fluid to the ground, and onsists of the resistane from the fluid to the tube wall, the tube resistane, and the resistane of the soil and grout. The parameter z' in equation 3.2b refers to the axial position of the fluid node that exhanges heat with the node at z. For example, in Figure 3.3 T(1,k) exhanges heat with T(6,k), therefore, in this ase z=1 and z'=6.

38 38 Fluid Inlet Fluid Exit Fluid Inlet T(i,1,k) T(i,6,k) T(i,1,k) T(i,6,k) T(i,2,k) T(i,5,k) T(i,2,k) T(i,5,k) T(i,3,k) T(i,4,k) T(i,3,k) T(i,4,k) T(i,3,k) T(i,4,k) T(i,2,k) T(i,5,k) T(i,1,k) T(i,6,k) Fluid Exit Figure 3.3 Single tube approximation using the shape fator{ TC "Figure 3.3 Single tube approximation using the shape fator" \l 8 } ( ) T ( ρv) f (z,k +1) T f (z, k) f t ( m Ý ) f ( T f (z 1,k) T f (z,k)) ( Q out + Q oupling ) = (3.2) ( Qout = T f (z, k) T s (i,z,k)) R fs (3.2a) Q oupling = ( Shape k s )( T f (z, k) T f (z',k )) (3.2b) This model performed the simulations rapidly, but using the shape fator overpredited the amount of oupling. The model geometry is also limited by the shape fator beause it beomes inaurate

39 39 if L is less than 3 times r. Beause of this and the single tube approximation required to alulate the ground temperature, this model is not aurate enough to be useful. 3.5 Retangular Finite Differene Grids{ TC "3.5 Retangular Finite Differene Grids" \l 2 } This model simulates the thermal system of ground oupled u-tube heat exhangers as desribed in setion 3.2. The model uses Cartesian oordinates at eah axial setion of the ground storage volume as shown in Figure 3.4. Eah setion represents the ground at a speifi depth. The model allows heat transfer to our in the x-y plane, but not axially. The assumption of no axial ondution is one again justified beause of the large distanes between the axial nodes relative to their temperature differene. The heat transfer above and below the heat exhanger was again negleted. Beause of the omplex geometry of a u-tube heat exhanger, the irular tubes are approximated with retangles. The distane between the nodes along the x-axis (x) and the distane between nodes along the y-axis (y) are suh that the perimeter of the retangular and irular tubes are equal and the error in the ross setional areas minimum. The retangular grids are then used at eah axial setion to reate a three dimensional model as shown in Figure 3.5. The fluid enters at the surfae (z=1), travels through to the bottom of the grid (z=l), bak up the return tube, and exits again at the top (z=1).

40 40 Constant Temperature Boundary Constant Temperature Boundary y x x = 1 y = 1 Nodal Area Adiabati Boundary Tube Nodes Farfield Radius Figure 3.4 Retangular mesh for u-tube model{ TC "Figure 3.4 Retangular mesh for u-tube model" \l 8 } Setion of retangle Grid Fluid Inlet Fluid Exit Z = 1 (Ground Surfae) ² Z Z = 2 Z = L (Bottom of Bore) Figure 3.5 Retangular grids are ombined to make a three dimensional model{ TC "Figure 3.5

41 Retangular grids are ombined to make a three dimensional model" \l 8 } 41 This approah aurately modeled the system, but is extremely slow. An annual simulation would require approximately a week to omplete. The model is slow beause of the large number of nodes it used (approximately 20,000). Sine typial u-tube diameters are 1" and the farfield radius is 8' this number of nodes is required for eah axial setion. The spaing needs to be small lose to the tubes beause of the large temperature gradients, but it an be inreased as the radial distane from the tubes inreases. Unfortunately, this is not easily implemented into a retangular finite differene mesh. An attempt to inrease the speed of a retangular finite differene mesh was made by employing the assumption that the heat transfer is one dimensional far from the u-tubes. Two dimensional heat transfer (in the x-y plane) is alulated near the tubes, but farther from the tubes the heat transfer is approximated as only radial. The distane from the tubes at whih the heat transfer ould be approximated as radial only was determined to be around 2 feet. This dramatially redued the number of nodes, and inreased the speed of the model. However, a more flexible model with fewer assumptions was reated, and is desribed in Chapter 4.

42 42 Chapter 4 U-Tube Geothermal Heat Exhanger Model{ TC "U-Tube Geothermal Heat Exhanger Model" \l 1 } This hapter gives a detailed desription of the approah used to model a vertial u-tube ground oupled heat exhanger. It begins with a brief desription of the finite differene numerial method, then desribes the model. The hapter ontinues with the results for testing and validation of the method, and onludes by omparing it to the model reated by the University of Lund. 4.1 Finite Differene Method{ TC "4.1 Finite Differene Method" \l 2 } The final model employs a finite differene method (i.e. expliit euler) whih is a numerial tehnique used to disretize heat transfer problems and transform the differential equations into a system of algebrai equations. The system is divided into points referred to as nodes, and eah node represents a volume of material. The temperature of the volume is speified by its node. An entire system of nodes make up a finite differene grid or mesh. The energy transfer is driven by the temperature differene between the nodes, and obeys Fourier's law of heat transfer. An energy balane is applied to eah node to obtain its temperature. Finite differene models simulate the transient response of a system by updating the temperature of the nodes with time, whih is also divided into disrete steps. The temperatures of the nodes are updated after eah time step, and the alulations ontinue until reahing the speified finishing time. There are two tehniques to solving finite differene numerial equations. Expliit methods use the temperatures alulated at the

43 previous time step to obtain the new temperatures. These are onstrained to a ritial time step, 43 and any time step larger than this auses the simulation to beome unstable. The value for the time step is found by dividing the sum of the thermal apaity of the node by the sum of its surrounding thermal resistanes as given in equation 4.1. ti = mass i p,i 1 R i (4.1) Impliit methods use temperatures from the previous and new time step to obtain the new temperature of the fluid. Impliit methods do not have a time step onstraint and are unonditionally stable. The numerial tehnique used in this model is the Expliit Euler method. 4.2 Modeling of U-Tube Heat Exhanger{ TC "4.2 Modeling of U-Tube Heat Exhanger" \l 2 } The thermal system of a u-tube heat exhanger is desribed in setion 3.1. The model is similar to the retangular model desribed in setion 3.4, but it uses ylindrial oordinates. The reason behind this grid geometry is to reate a irular grid outside the bore hole. This allows for inreasing the nodal spaing in the radial diretion to minimize the total number of required. The ground storage volume is divided axially into setions, and eah setion represents the ground at a speifi depth as shown in Figure 4.1. The model developed allows heat transfer in the ground to our radially and irumferentially, but not axially. The assumption of no axial ondution along the tube length is justified beause of the large distanes between the axial nodes relative to their temperature differene. The assumption of no axial ondution is extended to the ground below the heat exhanger and the ground and/or air above the heat exhanger. A two dimensional ylindrial grid represents the fluid, tubes, grout, and soil at one axial setion. Beause of the omplex geometry (two tubes in lose proximity buried in a third material) of a u-tube heat exhanger the irular tubes