A practical heat transfer model for geothermal piles

Energy and Buildings 66 (2013) 470–479

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A practical heat transfer model for geothermal piles Omid Ghasemi-Fare 1 , Prasenjit Basu ∗ Department of Civil and Environmental Engineering, The Pennsylvania State University, United States

a r t i c l e

i n f o

Article history: Received 4 January 2013 Received in revised form 11 May 2013 Accepted 17 July 2013

a b s t r a c t Idealized heat source models, which assume constant heat flux along the entire length of heat sources, cannot be used for accurate quantification of ground temperature response during thermal operation of geothermal piles. This paper presents an annular cylinder heat source model that can realistically simulate heat transport by the fluid circulating through the tubes embedded in heat exchanger piles. A finite difference code is developed for simultaneous solution of partial differential equations, which describe both transient and steady-state heat transfer from a geothermal pile to the surrounding soil. Results show that the use of a constant heat flux along the entire length of a heat exchanger pile may significantly misinterpret thermal response over time after the start of heat exchange operation. The impact of different model parameters on the performance of a geothermal pile is investigated through a sensitivity study. Based on the results from sensitivity study, initial temperature difference between ground and circulation fluid, thermal conductivity of soil, and radius of circulation tube are identified to be the most important parameters that affect thermal efficiency of a geothermal pile. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Seasonal variation of ground temperature is insignificant below a shallow depth (usually couple of meters) from the ground surface and thus, deep foundations are good candidates for harvesting geothermal energy through heat exchange operation. Heat can be transported by circulating heat carrier fluid through a closed loop embedded within concrete piles. Such piles are commonly known as geothermal piles, heat exchanger piles or energy piles. The great potential of environmental, social and economic benefits of utilizing geothermal energy has made the use of geothermal piles quite popular in different parts of the world. The use of ground-source heat pump (GSHP) systems results in a higher coefficient of performance (COP) compared with the use of air-source heat pump (ASHP) systems because the temperature of the ground (used as a heat source or sink) is relatively stable compared with air temperature. Several research articles indicate that the use of GSHP and ground-water heat pump (GWHP) can result in a cost savings of 18–56% and in a reduction in carbon dioxide emission by 45–80% as compared to the use of ASHP and other conventional sources of energy (e.g., coal, petroleum and natural gas) in residential and commercial buildings [1–10]. In fact, the use of geothermal piles as heat exchangers in GSHP systems further

∗ Corresponding author. Tel.: +1 814 863 4010; fax: +1 814 865 9668. E-mail addresses: [email protected] (O. Ghasemi-Fare), [email protected] (P. Basu). 1 Tel.: +1 814 865 9675; fax: +1 814 8659668. 0378-7788/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.enbuild.2013.07.048

reduces the initial cost of GSHP system installation [11–14]. Therefore, for cost-efficient design of GSHP systems with heat exchanger piles, accurate quantification of heat transfer through these piles should be an integral part of the design. Moreover, it is important to identify and characterize different design, operational and site-specific variables (e.g., radii of pile and circulation tube, fluid circulation rate, and thermal conductivity of soil and pile material) that may significantly affect heat transfer through geothermal piles. This paper presents an annular cylinder heat transfer model that realistically simulates heat exchange between a geothermal pile and surrounding soil. A finite difference code is developed for solving a system of partial differential equations which describe heat flow through heat carrier fluid and heat conduction in soil and concrete. The proposed model can capture the effects of different design, operational and site-specific variables on time-dependent variation of ground temperature. A sensitivity study is performed to identify key parameters that may significantly affect heat transfer efficiency of geothermal piles. 2. Idealized heat source models Researchers from petroleum engineering analyzed heat transfer through geothermal boreholes as early as in 1947. Since then, several researchers have developed and modified idealized heat transfer models for predicting heat exchange between geothermal heat exchangers and ground. Different idealized heat source models were analyzed over the last two decades to estimate variation of ground temperature due to the presence of a finite or infinite

O. Ghasemi-Fare, P. Basu / Energy and Buildings 66 (2013) 470–479

Nomenclature A Cp E Fo i, j k Lb L ˙ m q q˙ l q¨ r rb rt rp Rb t T Tf Tg Tinlet Toutlet

v z

Area (m2 ) Specific heat (J kg−1 C−1 ) Energy output (W) Fourier number Node index Thermal conductivity (W m−1 C−1 ) Length of an idealized heat source (m) Pile length (m) Mass flow rate (kg s−1 ) Heat flow rate (W) Heat flux per unit length (W/m) Heat flux (W m−2 ) Radial coordinate (m) Radius of idealized heat source (m) Radius of circulation tube (m) Pile radius (m) Normalized radius Time (s) Temperature (◦ C) Fluid temperature (◦ C) Ground temperature (◦ C) Temperature at circulation tube inlet (◦ C) Temperature at circulation tube outlet (◦ C) Circulation velocity of heat carrier fluid (m s−1 ) Depth (m)

Greek symbols ˛ Thermal diffusivity (m2 /s) Initial temperature difference between fluid inlet  point and ground (◦ C) Mass density (kg m−3 )  Subscripts c Concrete f Fluid p Pile s Soil Circulation tube t

heat source within the ground. Available heat source models can broadly be divided into three main categories: (i) infinite and finite line sources, (ii) hollow cylinder source and (iii) one- and twodimensional solid cylinder sources. Simplified assumptions were made in each of these models in order for the analytical solutions to be possible. Vertical heat exchanger boreholes were first modeled as infinite line and infinite hollow cylinder heat sources with constant heat flux along the length. Carslaw and Jaeger [15] and Ingersoll et al. [16] provided analytical solutions for heat transfer through infinite hollow cylinder and infinite line sources, respectively. Analytical solutions for heat conduction in soil surrounding a finite line heat source with constant heat flux were proposed by Eskilson [17] for steady state condition and by Zeng et al. [18] and Lamarche and Beauchamp [19] for transient condition. Cui et al. [20] and Lamarche [21] provided analytical solutions for transient ground temperature response caused by a single inclined line source. Lamarche and Beauchamp [22] studied ground temperature distribution around a heat exchanger borehole using infinite solid cylinder heat source model with two different boundary conditions: (i) constant heat flux and (ii) constant mean temperature for the heat carrier fluid (or grout). Man et al. [23] developed analytical solutions for heat conduction through one- and two-dimensional

471

solid cylinder sources using Green’s function. Researchers have also suggested the use of spiral heat source model for heat exchanger elements with spiral heat sources [24,25]. Table 1 summarizes available solutions for the idealized heat source models. Note that previous theoretical studies used idealized heat source models to quantify temperature distribution in the surrounding medium. Circulation of the heat transporting fluid within a geothermal heat exchanger element (such as borehole heat exchanger or geothermal pile) was not modeled in those studies; a constant value of either temperature or heat flux along the entire length of a geothermal heat exchanger was assumed, which is far from being practical [26]. Therefore, the effect of fluid circulation rate on heat transfer efficiency of a geothermal pile cannot be quantified using solutions obtained from idealized heat source models available in literature. Such solutions cannot also predict the time-dependent evolution of heat flux and thus, would lead to an inaccurate estimation of temperature distribution in soil surrounding geothermal piles.

3. Annular cylinder heat source model with heat carrier fluid 3.1. Model development Heat transfer through a concrete geothermal pile with an embedded U-shaped circulation tube is modeled in this study. Half of the pile is modeled exploiting the approximately axisymmetric heat flow condition in the medium surrounding the pile (Fig. 1a). Note that the location and arrangement of the circulation tubes within a geothermal pile does not strictly satisfy the condition of an axisymmetric geometry. However, the diameter of the circulation tube (heat source) is two orders of magnitude smaller than the expected thermal influence zone surrounding the pile. Therefore, the assumption of axisymmetric heat conduction in the media (i.e., concrete and soil) surrounding the heat source is not far from reality. Interaction between two vertical limbs of the U-tube is not considered in this study. The wall thickness of circulation tube is assumed to be zero; thus, possible heat loss within the tube wall is neglected. This is a reasonable assumption because the thickness of circulation tubes used in practice is often in the order of couple of millimeters only. Heat transfer from the heat carrier fluid to the surrounding media is analyzed by coupling heat conduction and heat balance equations. Time-dependent evolution of temperature T (r, z, t) due to heat conduction within an axisymmetric domain can be expressed as: 1 ∂T ∂2 T ∂2 T 1 ∂T = + 2 + ˛ ∂t r ∂r ∂z 2 ∂r ˛=

k Cp

(1a)

(1b)

where ˛, k,  and Cp are, respectively, thermal diffusivity, thermal conductivity, mass density and specific heat capacity of the heat conduction medium and t is time. Radial distance r and depth z are measured from the origin O (Fig. 1b). Eq. (1) alone cannot describe heat transfer through a geothermal pile because it does not capture heat flow within the circulation tube. Considering that average temperature of an element A (Fig. 1a) within the circulation tube increases by an amount dT over time dt and assuming an average heat flow rate q (from element A to concrete pile) over the length dz, the heat balance equation for element A can be written as: ˙ pf dtdT e = qdt + f rt2 dzCpf dT mC

(2)

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Table 1 Available solutions for idealized heat source models. Idealized heat source models

Analytical solution

2

q˙

l T (r, t) = − 4k Ei( −r ) 4˛t

Infinite line (Ingersoll et al. [16])

⎧  ⎫  L ⎨ r 2 +(z−z  )2 r 2 +(z+z  )2 ⎬ √ √ erfc( erfc( ) ) q˙ l  2 ˛t T (r, z, t) = 4k −  2 ˛t dz  ⎩ ⎭ r 2 +(z−z  )2 r 2 +(z+z  )2

Finite line (Zeng et al. [18])

0

⎡∞ ⎤  ⎣ (e−˛u2 t − 1) J0 (ur)Y 1 (ur0 )−Y0 (ur)J1 (ur

0 ) du⎦

q˙ l

Infinite hollow cylinder (Carslaw and Jaeger [15])

T (r, t) =

Infinite solid cylinder (Man et al. [23])

l T (r, t) = − 4k

2 kr0

J 2 (ur0 )+Y 2 (ur0 )

u2

1

1

0

 q˙

 1 Ei 

−

r 2 +r 2 −2rr0 cos ϕ



0

dϕ

4˛t

0

T (r, z, t) =

Finite solid cylinder (Man et al. [23])

t L

√

q˙ − clp

8 0

1

˛(t−t  )

3 I0



rr0 2˛(t−t  )

T (r, ϕ, z, t) = −

d= d =

×



exp −

r 2 +r 2 +(z  −z )

2



0

4˛(t−t  )

 − exp −

r 2 +r 2 −(z  +z ) 0

4˛(t−t  )

2

 dz  dt 

0

2n Spiral line model (Li and Lai [25])



q˙ l b

4

√



˛x ˛y

 √d

1 erfc d

2

˛z t

 −

1 d

erfc

√d

2





˛z t

dˇ

0 k 2 2 2 z (r cos ϕ − r0 cos ˇ) + kkz (r sin ϕ − r0 sin ˇ) + (z − bˇ) kx y    kz kx

2

(r cos ϕ − r0 cos ˇ) +

kz ky

2

(r sin ϕ − r0 sin ˇ) + z + bˇ

2

;b =

L 2n

Cp , specific heat; k, thermal conductivity; L, length of the heat source; r0 , cylinder radius; r, z, and ϕ, Cylindrical coordinate system; ˇ, integration parameter; t, time; T, temperature; ˛, thermal diffusivity; , mass density; J0 , Y0 , and I0 , Bessel’s functions; x, y, and z, Cartesian coordinate system

x

∞

eu u

Ei (x) =

du;

−∞

erfc (x) =



2

e−u du;

√2  x

I0 (x) =

1 





exp xcosˇ dˇ 0

where dTe is the temperature difference (over the length dz) ˙ and Cpf are, respecbetween top and bottom of element A, m tively, mass flow rate and specific heat capacity of heat carrier fluid circulating through the tube, and rt is radius of the circulation tube. Radial variation of fluid temperature at a particular depth is neglected in Eq. (2). Heat flow rate q can be related to heat flux q¨ as: ¨ q = qdA

(3)

where dA (=2␲rt dz) is the surface area available for heat transfer from element A to the concrete pile. Heat flux q¨ is further defined as: ¨ q(z, t) = −kc

∂T ∂r

(4)

where kc is thermal conductivity of concrete. Using the definition of ˙ and replacing Eqs. (3) and (4) in Eq. (2), mass transfer (flow) rate m

the heat balance equation expressed through Eq. (2) can be written as:

 f vrt2 Cpf

e

dtdT =

f rt2 Cpf dzdT

− 2kc rt dzdt

∂T ∂r

 (5)

where f is mass density of circulation fluid and v is fluid circulation velocity. Rearrangement of Eq. (5) yields the partial differential equation (PDE) of heat transport by the heat carrier fluid flowing through circulation tubes embedded in a geothermal pile. ∂T ∂T 2kc ∂T =v + f Cpf rt ∂r ∂t ∂z

(6)

Simultaneous solution of Eqs. (1) and (6) under different boundary and initial conditions will provide time-dependent evolution of temperature within a geothermal pile and that in the soil surrounding the pile.

O. Ghasemi-Fare, P. Basu / Energy and Buildings 66 (2013) 470–479

473

Fig. 1. Annular cylinder heat source model proposed in this study (a) isometric and plan view and (b) finite difference grid and boundary conditions.

3.2. Finite difference formulation for the proposed model

T = Tinitial

A finite difference (FD) code is developed to obtain simultaneous solution from Eqs. (1) and (6). The developed FD code uses an explicit solution scheme. A schematic FD grid and boundary conditions used for the analyses presented in this paper are shown in Fig. 1(b). Using explicit FD definitions, the following expressions are obtained, respectively, for Eqs. (1) and (6):

In addition to the boundary and initial conditions specified by Eqs. (10) and (11), a heat flow continuity condition is used at the pile-soil interface (i.e., at r = rp ).



t+1 t Ti,j = ˛t − Ti,j

 +

t+1 t − Ti,j =v Ti,j

t t + Tt Ti−1,j − 2Ti,j i+1,j

ri ri+1 t Ti,j−1

t − 2Ti,j

t + Ti,j+1





t t 1 (Ti+1,j − Ti,j ) + ri ri+1

zi zi+1

t  zj



t t Ti,j−1 − Ti,j +

(7)

r

v z

z 1

+

2kc t t (T t − Ti,j ) f Cpf ri ri i+1,j

(8)

rt r

(9)

2kc f Cpf rt r

The boundary conditions shown in Fig. 1(b) and the initial condition used in the analyses are: T = Tinitial

for r = R, z≥0; r≥2rt , z = 0 and r≥0, z = Z

∂T = 0 for r = 0, z≥0 ∂r

1 = ri



⎧ 1 ⎪ ⎪ ⎪ 2˛ 2˛ 2˛ ⎪ ⎨ + + 2 2 ⎪ ⎪ ⎪ ⎪ ⎩

T t+1 − T t t



(10a) (10b)

×



ks ri+1

 Tt

i+2

−T t



i+1



ri+2 −ri+1

2 − r2 ri+1 i−1

− kc ri−1



t t + Tt Ti,j−1 − 2Ti,j i,j+1



t t + Tt − 2Ti,j Ti,j−1 i,j+1

zi zi+1



(11)





T t −T t i

⎤

i−1

ri+2 −ri+1

(ri+1 − ri−1 )

zi zi+1

 ×

⎢ ⎣



2 2 − r2  C ri2 − ri−1 c Cpc + ri+1 s ps i



⎡

Stability of FD solutions presented in this paper is ensured by selecting a time step t that is small enough to satisfy the Courant–Friedrichs–Lewy condition [27]. For simultaneous solution of Eqs. (7) and (8), the time-step stability criterion is expressed as:

t ≤ min



for t = 0; 0 ≤ r ≤ R and 0 ≤ z ≤ Z

2  2 ⎥  ri+1 − ri  ks ⎦+ 2 2 ri+1 − ri−1

2 2  r − ri−1  kc + i 2 2 ri+1 − ri−1

(12)

The continuity condition expressed through Eq. (12) is required to obtain realistic solution for heat transfer from a heat exchanger pile to the surrounding soil because the values of thermal diffusivity for concrete and soil are likely to be different for practical purposes. Most of the available idealized heat transfer models, except the ones developed by Hellstrom [28] and Lamarche and Beauchamp [19], assume a single homogeneous medium surrounding a heat source. Therefore, such idealized models cannot accurately quantify the variation of temperature in two different media (concrete and soil) surrounding a heat source. Moreover, none of the idealized models can capture variations of heat flux and fluid temperature along the length of a circulation tube; hence, cannot quantify the effects of these variations on heat transfer efficiency of geothermal piles.

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32

68

k=1.5 W/mC Tinitial=18 °C rb=0.025 m Lb=5 m z=2.5 m Rb=rb/Lb= 0.005

58

48

38 Analytical (steady satate) solution for finite line source [18] Finite difference solution

28

Temperature T (°C) adjacent to the heat source

Temperature T (°C) adjacent to the heat source

78

30

28

k=2.3 W/mC Tinitial=18 °C rb=0.3 m Lb/rb=100

26

24

22 Analytical (transient) solution for infinite hollow cylinder source [15] Finite difference solution

20

18

18 0

4000

8000

Normalized time Fo = t/rb

12000

0

4

8

12

Normalized time Fo = t/rb2

2

(a)

(b)

Fig. 2. Comparison between analytical solutions and results obtained using the developed finite difference code (with appropriate modifications) for (a) finite line heat source (steady-state solution) and (b) infinite hollow cylinder heat source (transient solution).

3.3. Validation of the developed finite difference code The FD code is developed for solving PDEs associated with the proposed annular cylinder heat source model, but with certain adjustments in boundary and initial conditions, this code can also produce solutions for idealized heat source models available in literature. We verify the FD code by comparing available analytical solutions for finite line and infinite hollow cylinder heat sources with the respective solutions obtained using the developed code. Note that both finite line and infinite hollow cylinder heat source models use constant heat flux (an input parameter for these models) along the entire length of the heat source. Additionally, both of these models consider a single value of thermal conductivity k for the homogeneous medium around the heat source. Hence, the following modifications are required in order for the developed FD code to capture the constant-heat-flux condition at r = 2rt :

"

∂T "" ∂r "

=− r=2rt

"

∂2 T "" ∂r 2 "

=

q˙ l 2rt k (∂T/∂r)i+1 − (∂T/∂r)r=2rt ri+1

r=2rt

=

(13)

[(Ti+1,j − Ti,j )/ri+1 ] + (q˙ l /2rt k) ri+1

(14)

where q˙ l is the constant heat flux per unit length of the heat exchanger. Fig. 2 shows that the developed FD code can successfully predict analytical heat transfer solutions for finite line source (q˙ l = 100 W/m, rb = 0.025 m, Lb = 5 m and Rb = rb /Lb = 0.005; rb and Lb are, respectively, radius and length of the idealized heat source) and infinite hollow cylinder source (q˙ l = 100 W/m, rb = 0.3 m and Lb /rb = 100).

Z = 35 m is considered around the pile. Thermal properties for concrete and soil, as assumed for the analyses, are given in Table 2; specific heat of the heat carrier fluid Cpf is assumed to be equal to 4190 J kg−1 C−1 . Few additional analyses are performed to identify the effects of some important input variables on thermal efficiency of heat exchanger piles and on time-dependent evolution of ground temperature Tg . Fig. 3(a) shows that the thermal influence zone around the heat exchanger pile extends approximately up to a radius of 160rt (=3.2 m ≈ 11rp ) after 60 days of heat rejection from the pile to the ground (a thermal loading condition that simulates operation of a geothermal pile during summer). Note that the thermal influence zone continuously grows with time after heat rejection starts. However, two months of continuous heat rejection from a geothermal pile to the ground (as simulated in this analysis) can be considered as an extreme scenario for thermal operation of such a pile during summer in most part of the world, and thus, 160rt (≈11rp ) would practically be an upper bound of thermal influence zone around a heat exchanger pile. Except in the vicinity of pile head and base, radial heat transfer is observed for the entire length of the pile. Such radial heat transfer is also observed in previous numerical studies of heat exchanger piles [29,30]. Even after 60 days of heat exchange operation, change in ground temperature is negligible (less than 1 ◦ C) beyond a depth of 6rp below the pile base (Fig. 3). Thermal conductivity of soil ks (and consequently, thermal diffusivity ˛s ) depends on various factors such as dry density, water content, and soil texture. For coarse- and fine-grained soils, the range of ks varies, respectively, from 0.9 to 4.2 W/mC and from 0.3 to 2.1 W/mC [8]. The value of ks reduces with decrease in soil water content; ks is minimum for dry soil (usually 0.2–0.4 W/mC; [31]). Soil near the ground surface is often not fully saturated and a low value of ks (and thus ˛s ) is expected within this desiccated zone. Heat transfer performance of a geothermal pile is investigated in the presence of a 5 m desiccated zone of soil (with ks = 0.38 W/mC

4. Heat transfer analyses 4.1. Analysis result Analyses are performed using the developed FD code to quantify heat transfer through a 30-m-long geothermal pile under different thermal loading. A soil domain with radius R = 10 m and height

Table 2 Thermal properties of concrete and soil used in the analyses. Thermal properties

Concrete

Soil

Diffusivity ˛ (m2 /s) Conductivity k (W/mC)

˛c = 0.66 × 10−6 kc = 1.5

˛s = 1.02 × 10−6 ks = 2.3

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Fig. 3. Ground temperature (◦ C) profile around a geothermal pile after 60 days of heat rejection for (a) homogeneous ground and (b) ground with a top 5 m desiccated zone.

and ˛s = 1.7 × 10−7 m2 /s) just below the ground surface. Fig. 3(b) shows that the thermal influence zone is smaller within the top desiccated soil layer; however, increase in ground temperature Tg adjacent to the pile is greater in the desiccated soil layer with lower value of ˛s than that in the soil layer with higher value of ˛s . The effects of initial temperature difference  (=Tinlet –Tinitial ) and fluid circulation velocity v on ground temperature Tg is shown in Fig. 4. It is observed that at any given time t after the start of the heat transfer operation, the thermal influence zone is independent of  and v. Ground temperature Tg within the thermal influence zone increases with increase in both  and v. Fig. 5 shows (for v = 0.02 and 0.1 m s−1 ) the variation of temperature T along depth z at different radial distances; temperature gradient along depth (dT/dz) increases as v decreases.

The time-dependent evolution of heat flux (per unit length) q˙ l along the length of the circulation tube is shown in Fig. 6. Heat flux q˙ l decreases linearly along the length of the circulation tube. Over a heat rejection period of 60 days, q˙ l at the middle of the pile (i.e., at z = 15 m) reduces by almost 30% from its value at the end of the first day of operation. Therefore, the use of idealized heat transfer models with constant values of q˙ l along the entire length of the heat source would introduce significant errors in the quantification of heat transfer through a geothermal pile. Transient variation of fluid temperature Tf along the length of the circulation tube is shown in Fig. 7. Only few minutes after the heat transfer starts, Tf varies linearly with depth z. From in situ performance tests on geothermal piles, Gao et al. [32,33] observed similar linear distribution of fluid temperature along the length of

Fig. 4. Variation of ground temperature Tg for different values of (a) initial temperature difference  (= Tinlet −Tinitial ) and (b) fluid circulation velocity v.

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0

0 r=2rp

r=0.5rp

r=rp

5

10

20

Depth z (m)

Depth z (m)

10

Tinlet=37 °C Tinitial=18 °C rt=0.02 m rp=0.3 m L=30 m z=L/2 ks=2.3 W/mC kc=1.5 W/mC

30 v=0.02 m/s, r=rp

200 seconds 20 minutes 60 minutes 1 day 12 days 60 days

Tinlet=37 °C Tinitial=18 °C rt=0.02 m rp=0.3 m L=30 m ks=2.3 W/mC kc=1.5 W/mC v=0.1 m/s

15

20

25

v=0.1 m/s, r=rp 30 16

40 18

22

26

20

24

28

32

36

40

Fluid temperature Tf (°C)

30

Temperature T (°C)

Fig. 7. Variation of fluid temperature Tf along the length of the circulation tube.

Fig. 5. Effect of fluid circulation velocity v on temperature T along depth z.

circulation tubes. The distribution of Tf along the length of the circulation tube stabilizes (i.e., reaches steady state) after 12 days of heat exchange operation. In order to investigate the effect of variable heat flux on evolution of temperature within the heat exchanger pile and that in soil surrounding the pile, result obtained using the proposed annular cylinder heat source model is compared with finite line source solution available in literature (Fig. 8). For such a comparison, a constant value of heat flux q˙ l needs to be assigned for the finite line source. However, the choice of q˙ l for use in the finite line source model introduces significant uncertainty in the prediction because q˙ l varies along the length of a real geothermal pile and such variation of q˙ l changes with time during heat exchange operation (Fig. 6). The values of q˙ l used for finite line source solutions plotted in Fig. 8 are the maximum and minimum heat flux values (i.e., q˙ l, max and q˙ l, min , respectively at points near the top and bottom of the circulation tube) obtained from simulations of one hour, one day, and one week of heat exchange operation using the proposed annular cylinder model. It is observed that finite line source solutions (i.e., 0

1 day 12 days 35 days 60 days

Tinlet=37 °C Tinitial=18 °C rt=0.02 m rp= 0.3 m L=15 m ks=2.3 W/mC kc=1.5 W/mC v=0.1 m/s

Depth z (m)

10

20

30 30

35

40

45

50

the use of a constant value of q˙ l along the entire length of the heat source) can significantly misinterpret the increase in temperature within both pile and soil. The maximum difference between predictions using the proposed annular cylinder model and the idealized finite line source model can be as high as 17 ◦ C at a point adjacent to the heat source and 12 ◦ C at pile-soil interface. While the use of finite line source model with high values of constant q˙ l would result in significant overprediction for pile and soil temperature, the use of low values of q˙ l in finite line source model may consistently underpredict such temperature (Fig. 8). The effect of thermal cycles (i.e., successive heat injection and extraction) on thermal efficiency of a heat exchanger pile is also investigated in this study (Fig. 9). Energy output E per unit length of the heat exchanger pile, as plotted in Fig. 9(a), is calculated as: ˙ pf T f vrt2 Cpf (Tinlet − Toutlet ) mC E = = L L L

(15)

For the same values of analysis parameters, energy output (or thermal efficiency) of the heat exchanger pile does not change due to individual equivalent cycles (with same ) of heat injection and extraction (Fig. 9a). However, if a heat extraction cycle follows a heat injection cycle, thermal efficiency of the heat exchanger pile increases during heat extraction. This is because heat energy injected into the ground during the preceding heat injection operation creates a higher temperature gradient between pile and soil as soon as the following heat extraction operation starts. Fig. 9(b) shows ground temperature response due to individual 60 days cycles of heat injection and extraction and a combined 120 days injection-extraction cycle (heat extraction follows heat injection). It is observed that compared to a sole heat extraction cycle, ground temperature Tg is always higher at any time during a heat extraction cycle that follows a heat injection cycle. 4.2. Sensitivity of different analysis parameters

55

60

Heat flux per unit length of circulation tube (W/m) Fig. 6. Variation of heat flux q˙ l (per unit length) with depth z at different instants of heat rejection operation.

Sensitivity analysis is performed to investigate the effects of important analysis parameters on thermal efficiency (expressed in terms of energy output) of geothermal piles and on ground temperature increment at pile-soil interface. Results from this sensitivity study is presented in the form of Tornado diagrams (Fig. 10), which show the relative influences of important model parameters on energy output from a heat exchanger pile and on ground temperature increment. The vertical dashed lines in Fig. 10 show the values of desired output (i.e., energy output and ground

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477

Fig. 8. Effect of variable heat flux on temperature within pile and soil at different times after the start of heat exchange operation for (a) t = 4 days, (b) t = 12 days, (c) t = 35 days and (d) t = 60 days.

Fig. 9. Effect of thermal loading cycles on (a) energy output (heat transfer efficiency) of a geothermal pile and (b) ground temperature response.

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O. Ghasemi-Fare, P. Basu / Energy and Buildings 66 (2013) 470–479

Fig. 10. Effects of important model parameters on (a) efficiency of a geothermal pile (energy output) and (b) ground temperature increment at pile-soil interface.

temperature increment) for an expected set of input parameters. The horizontal bars, known as swings of Tornado diagram, represent the variation of a desired output due to the expected variation of individual input parameters considered one at a time. Longer the swing, higher the influence of the corresponding input parameter on an output parameter. The initial temperature difference  (=Tinlet − Tinitial ), soil thermal conductivity ks , and radius of circulation tube rt are, sequentially, the three most important parameters affecting thermal efficiency of a heat exchanger pile (Fig. 10a). ks is the most sensitive parameter for ground temperature increment (Fig. 10b); it has reverse effects on thermal efficiency of a heat exchanger pile and on the ground temperature increment.

using the proposed model and prediction using the idealized finite line source model, it can be concluded that the use of a constant heat flux along the entire length of a geothermal pile may significantly misinterpret time-dependent evolution of temperature. Elevated ground temperature caused by the operation of a heat exchanger pile in summer helps in increasing thermal efficiency of the pile in winter. Sensitivity analysis performed with important analysis parameters reveals that the initial temperature difference between ground and circulation fluid, thermal conductivity of soil, and circulation tube radius are, sequentially, the most important parameters affecting thermal efficiency of a heat exchanger pile.

4.3. Comparison with field test result

The authors gratefully acknowledge the financial support (Grant No. 415-77 76R20) provided by the Mid-Atlantic Universities Transportation Center (MAUTC).

The proposed annular cylinder heat transfer model is used to predict the temperature variation along one branch of a U-shaped circulation tube (rt = 0.01 m) embedded in a heat exchanger pile (L = 25 m, rp = 0.3 m) installed in field [32,33]. For this comparison, boundary conditions and values of different input parameters (ks = 1.3 W/mC, ˛s = 5.86 × 10−7 m2 /s, kc = 1.63 W/mC and ˛c = 7.78 × 10−7 m2 /s, Tinitial = 18.2 ◦ C, Tinlet = 35.13 ◦ C, t = 3 h) are adopted from Gao et al. [32,33]. At the bottom of the circulation tube, fluid temperature obtained from the proposed analysis (=32.18 ◦ C) compares reasonably well with the temperature (=33.14 ◦ C) reported by Gao et al. [32,33]. 5. Conclusions An annular cylinder heat transfer model is proposed for analyzing heat transfer through geothermal piles. A finite difference code is developed for simultaneous solution of PDEs describing heat conduction within soil and concrete and heat flow through heat carrier fluid. Results from analyses using the proposed model confirm that heat transfer through a geothermal pile is mostly a radial phenomenon. Temperature of the heat carrier fluid decreases linearly along the length of the circulation tube and reaches to a steady state within a few days after the beginning of heat exchange operation. Based on a comparison of result obtained

Acknowledgment

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