Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2010, Article ID 784794, 13 pages
doi:10.1155/2010/784794
Research Article
Calculation of Some Integrals Arising in
Heat Transfer in Geothermics
J. L. G. Santander,1 P. Castañeda Porras,2 J. M. Isidro,3
and P. Fernández de Córdoba3
1
Departamento de Ciencias Experimentales y Matemáticas, Universidad Católica de Valencia,
46002 Valencia, Spain
2
Departamento de Matemáticas, Universidad de Pinar del Rı́o, 20200 Pinar del Rı́o, Cuba
3
Instituto Universitario de Matemática Pura y Aplicada, Universidad Politécnica de Valencia,
46022 Valencia, Spain
Correspondence should be addressed to J. L. G. Santander, martinez.gonzalez@ucv.es
Received 28 September 2010; Accepted 29 November 2010
Academic Editor: Miguel Cerrolaza
Copyright q 2010 J. L. G. Santander et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We calculate some integrals involved in the temperature field evaluation of the ground, heated
by a borehole heat exchanger. This calculation allows a faster computation of that component of
the temperature field which involves the periodic oscillation of the ambient temperature or the
ambient heat flux.
1. Introduction
Ground-coupled heat pumps GCHPs are an attractive choice of system for heating
and cooling buildings 1. By comparison with standard technologies, these pumps offer
competitive levels of comfort, reduced noise levels, lower greenhouse gas emissions, and
reasonable environmental safety. Furthermore, their electrical consumption and maintenance
requirements are lower than those required by conventional systems and, consequently, have
lower annual operating cost 2.
A GCHP exchanges heat with the ground through a buried U tube loop. Since this
exchange strongly depends on the thermal properties of the ground, it is very important
to have a knowledge of these properties when designing GCHP air-conditioning systems.
Methods to estimate them include literature searches, conducting laboratory experiments on
soil/rock samples, and/or performing field tests. The in situ tests are based on studying the
thermal response of the borehole heat exchanger to a constant heat injection or extraction. The
outputs of the thermal response test are the inlet and outlet temperature of the heat-carrier
2
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fluid as a function of time. From these experimental data, and with an appropriate model
describing the heat transfer between the fluid and the ground, the thermal conductivity of
the surroundings is inferred.
The results presented in this study are based on a model of the underground heat
transfer due to a borehole heat exchanger assuming that it works as a line heat source of
finite depth inside the ground. We assume as well a constant power of length unit for the
heat source considered and a periodic oscillation of the ambient temperature or heat flux on
the ground surface.
This paper is organized as follows. Section 2 describes the equations that model the
heat transfer inside the ground due to a borehole heat exchanger. Section 3 solves the integrals
given in literature for periodic oscillations of the ambient temperature or heat flux, on the
ground surface. Section 4 shows that our results agree with the literature in some particular
cases, such as the solution on the ground surface and the solution of the quasistationary
regime.
2. The Equations
Let us consider the heat equation in cylindrical coordinates, in which Tr, z, t is the
temperature field,
Qz
∂T
− k∇2 T δrHz − Hz − L.
∂t
C
2.1
On the one hand, the thermal parameters of 2.1 are the ground thermal diffusivity km2 s−1 ,
the heat flow per length unit Qz Wm−1 , and the volumetric heat capacity of ground
CJm−3 K −1 . On the other hand, the geometry of the heat source, depicted in Figure 1, has
been taken into account introducing the Heaviside function H for its finite length L and the
Dirac delta distribution δ for its infinitely thin width. Equation 2.1 is subject to to the initial
condition
Tr, z, 0 T0 kgeo z,
z ≥ 0,
2.2
where T0 is the undisturbed ground temperature and kgeo Km−1 is the geothermal gradient.
We may consider that 2.1 is subject to a Dirichlet boundary condition, which represents a
time-dependent temperature on the surface z 0,
Tr, 0, t ψt,
t ≥ 0,
2.3
or to a Neumann boundary condition, which represents a time-dependent heat flux on the
surface z 0,
−λ∂z Tr, 0, t ψt,
t ≥ 0,
2.4
Mathematical Problems in Engineering
3
δr
z0
L
Qz
Figure 1: Borehole heat exchanger model.
where λWm−1 K −1 is the ground thermal conductivity. We split the Cauchy problem posed
in 2.1, 2.2, and 2.3 or 2.4 in three terms 3, so that
Tr, z, t vd r, z, t v0 z, t vs z, t,
2.5
and where the functions v0 , vs and vd satisfy the Cauchy problems presented below.
2.1. The Problem for vd
The partial differential equation that vd r, z, t satisfies is
∂vd
Qz
− k∇2 vd δrHz − Hz − L,
∂t
C
2.6
subject to a homogeneous initial condition,
vd r, z, 0 0,
z ≥ 0,
2.7
and to a homogeneous Dirichlet boundary condition,
vd r, 0, t 0,
t ≥ 0,
2.8
or to a Neumann boundary condition, homogenous as well,
∂z vd r, 0, t 0,
t ≥ 0.
2.9
4
Mathematical Problems in Engineering
The solution to 2.6, 2.7, and 2.8 is 4
vd r, z, t e−u
Qz ∞
du
8πλ r 2 /4kt
u
L − z√
L z√
z√
u erf
u − erf
u ,
× 2 erf
r
r
r
2.10
and the solution to 2.6, 2.7, and 2.9 is
vd r, z, t Qz
8πλ
∞
r 2 /4kt
z − L√
z L√
e−u
u − erf
u du,
erf
u
r
r
2.11
that we will present in a future work.
2.2. The Problem for v0
The partial differential equation that v0 z, t satisfies is
∂v0
∂2 v0
k 2 ,
∂t
∂z
2.12
subject to the initial condition
v0 z, 0 T0 kgeo z,
z ≥ 0,
2.13
and to a Dirichlet boundary condition,
v0 0, t 0,
t ≥ 0,
2.14
or to a Neumann homogeneous boundary condition
∂z v0 0, t 0,
t ≥ 0.
2.15
The solution to 2.12, 2.13, and 2.14 is 5, Equation 2.4.13,
z
v0 z, t T0 erf √
kgeo z,
2 kt
2.16
and the solution to 2.12, 2.13, and 2.15 is 5, Equation 2.9.7
v0 z, t T0 kgeo
z
z erf √
2 kt
√
2 kt
z2
,
√ exp −
4kt
π
2.17
Mathematical Problems in Engineering
5
where the error function is defined as
z
2
erfz : √
π
e−u du.
2
2.18
0
2.3. The Problem for vs
The partial differential equation that vs z, t satisfies is
∂vs
∂2 vs
k 2 ,
∂t
∂z
2.19
subject to an homogeneous initial condition
vs z, 0 0,
z ≥ 0,
2.20
and to the time-dependent Dirichlet boundary condition
vs 0, t ψt,
t ≥ 0,
2.21
or to the time-dependent Neumann boundary condition
−λ∂z vs 0, t ψt,
t ≥ 0.
2.22
The solution to 2.19, 2.20 and 2.21 is 5, Equation 2.5.1.
2
vs z, t √
π
∞
z2
√ ψ t−
4ku2
z/2 kt
e−u du,
2
2.23
and the solution to 2.19, 2.20 and 2.22 is 5, Equation 2.9.9
√ t
ψt − u −z2 /4ku
k
vs z, t √
du.
e
√
λ π 0
u
2.24
3. Harmonic Analysis of the Boundary Condition
We may consider that on the ground surface we have a periodic temperature or heat flux
oscillation, due to daily or annual cycles. Therefore, ψt is a periodic function of a certain
period τ, continuous and bounded, that we may expand in Fourier series
ψt ∞
a0 an cosnωt bn sinnωt,
2 n1
3.1
6
Mathematical Problems in Engineering
where the frequency is ω 2π/τ, and the coefficients an and bn are given by
2
a0 τ
an 2
τ
bn 2
τ
τ/2
−τ/2
ψudu,
τ/2
−τ/2
ψu cosnωudu,
3.2
τ/2
−τ/2
ψu sinnωudu,
where n ∈ Æ .
3.1. Dirichlet Boundary Condition
Substituting 3.1 in 2.23, we obtain
vs z, t z
a0
erfc √
2
2 kt
∞
∞
2 nωz2
2
√
an
e−u du
√ cos nωt −
2
4ku
π n1
z/2 kt
∞
2
nωz2
bn
e−u du,
√ sin nωt −
2
4ku
z/2 kt
3.3
where erfcz 1 − erfz is the complementary error function. The integrals given in 3.3
for t ≥ 0 may be rewritten as
nωz2
−u2
inωt
,
e
cos
nωt
−
du
Re
e
I
t
z,
D,n
√
4ku2
z/2 kt
∞
nωz2
2
e−u du Im einωt ID,n z, t ,
√ sin nωt −
2
4ku
z/2 kt
∞
3.4
where we have defined
inωz2
2
ID,n z, t :
− u du.
√ exp −
4ku2
z/2 kt
∞
3.5
In order to calculate 3.5, let us apply the integral 6, Equation 7.4.33
√ √ √ √
√
√
√
π 2 ab
b
b
b
2
−2 ab
exp −ax − 2 dx √ e
e
const,
erf
ax erf
ax −
x
x
x
4 a
3.6
Mathematical Problems in Engineering
7
taking a 1, b inωz2 /4k, and z ≥ 0, so that
√
√ √
π
z
z inω
ID,n z, t erfc √ inωt
exp
√
4
k
2 kt
√
√
z
−z inω
erfc √ − inωt .
exp
√
k
2 kt
3.7
Finally, substituting 3.7 in 3.4, we obtain for z, t ≥ 0,
√
nωz2
π
−u2
inωt −
Re
e
du
e
A
t
A
t
,
z,
z,
n
n
√ cos nωt −
4
4ku2
z/2 kt
√
∞
π
nωz2
−u2
inωt
−
e
Im
e
sin
nωt
−
du
A
t
A
t
,
z,
z,
n
n
√
4
4ku2
z/2 kt
∞
3.8
where we have defined
√
√
z
z inω
erfc √ ± inωt .
: exp ± √
k
2 kt
A±n z, t
3.9
As far as we know, the integrals given in 3.8 are not reported in the literature.
3.1.1. Neumann Boundary Condition
Substituting 3.1 in 2.24, we obtain
√ 2
a0 k t e−z /4ku
vs z, t √
du
√
2λ π 0
u
√ ∞ t
k cosnωt − u −z2 /4ku
e
an
du
√
√
λ π n1
u
0
bn
t
0
sinnωt − u −z2 /4ku
e
du.
√
u
3.10
3.11
3.12
√
Integrating by parts and performing the substitution ζ z/2 ku, considering z, t ≥ 0, we
may calculate the integral given in 3.10
t
0
√
e−z /4ku
du 2 t exp
√
u
2
−z2
4kt
√
πz
z
− √ erfc √
.
k
2 kt
3.13
8
Mathematical Problems in Engineering
In order to calculate the integrals given in 3.11 and 3.12, let us perform the substitution
u v2 , considering t ≥ 0, so that
t
0
cosnωt − u −z2 /4ku
e
du 2
√
u
√t
2
2
cos nω t − v2 e−z /4kv dv
0
2 Re einωt IN,n z, t ,
3.14
and similarly
t
0
sinnωt − u −z2 /4ku
du 2 Im einωt IN,n z, t ,
e
√
u
3.15
where we have defined
IN,n z, t :
√t
z2
exp −inωv −
4kv2
2
0
dv.
3.16
Taking a inω, b z2 /4k, and z ≥ 0 in 3.6, we may rewrite 3.16 as
√
√
√
z
π
−z inω
IN,n z, t √
erfc √ − inωt
exp
√
4 inω
k
2 kt
√
√
z inω
z
− exp
erfc √ inωt .
√
k
2 kt
3.17
Finally, substituting 3.17 in 3.14 and 3.15, for z, t ≥ 0, we obtain
√
einωt −
cosnωt − u −z2 /4ku
π
Re √
e
An z, t − An z, t ,
du √
2
u
inω
0
t
√
einωt −
sinnωt − u −z2 /4ku
π
Im √
du e
An z, t − An z, t .
√
2
u
inω
0
t
3.18
As far as we know, the integrals given in 3.18 are not reported in the literature.
4. Particular Cases
As a consistency check, let us verify that in certain particular cases the new integrals given in
3.8 and 3.18 are reduced to integrals reported in the literature. These particular cases have
to do with the quasistationary regime and the solution on the ground surface.
Mathematical Problems in Engineering
9
4.1. Quasistationary Regime
Since there is a periodic oscillation on the ground surface, the surroundings of the borehole
heat exchanger never reach a stationary regime. However, according to 5, Section 2.6, we
may define a quasistationary regime in which the ground temperature field is stabilized
periodically. In this quasistationary regime, 3.8 becomes
√
π
nωz2
−u2
inωt
,
e
Re
e
cos nωt −
du
A
z
D,n
4
4ku2
0
∞
√
π
nωz2
−u2
inωt
,
Im
e
sin nωt −
du
A
e
z
D,n
4
4ku2
0
∞
4.1
4.2
where we have defined
AD,n z : lim An z, t A−n z, t .
t→∞
4.3
Similarly, 3.18 becomes
∞
0
∞
0
√
einωt
cosnωt − u −z2 /4ku
π
Re √
e
AN,n z ,
du √
2
u
inω
√
einωt
sinnωt − u −z2 /4ku
π
Im √
du e
AN,n z ,
√
2
u
inω
4.4
where we have defined
AN,n z : lim A−n z, t − An z, t .
t→∞
4.5
4.1.1. Dirichlet Case
Taking into account that erfc∞ 0 and erfc−∞ 2, we may calculate the limit given in
4.3, so that 4.1 becomes
∞
0
nωz2
cos nωt −
4ku2
where we have substitute
∞
0
e
−u2
√
√
π
−z inω
inωt
Re e
du exp
√
2
k
√
π −z√nω/2k inωt−z√nω/2k e
,
Re e
2
4.6
√
√
i 1 i/ 2. Thus, for z, t ≥ 0,
nωz2
cos nωt −
4ku2
e−u du 2
√
√ π −z√nω/2k
z nω
e
cos nωt − √
.
2
2k
4.7
10
Mathematical Problems in Engineering
Similarly, we may obtain
∞
0
nωz2
sin nωt −
4ku2
e−u du √
2
√ π −z√nω/2k
z nω
e
.
sin nωt − √
2
2k
4.8
Notice that 4.7 and 4.8 are sinusoidal in t, with an amplitude exponentially decreasing in
depth z. In the literature, we may find 7, Equation 3.928.1-2.
∞
e
−p2 x2 −q2 /x2
0
b2
sin a x 2
x
dx 2 2
√
π −2rs cosAB
e
2r
× sinA 2rs sinA B,
∞
e−p
x −q /x
2 2
2
0
2
b2
cos a2 x2 2
x
√
π −2rs cosAB
dx e
2r
4.9
× cosA 2rs sinA B,
where a2 p2 > 0 and
4
a4 p4 ,
a2
1
−1
,
A tan
2
p2
r
4
b4 q4 ,
1
b2
−1
B tan
,
2
q2
s
4.10
thus, taking p 1, q 0, and a 0,
√
√ π −√2b
dx e
e sin
sin 2b ,
2
0
∞
√
√ b2
π −√2b
2
e
e−x cos
cos 2b .
dx
2
2
x
0
∞
−x2
b2
x2
4.11
Therefore, rewriting the left side of 4.7 as
∞
0
nωz2
cos nωt −
4ku2
e
−u2
nωz2
2
du cosnωt
cos
e−u du
2
4ku
0
∞
nωz2
2
e−u du,
sinnωt
sin
2
4ku
0
∞
4.12
√
√
and applying 4.11, taking b nωz/2 k, z ≥ 0, we eventually get the same result as
4.7. Similarly, we may check the result given in 4.8.
Mathematical Problems in Engineering
11
4.1.2. Neumann Case
In a similar way as in the previous subsection, we may calculate 4.4 for z, t ≥ 0, arriving at
∞
0
∞
0
√
√
πe−z nω/2k
z nω π
,
−
cos nωt − √
√
4
nω
2k
√
√ −z√nω/2k
z nω π
sinnωt − u −z2 /4ku
πe
e
sin nωt − √
.
du −
√
√
4
u
nω
2k
cosnωt − u −z2 /4ku
e
du √
u
√
4.13
4.14
Once again, 4.13 and 4.7 are sinusoidal in t, with an amplitude exponentially decreasing
in depth z. In the literature, we may find 7, Equation 3.957.1-2
∞
μ √ √ β
x e
cos ax dx e−iπμ/4 Kμ βeiπ/4 a eiπμ/4 Kμ βe−iπ/4 a ,
√
2 a
0
4.15
∞
μ √ √ β
μ−1 −β2 /4x
−iπμ/4
iπ/4
iπμ/4
−iπ/4
e
x e
sin ax dx i √
Kμ βe
a −e
Kμ βe
a ,
2 a
0
μ−1 −β2 /4x
√
where Re β > 0, Re μ < 1 and a > 0. Therefore, taking in 4.15, μ 1/2, a nω, β z/ k,
z ≥ 0, and knowing that the Macdonald function of order 1/2 is 8, Equation 5.5.5,
K1/2 z π −z
e ,
2z
4.16
we eventually get the same results as 4.13 and 4.14.
4.2. Solution on the Ground Surface
Notice that in the special case z 0, 3.8 becomes trivial,
∞
√
π
cosnωt,
2
0
∞
√
2
π
sinnωt,
sinnωte−u du 2
0
4.17
√
√
einωt
cosnωt − u
du π Re √
erf inωt ,
√
u
inω
0
t
√
√
sinnωt − u
einωt
du π Im √
erf inωt .
√
u
inω
0
4.18
cosnωte
−u2
du while 3.18 are reduced to
t
12
Mathematical Problems in Engineering
Since 8, Equation 2.4.1
erf
√ ⎞
⎛
⎞⎤
⎡ ⎛
ix
√
2
2
x⎠ − iS⎝
x⎠⎦,
2⎣C⎝
√
π
π
i
x ∈ Ê,
4.19
where
z
Cz :
cos
0
z
Sz :
0
πt2
dt,
2
4.20
πt2
dt
sin
2
are the Fresnel integrals; then 4.18 may be expressed as
t
0
t
0
cosnωt − u
du √
u
sinnωt − u
du √
u
⎛
⎡
⎞
⎛
⎞⎤
2π ⎣
2nωt
2nωt
⎠ sinnωtS⎝
⎠⎦,
cosnωtC⎝
nω
π
π
⎛
⎡
2π ⎣
sinnωtC⎝
nω
⎞
⎛
2nωt ⎠
− cosnωtS⎝
π
⎞⎤
4.21
2nωt ⎠⎦
.
π
The results given in 4.21, agrees with 7, Equation 2.653.1-2
√
!√ "
sin x
√ dx 2π S x ,
x
√
!√ "
cos x
√ dx 2π C x .
x
4.22
Acknowledgments
The authors wish to thank the financial support received from Generalitat Valenciana under
Grant no. GVA 3012/2009 and from Universidad Politécnica de Valencia under Grant no.
PAID-06-09. This work has been partially supported by the Structural Funds of the European
Regional Development Fund ERDF.
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Mathematical Problems in Engineering
13
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