1. Performance Sensitivity in Vertical U-tube Geothermal Energy Harvesting Systems
Sophia Novitzky, Keenan Hawekotte, Samuel Naden, Mahalia Sapp
Research Experience for Undergraduates 2014 in Industrial Mathematics and Statistics at Worcester Polytechnic Institute
Advisors: B.S. Tilley1
, S.L. Weekes1
, C. Lehrman2
, P. Ormond2
1
Center for Industrial Mathematics and Statistics, Department of Mathematical Sciences, Worcester Polytechnic Institute
2
New England Geothermal Professional Association, Haley & Aldrich, Inc.
Geothermal Systems
Figure 1: Fossil fuels are the source of 74% of the energy used for heating and cooling
buildings in the U.S.[2]
To minimize fossil fuel consumption, we
consider geothermal energy - an
eco-friendly, renewable alternative.
Residential ground source heat pump
(GSHP) systems are geothermal systems
that utilize the soil’s constant temperature
(below 15 m) to heat homes during the
winter and cool homes during the summer.
blank line
Figure 2: Soil’s constant
temperature [3]
Figure 3: Heating [4] Figure 4: Cooling [4] Figure 5: GSHP types [4]
In our research, we consider the vertical
GSHP U-tube configuration.
• Heat transfer occurs between the fluid
and grout along the pipes
• Heat transfer occurs between the grout
and soil at the borehole wall
• Thermal theft occurs when heat is
exchanged between the two pipes
• Typical borehole depths: 100 − 150m
• hello
Figure 6: Example of a U-Tube
GSHP (Image courtesy of NEGPA)
Project Goals
• Create a first principles mathematical model of heat transfer in the
U-tube system
– Analytical model
– Numerical model
• Investigate the relationship between system efficiency and depth of the
borehole
• Model the temperature effects of the borehole on the surrounding soil
• Consider the impact of input fluid temperature as a function of time
•
Assumptions
• All heat transfer between the fluid and the grout occurs above the U-
bend
• The pipes are straight, vertical, and symmetric about the center of the
borehole
• Fluid undergoes laminar flow
• Temperature profiles are symmetric about the vertical plane through the
borehole’s and the pipes’ centers
• Perfect thermal contact between the fluid and the grout
• Grout is a homogeneous solid
• Heat transfer between grout and soil is governed by Newton’s Law of
Cooling
• Perfect insulation at z = 0 and z = L of borehole
z = L
z = 0
Figure 7: U-tube GSHP design
T∗
g
T(1)∗ T(2)∗
DD
R∗
p R∗
p
R∗
b
Ts
θ
y
x
S∗
1 S∗
2
S∗
g
Figure 8: Cross-sectional view
Analytical Model
We desribe our model through the heat equations. Blue and purple terms
represent convection and conduction, respectively.
For t∗ > 0, and 0 < z∗ < L,
ρfcp,f
∂T(1)∗
∂t∗ + w∗
1(r∗, θ)
∂T(1)∗
∂z∗ = kf
2T(1)∗, where (r∗, θ) ∈ S∗
1
ρfcp,f
∂T(2)∗
∂t∗ + w∗
2(r∗, θ)
∂T(2)∗
∂z∗ = kf
2T(2)∗, where (r∗, θ) ∈ S∗
2
ρgcp,g
∂T∗
g
∂t∗ = kg
2T∗
g , where (r∗, θ) ∈ S∗
g
Nondimensionalization
We apply the following scales to nondimensionalize our equations:
r =
r∗
R∗
p
z =
z∗
L
t =
t∗
R∗2
p /αg
T∗
= Ts(1 + T)
Pe =
V0R∗2
p
αfL
= O(103
) α =
αg
αf
= O(1) k =
kg
kf
= O(1) δ =
R∗
p
L
= O(10−4
)
The terms with the δ2 factor become negligible.
α
∂T(1)
∂t
+ w1(r, θ)Pe
∂T(1)
∂z
=
1
r
∂
∂r
r
∂T(1)
∂r
+
1
r2
∂2
T(1)
∂θ2
+ δ2∂2
T(1)
∂z2
, (r, θ) ∈ S1 0 < z < 1 t > 0
α
∂T(2)
∂t
+ w2(r, θ)Pe
∂T(2)
∂z
=
1
r
∂
∂r
r
∂T(2)
∂r
+
1
r2
∂2
T(2)
∂θ2
+ δ2∂2
T(2)
∂z2
, (r, θ) ∈ S2 0 < z < 1 t > 0
∂Tg
∂t
=
1
r
∂
∂r
r
∂Tg
∂r
+
1
r2
∂2
Tg
∂θ2
+ δ2∂2
Tg
∂z2
, (r, θ) ∈ Sg 0 < z < 1 t > 0
Laplace Transform and Separation of Variables
αs ˜T(1)
+ w1Pe
∂ ˜T(1)
∂z
=
1
r
∂
∂r
r
∂ ˜T(1)
∂r
+
1
r2
∂2 ˜T(1)
∂θ2
, (r, θ) ∈ S1 0 < z < 1
αs ˜T(2)
+ w2Pe
∂ ˜T(2)
∂z
=
1
r
∂
∂r
r
∂ ˜T(2)
∂r
+
1
r2
∂2 ˜T(2)
∂θ2
, (r, θ) ∈ S2 0 < z < 1
s ˜Tg =
1
r
∂
∂r
r
∂ ˜Tg
∂r
+
1
r2
∂2 ˜Tg
∂θ2
, (r, θ) ∈ Sg 0 < z < 1
where ˜T(r, θ, z, s) = L {T(r, θ, z, t)}. Separating variables, we find



˜T(1)
˜T(2)
˜Tg


 = e−λz/Pe



ˆT(1)(r, θ, s)
ˆT(2)(r, θ, s)
ˆTg(r, θ, s)


 ,
which leads to the eigenvalue problem
αs ˆT(1) − λw1
ˆT(1) = 2 ˆT(1),
αs ˆT(2) − λw2
ˆT(2) = 2 ˆT(2),
s ˆTg = 2 ˆTg.
• λ is the temperature decay rate in the z direction
• Smallest λ corresponds to the eigenfunction that decays the slowest
Numerical Model
We use the finite element method in two dimensions to solve for the eigen-
value problem above.
To find the weak formulation, we define the piecewise constant functions
k(x, y), α(x, y), and w(x, y) to represent constants k, α, and w on the S1,
S2, and Sg regions. The function T(x, y) describes the temperature on the
entire domain:
k(x, y)α(x, y)sT(x, y) = k(x, y) 2T(x, y) + λw(x, y)T(x, y)
k(x, y) =
1, in S1, S2
k, in Sg
w(x, y) =



w1, in S1
w2, in S2
0, in Sg
α(x, y) =
α, in S1, S2
1, in Sg
T(x, y) =



T(1), in S1
T(2), in S2
Tg, in Sg
Multiplying by a continuous function φj(x, y) and integrating over the
domain:
λ Ω(wTφj)dA = Ω(k T · φj + skαTφj)dA.
Let T ≈ N
i=1 γiφi(x, y), where
φi(xj, yk) =
1 if i = j = k
0 if i = j or i = k.
Figure 9: The function φi.
Figure 10: Discretized domain - N points.
λ
N
i=1 Ω
(wφiφj)dA γi =
N
i=1 Ω
(k φi · φj + skαφiφj)dA γi
⇒ λA−→γ = B−→γ .
Numerical Results
We use COMSOL to:
• Model the original dimensional
equations
• Provide a comparison with the
egenvalue problem
Quantitative analysis:
We consider ratio of COMSOL
temperatures at (x, y, z1, s) and
(x, y, z2, s): Figure 11: COMSOL mesh
T1(x, y, z1, s)
T2(x, y, z2, s)
=
ˆT(x, y, s)e−λz1/Pe
ˆT(x, y, s)e−λz2/Pe
= e
λ
Pe(z2−z1).
We compare λ to the values found in the eigenvalue problem:
λ =
Pe ln T1
T2
z2 − z1
.
Figure 12: MATLAB result Figure 13: COMSOL result
Qualitative analysis:
In the MATLAB code, the two pipes have contrasting temperatures and
no heat is lost at the borehole wall boundary. In COMSOL, the two pipes
also have contrasting temperatures, but the grout is relatively hotter than
both of the pipes.
Acknowledgements
We gratefully acknowledge support from the National Science Founda-
tion Award DMS-1263127. We thank the New England Geothermal Pro-
fessional Association and our liaisons Paul Ormond and Chis Lehrman.
We thank Professor Sarkis and Dan Brady for the helpful discussions re-
garding this project.
References
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able Energy Reviews 12 (2008), no. 9, 2265 2300.
[2] US Department of Energy, Buildings Energy Data Book, http://buildingsdatabook.eren.doe.gov.
[3] J. Busby, M. Lewis, H. Reeves, and R. Lawley, Initial geological considerations before installing
ground source heat systems, Quart. J. Eng. Geology Hydrogeo., vol 42: 295-306 (2009).
[4] W.A. Duffield, J.H. Sass, Geothermal Energy - Clean Power From the Earths Heat, U.S. Department
of the Interior and U.S. Geological Survey.
[5] Jochen Alberty, Carsten Carstensen, and StefanA. Funken, Remarks Around 50 Lines of Matlab: Short
Finite Element Implementation, Numerical Algorithms 20 (1999), no. 2-3, 117137 (English).
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