Project report
MVK160 Heat and Mass Transfer
May 08, Lund, Sweden
Heat transfer in Ground Heat Exchangers (GHEs)
Audrey Chesneau
Dept. of Energy Sciences, Faculty of Engineering,
Lund University, Box 118, 22100 Lund Sweden
ABSTRACT
The report describes heat transfer process
within geothermal heat exchangers. An
analytical model, which couples a model
outside the borehole with another one inside,
is presented.
The aim of this study is to understand how
heat transfer works inside ground-coupled
heat exchangers and to show how difficult it is
to size a GHE due to the numbers of
parameters which are involved in the heat
transfer process.
The report will only focus on one type of
analytical model, which is the infinite source
line model. This model is applied to only one
type of GSE, the vertical ground-source
exchangers. This report also discusses the
limitations of this model.
NOMENCLATURE
rb
borehole radius (m)
Rb
borehole thermal resistance (m.K/m²)
Rf
convective resistance of the fluid in
one pipe (m.K/m²)
Rp
conductive resistance (m.K/m²)
Rg
conductive resistance of the grout
(m.K/m²)
To
initial temperature (°C)
r
radius (m)
Tg
ground temperature at radius r (°C)
Tf
fluid temperature (°C)
Tb
borehole temperature (°C)
Q
heat flux per unit length (W/m)
k
thermal conductivity (W/mK)
α
thermal ground diffusivity (m²/s)
t
time (s)
γ
Euler constant
INTRODUCTION
It has become globally important to change
the way we produce energy and to reduce
emissions of carbon dioxide and the other
greenhouse gases. That is why sustainable
energies have kept improving.
In 2010, 0.4% of annual global needs was
covered by geothermal energy but now using
geothermal energy has kept becoming very
interesting and it takes part in an economical
strategy of a sustainable development (IEA).
Indeed, geothermal energy could account for
around 3.5% of annual global electricity by
2050 (IEA).
Ground heat exchangers (GHEs) are mainly
used in ground-coupled heat pumps (GCHPs)
and these geothermal heat pumps are the
most efficient heating and cooling technology
since they use 25% to 50% less electricity than
other heating and cooling systems (Iona
Sarbu, 2014).
However, a proper design of the borehole is
essential to ensure a good efficiency and a key
prerequisite for this optimization is to analyse
the temperature of the circulating fluid inside
the borehole.
A coupled-ground heat exchanger (GHE) is a
heat exchanger situated in the ground in
which a heat carrier fluid circulates into Utubes to extract or/and reject heat from/to
the ground. They are composed of HDPE
(high-density polyethylene which is efficient
for heat transfer) pipes either in vertical or in
horizontal boreholes. Their design is critical to
the
performance
of
ground-coupled
geothermal pumps. That is why, it is very
important to analyse the GSEs.
PROBLEM STATEMENT:
Vertical ground-source exchangers are the
commonest form of GHEs used for GCHP
systems. They are also known as Borehole
Heat Exchangers (BHEs). The BHEs are usually
used for large buildings with limited area since
they require a smaller area in comparison with
horizontal trenches. The BHEs also provide an
ability to reduce the influence of the ground
temperature variability on its performance.
However, they still remain more expensive
than other heat exchangers.
Throughout this report, a vertical ground heat
exchanger with a fluid ascending in one pipe
and descending in the other is considered.
Figure 1 and Figure 2 show the system.
Figure 2: Single U-tube from a BHE
There are three components in these systems
to consider. Firstly, the convection between
the fluid and the inner pipe wall, then the
conduction through the pipe and grout and
finally the conduction through the ground.
In this report, heat transfer will be analysed in
the aim of determining the temperature of the
heat carried fluid. This kind of studies are very
important in sizing the exchangers for
instance. For that, a lot of models already
exist.
Figure 1: Cross section of a BHE
LITERATURE SURVEYS
Many models have been developed to do a
thermal analyse of BHEs; there are analytical
and numerical models. But in this report, only
analytical models will be studied. The earliest
approach came from Kelvin in 1882: it
provided the ability to calculate the
temperature in the ground due to a constant
heat rate. His approach was reiterated and
improved in the infinite line source model
(Ingersoll and Plass 1948) and in the cylindrical
heat source model (Carslaw and Jaeger 1947,
Hellstrom 1991, Bernier 2001). Other
analytical solutions (Hellstrom 1991, Sutton,
et al. 2002) led to sizing the vertical GHEs.
PROJECT DESCRIPTION
The heat transfer is complicated in these
systems, thus it will be studied in two
separates regions: the first one being inside
the borehole and the second one outside it.
Finally, both analyses could be coupled with
the temperature wall borehole.
Heat transfer outside borehole:
The heat transfer process is typically a threedimensional transient process but in view to
the thickness of the borehole, the axial effect
can be neglected. And as the model considers
one infinitely long pipe, T only depends on the
radius r and on the time: T(r,t). The line source
model which was developed based on Kelvin’s
line source theory (1882) will be used to
obtain the temperature of the ground
surrounding the borehole. The model ignores
the thermal properties of the borehole
(thermal mass of the fluid, pipe and grout) and
considers a constant heat flux per unit depth
at zero radius.
The system is represented by these equations,
where the subscripts 1, 2 represent grout, and
soil since the radial heat equation must be
satisfied in both the grout and the ground
regions.
𝝏²𝑻𝒊 𝟏 𝝏𝑻𝒊 𝟏 𝝏𝑻𝒊
+
−
=𝟎
𝝏𝒓² 𝒓 𝝏𝒓 𝜶𝒊 𝝏𝒕
The region 1 is for 𝑟𝑝 < 𝑟 < 𝑟𝑏 and the region
2 is for 𝑟𝑏 < 𝑟.
For both regions, the heat flux can be defined
by:
−𝒌𝒊
𝝏𝑻
(𝟐𝝅𝒓) = 𝑸(𝒓, 𝒕)
𝝏𝒓
With the boundary conditions below:
•𝑻(𝒓, 𝟎) = 𝑻𝒐
•As the heat flux at the ground/soil interface
is continuous:
−𝒌𝒈
𝝏𝑻
𝝏𝑻
= −𝒌𝒔
𝝏𝒓 |𝒓=𝒓𝒃−𝜺
𝝏𝒓 |𝒓=𝒓𝒃+𝜺
From now, the heat is assumed to be rejected
into the ground so the heat flux is positive.
Carslaw and Jaeger (1947) used the
exponential integral E1 to approximate the
solution of the Kelvin source model and they
obtained this line source equation below:
∞
𝑸
𝒆−𝒖
𝑸
𝒓𝟐
𝑻 − 𝑻𝒐 =
∫
𝒅𝒖 =
𝑬𝟏(
)
𝟒𝒌𝝅
𝒖
𝟒𝒌𝝅
𝟒𝜶𝒕
𝒓𝟐
𝟒𝜶𝒕
The exponential integral can be approximate
by:
𝒓𝟐
𝟒𝜶𝒕
𝑬𝟏 (
) ≅ 𝒍𝒏 𝟐 − 𝜸
𝟒𝜶𝒕
𝒓
Where γ is the Euler constant (around
0.5772…).
Consequently, the distribution of
temperature outside the borehole is:
𝑻 = 𝑻𝒐 +
the
𝑸
𝟒𝜶𝒕
(𝒍𝒏 𝟐 − 𝜸)
𝟒𝒌𝝅
𝒓
And in particular, the temperature at the
borehole wall is:
𝑻𝒃 = 𝑻𝒐 +
𝑸
𝟒𝜶𝒕
(𝒍𝒏 𝟐 − 𝜸)
𝟒𝒌𝝅
𝒓𝒃
Now the temperature at the borehole is
determined and could be used to obtain the
thermal resistance for instance.
This model uses many approximations so it
has some limitations. For instance, it ignores
the geometry of the borehole as well as the
thermal properties within the borehole.
Consequently, it is widely used in analytical
design methods; however, it is not suitable for
short timescale applications.
Heat transfer inside the borehole:
Heat transfer inside the borehole is
characterized by the heat transfer resistance.
The purpose of this analyse is to obtain the
temperature of the circulating fluid in the
borehole owing to the borehole wall
temperature, the heat flow and the thermal
resistance.
Within the borehole, heat transfer is
considered as a steady-state process and a
constant heat flux is assumed.
Once again, Carslaw and Jaeger had been
based on the Kelvin’s line theory and they had
found a relationship between the mean fluid
temperature and the borehole temperature.
𝐓𝐟 = 𝐓𝐛 + 𝐐. 𝐑𝐛
↔ 𝐓𝐟 = 𝐓𝐨 + 𝐐[𝐑𝐛 +
𝟏
𝟒𝛂𝐭
(𝐥𝐧 𝟐 − 𝛄)]
𝟒𝐤𝛑
𝐫𝐛
With an electrical analogy,
resistance can be defined by:
𝑹𝒃 =
a
thermal
𝑻𝒇 − 𝑻𝒃
𝑸
A steady state borehole model is rightly to
calculate the borehole thermal resistance. In
view to the three components to consider for
the system, the borehole thermal resistance
can be defined by:
𝑹𝒃 =
𝑹𝒇 − 𝑹𝒑
+ 𝑹𝒈
𝟐
Once again, a lot of models exist to express
every resistance but the report do not deal
with it.
CONCLUSION
As a conclusion, the report only proposed a
basic analyse of heat transfer for BHEs which
is simplified as a one-dimensional model.
However, this simple model is often used for a
quick design of ground-coupled heat pumps
and many numerical models are based on it.
Moreover, analysing heat transfer process can
also be more difficult if the borehole heat
exchanger is composed of more than one
tube.
Other crucial parameters are necessary to
design a proper borehole, such as the
borehole resistance, but other models are
required
to
find
them.
Finally, this kind of analysis is often studied in
order to improve the efficiency and to reduce
the cost of taking part in a sustainable
challenge, which is led by ground-coupled
heat pumps.
REFERENCES
[1]
Saqib Javed, 2012, “thermal modelling
and evaluation of borehole heat transfer”,
thesis, journal article
[2]
U.S Department of Energy, 2012,
“geothermal heat pumps”, article on the
Internet
http://energy.gov/energysaver/articles/geoth
ermal-heat-pumps
[3]
U.S Department of Energy , 2012,
“geothermal heat pumps”, article on the
Internet
http://www.wbdg.org/resources/geothermalh
eatpumps.php
[4]
Ioan Sarbu, Calin Sebarchievici, 2013,
“general review of ground-source heat pump
systems for heating and cooling of buildings”,
journal article
[5]
Miaomiao He, 2012, “Numerical
modelling of geothermal borehole heat
exchanger systems”, thesis, journal article
[6]
International
Energy
Agency
http://www.iea.org/ articles on the Internet