Claesson and Eskilson's theoretical steady-state solution
Early phase of heat abstraction
Swedes Johan Claesson and Per Eskilson (1987ab), via numerical modelling, derived a curve matching Figure 1b, and developed principle equations that govern the radial conduction of heat towards a line “sink” in the earth.
The change of average temperature of the carrier fluid (a) with time and (b) with log10 (time) (Borehole diameter, length=126 mm,100 m; heat extraction rate=2 k; Rb= 0.12 KmW−1; ground thermal conductivity 2.48 WmK−1, SVC= 2.4 MJm−3 K−1; ground temperature=11◦C)
The equation, in radial coordinates (r), is:
with the assumptions that they made to simplify the equation:
- Ground temperature is uniform, neglecting the geothermal gradient
- Heat flow other than from the ground is negligible
The evolution of the ‘thermal front' around a close-loop borehole (a) The contours represent the loci of a 1◦C temperature drop (heat extraction rate=22Wm−1; thermal conductivity=3.5Wm−1 K−1; diffusivity=1.62×10−6 m2 s−1; arrows=lines of flux after operation years; (b) The contours represent ground temperature around a BHE in Germany, after 7 months of heat operation.
The boundary conditions are as follow:
- at t = 0, θ = θofor all values of r and z,
- as r → ∞, θ = θofor all values of t, giving
E(u) is a Theis-type polynomial expression
u = (rb2SVC)/(4λt); θbis the average temperature of the antifreeze solution in the borehole at time t. θb is assumed to be similar to the temperature at the wall of the borehole - at a radius rbfrom the centre of the line heat sink.
θo− θb = temperature ‘drawdown' or displacement (K)
q = rate of heat extraction per metre of effective borehole depth (Wm−1);
λ = thermal conductivity of the rock (Wm−1 K−1)
rb= borehole radius (m)
γ = Euler's constant = 0.5772
Using the Cooper-Jacob expression, , the equation can be simplified for low values of u (high values of t) giving
This approximation, implies that the temperature of the carrier fluid decreases with the logarithm of time, is only valid for values of t within the range of
The mathematical assumption underlying the approximation breaks down when t is too small. If t is beyond the upper constraint, the log-linear relationship between time and θbbegins to diverge slightly from the real curve (Figure 1) as the system starts to induce heat flow from the ground surface to gradually reach a steady state which means the assumption 2 is not obeyed or valid anymore.
Steady state phase of heat extraction
The time ts, after which ‘steady state' begins to be a better description of the temperature evolution, is given by Claesson and Eskilson (1987a) as
where Euler's constant (γ )= 0.5772, and D = borehole depth (m).
As the real temperature curve converges, the carrier fluid's steady-state temperature is given by
where θs,b= carrier fluid's steady-state temperature in the ideal borehole (K),which is also the steady-state temperature of the borehole walls.
Analytical Computer Modelling Simulation
Computer programs such as Earth Energy Designer (EED) and Ground Loop Heat Exchanger Profesisonal (GLHEPro) are usually used for designing large ground source heating and cooling scheme with complex borehole array geometry. These programs are capable of simulating the development of temperature of the carrier fluid and and both heating and cooling effect. In this research, EED is employed to study how the key properties such as the ground, borehole, heat exchanger and carrier fluid affect the energy balance. This is further discussed in Chapter ?? .
Long term ground temperature change
heat from an aestifer, showing typical boundary conditions. θo = initial ‘far-field' temperature.
Ground temperature changes gradually resulting from long-term net extraction from or rejection of heat into the ground and will have detrimental effect on the performance and efficiencies of the entire system. The ground may not begin to recover its steady state temperature until after 30 years or more have elapsed. The solar and atmospheric energy accounts for the energy balance by contributing hundreds of Watts. Heat conduction from surrounding areas and groundwater flow play a role in heat balance restoration too, aided by geothermal heat flux (<0.4 W/m2).
However, the more systems there are operating in close proximity, the higher the risk of thermal interference and thermal overload of the ground. For instance, long-term ‘below ground global warming' is happening in the Stockton College of New Jersey due to overwhelming cooling load in comparison to heating load. Temperatures can boost up to 90°C from heat rejection into the ground and rigorous consideration of materials used in the system is necessary. These phenomena can be avoided if annual totals of heating energy and cooling energy are balanced. But it is unlikely that the annual thermal loads would exactly balance out each other. The effects can be reduced by providing both heating and cooling, to reach approximate annual balance between summer and winter by conscientious system control. This concept is known as Underground Thermal Energy Storage (UTES).
Underground Thermal Energy Storage (UTES)
UTES uses the concept that, during the annual energy cycle, heat energy is stored in during summer when cooling demand is higher and then recovered from the ground during winter for heating purposes. When the system is operated in cooling mode, more heat is deposited into the ground than extracted in the winter causing ground temperature to steadily creep up with time. By having integrated conventional heating or cooling to absorb any excess heating or cooling load, the energy or heat balance can be achieved to reach the steady-state condition of the carrier fluid's temperature, hence providing a sound UTES scheme.
Factors that influence the pace of the carrier fluid's temperature to reach steady-state condition:
- The larger the site surface area, the more heat flow induced from the surface, the quicker the temperature reaches a dynamic steady-state condition
- Groundwater flow through the borehole array will be beneficial for replenishing or transporting heat away from the site.
- Borehole spacing (of at least 10m) is important in prevention of thermal interference.
- A less closely-packed borehole array will allow better heat exchange with the ground.
However, a system that utilises UTES scheme need not consider the factors mention above; as a matter of fact, they need to be inversed to optimise the efficiency of the entire system. This is because temperatures in this type of system reach the steady-state condition relatively fast varying over the annual cycle. Moreover, since UTES incorporates heat storage in the ground, leakage of heat is unprofitable and thus a more tightly-spaced borehole array is preferred. Having large groundwater flows and large surface area will only cause more and quicker heat loss from the subsurface store, hence an equidimensional hexagonal or cylindrical borehole (smaller area to volume ratio) and lower hydraulic conductivity or gradient are more suited to heat storage. UTES's heat storage scheme weighs down the importance of the risk of thermal interference between boreholes and closer spacing between boreholes will not have any detrimental effect on the system.
Further solutions to long term change in ground temperature:
- diversifying GSHP-powered building profiles in a local area i.e. commercial, domestic etc.
- when cooling load exceeds heating load, adding pre-heating of domestic hot water (back-up plant) will reduce the imbalances in demand
- cooling of the ground when the external ambient temperature is lower using dry coolers
- waste heat recharge of the ground when heating demand is imbalanced