There exist as analytical as well as numerical methods of modelling to calculate such earth heat exchangers, but usually without the heat pump circuit.

Fundamental for all models is the Fourier differential equation

dt	c • P	(4.1)
with a :	Temperature Conductivity [m2/s]
c :	Specific Heat Capacity [J/(kgK)]
T :	Temperature [K]
W :	Heat Power Density [W/m3]
A :	Laplace-Operator
P :	Media Density [kg/m3]

A useful analytical model is the geometry shape factor model described by K. J. Albers 1991 [2]. This method neglects the Fick’s diffusion and Darcy streams, uses a simplified heat transfer equation and makes a superposition of temperature fields of the undisturbed earth (nonsteady) and the disturbed earth with the concrete pipes (steady).

The temperature curvature over the period of one year is given by a cosine function, which leads in the undisturbed earth to an exponentially damped temperature dependence on the deepness.

with x : Position under the Earth Surface [m]

t : Time [s]

a : Temperature Conductivity [m2/s]

Air temperature:

Tl = Tm + (Tmal — Tm) cos(2n—-p)

t0

Tm : Annually Averaged Air Temperature [°C]

Tmax : Maximum Average Month Temperature [°C] t0 : Time Period, here one year [s]

t : Time [s]

p0 : Phase Shift

(4.3)

with

For the temperature filed of the undisturbed earth, the following simplified equation is used

d 2T d2T

V2T = — + — = 0

TOC o "1-5" h z

(4.4)

Earth temperature:

Eyndisturbed

(x, t) = Tm + (T

2-n • ( )-x•

(4.2)

dx2 dy2

which lead to an isothermal solution with a shape (geometry) factor S

Q = A — S — (Ti — T2) (4.5)

The final result is the superposition of an unsteady and a steady temperature field

T(X y, t) = TE, stat (x> У) + TEinstaAX t) (4.6)

In order to integrate inner heat sources like the heat pump pipes, Mr. Jorn Herz extended this shape factor model. For this purpose he made the assumption, with respect to an analytical solution, that the heat power density is alternatively independent or linearly dependent from the temperature.

This would be possible, because of the dependence of the enthalpy flux on the earth temperature, which is at the inlet higher than at the outlet, with respect to the higher temperature difference. For simplification, the temperature decrease between in — and outlet is considered to be linear with the pipe length.

Figure 5 shows the geometric conditions of the temperature and a comparison (upper picture) to the heating circuit in the earth with constant difference.

In a next step, the linear curvature (red) will be segmented in constant average values (green), in order to use the shape factor model mathematically in the usual way.

Figure 5: Linear Temperature Dependence between In — and Outlet of the Single Heat Circuits

Further, Mr. Herz dealt with the heating circuits a) as line shaped (one dimensional) heat sources and b) cylinder like heat sources (heat source and drain method), using the formulas

T (r, t) = —

О? 0

4 n-A-L

Ei

f — r2 ^ 4 — a-1 V У

(4.7)

a)

u

with the exponential integral Ei(-Z) = J——————— du

Z u

T (r) = To +

Q o 2-n-A-L

-ln

r r0

У r )

(4.8)

b)

The total temperature field of an earth pipe with the heat sources of the heat pump results as a superposition of all single calculated heat circuits with different segmented heat fluxes Q and the temperature field of the heat drains.

For further details and explanations and insight see the diploma thesis of Jorn Herz [1] and the dissertation thesis of J. K. Albers [2].

1. Measurements

All Measurements were done manually, because the automatic measurement system is already not completed.

Manual temperature measurements were done on

• the heated earth area

• the surfaces of the concrete pipes

• inner temperature of the concrete pipe (air flow)

with respect to the working status of the heat pump.