impure functiontemperatureDrop

Information

This function calculates the temperature drop of the soil at the outer boundary of the cylinder. The analytical formula of Hart and Couvillion (1986) for constant heat extraction is adapted to a non-constant heat flux. To adapt the formula for a variable rate of heat extraction, different constant heat extraction rates, starting at different time instances, are super-imposed. To obtain the temperature drop at the time t=n*Δt, the effects of constant rate of heat extractions are super-imposed as

ΔT ( r , t=n Δt )= 1 ⁄ ( 4 π k ) ∑ W(u(r, t= i Δt)) (q_n-i+1-q_n-i),

where r is the radius for which the temperature is computed, k is the thermal conductivity of the material, W is a solution of the heat conduction in polar coordinates and q_i=Q_i/h is the specific rate of heat extraction per unit length at time t=i Δt. The value of W is obtained using

W(u)=[-0.5772 - ln(u) + u - u²/(2   2!) +u³/(3   3!) - u⁴/(4   4!) + ....].

where u(r,t)= c ρ r² ⁄ (4 t k) , ρ is the mass density and c is the specific heat capacity per unit mass.

Implementation

The rate of heat flow Q_i is obtained from the function Buildings.Fluid.Geothermal.Boreholes.BaseClasses.exchangeValues.

References

Hart and Couvillion, (1986). Earth Coupled Heat Transfer. Publication of the National Water Well Association.

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Outputs

Revisions

• September 11, 2026, by Michael Wetter:
  Declared function as impure as it calls the impure function exchange. This avoids a warning in OpenModelica.
• July 27, 2011, by Pierre Vigouroux:
  First implementation.