.Buildings.Fluid.Geothermal.Boreholes.BaseClasses.temperatureDrop

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.

Interface

function temperatureDrop
  input ExtendableArray table "External object that contains the history terms of the heat flux";
  input Integer iSam(min = 1) "Counter for how many time the model was sampled. Defined as iSam=1 when called at t=0";
  input Modelica.Units.SI.HeatFlowRate Q_flow "Heat flow rate to be stored in the external object";
  input Modelica.Units.SI.Time samplePeriod "Period between two samples";
  input Modelica.Units.SI.Radius rExt "External radius of the cylinder";
  input Modelica.Units.SI.Height hSeg "Height of the cylinder";
  input Modelica.Units.SI.ThermalConductivity k "Thermal conductivity of the soil";
  input Modelica.Units.SI.Density d "Density of the soil";
  input Modelica.Units.SI.SpecificHeatCapacity c "Specific heat capacity of the soil";
  output Modelica.Units.SI.TemperatureDifference dT "Temperature drop of the soil";
end temperatureDrop;

Revisions

• July 27, 2011, by Pierre Vigouroux:
  First implementation.