.BuildingSystems.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.Cylindrical

Information

Model for radial heat transfer in a hollow cylinder.

If the heat capacity of the material is non-zero, then this model computes transient heat conduction, i.e., it computes a numerical approximation to the solution of the heat equation

ρ c ( ∂ T(r,t) ⁄ ∂t ) = k ( ∂² T(r,t) ⁄ ∂r² + 1 ⁄ r   ∂ T(r,t) ⁄ ∂r ),

where ρ is the mass density, c is the specific heat capacity per unit mass, T is the temperature at location r and time t and k is the heat conductivity. At the locations r=r_a and r=r_b, the temperature and heat flow rate are equal to the temperature and heat flow rate of the heat ports.

If the heat capacity of the material is set to zero, then steady-state heat flow is computed using

Q = 2 π k (T_a-T_b)⁄ ln(r_a ⁄ r_b),

where r_a is the internal radius, r_b is the external radius, T_a is the temperature at port a and T_b is the temperature at port b.

Implementation

To spatially discretize the heat equation, the construction is divided into compartments with nSta ≥ 1 state variables. The state variables are connected to each other through thermal conductors. There is also a thermal conductor between the surfaces and the outermost state variables. Thus, to obtain the surface temperature, use port_a.T (or port_b.T) and not the variable T[1].

Revisions

• January, 2014, by Damien Picard:
  Modify the discretization of the cilindrical layer so that the first three layers have an equal thickness the following an exponentionally growing thickness. This follows the guidelines of Eskilson (P. Eskilson. Thermal analysis of heat extraction boreholes. PhD thesis, Dep. of Mathematical Physics, University of Lund, Sweden, 1987).
• March 9, 2012, by Michael Wetter:
  Removed protected variable der_T as it is not required.
• April 14 2011, by Pierre Vigouroux:
  First implementation.