modelThermalResistorRadial_withDispersion

model of thermal resistor can consider dispersion

Diagram of ThermalResistorRadial_withDispersion

Extends from Modelica.Thermal.HeatTransfer.Interfaces.Element1D (Partial heat transfer element with two HeatPort connectors that does not store energy).

Information

1. Purpose of model

Quasi-static radial thermal resistor for heat conduction in cylindrical geometry between two neighbouring aquifer control volumes. Optionally adds hydrodynamic dispersion to the effective thermal conductivity based on the local radial Darcy velocity.

2. Level of detail, physical effects considered, and physical insight

L1: Quasi-static 1D heat transport in radial direction. No thermal storage within the resistor. Thermal resistance is computed from geometry and effective conductivity. Dispersion increases effective conductivity proportionally to the magnitude of the Darcy flux.

3. Limits of validity

- No thermal mass inside the resistor (instantaneous conduction).

- Dispersion term assumes a linear relationship between Darcy velocity and dispersivity.

4. Interfaces

Interface diagram: port_a (heat in, left), port_b (heat out, right), q_r (Darcy velocity signal, left)

port_a: inner heat port — heat inlet (Heat, inherited from Element1D)

port_b: outer heat port — heat outlet (Heat, inherited from Element1D)

q_r: radial Darcy velocity input for dispersion calculation (Signal)

5. Nomenclature

	R	radial thermal resistance [K/W]
$\lambda_\mathrm{eff}$	lambda_overall	effective thermal conductivity incl. dispersion [W/(m·K)]
$\lambda_\mathrm{disp}$	lambda_disp	dispersion contribution to thermal conductivity [W/(m·K)]
	q_r	radial Darcy velocity [m/s]
$r_\mathrm{outer}$	r	outer edge radius of control volume [m]
	r_r	centre radius of control volume [m]
	disp_length	dispersivity length [m]
	C_f	volumetric heat capacity of fluid [J/(m³·K)]
$\dot{Q}$	Q_flow	heat flow rate [W] (inherited from Element1D)
$\Delta T$	dT	temperature difference across resistor [K] (inherited from Element1D)

6. Governing Equations

Effective thermal conductivity with optional dispersion:

lambda_disp = d_L * |q_r| * C_f

Radial thermal resistance for cylindrical geometry:

R_r = (ln(r_outer) - ln(r_r)) / (lambda_eff * 2*pi*z)

Heat transport (inherited from Element1D):

dT = R * Q_flow

7. Remarks for Usage

Connect q_r to the q_long output of the corresponding ControlVolumeL4 cell. Set dispersion = false (default) to use pure conduction.

8. Validation

(no remarks)

9. References

(no remarks)

10. Version History

Model created by Markus Gillner (markus.gillner@tuhh.de), May 2026

Parameters

Type	Name	Default	Description
SI.ThermalConductivity	lambda		Heat Capacity of the aquifer
SI.Length	r_r		radius from well to center of control volume
SI.Length	z		height of control volume
SI.Length	r		Radius of thermal resistance
SI.Length	disp_length	0.7	dispersion length
Real	C	nC_f + (1 - n)C_s	volumetric heat capacity of stone and fluid
Real	C_f	4.3*10^6	volumetric heat capacity of fluid
Real	C_s	1.81*10^6	volumetric heat capacity of stone
Real	n	0.25	porosity of underground
Boolean	dispersion	false	whether dispersion is considered or not

Connectors

Type	Name	Description
HeatPort_a	port_a (from Element1D)
HeatPort_b	port_b (from Element1D)
Modelica.Blocks.Interfaces.RealInput	q_r	darcy velocity in radial direction

Components

Type	Name	Description
SI.HeatFlowRate	Q_flow (from Element1D)	Heat flow rate from port_a -> port_b
SI.TemperatureDifference	dT (from Element1D)	port_a.T - port_b.T
SI.ThermalConductivity	lambda_overall	overall effective thermal conductivity
SI.ThermalResistance	R	Thermal resistance of material
SI.ThermalConductivity	lambda_disp	dispersion depending part of thermal conductivity