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Validation Case: Conjugate Heat Transfer – Rectangular Fins

This validation case belongs to fluid dynamics. The aim of this validation case is to validate the conjugate heat transfer (CHT) v2.0 analysis implemented in SimScale.

The simulation results of SimScale were compared to the results presented in the study titled “Unshrouded Plate Fin Heat Sinks for Electronics Cooling: Validation of a Comprehensive Thermal Model and Cost Optimization in Semi-Active Configuration”\(^1\) written by L. Ventola, G. Curcuruto, et. al.

Geometry

The geometry used for the case has three major regions, which is as follows:

1. Enclosure – Fluid Region

The length of the fluid region upstream of the heat sink is 6 times the length of the heat sink \((L)\) and the fluid region length downstream is 15 times \(L\) visualized in the figure below:

dimension of enclosure for conjugate heat transfer validation case with the upstream length 6 times the length of the heat sink and the downstream 15 times the length of the heat sink
Figure 1: Dimension of enclosure based on the length of the heat sink

2. Heat Sink

The model of the heat sink with its dimensions can be seen in the figure and table below:

model of heat sink used for conjugate heat transfer validation case
Figure 2: Heat sink model
h \([mm]\)W \([mm]\)L \([mm]\)t_b \([mm]\)t \([mm]\)p \([mm]\)N [-]
21.841.457.28.412.114
Table 1: Dimension of the heat sink

Where:

  • h: height
  • W: width
  • L: length
  • t: the thickness of a fin
  • p: spacing between adjacent fins
  • tb: base height
  • N: number of fins

3. Heat Source

The heat source is a cube with an assumed length \((a)\) of 12.45 \(mm\) and has a contact surface area \((A_s)\) with the heat sink of 1.555 \(cm^2\).

cube heat source used in conjugate heat transfer validation case
Figure 3: Cube heat source used in validation case

Analysis Type and Mesh

Tool Type: OpenFOAM®

Analysis Type: Conjugate Heat Transfer v2.0

Mesh and Element Types:

The mesh was generated using the Standard meshing algorithm. The following table provides the details of the mesh:

Mesh TypeNumber of CellsElement Type
Standard2.2 million3D Tetrahedral/Hexahedral
Table 2: Details of generated mesh
mesh of fluid domain
Figure 4: Generated mesh of enclosure

A region refinement was added to the heat source and the heat sink and the area close to the heat sink to be able to accurately capture the results. The mesh of the heat sink and heat source can be seen in the figure below:

mesh of heat sink and heat source with region refinement
Figure 5: Mesh of heat sink and heat source with region refinement with a maximum edge length of 0.001 \(m\)

Simulation Setup

Material:

  • Fluid:
    • Air
      • Kinematic viscosity \((\nu)\): 1.529e-5 \(m^2/s\)
      • Density \((\rho)\): 1.196 \(kg/m^3\)
      • Thermal expansion coefficient: 3.43e-3 \(1/K\)
      • Reference temperature \((T_0)\): 273.1 \(K\)
      • Laminar Prandtl number \((Pr_{lam})\): 0.713
      • Turbulent Prandtl number \((Pr_{turb})\): 0.85
      • Specific heat: 1004 \(J/(kg\ K)\)
  • Solid:
    • Heat sink – Aluminium:
      • Thermal conductivity \((\kappa)\): 209 \(W/(m\ K)\)
      • Specific heat: 897 \(J/(kg\ K)\)
      • Density \((\rho)\): 2700 \(kg/m^3\)
    • Power Source:
      • Thermal conductivity \((\kappa)\): 80 \(W/(m\ K)\)
      • Specific heat: 1004 \(J/(kg\ K)\)
      • Density \((\rho)\): 1.28 \(kg/m^3\)
    • Thermal resistance between the heat sink and power source \((R_{jc})\)) is 0.5 \(K/W\).
    • Since the material properties of the heat source were not provided, the conductivity of the heat source was calculated to be 80 \(K/W\) with the following formula:

$$\kappa_{hs} = \frac{\frac{1}{2}a}{R_{jc}A_s} \tag{1}$$

where:

  • \(\kappa_{hs}\): conductivity
  • \(a\): side length of heat source
  • \(R_{jc}\): junction-to-air resistance
  • \(A_s\): contact surface area

Initial Conditions:

The velocity \((U)\) and temperature \((T)\) are given an initial condition the same as for the boundary conditions to allow for faster convergence of the simulation.

Boundary Conditions:

boundary condition overview of rectangular fin validation case
Figure 6: Overview of boundary conditions

The simulation was run with 7 different velocities with each velocity having its corresponding inlet temperature and heat source, as seen in the table below:

\(U\,[m/s]\)\(T\,[K]\) Heat Flux \([W]\)Pressure Outlet \([Pa]\)WallsHeat Sink
5.6296.956.640No-slipNo-slip
7.2297.471.40No-slipNo-slip
8.8297.982.360No-slipNo-slip
10.2298.387.320No-slipNo-slip
11.5298.985.070No-slipNo-slip
12.8299.376.30No-slipNo-slip
13.9299.660.240No-slipNo-slip
Table 4: Inlet velocity with its corresponding inlet temperature

Note

The walls of the fluid domain and the heat sink were automatically assigned as no-slip walls with temperature as zero gradient. This assignment cannot be seen under Boundary conditions in the attached project.

Reference Solution

The overall heat resistance between the heat source and the ambient air (heat sink) \((R_{ja})\) as calculated in the analytical and experimental results from the reference study\(^1\) are explained as follows. The complete array of thermal resistances between the heat source and the ambient air can be seen in the figure below:

thermal resistances between the heat source and the ambient air
Figure 7: Illustration of thermal resistances between the heat source and the ambient air (Ventola, et.al. 2016)\(^1\)

1. Analytical

The analytical solution for the junction-to-air thermal resistance is given by:

$$R_{ja,t} = R_{jc}+R_{cs}+R_{sa}+R_{spr} \tag{2}$$

where:

  • \(R_{ja,t}\): analytical junction-to-air thermal resistance
  • \(R_{jc}\): junction-to-case thermal resistance
  • \(R_{cs}\): case-to-sink thermal resistance
  • \(R_{sa}\): sink-to-air thermal resistance
  • \(R_{spr}\): spreading thermal resistance

2. Experimental

The experimental solution is gained by doing an experiment with the scheme explained in the figure below:

experimental setup of measuring junction to air thermal resistance
Figure 8: Experimental setup of heat sink thermal model validation (Source: Ventola, et.al, 2016)\(^1\)

The air at ambient temperature flows into the rig and the airflow is measured with an orifice plate method where the orifice plate is in the inlet pipe. Then, the air flows into a plenum chamber and finally passes through a feeding branch and enters the HVAC. The HVAC fan flows the air through the experimental rig where the heat sink is placed.

The transistor voltage drop (\(V\)), electric current (\(I\)), junction temperature (\(T_j\)) and the temperature of air approaching the heat sink (\(T_a\)) are measured with a GL220 data logger (Graphtec™ Digital Solutions, Plano, TX, USA)\(^1\). The junction temperature (\(T_j\)) is measured at the interface between the transistor and the heat sink and the approaching air temperature (\(T_a\)) is measured at the inlet of the HVAC. Finally, the overall junction-to-air thermal resistance is calculated with the formula below:

$$R_{ja,e} = \frac{T_j-T_a}{P} \tag{3}$$

where:

  • \(R_{ja,e}\): thermal resistance between the heat source and the heat sink from experimental results
  • \(T_j\): measured junction temperature
  • \(T_a\): ambient temperature
  • \(P\): thermal power of the heat source

Result Comparison

The junction-to-air thermal resistance (\(R_{ja}\)) obtained from SimScale were calculated with the formulae below:

$$T_{j,s} = T_{interface} + P.R_{jc} \tag{4}$$

$$R_{ja,s} = \frac{T_{j,s}-T_a}{P} \tag{5}$$

where:

  • \(T_{j,s}\): junction temperature of simulation
  • \(T_{interface}\): the temperature at the interface of the heat source and the heat sink
  • \(P\): thermal power of the heat source
  • \(R_{j,c}\): thermal resistance between the heat source and the heat sink
  • \(R_{ja,s}\): junction-to-air thermal resistance obtained from simulation
  • \(T_a\): ambient temperature

The comparison of the junction-to-air thermal resistance (\(R_{ja}\)) between the simulation results and the results in the reference study is given in Table 5 and Figure 9:

\(U\) \([m/s]\)\(T_a\) \([K]\)P \([W]\)\(T_{j,e}\) \([K]\)\(R_{ja,t}\) \([K/W]\)\(R_{ja,e}\) \([K/W]\)\(T_{j,s}\) \([K]\)\(R_{ja,s}\) \([K/W]\)Error – Simulation to Analytical [%]Error – Simulation to Experiment [%]
5.656.6456.64371.251.2031.312366.0631.2302.26-6.23
7.271.471.4384.151.1421.216379.091.1450.25-5.85
8.882.3682.36391.851.11.140386.1451.071-2.65-6.07
10.287.3297.32395.051.0711.109392.9681.0851.28-2.19
11.585.0785.07391.151.051.084390.2091.0732.17-1.04
12.876.376.3377.451.0311.024379.6631.0542.22.82
13.960.2460.24357.551.0170.961362.2891.0402.248.20
Table 5: Comparison of results between analytical, experimental and simulation
plot of thermal resistance obtained from analytical, experimental and simulation
Figure 9: Graphical visualization of results between the analytical, experiment, and the simulation.

The temperature distribution obtained from the simulation when the velocity is 5.6 \(m/s\) is as below:

temperature distribution of heat sink and flow region
Figure 10: Temperature distribution of power source and heat sink

Note

If you still encounter problems validating you simulation, then please post the issue on our forum or contact us.

Last updated: June 14th, 2021

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