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) 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.

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Geometry

The geometry used for the case has three regions, which are 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:

Validation Heat Sink Windtunnel Dimensions — 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

Dimension $[mm]$	Value
h (height)	21.8
W (width)	41.4
L (length)	57.2
t_b (base height)	8.4
t (thickness of fin)	1
p (spacing between adjacent fins)	2.1
N (number of fins)	14

Table 1: Dimensions of the heat sink

3. Heat Source

The heat source is a chip with the dimensions shown in the table below and has a contact surface area $(A_s)$ with a heat sink of 1.555 $cm^2$.

Validation Heat Sink Chip Dimensions — Figure 3: Chip heat source used in validation case

Dimension $[mm]$	Value
a	15.5
b	10
c	4.5
d	9
e	1.3

Table 2: Chip dimension in $mm$

Chip Model

STP130NS04ZB by STMicroelectronics

Analysis Type and Mesh

Tool Type: OpenFOAM®

Analysis Type: Conjugate Heat Transfer

Mesh and Element Types:

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

Mesh Type	Number of Cells	Element Type
Standard	603883	3D Polyhedral

Table 3: Details of the generated mesh

CHT fins polyhedral mesh simscale — Figure 4: Standard mesh of enclosure with 3D polyhedral elements

A region refinement was added to the heat source, heat sink, and the area close to the heat sink to be able to accurately capture the features of the heat sink as well as the wake region. In order to reduce the cell count and therefore decrease the run time, a symmetry plane in the XZ-Plane is used. A local element refinement was further used to increase the fineness around the fins of the heat sink. The mesh of the heat sink, heat source, and symmetry plane can be seen in the figure below:

Validation Heat Sink Chip Heat Sink Mesh — Figure 5: A close view of the mesh on the flow region, fins, and the chip. Volume and surface custom sizing were added to refine the mesh in the wake region and on the fins.

Simulation Setup

Material:

Fluid:
- Air
  - Kinematic viscosity $(\nu)$: 1.529e-5 $m^2/s$
  - Density $(\rho)$: 1.179 $kg/m^3$
  - 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: 1013 $J/(kg\ K)$
Solid:
- Heat sink – Aluminium:
  - Thermal conductivity $(\kappa)$: 209 $W/(m\ K)$
  - Specific heat: 897 $J/(kg\ K)$
  - Density $(\rho)$: 2700 $kg/m^3$
- Chip Material
  - Thermal conductivity $(\kappa)$: 38.6 $W/(m\ K)$
  - Specific heat: 705 $J/(kg\ K)$
  - Density $(\rho)$: 2330 $kg/m^3$
- Thermal resistance between the heat sink and power (heat) 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 38.6 $K/W$ 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-case resistance
$A_s$: contact surface area

Initial Conditions:

The velocity $(U)$ and temperature $(T)$ are given an initial condition, based on the experimental results, the same as for the boundary conditions to allow for faster convergence of the simulation.

Boundary Conditions:

Surface	Boundary Condition
1	Volumetric Flow Inlet
2	Pressure Outlet
3	Symmetry Plane
4	Symmetry Plane Heat Sink
5	Symmetry Plane Chip

Table 4: Boundary Condition for individual surfaces

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

Approach (Inlet) Velocity $[m/s]$	Volumetric flow rate $\dot{U}\,[m^3/s]$	$T\,[K]$	Absolute Power Source (chip) $[W]$	Pressure Outlet $[Pa]$	Walls	Heat Sink
5.47	0.027	296.9	56.64	0	No-slip	No-slip
7.03	0.0352	297.4	71.4	0	No-slip	No-slip
8.59	0.04295	297.9	82.36	0	No-slip	No-slip
9.96	0.0498	298.3	87.32	0	No-slip	No-slip
11.23	0.0562	298.9	85.07	0	No-slip	No-slip
12.50	0.062	299.3	76.3	0	No-slip	No-slip
13.57	0.06783	299.6	60.24	0	No-slip	No-slip

Table 5: Inlet volumetric flow rate with its corresponding inlet temperature

Note

The walls of the heat sink were automatically assigned as 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:

conjugate heat transfer validation 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:

conjugate heat transfer validation 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:

$R_{ja,e}$: junction-to-air thermal resistance 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 was calculated using the equation below:

$$R_{ja,s} = \frac{T_{j,s}-T_a}{P} \tag{4}$$

Where:

$T_{j,s}$: junction temperature from simulation (temperature at the bottom face of the chip)
$P$: thermal power of the heat source
$R_{ja,s}$: 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:

Approach (Inlet) Velocity $[m/s]$	Volumetric Flow Rate $\dot{U}\,[m^3/s]$	$T_a$ $[K]$	P $[W]$	$T_{j,e}$ $[K]$	$T_{j,s}$ $[K]$	$R_{ja,t}$ $[K/W]$	$R_{ja,e}$ $[K/W]$	$R_{ja,s}$ $[K/W]$	Error – Simulation to Analytical [%]	Error – Simulation to Experiment [%]
5.47	0.027	296.95	56.64	371.25	370.84	1.203	1.312	1.304	8.44	-0.567
7.03	0.0352	297.35	71.4	384.15	384.03	1.142	1.216	1.214	6.30	-0.164
8.59	0.04295	297.95	82.36	391.85	392.89	1.1	1.140	1.152	4.79	1.11
9.96	0.0498	298.25	97.32	395.05	406.4	1.071	1.109	1.11	3.76	0.205
11.23	0.0562	298.95	85.07	391.15	391.146	1.05	1.084	1.083	3.215	-0.02
12.50	0.062	299.25	76.3	377.45	380.435	1.031	1.024	1.064	3.20	3.908
13.57	0.06783	299.65	60.24	357.55	362.87	1.017	0.961	1.049	3.19	9.205

Table 6: Comparison of results between analytical, experimental, and simulation

conjugate heat transfer validation Heat Sink Thermal Resistance [K/W] — Figure 9: Graphical visualization of results between the analytical, experimental, and simulation against approach velocity.

The temperature distribution on the heat sink and velocity distribution on chip obtained from the simulation when the approach velocity is 13.57 $m/s$ are as below:

conjugate heat transfer validation Heat Sink Velocity Temperature V = 13.7 — Figure 10: Temperature and velocity distribution over the chip and flow region, respectively.

References

Ventola, L., Curcuruto, G., Fasano, M., Fotia, S., Pugliese, V., Chiavazzo, E. and Asinari, P., Unshrouded Plate Fin Heat Sinks for Electronics Cooling: Validation of a Comprehensive Thermal Model and Cost Optimization in Semi-Active Configuration, Energies 8 (9), 2016, DOI:10.3390/en9080609