COMSOL Multiphysics 4.3 is used to perform finite element analysis for the heat sink.
All numerical computer simulations are carried out using the in-built Heat Transfer module. The simulations are run on a computer with eight CPU dual - cores (2 × Intel Xeon X5570) with 24 GB RAM and 2.93 GHz processor clock speed. Due to the system symmetry, only half of the heat sink is simulated to reduce the total of number of finite elements and thus reduce the overall computational time to arrive at steady state results. In the course of laminar forced convection flow model development, two points are of great interest in order to achieve the final objective:
1. To determine the ideal number of fins for a given overall width of the base plate of the heat sink to have the best balance between heat transfer enhancement and pressure losses.
This is done with the help of a two-dimensional (2D) steady state model of the heat sink.
2. To validate the simulation results with experimental test runs on the prototype of the heat sink. For this, a three dimensional (3D) model is computed to give a fully coupled analysis of the fluid flow and heat transfer mechanism using the design parameters obtained in 2D analysis and the space constraints associated with the prototype. The 3D model gives a more accurate representation of the actual flow field in order to validate it with experimental results.
17 2.2.1 Verification Cases
Flow and heat transport phenomena in heat sinks of various types have been thoroughly studied; theoretically, experimentally and numerically, mostly because of the common occurrence of such devices in various thermodynamics systems. However, before any FEM model, is deemed suitable for further simulations and analysis; the process of FEM modeling used by the author, should be thoroughly verified and if possible validated with known experimental and analytical results. Here two verification cases have been presented. They are compared with results obtained from the FEM model developed for the system represented by these cases. These specific verification cases closely relate to work done in this thesis. The results for both the experimental case and the analytical case match with those obtained with the FEM model. Since COMSOL is the FEM modeling tool used here, the match indicates the ability of the author of this thesis to correctly discretize the system into various elements and apply the relevant boundary conditions to arrive at steady state solutions.
2.2.1.1 Verification with experimental data. Fehle et al. (21) conducted a study aiming at enhancing the heat transfer in a compact heat exchanger. In order to have the exact knowledge of the temperature distribution in the heat exchanger, they applied holographic interferometry to visualize the temperature field. Figure 2.1 shows the heat exchanger prototype along with the necessary dimensions, as given by the author, shown in Table 2.1. The flow conditions for air and the thermal and fluid boundary conditions are applied using the in-built module for the geometry as described above. The interferograms produced are processed using a digital image processing system. The interference lines in the interferograms approximately resemble isotherms of the investigated duct flow.
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Parameter Dimension (mm)
Height of fins, e 10
Width of the duct, b 10
Length, lr 300
Fin thickness, tf 2
Radius, r 1
Table 2.1 Dimension of plate fin arrangement
Figure 2.1 Plain fin arrangements in a compact plate heat exchanger
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Governing Equations: The governing energy equation for heat exchanger solid domain is as follows:
( ) (2.14)
The governing equations for incompressible fluid flow domain are:
( ) ( ( ) ) ( ) (2.15)
( ) (2.16)
( ) (2.17)
where the symbols stand for their usual meanings. The dependent variables in this type of analysis are T, temperature, P, pressure and u, velocity.
Boundary Conditions:
1. Initial Temperature Guess for the Non-Linear Solver: T= 298.13 K 2. Inlet Temperature: 298.13 K
3. Inlet Flow Reynolds Number: 500 with Laminar inflow 4. Air Inlet Pressure: 1 atm
5. Temperature Boundary Conditions: Applied at top and bottom surfaces maintained at constant temperature by heating plates
6. Thermal Insulation: Applied to all solid and liquid interfaces not covered by other boundary conditions.
7. No slip: This condition prescribes that the fluid at the wall is not moving.
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The test section is itself supplied with six water-supplied heating plates. The temperature of each plate is measured by thermocouples in order to maintain a uniform test surface temperature. Figure 2.2 show the close resemblance for the results obtained by holographic interferometry by the author and COMSOL. Fehle et al. report that the temperature difference between two neighboring isotherms is approximately 2.3 K. This particular observation can also be seen in the results obtained with the FEM model developed for this particular system.
Figure 2.2 (a) Interferograms for plain fin arrangements for r =1 mm, Re=500 (b) Results obtained using COMSOL with isotherm temperatures in K
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2.2.1.2 Verification with known analytical results. A known problem in the field of heat transfer is temperature distribution in the flow of a fluid stream inside a solid object.
Because the internal flow is completely enclosed, an energy balance is applied to determine how the mean fluid temperature, Tm, and the solid surface temperature, Ts, vary with position along the enclosed space, a tube in this case. The solution to this problem with constant surface heat flux is given by Equation 2.2 and shown in Figure 2.3 (20).
( )
̇ ( ) (2.2)
Governing Equations: The governing equations for the incompressible internal fluid flow domain are:
Figure 2.3 Axial temperature variations for heat transfer in a tube
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( ) ( ( ) ) ( ) (2.15)
( ) (2.16)
( ) (2.17)
Boundary Conditions:
1. Initial Temperature Guess for the Non-Linear Solver: T= 298.15 K 2. Inlet Temperature: 298.15 K
3. Mass flow rate: 0.1 kg/s
4. Total Heat Flux: Enters the total heat flux across the boundaries where the node is active. In this case, applied to pipe surface, = 2000 W
5. No slip: This condition prescribes that the fluid at the wall is not moving.
The mean temperature thus varies linearly along the tube and the temperature difference (Ts-Tm) also varies along the length. This difference is initially small but increases due to decrease in h (convection heat transfer co-efficient) in the entrance region. However, in the fully developed region, h is constant and hence, the difference remains the same. In order to simulate this problem using FEM modeling technique, a system of heating water from an inlet temperature of 298.15 K is considered. The water passes through a thick walled tube of inner and outer diameters of 20 mm and 40 mm respectively. It is assumed that the outer surface of the tube is well insulated and electrical heating provides a constant heat flux distributed uniformly over the entire tube periphery. For a water mass flow rate of 0.1 kg/s, Figure 2.4 shows the results obtained during this analysis. The same trend in temperature variation along the axis is seen as explained by the analytical solution.
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The analytical and experimental cases are considered for simply verifying the proper use of the software in general and the heat transfer module in particular to develop an FEM model for various systems with known, reported results. Chapters 2 and 3, further talk about validating the FEM model for superconducting cable termination with results obtained from the experimental setup consisting of the prototype heat sink manufactured specifically for this purpose.
Figure 2.4 Axial temperature variations obtained using FEM technique
24 2.2.2 Two-dimensional FEM model
A superconducting cable termination essential consists of, for the sake of simplicity and the purview of this thesis, a heat sink that is required to intercept the heat leak from the room temperature components to the cryogenic temperature components of the superconducting cable.
In order to validate the computational FEM models, a prototype heat sink is manufactured as shown in Figure 2.5. This is a scaled down version of the actual heat sink required to be installed in the superconducting cable system. The heat sink designed and modeled in this study is made of copper and features 18 fins of 10 cm length (22). It is integrated in a cylindrical copper tube, flattened on the bottom side. The flat surface allows for known and variable thermal load to be applied to the heat sink. Cryogenic helium gas is injected at high pressure by an external helium circulation system (15). The temperature and pressure at the inlet to the heat sink can be adjusted depending on cooling requirements. The assembled heat sink was wrapped in Aluminized Mylar foil and enclosed in a vacuum chamber to reduce the conduction and radiation heat inleaks into the system.
The simplest Finite Element Method model is designed to determine the optimum number of fins required for the heat sink under the constraint that the overall base width and fin geometrical parameters remain constant. The model takes a vertical cross section of the heat sink as shown in Figure 2.6, and hence the focus of 2D numerical simulation is solely on the heat transfer mechanism. The pressure loss across the length is calculated analytically taking into account the flow between the parallel fins (plates) of the heat sink along with the entrance-exit and acceleration-deceleration effects.
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The density of helium gas varies largely with temperature and pressure. Hence, average density is calculated and used based on the inlet and outlet temperatures of the fluid. Heat Transfer in Solids (ht) module available in COMSOL is used over a parameterized geometry so as to easily allow sweeping over a number of fins. The software calculates the properties of copper, namely thermal conductivity, κ; specific heat at constant pressure, ϲp, and density, ρ, which are temperature dependent as given by (23).
The helium properties are calculated using Engineering Equation Solver (EES) software package at the required temperature and pressure. The idea of using EES is to find the value of convective heat transfer co-efficient, һ, which is used for applying the convective cooling boundary condition in the 2 dimensional models. This co-efficient is found using inbuilt EES
Figure 2.5 Design of the prototype cable termination (total view) and cut views to show the internal fin structure (vertical and horizontal cut)
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functions and thermophysical property tables for gaseous Helium. EES uses an implementation of (24), (25) for calculating thermophysical properties except for thermal conductivity which is computed using (26). Reynolds number ( ) is calculated using the hydraulic diameter concept for parallel plate fins consistent with the geometry and mass flow rate assumed. The calculations are carried out for various mass flow rates so that flow stays mostly in the laminar regime. Correlations for both laminar and turbulent flow (if any) between smooth parallel plates, as given in (20), are used to find the heat transfer co-efficient and pressure losses inside the heat sink.
( ) (2.3)
( ) (2.4)
For a smooth surface the friction factor for laminar and turbulent regime respectively is given by
�=96��� (Laminar Flow) (2.5)
( ) (2.6)
The pressure loss due to drag experienced by fluid flow is estimated as
(2.7)
where fin s acing mean velocity of gHe ength of heat sin usselt umbe
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Figure 2.6 shows an instance of density variations across various temperature domains for gaseous Helium under 8 bar pressure obtained from RefProp which a standard software to calculate thermophysical properties of various fluids at different temperatures and pressures.
From this figure, with a 20 K increase in the temperature of gaseous Helium, its density reduces by approximately 33%. Hence in order to account for the pressure drop due to acceleration and deceleration of the fluid stream due to density variations, an additional term is evaluated as given by (27).
Figure 2.6 Variation of density of gaseous Helium with temperature at constant pressure
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where G is the mass velocity of the stream. ̇ ; A = channel cross sectional area and ̇ = gaseous Helium mass flow rate.
In order for the two dimensional model to be more precise in calculating the pressure drop across the channel, entrance and exit effects due to sudden expansion and contraction are included as given by (28).
[ ] respectively; d, D are the smaller and larger diameters of the connecting pipes
Total pressure drop across the heat sink for a given flow parameter is given by
(2.11)
For the various cases of different mass flow rates and different geometries, the helium flow is found to be in laminar regime mostly and is modeled in COMSOL by providing the convective cooling boundary on the fin walls. In the case of the model with 9 fins, for example, using the above relations, an effective heat transfer coefficient value of h = 90.25 W/ (m2K) for the convective boundary condition is calculated. Separate values of h are calculated for different flow conditions and geometries. All the calculations for h and are performed using EES and the simulations are carried out in COMSOL.
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Governing Equation: The steady state heat transfer equation for heat flow in the solid block of copper is governed by
( ) ( ) (2.12)
where f
and , convective heat transfer coefficient.
Boundary Conditions:
1. Initial Temperature Guess for the Non-Linear Solver: 50 K
2. Heat Flux: A heat influx boundary condition of 50 W is applied at the base of the 2D model appearing as a line at the bottom in the front view as shown in Figure 2.7 3. Convective Cooling: It adds the convective term of Equation (2.12) to the
boundaries wherein h is defined using the correlations discussed above.
4. Thermal Insulation: The thermal insulation condition is applied across all other boundaries to mimic the experimental setup. It follows the following equation indicating no heat flux crosses the boundary.
( ) (2.13)
The mesh size is chosen as “no mal” with the default values as available in the gene al physics category. The normal mesh with 2986 elements is sufficient to satisfy mesh independence. Stationary Linear Solver produced the results as shown below in Figure 2.8 which indicates the surface temperature distribution across a vertical cross-section of the heat sink for a particular case with 9 fins in one half of the heat sink.
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The objective of the 2D computation was to find the number of fins required for optimum performance of the heat sink, i.e., a best tradeoff between temperature gradient and pressure drop across the heat sink. The entire heat sink is modeled for varying mass flow rates and varying number of fins (incremented in steps of 3) for a fixed fin thickness. It can be seen from Figure 2.9 that the 9 fin heat sink model (corresponding to half of the actual design) provides a good system balance for the heat sink performance. This primary two dimensional study forms the basis for further detailed three dimensional analyses and experimental validation.
Figure 2.7 2-Dimensional FEM model of the heat sink with important boundary conditions
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Figure 2.8 Surface temperature distribution (in Kelvin) with h = 90 W/m2K and ṁ = 1.5 g/s
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