2. SIMULATION OF SUPERCONDUCTING CABLE TERMINATION FOR LAMINAR
2.3 Experimental Setup and Results
2.3 Experimental Setup and Results
An experiment, conducted at the Center for Advanced Power Systems (CAPS), as described in detail in (22) is used to validate the simulation model. The prototype heat sink consists of four parts: The base block with fins, two end plates, and the surrounding enclosure (partially shown in Figure 2.10). All parts except for the cuts between the fins are machined
Figure 2.9 Tpeak and Δp curves for various mass flow rates
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discharge machining (EDM). The four parts are joined through silver brazing for maximum conductivity and excellent structural strength. The supply tubes are soldered to the end plates using tin-lead solder. A heater with a nown esistance of 10.09615 Ω is attached to the bottom plate. Two temperature sensors are attached to the side walls of the heat sink to determine the temperature of the solid.
Heat Sink Geometrical Parameters Dimensions (inch)
Total Length 6
Two additional temperature sensors are attached to the supply and exit tubes to measure the inlet and outlet fluid temperatures. An adjustable DC voltage source is used to control the heat influx to the heat sink. The helium circulation system allows adjusting the pressure and temperature of the helium flow at the inlet of the experimental setup. A differential pressure gauge is used to measure the pressure drop across the heat sink. The experimental setup with the gaseous Helium flow tubing, sensors attachments and heater wire within the cryostat is as shown in Figure 2.11. The heat sink, wrapped in aluminized Mylar (multi-layer insulation, MLI), is shown in Figure 2.12.
Table 2.2 Prototype heat sink dimensions
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Figure 2.10 Copper heat sink prototype used for experimentation
Figure 2.11 Experimental setup with flow lines and heat sink with heater attached
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The experiment is conducted for different gas mass flow rates entering the system at different temperatures and pressures. Three different total heat flux values of 30 W, 50 W and 100 W are applied at the bottom of the heat sink in order to obtain a good range of experimental data in both laminar and turbulent regime. The pressure drop across the heat sink is located at the cryocooler. Table 2.3 shows the configuration settings for the heater wire in order to achieve different thermal load values whereas Table 2.4 shows the experimental data obtained by operating the heat sink at various temperatures and pressures and under different flow conditions.
The entire experimental setup is initially cooled to the inlet temperature specified in the table before starting the experiment.
Figure 2.12 Heat sink wrapped in MLI before insertion into the cryostat
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Thermal heat load (W) Current (A) Voltage (V) Resistance Ω
30 1.72 17.4
10.09615
50 2.23 22.5
100 3.15 31.77
An essential first step in any convection problem is to determine whether the flow is laminar or turbulent. Surface friction (hence pressure drop) and the convection heat transfer rates depend strongly on which of these conditions exists. Table 2.5 relates the experimentally measured gaseous Helium flow rates with the corresponding Reynolds number calculated using the standard definition as given by Equation 2.1. The wide range of Reynolds number indicates the range of experimental data available for further validation with simulation results. This chapter only focuses on developing a FEM model for gaseous Helium flows through the heat sink in the laminar regime. The turbulent flows result in fluctuations that enhance the heat transfer rates and lead to increase in pressure drop. These fluctuations have to be dealt with differently and thus the turbulent flow FEM model for cryogenic heat sink is presented in the next chapter.
Table 2.3 Heater wire configuration to obtain various heat load values
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Measure pressure drop (mbar) 0.39 0.51 2.94 2.97 1.83 1.85 3.34 3.34
Approxim ate mass flow rate (kg/s) 0.18 0.24 1.5 1.7 1.26 1.17 2.50 2.39
Volume Flow rate (m3 /h) 0.115 0.144 0.844 0.890 0.666 0.712 0.680 0.702
T peak (K) 82.0 88.0 74.8 81.8 58.0 70.8 55.0 62.8
Tout (K) 64.5 75.4 62.0 71.7 45.6 57.6 44.8 52.3
Tin (K) 56.8 61.3 58.6 65.5 42.3 50.3 42.8 48.0
Inlet Pressure (bar) 9.3 10.7 8.57 9.73 5.98 7.18 10.49 12.01
Applied Heat Load (W) 50 100 50 100 50 100 50 100
Case No. 1 2 3 4 5 6 7 8
Table 2.4 Experimental results for heat sink setup
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Measure pressure drop (mbar) 1.02 1.02 3.12 3.17 25.56 25.59
Approxima te mass flow rate (kg/s) 0.98 0.98 2.29 2.38 8.56 8.35
Volume Flow rate (m3 /h) 0.307 0.307 0.699 0.712 1.688 1.664
T peak (K) 60.4 62.5 47.6 50.7 45.2 46.6
Tout (K) 53.5 56.4 43.4 45.7 40.8 41.5
Tin (K) 48.4 50.4 41.1 42.6 39.7 40.3
Inlet Pressure (bar) 12.06 12.5 8.85 9.28 12.14 12.39
Applied Heat Load (W) 30 50 30 50 30 50
Case No. 9 10 11 12 13 14
Table 2.4 - continued
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Table 2.5 Flow conditions for various experimental cases
Laminar Flow
Turbulent Flow
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2.4 Laminar 3 Dimensional FEM Model
Cryogenic circulation systems have been typically limited by the pressure loss handling capabilities. Two-dimensional analysis gives us a rough estimate of the pressure losses inside the heat sink. But, the fluid flow behavior could not be predicted accurately and can only be estimated in two dimensional studies since the entrance and exit chambers substantially impact the flow pattern. Therefore a three dimensional FEM model has been developed using the Conjugate Heat Transfer physics in COMSOL Multiphysics 4.3 to simulate a steady state, three dimensional fluid flow and determine its effects on the thermal performance of the heat sink. The fluid velocity field in certain cases is low enough to be assumed as laminar flow. Steady state results, assuming laminar flow consistent with experimental mass flow rates, are obtained.
The entire geometry is divided into solid (copper) domain and fluid (helium gas) domain.
The copper and helium properties are temperature and/or pressure dependent. The property functions are implemented in COMSOL using (23), (29), (30), (31). For temperatures below 140K, COMSOL does not provide any temperature dependence density function in the module present with us. For this, EES was used to calculate a piecewise function for temperature dependent Helium properties. Helium density is evaluated at average fluid operating temperature and provided as input to COMSOL.
Governing Equations: The governing energy equation for copper domain is as follows:
( ) (2.14)
The governing momentum and energy equations for fluid flow domain are:
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( ) ( ( ) ) ( ) (2.15)
( ) (2.16)
( ) (2.17)
where the symbols stand for their original meanings as explained earlier. The dependent variables in this type of analysis are T, temperature, P, pressure and u, velocity.
Boundary Conditions:
8. Initial Temperature Guess for the Non-Linear Solver: T= 58.6 K 9. Inlet Temperature: 58.6 K
10. Inlet Flow Rate: 0.78 g/s with Laminar inflow 11. Helium Outlet Pressure: 857 kPa
12. Total Heat Flux: Enters the total heat flux across the boundaries where the node is active. In this case, = 50 W
13. Thermal Insulation: As described earlier, the thermal insulation boundary condition is applied to all solid and liquid interfaces not covered by other boundary conditions.
14. No slip: This condition prescribes that the fluid at the wall is not moving.
Due to the laminar inflow assumption, a parabolic velocity distribution at the entrance is assumed, and a surface heat flux at the bottom is used to simulate the heat influx into the heat sink from the ambient. The meshing is carried out with an aim to keep the computational time as sho t as ossible and yet not affect the esults thus obtained. The solid is meshed by a “no mal”
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sized mesh whereas the fluid and its interface with the solid a e meshed with a “fine ” mesh wherein the meshing technique chosen is default as provided in the software.. The heat sink geometry consists of 9 fins, which are very closely spaced, forming 18 boundary layers on either side. In order to capture all effects, the meshing density is higher than usual.
This amounts to a total of 1.54 million elements taking 156 minutes for convergence. A non-uniform mesh with higher mesh density towards the fluid/solid interfaces is chosen to ensure greater computational accuracy. The results are computed using the stationary solvers, which incorporate a GMRES solver and a non-linear solver at default settings. GMRES required 240 iterations and the non-linear solver 45 iterations to arrive at steady state results as shown in Figure 2.14 and Figure 2.15 respectively.
Figure 2.13 Mesh structure for the laminar FEM model
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Figure 2.14 GMRES solution curve for each iteration
Figure 2.15 Error curve for non-linear solver using default settings
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In both the curves, steady state is achieved when the relative error is less than 10-3. The temperature and velocity fields (shown by arrow heads) obtained from the computation are shown in Figure 2.16. The thicknesses of the arrow indicate the magnitude of the velocity and the arrow heads indicate the direction.
Figure 2.16 clearly show the effect that entrance and exit chambers have on the nature of fluid flow as predicted and calculated by 2 dimensional analyses. The heat sink heats up as you go downstream. The results obtained by the laminar model, as explained above, are presented in
Figure 2.16 Heat sink surface temperature (in Kelvin) with the fluid velocity field shown by black arrow heads
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the next chapter. After formulating a 3 dimensional turbulent model, all simulations results from both the models are validated with the experimental results for the prototype heat sink. In order to increase the effectiveness of the heat sink, geometrical optimization studies are carried out by keeping the inlet fluid flow conditions and overall dimensions fixed.
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CHAPTER THREE
SIMULATION, MODEL VALIDATION AND OPTIMIZATION OF CABLE TERMINATION GEOMETRY
3.1 Introduction
The results, obtained from the experimental setup, consist of flows in laminar or turbulent regimes. In laminar flow regime fluid motion in highly ordered whereas for turbulent regimes the fluid motion is highly irregular and is characterized by velocity fluctuations. These fluctuations have two effects:
1. They enhance the transfer of momentum and energy thereby increasing the convection heat transfer rates.
2. They also increase the surface friction resulting in higher resistance to flow and hence higher pressure drop across the device.
Generally, the Navier-Stokes equations can be used for turbulent flow simulations, although this would require a large number of elements to capture the wide range of scales in the flow. An alternative approach, widely used, is to divide the flow in large resolved scales and small unresolved scales. The small scales are them modeled using a turbulence model with the goal that the model is computationally less time consuming and hence less expensive. Different turbulence models invoke different assumptions. COMSOL has a turbulence interface using k-ε model in the heat transfer module. This model includes Reynolds-averaged Navier-Stokes
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(RANS) most commonly used in industrial flow application (32). The RANS model divides the flow quantities into
̅ (3.1)
where ̅ is a mean value of a scalar quantity of flow obtained by time averaging over a long time and is the fluctuating component that averages to zero over time as shown in Figure 3.1. The Turbulent Flow, κ-ε interface uses a RANS turbulence model type as explained in the next section.
3.2 Turbulent 3-D Simulation using κ-ε Model
Various experimental cases, reported here, lie in the turbulent flow regime. The corresponding FEM model developed ovides fo a tu bulent κ-ε model as desc ibed in (33).
This model assumes that the flow is incompressible and Newtonian and the Navier stokes equation is as given below. Also, for κ-ε model two additional transport equations and two de endent va iables a e added: the tu bulent inetic ene gy κ and the dissi ation ate of
Figure 3.1 The average velocity component and the fluctuating velocity component (32)
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tu bulence ene gy ε . The equations fo tu bulent viscosity μT, transport equation fo κ and t ans o t equation fo ε ead as given below.
Dependent Variables: T, P, u, κ and ε
Governing Equations:
(3.2)
( ) ( ( ) ) (3.3)
where the symbols have their usual meanings as explained earlier.
(3.4)
(( ) ) (3.5)
(( ) ) (3.6)
( ( ( ) ) ( ) ) (3.7)
where , , are constants obtained from experimental data (33).
Boundary Conditions: The same boundary conditions as described in the laminar case are applicable for the turbulent case with specific changes pertaining to turbulence.
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The three dimensional FEM model, developed using the above equations, is integrated with the help of the turbulent non-isothermal flow model. The values of various constants, turbulent length scale and intensity are default as provided by the interface. Helium density averaged over the entire operating temperature range is again given as input and the boundary conditions remain the same as explained in the laminar case. A different meshing technique has been carried out in order to obtain the same accuracy with a lesser number of elements and hence lesser computational time. Firstly, the interior interface boundaries between the solid and fluid domain are meshed. Then the fluid domain is meshed and finally the solid domain is discretized.
The co e domain is meshed with a “no mal” sized mesh whe eas the fluid is meshed with a
“fine ” mesh size. The inte face between the two domains is meshed with a t iangula mesh of default “no mal” size. This esults in limiting the total number of elements in the mesh structure to 371842, as shown in Figure 3.2. The total computational time for any turbulent case is approximately 185 min in order to arrive at steady state results. Figure 3.3 shows the convergence curve for the simulation case. Segregated group 1 solves for velocity, temperature and pressure at all the modes/grid points whereas segregated group 2 solves for turbulent kinetic ene gy κ and dissi ation ate of tu bulence ene gy ε.
The results obtained, for case no. 6, for temperature and velocity field as as shown in Figure 3. 4 and 3.5. The streamline velocity profile clearly indicates a little bit of separation at the entry chamber of the heat sink and highly turbulent mixing at the exit chamber of the heat sink. The flow remains almost laminar between the fins due to small gap between each fin. The temperature profile for case no. 6 is shown here.
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Figure 3.2 Mesh structure obtained by separately meshing the domains and the interface boundaries
Figure 3.3 Convergence curve for stationary turbulent flow solver
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Figure 3.4 Streamline velocity field in the heat sink for case no. 6
Figure 3.5 Temperature profile in the heat sink for case no.6
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3.3 Model Validation with Experimental Results
The experimental results, as reported earlier, are used to validate both the simulation models. In the experimental setup, the Helium gas pressure drop, reported, is measured at the cryocooler end as shown in Figure 3.6. This cryocooler supplies helium gas at 40-70 K to the experimental setup with the help of special cryogenic pipes under vacuum. The total length of these 1 inch cryopipes is about 10 ft. Also the pressure drop across the 25cm long, 0.4 inch diameter copper tubing, for inflow and outflow of gas through the heat sink, needs to be taken into consideration. Hence, an additional pressure drop term is calculated and added to the simulation results in order to compensate for the same. The additional term is calculated using Equations 2.4-2.6 and using the temperature and pressure dependent properties of fluid flow as given by ΔPflow_system in Table 3.1.
Figure 3.6 Schematic diagram for gHe flow system
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The resulting temperature of gaseous Helium at the outlet of heat sink, Tout, temperature of the solid copper block, Tpeak and essu e d o ac oss the heat sin ΔP a e lotted as shown pressure gauges, flow meters, etc. uncertainty analysis cannot be carried out. However, as a rule of thumb, the uncertainty of a measuring device is 20 % of the least count (34). Hence error bars, accordingly, have been plotted on the graphs using this rule.
Table 3.1 Pressure drop validation results with experimental data
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Figure 3.7 Comparison of model results with experimental data for fluid outlet temperature
Figure 3.8 Model validation for peak temperature of solid copper block
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The data plotted in these Figures do not show any particular trend or transition from laminar to turbulent regimes. This is so because each and every case has a unique set of pressure and flow conditions and also a unique matching value of heat flux is applied to each case as reported earlier in Table 2.4. The maximum relative error for Tout, Tpeak and ΔP are 1.97%, 6%
and 17.94% respectively. These error percentages show good agreement of the numerical results with the experimental data. The numerical results mostly lay within the experimental measurement error bars as shown in the Figures.
0
Figure 3.9 Model validation results for pressure drop across the experimental setup
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3.4 Geometric Optimization of the Heat Sink
After validating the model, optimization studies are carried out for various geometrical parameters of the 9 fin (half section) model. Many optimization studies can be carried out since this heat sink incorporates a vast variety of variables such as geometrical parameters, gaseous helium mass flow rate, thermal mass of copper, fluid operating pressure, etc. Here, the focus is put on geometrical parameters, particularly on spacing between fins, keeping all the other input parameters constant.
Objective Function: To minimize the pressure drop across the system and the peak temperature acquired by the heat sink.
Constraints: Fixed overall base width and fin thickness of the heat sink; fixed flow parameters such as gHe mass flow rate, inlet temperature and pressure; fixed heat influx; fixed overall heat sink geometry. The corresponding values are as given below:
Heat influx = 100 W gHe ṁ = 0.54 g/s, w = 17.8 mm, t = 0.79 mm, Helium density = 6.3 kg/m3.
Problem Formulation: The various important geometrical parameters are shown in Figure 3. 9. Equation 3.8 binds all of them together.
(3.8)
where
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Non-dimensionalizing the above equation with respect to d3 and ee ing ‘w’ as a constant known value we get,
(3.9)
Keeping the thickness constant, the value of d3 can be determined for various values of x = d1/d3 and y = d2/d3 thereby satisfying the overall constraint on the heat sink geometry.
Keeping all the other input parameters constant, optimization studies are performed plugging in Figure 3.10 Vertical cut section showing geometrical optimization parameters considered for
the study d2
d3
d1
w
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the values of d1, d2 and d3 in each case into the validated 3D COMSOL model. Initial studies pointed out the worst case scenario when x ≥ y ≥ 1. Hence, the further studies concentrated efforts on x ≤ y ≤ 1. Figure 3.11 and Figure 3.12 show the optimization curves for a given set of input conditions as described earlier. The contour plots help in determining the minimization values for the objective function.
The corresponding optimum values are shown in Table 3.2. The results show that the performance of the heat sink can be increased by incorporating an unequally spaced fin structure that well distributes the coolant flow evenly across the fin structure in the design. This performance increase will be considerable for operating conditions and system parameters much higher than those considered here and in operation of the test apparatus.
Parameter Optimum Values for
Table 3.2 Optimum values for objective function under given constraints
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Figure 3.11 (a) Variation of heat sink peak temperature with fin spacing
(b) Contour plot showing optimized geometry for minimum peak temperature (a)
(b)
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Figure 3.12 (a) Variation of pressure loss across the heat sink with fin spacing (b) Contour plot showing optimized geometry for minimum pressure
(a)
(b)
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CHAPTER FOUR
CONCLUSION
The results presented in this thesis have some important applications to the design of cryogenic heat sink cooled by gaseous Helium for superconducting power device applications and to the basic understanding of heat transfer and fluid flow phenomena in forced convection type heat exchange devices. After summarizing the results here, suggestions for future work including integrating the present work with on-going research efforts at CAPS, Florida State University are given.
4.1 Summary of Research Efforts
A FEM model approach is used to simulate and optimize the problem presented in this thesis. COMSOL software package is used as the simulations tool. Before using COMSOL for model development, standard verification problems are presented with known and reported analytical and experimental results. The numerical results match with those obtained by the respective reporting agencies. A two dimensional model is developed in order to determine the ideal number of fins required to be made inside the prototype heat sink. It follows that a prototype heat sink with 18 fins has a good balance between the fluid pressure loss and the thermal performance of the heat sink.
A three dimensional FEM, laminar and turbulent, models are developed and appropriate
A three dimensional FEM, laminar and turbulent, models are developed and appropriate