Thermal analysis and Design optimization using

Pro Mechanica

Sasi Bhushan # 35763829Srikanth # 35762927

CAD Applications (MAE 477/577) - Final Project

ObjectiveExplore the capability of Mechanica to study the

conduction/convection heat transfer analysis by considering simple examples and quantifying the ProM results with analytical calculations.

And the second part of the study involves exploring the design optimization capabilities of Mechanica.

The Game Plan

For the first part of the study we consider simple geometries like:

Flat PlateLong CylinderTwo Cylinders of different cross sections

For the second part of the study, we considered a Finned Plate and optimized its shape and size for a given set of constraints.

Case Study 1: Flat Plate

Material: CopperDimensions:l = 95 mm, b = 100 mm, t = 10 mmheat transfer coefficient of air:H = 0.01 N/mm Sec SThermal Conductivity of Copper:K = 121.223 N/Sec C

The above plate is subjected to a uniform temperature of 100 C on one surface and the other surface is exposed to atmosphere at a temperature of 25 C. Therefore, there is convective heat transfer from the surface.

Steady State Analysis

Temperature distribution in Flat Plate

Analytical Calculations:Temperature of hot surface: T1 = 100 CTemperature of Surroundings: 25 CArea of the plate: A = l*b = 9.5E-03 m^2Heat flux across the plate:q = k*A*(T1-T2)/t a) Conduction equationHeat transfer to the environment:q = h*A*(T2-Tatm) b) Convection equationAt steady state both conduction and convection should be equal. Solving the above two equations we get T2 = 99.9 C.

ProM Analytical

Max Temp 100 C 100 C

Min Temp 99.93818 C 99.9 C

Heat Flux 7.49E-01 0.749

Analysis 2: Transient

One end maintained at 100 C and other end exposed to air at temp of 25 C through a convection heat transfer coefficient of 0.01 N/mm Sec C. The primary question we want to answer is to find the eventual temperature of the cold end of the plate, and the time it takes to reach the steady state.

Analysis 3: Thermally Induced stresses

By fixing all degrees of freedom except displacement in X and Y direction on the hot surface we get : Maximum Stress: = 9.351600e+12 N/mm2

VonMises stresses induced

Case Study 2: Cylindrical Rod

The above cylindrical rod is given a uniform heat flux of 100 mW on end and the other is exposed to convection to the atmospheric air.

Material Cu

Height 300 mm

Diameter 30 mm

Heat Source 100 mW

Boundary Conditions

h=0.01 N/mm SecC

Analytical calculations for Flat Plate study: Heat transfer coefficient of air :h = 0.01 N/mm Sec

C Specific heat capacity: 3.77186e+08 mm^2/sec^2 Thermal Conductivity: K =121.223 N/Sec C Density : 8.21395e-06 kg/m^3 Poisson’s ratio: 0.35

At Surface 1 Heat Flux Q: k*A*(T1-T2)/L a) (Conduction) k= 121.223 N/Sec C A = π*r^2 = 7.065E-04 m^2

At Surface 2 Heat Flux Q: h*A*(T2-25) b) (Convection) Q = 0.1 WTherefore, solving the above two equations for T1 and

T2 : T1 = 39.5 C T2 = 39.15 C

ProM results: Steady State AnalysisTemperature distribution

Heat Flux

ProM Analytical

Max Temp 39.51 C 39.5 C

Min Temp 39.91 C 39.15 C

Analysis 2: Transient

As expected the temperatures at both the ends come down to 25 C as the heat load is shut off. Time = 1.06896e+07 sec

Analysis 3: Thermally Induced stresses

By fixing all degrees of freedom except displacement in X and Y direction on the hot surface we get : Maximum Stress: = 1.970081e+02 N/mm2

VonMises stresses induced

Case Study 3: 2 Cylindrical Rods with interface

The combined cylindrical rod is given a uniform heat flux of 100 mW on end and the other is exposed to convection to the atmospheric air.

Rod 1 Rod 2

Material Steel Copper

Height 300 mm 400 mm

Diameter 30 mm 20 mm

Heat Source 100 mW

Boundary Conditions

Insulated h=0.01 N/mm Sec C

Steady State Analysis:• Convection Coefficient: h = 0.01 N/mm

Sec C• Heat load: q = 100 mW• Area of Rod1: A1 = 7.065E-04 m^2• Area of Rod2: A2 =3.14E-04 m^2• Length of Rod1:L1 = 0.3 m• Length of Rod2:L2 = 0.4 m• Tatm = 25 C

Steady state equations:At surface 1:

• q = k1*A1*(T1-T2)/L1 a)At Surface 2:

• q = K2*A2*(T2-T3)/L2 b)At Surface 3:

• q = h*A2*(T3-Tatm) c)Solving the above three equations we get,

• T1 = 58.87 C• T2 = 57.89 C• T3 = 56.84 C

Temperature distribution

ProM Analytical

Max Temp 5.89E+01 58.87 C

Interface Temp

5.76E+01 57.87 C

Min Temp 5.68E+01 56.84 C

Analysis 2: Transient

Heat Load is turned on. And we want to find the time to reach the steady state temperatures.

Analysis 3: Thermally Induced stresses

By fixing all degrees of freedom except displacement in X and Y direction on the hot surface we get : Maximum Stress: = 4.449016e+02N/mm2

VonMises stresses induced

Finned Plate

In this study we will add fins for the flat plate discussed earlier and determine the optimum height and thickness of the fins to maximize heat transfer and minimize mass along with the steady state and transient thermal analysis.

Finned Plate

Dimensions of the plate l = 95 mm, b = 100 mm, t = 10 mm

Dimensions of the Fins t= 5 mm, b = 100 mm, h = 5 mm

Steady State AnalysisFor the given boundary conditions the temperature distribution in the fin is given by the equation:

ProM Analytical

Max Temp 100 C 100 C

Min Temp 95.6C 96.82C

Sensitivity Analysis

Variation of temperature with respect to thickness and height of the fins

We run the global sensitivity analysis to determine the variation of Min Temperature and Mass with respect to the design variables.

Optimization AnalysisThe optimization problem can be formulated as shown below:Objective function: f = MassConstraints: g = Min Temp <= 80 CDesign variables:height of the fins: 20<=h<=100thickness: 2<=t<=8Optimized vs. Initial Design

Results

Initial Design Optimized design

Fin Height 50 mm 85.0972 mm

Fin Thickness 5 mm 2 mm

Mass 2.83E-03 2.1783E-03

Max Temp 100E+02 100E+01

Min Temp 9.56E+01 8.0000E+01

Conclusions

In this project, we explored the design optimization, thermal stress, conduction/convection heat transfer analysis capabilities of ProE. We went through this process by taking a few simple examples for steady state/transient analysis and validating the obtained ProM results with analytical calculations. We can infer from the coincidence of the final results that our analysis calculations are true and thus validate the overall process.

The next step was taking up a more complex problem and exploring the design optimization capabilities of ProM. The finned plate problem is taken up and the whole previous process is went through to obtain thermal analysis data, which is used for optimizing the plate design for particular parameters. The thermal analysis of the final optimized design shows the improved performance with better design parameters.