4 - CHG 2314 Heat Transfer Part 2c Two-Dimensional...

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University of Ottawa, CHG 2314, B. Kruczek 1 CHG 2314 Heat Transfer Part 2c Two-Dimensional Steady-State Conduction - Analytical solution - Graphical method and shape factors - Dimensionless conduction rate
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University of Ottawa, CHG 2314, B. Kruczek 2 Multidimensional effects | Surface area for conduction changes in the direction of heat flow in a Cartesian system z Long prismatic solid with two surfaces insulated and the other two maintained at different temperatures z “End effects” – thick wall oven | Inherently multidimensional systems z Temperature gradients exist in more than one direction
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University of Ottawa, CHG 2314, B. Kruczek 3 The heat equation and methods of solution ( ) 22 0 qx ,y TT k xy ∂∂ + += ± | Methods of solution z Exact/Analytical: Separation of Variables (Section 4.2) ¾ Limited to simple geometries and boundary conditions z Approximate/Graphical: Flux plotting (Sections 4.S.1 - supplemental material) ¾ Requires zero heat generation | We will limit our discussion to two-dimensional conduction. z For constant thermal conductivity, 2-D heat diffusion equation in Cartesian coordinates becomes: z Approximate/Numerical: Finite Difference, Finite Element or Boundary Element Method (Sections 4.4 and 4.5) – not required ¾ Of limited value for quantitative considerations but a quick aid to establishing physical insights ¾ Most useful approach and adaptable to any level of complexity
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University of Ottawa, CHG 2314, B. Kruczek 4 Exact/Analytical solutions | Analytical solutions can be obtained for a limited number of multi-D problems z Consider two-dimensional conduction in a thin rectangular plate with the edges maintained at constant temperatures as shown: z Governing differential equation: z Boundary conditions: 0 2 2 2 2 = + y x θθ 1 21 where: TT θ = () ( ) ( ) 1 , 0 , 0 0 , 0 , 0 = = = = W x and y L x and y
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University of Ottawa, CHG 2314, B. Kruczek 5 Exact solutions z The analytical solution is obtained by using the method of separation of variables z The temperature profile and heat flow can then be plotted: The lines at θ = 0.1, 0.25, 0.5, 0.75 are isotherms, as are the edges at θ = 0 and θ = 1 The arrows are heat flow lines, which are perpendicular to the isotherms () ( ) = + = 1 1 sinh sinh sin 1 2 , n n L W n L y n L x n n y x π ππ θ
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University of Ottawa, CHG 2314, B. Kruczek 6 Graphical method | The objective of the graphical method is
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This note was uploaded on 05/30/2011 for the course CHG 2314 taught by Professor Kruz. during the Spring '10 term at University of Ottawa.

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4 - CHG 2314 Heat Transfer Part 2c Two-Dimensional...

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