Heat Transfer ME 3345
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This 0 page Class Notes was uploaded by Chloe Reilly on Monday November 2, 2015. The Class Notes belongs to ME 3345 at Georgia Institute of Technology - Main Campus taught by Staff in Fall. Since its upload, it has received 12 views. For similar materials see /class/234248/me-3345-georgia-institute-of-technology-main-campus in Mechanical Engineering at Georgia Institute of Technology - Main Campus.
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Date Created: 11/02/15
AO l39e ME 3345C Heat Transfer Spring 2006 11206 Objectives Develop Heat Diffusion Equation Discuss Initial and Boundary Conditions Heat Conduction Equation Thus far we have learned that heat diffuses in a stationary medium aCcording to Fourier s Law However only in speci c cases can this equationbe used to directly tell us the temperature distribution in a solid Example q KAZZ Plane Wall L If q K and A are constant q then ATL is constant Linear Temperature Distribution l L l We must develop an additional analysis method in order to predict the general temperature distribution in a solid undergoing conduction heat transfer Let s Look at a general case of conduction heat transfer in a plane wall This time include generation and energy storage How do we nd the temperature distribution Look At Differential Control Volume Ein E0ut Eg Est Si xdx qxdx Analysis of the Differential Control Volume leads to the following Expression 3k jC 61 ax ax 1 ppat This is the 1D Heat Diffusion Equation for Rectangular Coordinates Differential Control Volume in 3D qzdz The same analysis as Performed in 1D may Be used on a 3D qydy Control volume Note d qxdx q dx qxdx qx dx dq 61M qy d ydy dq X qzdz qz dz 3D Heat Diffusion Equation 6 6T 6 5T 6 6T 6T k k k qp 6x 8x 6y 6y 62 82 p at Simpli cations K nota function of position 62T 62T azT 8T k 2 2 2 q 6x 8y 82 p at or 62T 62T 62T 4 6T 2 2 2 ax ay 52 k w 62 1 05 Thermal diffusivity only arises in transient conduction analysis Simpli cations Continued SteadyState with Generation 62T 6 yr 4 8x2 6y2 822 k Steady State 2D Analysis With no energy generation Laplace Equation 62quot azr 0 8x2 62 For the 1D Heat Diffusion Equation if k is constant k 6 q p0 6T 8x2 p at We also need atO know the initial temperature and two boundary conditions in order to solve this secondorder partial differential equation In 3D we need the initial temperature and 6 boundary conditions to solve Cylindrical Coordinates Iz dz gr I39quot quotquot s I I x I warquot ran 1 J 4 dz 39 397 w l I I I i 39 39 Q I P I H mm 1quot 39 4 u p d x I w 39 9 quot 5quot quot39 y Spherical Coordinates amp rh9d 9 77039 9 9 3