2 D Steady State Conduction

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Two Dimensional Steady State Conduction

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Numerical methods • Analytical solutions that allow for the determination of the exact temperature distribution are only available for limited ideal cases. • Graphical solutions have been used to gain an insight into complex heat transfer problems, where analytical solutions are not available, but they have limited accuracy and are primarily used for two-dimensional problems. • Advances in numerical computing now allow for complex heat transfer problems to be solved rapidly on computers, i.e., "numerical techniques“. • Current numerical techniques include: finite-difference analysis; finite element analysis (FEA); and finite-volume analysis. • In general, these techniques are routinely used to solve problems in heat transfer, fluid dynamics, stress analysis, electrostatics and magnetics, etc. • Use of finite-difference analysis to solve conduction heat transfer problems. Department of Mechanical Engineering, NUST College of E&ME 3

• Local heat flux is a vector perpendicular to isothermal lines. • Heat flow lines represents the direction of heat flux vectors. • Heat flow lines refers to adiabats. • Appropriate form of the heat equation for two-dimensional steady state case with no heat generation Department of Mechanical Engineering, NUST College of E&ME 4

Finite-difference Analysis • Numerical techniques result in an approximate solution, however the error can be made very small. • Properties (e.g., temperature) are determined at discrete points in the region of interest-these are referred to as nodal points or nodes.

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Finite-difference Analysis Consider the finite-difference technique for 2-D conduction heat transfer: • In this case each node represents the temperature of a point on the surface being considered. • The temperature at the node represents the average temperature of that region of the surface. • Algebraic expressions are used to define the relationship between adjacent nodes on the surface –usually the boundary conditions are specified. • By increasing the number of nodes on the surface being considered it is possible to increase the spatial resolution of the solution and to potentially increase the accuracy of the numerical solution, however this increases the number of calculation is required to obtain a solution to the problem.

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EXAMPLE 4.2 Using the energy balance method, derive the finite-difference equation for the m, n nodal point located on a plane, insulated surface of a medium with uniform heat generation.

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Problem: Consider the square channel shown in the sketch operating under steady-state conditions. The inner surface of the channel is at a uniform temperature of 600 K, while the outer surface is exposed to convection with a fluid at 300 K and a convection coefficient of 50W/m2 .K. From a symmetrical element of the channel, a two-dimensional grid has been constructed and the nodes labeled. The temperatures for nodes 1, 3, 6, 8, and 9 are identified. (a) Beginning with properly defined control volumes, derive the finite-difference equations for nodes 2, 4, and 7 and determine the temperatures T2, T4, and T7 . (b) Calculate the heat loss per unit length from the channel.

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Consider steady-state conditions for which heat is uniformly generated at a volumetric rate q . due to passage of an electric current. Using the energy balance method, derive finite-difference equations for nodes 1 and 13. Department of Mechanical Engineering, NUST College of E&ME 20

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Homework Problems : 4.32, 4.34, 4.37, 4.41, 4.45. Assignment 2 (1) Problems : 4.33, 4.39, 4.46 (2) Derive nodal finite difference equations (case 3, 4, 5) Due on 16 Apr 2019

Text Book : Incropera/DeWitt/Bergman/Lavine, Sixth Edition.

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