1. Chapter 3 Steady-State Conduction Multiple Dimensions
HEAT TRANSFER LECTURE
CHAPER 3
Steady-State Conduction Multiple Dimensions
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2. 3-1 Introduction
In Chapter 2 steady-state heat transfer was calculated in systems in which the temperature gradient and area could be expressed in terms of one space coordinate. We now wish to analyze the more general case of two-dimensional heat flow. For steady state with no heat generation, the Laplace equation applies.
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(3-1)
The solution to this equation may be obtained by analytical, numerical, or graphical techniques.
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3. 3-1 Introduction
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The objective of any heat-transfer analysis is usually to predict heat flow or the temperature that results from a certain heat flow. The solution to Equation (3-1) will give the temperature in a two-dimensional body as a function of the two independent space coordinates x and y. Then the heat flow in the x and y directions may be calculated from the Fourier equations
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4. 3-2 Mathematical Analysis of Two-Dimensional Heat Conduction
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Analytical solutions of temperature distribution can be obtained for some simple geometry and boundary conditions. The separation method is an important one to apply.
Consider a rectangular plate.
Three sides are maintained at
temperature T1, and the upper
side has some temperature
distribution impressed on it.
The distribution can be a constant
temperature or something more
complex, such as a sine-wave.
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5. 3-2 Mathematical Analysis of Two-Dimensional Heat Conduction
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Consider a sine-wave distribution on the upper edge, the
boundary conditions are:
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6. 3-2 Mathematical Analysis of Two-Dimensional Heat Conduction
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Substitute:
We obtain two ordinary differential equations in terms of this constant,
where λ2 is called the separation constant.
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7. 3-2 Mathematical Analysis of Two-Dimensional Heat Conduction
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We write down all possible solutions and then see which one fits the problem under consideration.
This function cannot fit the sine-function boundary condition, so that the solution may be excluded.
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8. 3-2 Mathematical Analysis of Two-Dimensional Heat Conduction
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This function cannot fit the sine-function boundary condition, so that the solution may be excluded.
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9. 3-2 Mathematical Analysis of Two-Dimensional Heat Conduction
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It is possible to satisfy the sine-function boundary condition; so we shall attempt to satisfy the other condition.
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10. Apply the method of variable separation, let
3-2 Mathematical Analysis of Two-Dimensional Heat Conduction
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Let
The equation becomes:
Apply the method of variable separation, let
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11. And the boundary conditions become:
3-2 Mathematical Analysis of Two-Dimensional Heat Conduction
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And the boundary conditions become:
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12. 3-2 Mathematical Analysis of Two-Dimensional Heat Conduction
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Applying these conditions,we have:
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13. 3-2 Mathematical Analysis of Two-Dimensional Heat Conduction
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accordingly,
and from (c),
This requires that
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14. 3-2 Mathematical Analysis of Two-Dimensional Heat Conduction
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then
We get
The final boundary condition may now be applied:
which requires that Cn =0 for n >1.
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15. 3-2 Mathematical Analysis of Two-Dimensional Heat Conduction
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The final solution is therefore
The temperature field for this problem is shown. Note that the heat-flow lines are perpendicular to the isotherms.
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16. 3-2 Mathematical Analysis of Two-Dimensional Heat Conduction
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Another set of boundary conditions
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17. 3-2 Mathematical Analysis of Two-Dimensional Heat Conduction
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Using the first three boundary conditions, we obtain the solution in the form of Equation:
Applying the fourth boundary condition gives
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18. 3-2 Mathematical Analysis of Two-Dimensional Heat Conduction
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This series is
then
The final solution is expressed as
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19. 3-2 Mathematical Analysis of Two-Dimensional Heat Conduction
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Transform the boundary condition:
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20. 3-3 Graphical Analysis
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neglect
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21. 3-4 The Conduction Shape Factor
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Consider a general one dimensional heat conduct-
ion problem, from Fourier’s Law:
let
then
where：S is called shape factor.
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22. 3-4 The Conduction Shape Factor
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Note that the inverse hyperbolic cosine can be calculated from
For a three-dimensional wall, as in a furnace, separate shape factors are used to calculate the heat flow through the edge and corner sections, with the dimensions shown in Figure 3-4. when all the interior dimensions are greater than one fifth of the thickness,
where A = area of wall, L = wall thickness, D = length of edge
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23. 3-4 The Conduction Shape Factor
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24. 3-4 The Conduction Shape Factor
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25. 3-4 The Conduction Shape Factor
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26. 3-4 The Conduction Shape Factor
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27. 3-4 The Conduction Shape Factor
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28. 3-4 The Conduction Shape Factor
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29. 3-4 The Conduction Shape Factor
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30. Example 3-1
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31. Example 3-2
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32. Example 3-3
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33. Example 3-4
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34. 3-5 Numerical Method of Analysis
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The most fruitful approach to the heat conduction is one based on finite-difference techniques, the basic principles of which we shall outline in this section.
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35. 3-5 Numerical Method of Analysis
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1、Discretization of the solving
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36. 3-5 Numerical Method of Analysis
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2、Discrete equation
Taylor series expansion
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37. Differential equation for two-dimensional steady-state heat flow
3-5 Numerical Method of Analysis
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2、Discrete equation
Differential equation for two-dimensional steady-state heat flow
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38. 3-5 Numerical Method of Analysis
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2、Discrete equation
Discrete equation at nodal point (m,n)
no heat generation
Δx= Δy
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39. 3-5 Numerical Method of Analysis
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2、Discrete equation
Thermal balance
(1) Interior points
steady-state & no heat generation
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40. 3-5 Numerical Method of Analysis
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Thermal balance
(1) Interior points
Δx= Δy
steady-state with heat generation
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41. 3-5 Numerical Method of Analysis
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2、Discrete equation
Thermal balance
(2) boundary points
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42. 3-5 Numerical Method of Analysis
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Thermal balance
(2) boundary points
Δx= Δy
Δx= Δy
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43. 3-5 Numerical Method of Analysis
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Thermal balance
(2) boundary points
Δx= Δy
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44. 3-5 Numerical Method of Analysis
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3、Algebraic equation
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45. 3-5 Numerical Method of Analysis
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Matrix notation
Iteration
Simple Iteration & Gauss-Seidel Iteration
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46. Example 3-5
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Consider the square shown in the figure. The left face is maintained at 100℃ and the top face at 500℃, while the other two faces are exposed to a environment at 100℃. h=10W/m2·℃ and k=10W/m·℃. The block is 1 m square. Compute the temperature of the various nodes as indicated in the figure and heat flows at the boundaries.
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47. Example 3-5 [Solution] The equations for nodes 1,2,4,5 are given by
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[Solution]
The equations for nodes 1,2,4,5 are given by
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48. Example 3-5 [Solution] Equations for nodes 3,6,7,8 are
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[Solution]
Equations for nodes 3,6,7,8 are
The equation for node 9 is
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49. Example 3-5
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The equation for node 9 is
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50. Example 3-5
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We thus have nine equations and nine unknown nodal temperatures So the answer is
For the 500℃ face, the heat flow into the face is
The heat flow out of the 100℃ face is
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51. Example 3-5 The heat flow out the right face is
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The heat flow out the right face is
The heat flow out the bottom face is
The total heat flow out is
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52. 3-6 Numerical Formulation in Terms of Resistance Elements
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Thermal balance — the net heat input to node i must be zero
qi — heat generation, radiation, etc.
i — solving node
j — adjoining node
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53. 3-6 Numerical Formulation in Terms of Resistance Elements
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54. 3-7 Gauss-Seidel Iteration
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Steps
Assumed initial set of values for Ti；
Calculated Ti according to the equation；
—using the most recent values of the Ti
Repeated the process until converged.
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55. 3-7 Gauss-Seidel Iteration
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Convergence Criterion
Biot number
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56. Example 3-6
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Apply the Gauss-Seidel technique to obtain the nodal temperature for the four nodes in the figure.
[Solution]
All the connection resistance between the nodes are equal, that is
Therefore, we have
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57. Example 3-6
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Because each node has four resistance connected to it and k is assumed constant, so
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58. 3-8 Accuracy Consideration
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Truncation Error — Influenced by difference scheme
Discrete Error — Influenced by truncation error & △x
Round-off Error — Influenced by △x
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59. Summary Numerical Method Solving Zone Nodal equations
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Numerical Method
Solving Zone
Nodal equations
thermal balance method — Interior & boundary point
Algebraic equations
Gauss-Seidel iteration
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60. Summary (2)Resistance Forms (3)Convergence Convergence Criterion CUMT
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(2)Resistance Forms
(3)Convergence
Convergence Criterion
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61. Summary (4)Accuracy Truncation Error Discrete Error Round-off Error
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(4)Accuracy
Truncation Error
Discrete Error
Round-off Error
Important conceptions
Nodal equations — thermal balance method
Calculated temperature & heat flow
Convergence criterion
How to improve accuracy
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62. Exercises
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Exercises: 3-16, 3-24, 3-48, 3-59
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