Chapter4 - Chapter 4: Two-Dimensional Steady-State...

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ChEn 323 – Chapter 4 – Page 1 Chapter 4: Two-Dimensional Steady-State Conduction ¾ Solution alternatives ¾ Method of separation of variables ¾ Introduction to graphical method ¾ Finite Difference Form of the Heat Equation ¾ Energy Balance Form of the Finite Difference Heat Equation ¾ Solution methods for Finite Difference Equations ¾ Introduction to Transient Conduction ¾ Lumped Capacitance Method
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ChEn 323 – Chapter 4 – Page 2 ¾ Until now we have only worked in 1 dimension and frequently that is just fine. However, there are cases when that assumption will not get good answers --- thus 2-dimensions ¾ Solution alternatives. ± Cf: General equation from chapter 2: . ± So for steady state with no internal generation (2D); . ± Numerical solution by either finite element or finite difference methods – we will strive to do some of this later in the summer (3 weeks?). This approach is at discrete points & it is approximate. However, it is often the only way to do it. ± Analytical solution is normally done by separation of variables. t T C q dz dT k z dy dT k y dx dT k x p = + + + ρ & 0 2 2 2 2 = + y T x T
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ChEn 323 – Chapter 4 – Page 3 Method of Separation of Variables
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ChEn 323 – Chapter 4 – Page 4 Method of Separation of Variables Boundary Conditions
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ChEn 323 – Chapter 4 – Page 5 Method of Separation of Variables – Boundary Conditions (Continued)
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ChEn 323 – Chapter 4 – Page 6 Method of Separation of Variables – Demonstration of Orthogonal Function
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ChEn 323 – Chapter 4 – Page 7 Method of Separation of Variables -- Solution
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Chapter4 - Chapter 4: Two-Dimensional Steady-State...

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