2. We've previously studied heat conduction in one spatial dimension that included conduction and convection (through Newton's law of cooling) and saw that, using finite differences, we arrive at a linear system of equations. If we now include radiation, we'll have instead a nonlinear system of equations upon discretization. Consider the following model for one-dimensional heat transfer that includes conduction, convection, and radiation (20 pts). d²T dx2 + h(T∞ − T) + 0 (T - T4) = 0, 0 < x < L σ T(0) = Ta, T(L) = πb (a) By hand, discretize the equations, for an arbitrary number of points N, using a central difference method and write down the resulting equations in a form similar to the discretized heat equation in the notes (5 pts). (b) Explain why the resulting system of equations is now nonlinear (5 pts). (c) Instead of solving the nonlinear system of equations, we may linearize it. Employ a first-order Taylor series expansion to linearize the quartic term in the equation as σT¹ = σù +4σó(T – Ñ), - where T is a known base temperature about which the term is linearized (5 pts). (d) Substitute this linearized relationship above into the differential equation and write down the system of equations using the linearized form of the radiation term (3 pts). (e) Show that now we have a linear system of equations (2 pts).
2. We've previously studied heat conduction in one spatial dimension that included conduction and convection (through Newton's law of cooling) and saw that, using finite differences, we arrive at a linear system of equations. If we now include radiation, we'll have instead a nonlinear system of equations upon discretization. Consider the following model for one-dimensional heat transfer that includes conduction, convection, and radiation (20 pts). d²T dx2 + h(T∞ − T) + 0 (T - T4) = 0, 0 < x < L σ T(0) = Ta, T(L) = πb (a) By hand, discretize the equations, for an arbitrary number of points N, using a central difference method and write down the resulting equations in a form similar to the discretized heat equation in the notes (5 pts). (b) Explain why the resulting system of equations is now nonlinear (5 pts). (c) Instead of solving the nonlinear system of equations, we may linearize it. Employ a first-order Taylor series expansion to linearize the quartic term in the equation as σT¹ = σù +4σó(T – Ñ), - where T is a known base temperature about which the term is linearized (5 pts). (d) Substitute this linearized relationship above into the differential equation and write down the system of equations using the linearized form of the radiation term (3 pts). (e) Show that now we have a linear system of equations (2 pts).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:2. We've previously studied heat conduction in one spatial dimension that included conduction
and convection (through Newton's law of cooling) and saw that, using finite differences, we
arrive at a linear system of equations. If we now include radiation, we'll have instead a nonlinear
system of equations upon discretization. Consider the following model for one-dimensional
heat transfer that includes conduction, convection, and radiation (20 pts).
d²T
dx2
+ h(T∞ − T) + 0 (T - T4) = 0, 0 < x < L
σ
T(0) = Ta, T(L) = πb
(a) By hand, discretize the equations, for an arbitrary number of points N, using a central difference
method and write down the resulting equations in a form similar to the discretized heat equation in
the notes (5 pts).
(b) Explain why the resulting system of equations is now nonlinear (5 pts).
(c) Instead of solving the nonlinear system of equations, we may linearize it. Employ a first-order
Taylor series expansion to linearize the quartic term in the equation as
σT¹ = σù +4σó(T – Ñ),
-
where T is a known base temperature about which the term is linearized (5 pts).
(d) Substitute this linearized relationship above into the differential equation and write down the
system of equations using the linearized form of the radiation term (3 pts).
(e) Show that now we have a linear system of equations (2 pts).
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