The temperature of an exotic liquid in a pipe of length L = 2 can be described by the 1D non-dimensional heat equation ƏT (1) Ət with time t, temperature T, temperature dependent thermal diffusivity a(T), and a source term describing additional heating/cooling along the pipe, Q(x, t). On the left end of the pipe, the fluid has a temperature of sin(t). On the right end, the boundary is adiabatic At t=0 the temperature of the fluid is = = a (T): a(T) = T(x = 0, t) = 2 ƏT 0²T əx² ?x The thermal diffusivity of the fluid can be modeled as + Q(x, t) - (x = L, t) = 2 10 = 0. T(x, t0) = 2. = ao+a₁T+ a₂T² + a3T³. For a grid of M+ 1 equally spaced points along the pipe and h the spacing between points: Tasks: ล (3) (4) (5) a) write the equation for the temperature at the first point Tr using Eq. (2); b) write the equation for the temperature at the last point TM+1 as a function of the neighboring points using a second order accurate finite difference approximation to Eq. (3); c) write the equation for the right-hand-side fr of Eq. (1) at mesh point i with second order accurate central finite differences leaving Q(x, t) as Q(xi, t) or Q. Use only the above equaitons and not any specific parameter values given in later problems. In all answers do not forget the superscript n to indicate the discrete time level.
The temperature of an exotic liquid in a pipe of length L = 2 can be described by the 1D non-dimensional heat equation ƏT (1) Ət with time t, temperature T, temperature dependent thermal diffusivity a(T), and a source term describing additional heating/cooling along the pipe, Q(x, t). On the left end of the pipe, the fluid has a temperature of sin(t). On the right end, the boundary is adiabatic At t=0 the temperature of the fluid is = = a (T): a(T) = T(x = 0, t) = 2 ƏT 0²T əx² ?x The thermal diffusivity of the fluid can be modeled as + Q(x, t) - (x = L, t) = 2 10 = 0. T(x, t0) = 2. = ao+a₁T+ a₂T² + a3T³. For a grid of M+ 1 equally spaced points along the pipe and h the spacing between points: Tasks: ล (3) (4) (5) a) write the equation for the temperature at the first point Tr using Eq. (2); b) write the equation for the temperature at the last point TM+1 as a function of the neighboring points using a second order accurate finite difference approximation to Eq. (3); c) write the equation for the right-hand-side fr of Eq. (1) at mesh point i with second order accurate central finite differences leaving Q(x, t) as Q(xi, t) or Q. Use only the above equaitons and not any specific parameter values given in later problems. In all answers do not forget the superscript n to indicate the discrete time level.
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:The temperature of an exotic liquid in a pipe of length L
=
equation
ƏT
Ət
On the right end, the boundary is adiabatic
At t=0 the temperature of the fluid is
=
8²T
= a(T) + Q(x, t)
მ2
with time t, temperature T, temperature dependent thermal diffusivity a(T), and a source term describing additional
heating/cooling along the pipe, Q(x, t). On the left end of the pipe, the fluid has a temperature of
T(x=0, t) = 2
2 can be described by the 1D non-dimensional heat
ƏT
əx
2
10
The thermal diffusivity of the fluid can be modeled as
π
sin(t).
2
(x = L, t) = 0.
T(x, t0) = 2.
(1)
a(T) = ao+a₁T + a²T² + a3T³ .
For a grid of M+ 1 equally spaced points along the pipe and h the spacing between points:
Tasks:
(2)
(3)
(4)
(5)
M+1
a) write the equation for the temperature at the first point T
b) write the equation for the temperature at the last point T
order accurate finite difference approximation to Eq. (3);
c) write the equation for the right-hand-side fr of Eq. (1) at mesh point i with second order accurate central finite
differences leaving Q(x, t) as Q(xį, t") or Q. Use only the above equaitons and not any specific parameter
values given in later problems.
In all answers do not forget the superscript n to indicate the discrete time level.
using Eq. (2);
as a function of the neighboring points using a second
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