Homework Help is Here – Start Your Trial Now!
Engineering
Computer Science
Case 2:Temperature Distrbution In Thin Rod The differential equation derives from a heat balance for along thin rod as follow: d?T + h' (Ta – T) = 0.. .... eq(1) - dx2 Where T=temperature ("C), =distance along the rod (m), h'=a heat transfer coefficent between the rod and the ambient air (m2), and T=the temperature of the surrounding air (°C).This equation can be transformed into a set of linear algebraic equations by using a finite dividd difference approximation for the second derivative as follow: d?T Ti+1 - 27; + Ti-1 .. .eq(2) ......... dx? Δx2 Where Ti designates the temperature at node i. This approximation can be substituded into eq(1) to give: -Ti-1+ (2+ h'Ax?)Ti – Ti+1 = h'Ax2Ta. eq(3) This equation can be written for each of the interior nodes of the rod resulting in a tridiagonal system of equations. The first and the last nodes at the rods ends are fixed by boundary conditions. T(r=0)=200 °C, T(x=10)= 40 °C, h'=0.02 m?, T=10 °C, Ar=2m. If this system of linear equation can be solved using Gauss-Seidel and Jacobi iterations methods. Please answer the following: 1. Develop a MATLAB program to solve the system above using both Gauss-Seidel and Jacobi iteration methods after n of iterations. 2. Plot the values of each tempratures (T1,T2,T3,T4) versus iteration number in both solutions. Compare the values you get from both solution methods. 3. Plot the values of the relative approximate error of each temperatures (Ea) versus iteration number in both solutions. Compare the values you get from both solution methods. 4. Use LU decomposition to find the exact solution {Ti} and compare your exact values to those you got from point 1. 5. Plot the temperatures you got from point 4 along the rod.

Case 2:Temperature Distrbution In Thin Rod The differential equation derives from a heat balance for along thin rod as follow: d?T + h' (Ta – T) = 0.. .... eq(1) - dx2 Where T=temperature ("C), =distance along the rod (m), h'=a heat transfer coefficent between the rod and the ambient air (m2), and T=the temperature of the surrounding air (°C).This equation can be transformed into a set of linear algebraic equations by using a finite dividd difference approximation for the second derivative as follow: d?T Ti+1 - 27; + Ti-1 .. .eq(2) ......... dx? Δx2 Where Ti designates the temperature at node i. This approximation can be substituded into eq(1) to give: -Ti-1+ (2+ h'Ax?)Ti – Ti+1 = h'Ax2Ta. eq(3) This equation can be written for each of the interior nodes of the rod resulting in a tridiagonal system of equations. The first and the last nodes at the rods ends are fixed by boundary conditions. T(r=0)=200 °C, T(x=10)= 40 °C, h'=0.02 m?, T=10 °C, Ar=2m. If this system of linear equation can be solved using Gauss-Seidel and Jacobi iterations methods. Please answer the following: 1. Develop a MATLAB program to solve the system above using both Gauss-Seidel and Jacobi iteration methods after n of iterations. 2. Plot the values of each tempratures (T1,T2,T3,T4) versus iteration number in both solutions. Compare the values you get from both solution methods. 3. Plot the values of the relative approximate error of each temperatures (Ea) versus iteration number in both solutions. Compare the values you get from both solution methods. 4. Use LU decomposition to find the exact solution {Ti} and compare your exact values to those you got from point 1. 5. Plot the temperatures you got from point 4 along the rod.

C++ for Engineers and Scientists
C++ for Engineers and Scientists
4th Edition
ISBN:9781133187844
Author:Bronson, Gary J.
Publisher:Bronson, Gary J.
Chapter6: Modularity Using Functions
Section6.2: Returning A Single Value
Problem 8E
See similar textbooks
icon
Related questions
Question
100%
applied numericl methods
T= 10 To = 200 Ts = 40 Ax T = 10 X= 0 x = 10 %3D
Transcribed Image Text:T= 10 To = 200 Ts = 40 Ax T = 10 X= 0 x = 10 %3D
Case 2:Temperature Distrbution In Thin Rod The differential equation derives from a heat balance for along thin rod as follow: d?T +h' (Ta – T) = 0.. .eq(1) dx2 Where T=temperature (°C), r-distance along the rod (m), h'=a heat transfer coefficent between the rod and the ambient air (m2), and T=the temperature of the surrounding air ("C).This equation can be transformed into a set of linear algebraic equations by using a finite dividd difference approximation for the second derivative as follow: d?T Ti+1 - 2T¡ + Ti-1 .eq(2) *........... dx2 Δx2 Where Ti designates the temperature at node i. This approximation can be substituded into eq(1) to give: -Ti-1+ (2+ h'Ax?)T: – T+1 = hAx?Ta. eq(3) This equation can be written for each of the interior nodes of the rod resulting in a tridiagonal system of equations. The first and the last nodes at the rods ends are fixed by boundary conditions. T(x=0)=200 "C, T(x=10)= 40 °C, h'=0.02 m2, T,=10 °C, Ar=2m. If this system of linear equation can be solved using Gauss-Seidel and Jacobi iterations methods. Please answer the following: 1. Develop a MATLAB program to solve the system above using both Gauss-Seidel and Jacobi iteration methods after n of iterations. 2. Plot the values of each tempratures (T1,T2,T3,T4) versus iteration number in both solutions. Compare the values you get from both solution methods. 3. Plot the values of the relative approximate error of each temperatures (ɛa) versus iteration number in both solutions. Compare the values you get from both solution methods. 4. Use LU decomposition to find the exact solution {T} and compare your exact values to those you got from point 1. 5. Plot the temperatures you got from point 4 along the rod.
Transcribed Image Text:Case 2:Temperature Distrbution In Thin Rod The differential equation derives from a heat balance for along thin rod as follow: d?T +h' (Ta – T) = 0.. .eq(1) dx2 Where T=temperature (°C), r-distance along the rod (m), h'=a heat transfer coefficent between the rod and the ambient air (m2), and T=the temperature of the surrounding air ("C).This equation can be transformed into a set of linear algebraic equations by using a finite dividd difference approximation for the second derivative as follow: d?T Ti+1 - 2T¡ + Ti-1 .eq(2) *........... dx2 Δx2 Where Ti designates the temperature at node i. This approximation can be substituded into eq(1) to give: -Ti-1+ (2+ h'Ax?)T: – T+1 = hAx?Ta. eq(3) This equation can be written for each of the interior nodes of the rod resulting in a tridiagonal system of equations. The first and the last nodes at the rods ends are fixed by boundary conditions. T(x=0)=200 "C, T(x=10)= 40 °C, h'=0.02 m2, T,=10 °C, Ar=2m. If this system of linear equation can be solved using Gauss-Seidel and Jacobi iterations methods. Please answer the following: 1. Develop a MATLAB program to solve the system above using both Gauss-Seidel and Jacobi iteration methods after n of iterations. 2. Plot the values of each tempratures (T1,T2,T3,T4) versus iteration number in both solutions. Compare the values you get from both solution methods. 3. Plot the values of the relative approximate error of each temperatures (ɛa) versus iteration number in both solutions. Compare the values you get from both solution methods. 4. Use LU decomposition to find the exact solution {T} and compare your exact values to those you got from point 1. 5. Plot the temperatures you got from point 4 along the rod.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 4 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Knowledge Booster
Fundamentals of Boolean Algebra and Digital Logics
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, computer-science and related others by exploring similar questions and additional content below.
Similar questions
  • SEE MORE QUESTIONS
Recommended textbooks for you
C++ for Engineers and Scientists
C++ for Engineers and Scientists
Computer Science
ISBN:
9781133187844
Author:
Bronson, Gary J.
Publisher:
Course Technology Ptr
Operations Research : Applications and Algorithms
Operations Research : Applications and Algorithms
Computer Science
ISBN:
9780534380588
Author:
Wayne L. Winston
Publisher:
Brooks Cole
SEE MORE TEXTBOOKS