EBK NUMERICAL METHODS FOR ENGINEERS
7th Edition
ISBN: 9780100254145
Author: Chapra
Publisher: YUZU
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Chapter 29, Problem 13P
Develop a user-friendly computer program to implement Liebmann's method for a rectangular plate with Dirichlet boundary conditions. Design the program so that it can compute both temperature and flux. Test the program by duplicating the results of Examples 29.1 and 29.2.
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3. Using the trial function u¹(x) = a sin(x) and weighting function w¹(x) = b sin(x) find
an approximate solution to the following boundary value problems by determining the value
of coefficient a. For each one, also find the exact solution using Matlab and plot the exact
and approximate solutions. (One point each for: (i) finding a, (ii) finding the exact solution,
and (iii) plotting the solution)
a.
(U₁xx -2 = 0
u(0) = 0
u(1) = 0
b. Modify the trial function and find an approximation for the following boundary value
problem. (Hint: you will need to add an extra term to the function to make it satisfy
the boundary conditions.)
(U₁xx-2 = 0
u(0) = 1
u(1) = 0
I.C
02/A/ Use the Crank-Nicolson method to solve for the temperature distribution of a long thin rod
with a length of 10 cm and the following values: k = 0.49 cal/(s cm °C), Ax = 2 cm, and At =
st 0.1 s. Initially the temperature of the rod is 0°C and the boundary conditions are fixed for all times
at 7(0, t) = 100°C and 7(10, t) = 50°C. Note that the rod is aluminum with C = 0.2174 cal/g °C)
and p = 2.7 g/cm³. List the tridiagonal system of equations and determined the temperature up
to 0.1 s.
SOLVE IN DIGITAL FORMAT
Chapter 29 Solutions
EBK NUMERICAL METHODS FOR ENGINEERS
Ch. 29 - 29.1 Use Liebmann’s method to solve for the...Ch. 29 - 29.2 Use Liebmann’s method to solve for the...Ch. 29 - 29.3 Compute the fluxes for Prob. 29.2 using the...Ch. 29 - Repeat Example 29.1, but use 49 interior nodes...Ch. 29 - Repeat Prob. 29.4, but for the case where the...Ch. 29 - 29.6 Repeat Examples 29.1 and 29.3, but for the...Ch. 29 - Prob. 7PCh. 29 - 29.8 With the exception of the boundary...Ch. 29 - Write equations for the darkened nodes in the grid...Ch. 29 - 29.10 Write equations for the darkened nodes in...
Ch. 29 - Apply the control-volume approach to develop the...Ch. 29 - Derive an equation like Eq. (29.26) for the case...Ch. 29 - 29.13 Develop a user-friendly computer program to...Ch. 29 - Employ the program from Prob. 29.13 to solve...Ch. 29 - Employ the program from Prob. 29.13 to solve Prob....Ch. 29 - Use the control-volume approach and derive the...Ch. 29 - 29.17 Calculate heat flux for node in Fig. 29.13...Ch. 29 - 29.18 Compute the temperature distribution for...Ch. 29 - 29.19 The Poisson equation can be written in...
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- 3. Using the trial function uh(x) = a sin(x) and weighting function wh(x) = b sin(x) find an approximate solution to the following boundary value problems by determining the value of coefficient a. For each one, also find the exact solution using Matlab and plot the exact and approximate solutions. (One point each for: (i) finding a, (ii) finding the exact solution, and (iii) plotting the solution) a. (U₁xx - 2 = 0 u(0) = 0 u(1) = 0 b. Modify the trial function and find an approximation for the following boundary value problem. (Hint: you will need to add an extra term to the function to make it satisfy the boundary conditions.) (U₁xx - 2 = 0 u(0) = 1 u(1) = 0arrow_forwardThe steady-state distribution of temperature on a heated plate can be modeled by the Laplace equation, a²T ²T + a²x a²y If the plate is represented by a series of nodes as illustrated in Figure, centered finite-divided differences can be substituted for the second derivatives, which results in a system of linear algebraic equations. Use the Gauss-Seidel method to solve for the temperatures of the nodes in Figure. 0= Submission date: 09/01/2024 25°C T12 T₂2 250°C # T₁1 T₂1 250 CO 75°C 25°C 75°C 0°C 0°Carrow_forwardUse the Lax method to solve the inviscid Burgers' equation using a mesh with 51 points in the x direction. Solve this equation for a right propagating discontinu- ity with initial data u = 1 on the first 11 mesh points and u = 0 at all other points. Repeat your calculations for Courant numbers of 1.0, 0.6, and 0.3 and compare your numerical solutions with the analytical solution at the same time.arrow_forward
- 2. Solve the following ODE in space using finite difference method based on central differences with error O(h). Use a five node grid. 4u" - 25u0 (0)=0 (1)=2 Solve analytically and compare the solution values at the nodes.arrow_forwardQ2/A/ Use the Crank-Nicolson method to solve for the temperature distribution of a long thin rod C with a length of 10 cm and the following values: k = 0.49 cal/(s cm °C), Ax = 2 cm, and At = st 0.1 s. Initially the temperature of the rod is 0°C and the boundary conditions are fixed for all times C=0.2174 cal/g °C) at 7(0, t) = 100°C and T(10, t) = 50°C. Note that the rod is aluminum with and = 2.7 g/cm³. List the tridiagonal system of equations and determined the temperature up P to 0.1 s.arrow_forwardsolve by hand and explain step by step pls, thank you so mucharrow_forward
- i need the answer quicklyarrow_forwardDo not actually solve the problem numerically or algebraically, just pick the one equation and define the relevant knowns and single unknown. Don’t forget to include direction when called for by a vector variable 12) The air conditioner removes 2.7 kJ of heat from inside a house with 450 m3 of air in it. At a typical air density of 1.3 kg/m3 that means 585 kg of air. If the specific heat of air is 1.01 kJ/(kg oC), by how much would this cool the house if no heat got in through the rest of the house during that time?arrow_forwardQuestion 2: Air at the temperature of T1 is being heated with the help of a cylindrical cooling fin shown in the figure below. A hot fluid at temperature To passes through the pipe with radius Rc. a) Derive the mathematical model that gives the variation of the temperature inside the fin at the dynamic conditions. b) Determine the initial and boundary conditions to solve the equation derived in (a). air To Re Assumptions: 1. Temperature is a function of only r direction 2. There is no heat loss from the surface of A 3. Convective heat transfer coefficient is constantarrow_forward
- A// Use Implicit Method to solve the temperature distribution of a long thin rod with a length of 9 cm and following values: k = 0.49 cal/(s cm °C), Ax = 3 cm, and At = 0.2 s. At t=0 s, the temperature of the rod is 10°C and the boundary conditions are fixed dT (9,t) 1 °C/cm. Note that the rod for alltimes at 7(0,t) = 80°C and derivative condition dx is aluminum with C = 0.2174 cal/g °C) and p = 2.7 g/cm³. Find the temperature values on the inner grid points and the right boundary for t = 0.4 s.arrow_forwardWhich procedure provides a method that may be used to apply Cstigliano's second theorem?arrow_forwardQuestion 2: Air at the temperature of T1 is being heated with the help of a cylindrical cooling fin shown in the figure below. A hot fluid at temperature To passes through the pipe with radius Rc. a) Derive the mathematical model that gives the variation of the temperature inside the fin at the dynamic conditions. b) Determine the initial and boundary conditions to solve the equation derived in (a). air T1 Re Assumptions: 1. Temperature is a function of onlyr direction 2. There is no heat loss from the surface of A 3. Convective heat transfer coefficient is constantarrow_forward
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