Concept explainers
Repeat Prob. 27.28, but for the following heat source:
(a)
To calculate: The temperature distribution by shooting method for a heated rod with a uniform heat source given by Poisson equation,
, where heat source
, also
Answer to Problem 29P
Solution: The table of the solution of the boundary value problem is,
Explanation of Solution
Given:
A differential equation,
, where heat source
, also
Formula used:
Linear-interpolation formula:
Calculation:
Consider the following Poisson equation,
Since,
. Then,
Now, change the above boundary value problem into equivalent initial-value problem. Then,
But
Thus,
Use the shooting method in the above system of first order linear differential equation
Suppose,
Then, the system of system of first order linear differential equation with initial condition is,
Now, solve the above system of differential equation.
Thus,
Integrate on both the sides of the above differential equation, to get
Where
Now, use the initial condition,
. Thus,
Put the value of
Thus,
Since,
. But
Therefore,
Integrate on both the sides of the above differential equation, to get
Where
is constant of integration.
Use the initial condition,
. Then,
Put the value of
, then
Now, evaluate the above for
. Thus,
But the above of
Then, put another guess. Suppose
Then, the system of system of first order linear differential equation with initial condition is,
Now, solve the above system of differential equation. Then,
Integrate on both the sides of the above differential equation
Thus,
Where
Now, use the initial condition,
Therefore,
Put the value of
Since,
. But
Thus,
Integrate on both the sides of the above differential equation. Then,
Where
Use the initial condition,
. Thus,
Put the value of
Now, evaluate the above for
. Thus,
Since, the first guess value
corresponds to
and the second-guess value
corresponds to
Now, use these values to compute the value of
that yields
Then, by linear interpolation formula,
Therefore, the right value of
which yields
Then, the equivalent initial value problem corresponding to the boundary value problem is,
Now, use the fourth order RK method with step size
The RK method for above system of first order linear differential equation with initial condition is,
Where
And
And
Where
Then, for
And
Also,
Thus,
And
In the similar way, find the remaining
. Then,
And
Therefore, the table of the solution of the boundary value problem is
Hence, the graph of the temperature distribution is
(b)
To calculate: The temperature distribution by finite difference method for a heated rod with a uniform heat source given by Poisson equation,
, where heat source
, also
Answer to Problem 29P
Solution:
The table of the solution of boundary value problem is
Explanation of Solution
Given:
A differential equation,
, where heat source
, also
Formula used:
(1) The finite difference method is:
(2) The Gauss-Seidel iterative method is:
Calculation:
Consider the following Poisson equation,
Since,
Thus,
The finite difference method is given by,
Now, substitute the value of second order derivative in the boundary value problem.
Then, the boundary value problem becomes,
Or
Since,
. Then,
For the first node,
For the second node,
For the third node,
For the fourth node,
Then, write the system of equations in matrix form
Since, the coefficient matrix is tridiagonal matrix, then use Gauss-Seidel iterative technique
The Gauss-Seidel iterative method is,
Now, evaluate
by above Gauss-Seidel method
Then,
And
Then, the table of the solution of boundary value problem is
Therefore, the graph of temperature distribution is
Want to see more full solutions like this?
Chapter 27 Solutions
Numerical Methods for Engineers
Additional Math Textbook Solutions
Elementary Statistics: A Step By Step Approach
University Calculus: Early Transcendentals (4th Edition)
APPLIED STAT.IN BUS.+ECONOMICS
Probability And Statistical Inference (10th Edition)
College Algebra (Collegiate Math)
Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)
- 3. An engine has three cylinders spaced at 120° to each other. The crank torque diagram can be simplified to a triangle having the following values: Angle 0° Torque (Nm) 0 (a) What is the mean torque? 60° 4500 180° 180° to 360° 0 0 (b) What moment of inertia of flywheel is required to keep the speed to within 180 ± 3 rpm? (c) If one cylinder of the engine is made inoperative and it is assumed that the torque for this cylinder is zero for all crank angles, determine the fluctuation in speed at 180rpm for the same flywheel. (a) 3375 Nm (b) 50kgm (c) ±21 rpmarrow_forwardProb 5. Determine the largest load P that can be applied to the frame without causing either the average normal stress or the average shear stress at section a-a to exceed o-150 MPa and 1-60 MPa, respectively. Member CB has a square cross section of 25 mm on each side. 2 m FAC 1.5 m Facarrow_forwardDerive the component transformation equations for tensors shown below where [C] = [BA] is the DCM (direction cosine matrix) from frame A to B. ^B [T] = [C]^A [T] [C]^Tarrow_forward
- Calculate for the vertical cross section moment of inertia for both Orientations 1 and 2 of a 1 x 1.5 in. horizontal hollow rectangular beam with wall thickness of t = 0.0625 in. Use the equation: I = ((bh^3)/12) - (((b-2t)(h-2t)^3)/12)arrow_forwardPlease answer 'yes' or 'no' and 'is' or 'is not' for the following:arrow_forwardConsider a large 23-cm-thick stainless steel plate (k = 15.1 W/m-K) in which heat is generated uniformly at a rate of 5 x 105 W/m³. Both sides of the plate are exposed to an environment at 30°C with a heat transfer coefficient of 60 W/m²K. The highest temperature will occur at surfaces of plate while the lowest temperature will occur at the midplane. Yes or No Yes Noarrow_forward
- My answers are incorrectarrow_forwardPicturearrow_forwardWhat is the weight of a 5-kg substance in N, kN, kg·m/s², kgf, Ibm-ft/s², and lbf? The weight of a 5-kg substance in N is 49.05 N. The weight of a 5-kg substance in kN is KN. The weight of a 5-kg substance in kg·m/s² is 49.05 kg-m/s². The weight of a 5-kg substance in kgf is 5.0 kgf. The weight of a 5-kg substance in Ibm-ft/s² is 11.02 lbm-ft/s². The weight of a 5-kg substance in lbf is 11.023 lbf.arrow_forward
- Mych CD 36280 kg. 0.36 givens Tesla truck frailer 2017 Model Vven 96154kph ronge 804,5km Cr Powertrain Across PHVAC rwheel 0.006 0.88 9M² 2 2kW 0.55M ng Zg Prated Trated Pair 20 0.95 1080 kW 1760 Nm 1,2 determine the battery energy required to meet the range when fully loaded determine the approximate time for the fully-loaded truck-trailor to accelerate from 0 to 60 mph while Ignoring vehicle load forcesarrow_forward12-217. The block B is sus- pended from a cable that is at- tached to the block at E, wraps around three pulleys, and is tied to the back of a truck. If the truck starts from rest when ID is zero, and moves forward with a constant acceleration of ap = 0.5 m/s², determine the speed of the block at D the instant x = 2 m. Neglect the size of the pulleys in the calcu- lation. When xƊ = 0, yc = 5 m, so that points C and D are at the Prob. 12-217 5 m yc =2M Xparrow_forwardsolve both and show matlab code auto controlsarrow_forward
- Principles of Heat Transfer (Activate Learning wi...Mechanical EngineeringISBN:9781305387102Author:Kreith, Frank; Manglik, Raj M.Publisher:Cengage Learning