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The ends of a metal rod of length L = 100 cm and thermal diffusivity a...
This is a PDE problem. I need help witb a, b and c 4.1.1. Suppose the ends of a bar of length l and therrnal diffusivity γ = 1 are held fixed at respective temperatures 0° and 10°. (a) Determine the equilibrium temperature profile (b) Determine the rate at which the equilibrium temperature profile is approached (c) What does the temperature profile look like as it nears equilibrium? 4.1.1. Suppose the ends of a bar of length l and therrnal...
Let a metallic rod 22 cm in length be heated to a uniform temperature of 90°C. Suppose that at t-0, the ends of the bar are plunged into an ice bath at 0°C and thereafter maintained at this temperature, but that no heat is allowed to escape through the lateral surface Find the time that will elapse before the center of the bar cools to a temperature of 5°C if the bar is made of the following materials. (A computer...
Problem 1: Axial vibrations of a rod The rod of length L is fixed at ends x = 0 and x = L. The density of the rod is ρ(x), stiffness k(x) being subjected to a force f(x, t). Let's derive the equations for axial vibrations of a rod using almped model. We express the rod niy mol 41 in as a chain of masses m,mm, connected to each other through springs as shown in the figure. Let's say each...
For (1) – (3), the model is with regards to a rod of length L with thermal diffusivity k coinciding along the interval (0, L) on the z-axis. Set up the boundary-value problem for the temperature u(x,t). (1) The left end is insulated and the right end is held at a temperature of 0°. The initial temperature is 1° throughout. (2) The left end is at a temperature of 50e-t, the right end if held at zero, and there is...
A metal rod of length a cm has initial temperature function f(x) = 2 sin 3x and its two ends are held at temperature zero for all time t>O The heat equation is given as: ди au 4 for 0 < x < it and t > 0 at @x? Boundary conditions: u(0,t) = u(1,t)=0, Initial conditions: u(x,0) = 2 sin 3x By using the method of separation of variables, calculate the general temperature u(x,t) for all cases, k =...
A metal rod of length a cm has initial temperature function f(x) = 2 sin 3x and its two ends are held at temperature zero for all time t>O The heat equation is given as: ди au 4 for 0 < x < it and t > 0 at @x? Boundary conditions: u(0,t) = u(1,t)=0, Initial conditions: u(x,0) = 2 sin 3x By using the method of separation of variables, calculate the general temperature u(x,t) for all cases, k =...
2. Solve the one-dimensional heat equation problem for a unit length bar with insulated ends with a prescribed initial linear temperature distribution: c2uxx = 111 , l4 (0,t)-14 (l,t)-0, 0 < x < 1 20-x) , Last Name A-M 3x, Last Name N -Z u(x,0) = The general solution to this problem is given in Example 4, page 563 in the text in terms of a Fourier Cosine Series. Write out the solution steps and evaluate the Fourier coefficients by...
2. (16pts) Consider the following heat equation for a rod of length L=1 with constant thermal properties (assume k=1): - xe" (0 < x <l, t>0) 04 (0,1) = 0 (1,1)= 1 a) Determine the equilibrium temperature distribution, and plot it on the interval [0, 1] b) Where does the energy enter, and where does it leave the rod? Explain your answers.
a-j (10 points) A semi-precious alloy rod of length 25cm is used in laboratory experiments in a varying temperature regime. The metal has the thermal diffu- sivity coefficient1.156 (0.34)2cm2/sec. The temperature at time t 0 at point x, 0 < < 25, is described by the Heat-conduction PDE α2uzz-u, The temperature at the end-points of the rod is maintained to be T1 = 10 and T2 - -6, by using a special thermostatic device. Every morning a tech- nician switches...
3. 3 points A metal rod has a length of 10cm, and has its left end at zn0 and right end at z = 5. The temperature T(z,1) of the rod satisfies the heat equation T,r(z,t)-m(r,t), its ends are kept at 0 for all t, and its temperature distribution at time t0 is given by T(a,0)a. Find the temperature function T(,t) for the rod 3. 3 points A metal rod has a length of 10cm, and has its left end...