Determinc the equilibrium temperature distribution for a one-dimensional rod with constant thermal properties with the following...
(a) The temperature distribution u(x, t) of the one- dimensional silver rod is governed by the heat equation as follows. du a²u at ar? Given the boundary conditions u(0,t) = t?, u(0.6, t) = 5t, for Osts 0.02s and the initial condition u(x,0) = x(0.6 – x) for 0 SX s 0.6mm, analyze the temperature distribution of the rod with Ax = 0.2mm and At = 0.01s in 4 decimal places. (10 marks)
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.
Consider the one-dimensional heat equation for nonconstant thermal properties debelo - (Kolon with the initial condition u(x, 0) = f(x). [Hint: Suppose it is known that if u(x, t) = ??(x ) h(t), then 1 dh 1 d do Ko(c) = -1 h dt c(x)p(x)o dx dx You may assume the eigenfuctions are known. Briefly discuss limt- u(x, t). Solve the initial value problem: (a) with boundary conditions u(0, t) = 0 and u(L, t) = 0 au *(b) with...
A long solid rod of constant thermophysical properties and radius ro is initially at a uniform temperature Tj. At time t = 0, the temperature of the peripheral surface at r=r, is changed to Tw and is subsequently maintained constant at this value for t> 0. (a) Show the governing equation with the boundary conditions. (b) Redefine the temperature for the homogeneous boundary conditions. (c) Show the separation of variables. (d) Show how to obtain the eigenvalues. (e) Obtain an...
4. Which one of the following is the correct expression for one-dimensional constant properties, heat conduction equation for a cylinder with heat eady-e generation r dr qun 5. A 10 cm diameter sphere maintained at 30°C is buried in the earth at a place where the ·K. The depth to the centerline is 24 cm, and the thermal conductivity, k = 1.2 W/m earth surface temperature is 0°C. Calculate the heat loss from the sphere. (A) 25.3W (D)42.4W (E)20.0 w...
19. The temperature distribution in a plane wall will be during steady and one-dimensional heat transfer with non-constant wall thermal conductivity. a. Straight line b. Linear c. Non-linear
(1) A sphere of decaying radioactive material of radius ro produces heat at a rate of q"" (W/m3). The sphere is contained in a spherical shell of graphite of outside radius r1. The outside surface of the graphite is cooled uniformly by flowing air of temperature To. The heat transfer coefficient at the outside surface is h. The constant thermal conductivities of the radioactive material and the graphite are ko and ki, respectively. Densities and heat capacities are ρο, co...
(a) Consider the one-dimensional heat equation for the temperature u(x, t), Ou,02u where c is the diffusivity (i) Show that a solution of the form u(x,t)-F )G(t) satisfies the heat equation, provided that 护F and where p is a real constant (ii) Show that u(x,t) has a solution of the form (,t)A cos(pr)+ Bsin(p)le -P2e2 where A and B are constants (b) Consider heat flow in a metal rod of length L = π. The ends of the rod, at...
heat transfer
Consider a long solid rod of constant thermal conductivity k whose cross section is a sector of a circle of radius ro and the angle a as shown in the figure. A peripheral heat flux 9":falls onto the peripheral surface. The plane surface at - O is kept isothermal at the ambient temperature T.. The other plane surface at = a loses heat by convection to the ambient. The steady temperature distribution is a function of r and...
Determine the transient temperature distribution in a one-dimensional (1-D) fluid with a thermal diffusivity a=2 and velocity u = 0.1, if the initial temperature in the fluid is 0.0° and if at all subsequent times, the temperature of the left side is held at 0° while the right side is held at To=2º. The distance between the two walls is L=1 meter. T=0 TT 0 The governing differential equation is the 1-D heat convection diffusion equation at at 22T +...