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Let A be the unit disk. Assume it is made of a heat conducting material and...
3/5 25 pts.J A slab of thickness L, made of material with constant thermal conductivity k,...
3/5 25 pts.J A slab of thickness L, made of material with constant thermal conductivity k, is undergoing a 1-D, steady heat transfer. Its boundary surface at x 0 is insulated while the boundary surface at x= 1 is kept at constant temperature T= oc. Heat energy is generated within the slab at a rate of 2. qx)o cos(rx/2L) is the energy generation rate per unit volume (Wm) at x= 0. where qo a. Develop an expression for the steady-state...
(1) A sphere of decaying radioactive material of radius ro produces heat at a rate of q"" (W/m3)....
(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...
1. Consider the insulated heat equation up = cum, 0 <r<L, t > 0 u (0,t)...
1. Consider the insulated heat equation up = cum, 0 <r<L, t > 0 u (0,t) = u (L, t) = 0, t > 0 u(x,0) = f(2). What is the steady-state solution? 2. Solve the two-dimensional wave equation (with c=1/) on the unit square (i.e., [0, 1] x [0,1) with homogeneous Dirichlet boundary conditions and initial conditions: (2, y,0) = sin(x) sin(y) (,y,0) = sin(x). 3. Solve the following PDE: Uzr + Uyy = 0, 0<<1,0 <y < 2...
Question 1 [Total 20 marks] (a) [5 marks] In a steady-state two-dimensional heat flow problem, th...
Question 1 [Total 20 marks] (a) [5 marks] In a steady-state two-dimensional heat flow problem, the temperature, u, at any point in the domain (t, ) satisfies the differential equation u y(2-y) u= U0F With the given temperature boundary condition as follows: u(x, 0) = 0, u(x, 2) = x(4-x), 0 < x < 4 Calculate the temperature at the interior points a, b, and c using a mesh size h-1.
Question 1 [Total 20 marks] (a) [5 marks] In...
Ry 82. Let f(x, y) - 0 if (x, y)-0 The graph of f is shown on page 813. a. Show that j x,0) - x f...
ry 82. Let f(x, y) - 0 if (x, y)-0 The graph of f is shown on page 813. a. Show that j x,0) - x for all x, and a0, y) y for all y. b. Show that (0, 0) ахау (0, 0) The three-dimensional Laplace equation dy dx is satisfied by steady-state temperature distributions T-f(x, y, z) in space, by gravitational potentials, and by electrostatic poten- tials. The two-dimensional Laplace equation ar'ду obtained by dropping the a2f/az2 term...
The two-dimensional heat equation reduces to Laplace's equation to = 0 if the temperature u is...
The two-dimensional heat equation reduces to Laplace's equation to = 0 if the temperature u is steady-state. u(x, y) is defined in 0<x<2 and 0 Sys2 and satisfy u(x,0) = u(x, 2) = u(0, y) = 0 and u(2, y) = 80 sin my. Answer the following questions. (1) Obtain two ODES (Ordinary Differential Equations) by the method of separation of variables. (2) Find u(x, y) satisfying the boundary condition. (3) Obtain the value of u(1,5).
6. a) For a thin conducting rod of length L = π, the temperature U(x, t) at a point 0 Sx S L at timet>0 is determined by the differential equation U, Uxx with boundary data U(x, 0) fx) and U(0...
6. a) For a thin conducting rod of length L = π, the temperature U(x, t) at a point 0 Sx S L at timet>0 is determined by the differential equation U, Uxx with boundary data U(x, 0) fx) and U(0,) UL, t)- 0 for all0. Show that for any positive integer k, the function U(x, t)- exp (-ak21) sin kx is a solution. It follows that Σ exp (-ak2 t) Bk sin kx is the general solution where Σ...
A sheet of metal coincides with the square in the xy-plane whose vertices (0, O), (1...
A sheet of metal coincides with the square in the xy-plane whose vertices (0, O), (1 0), (1, 1) and (0, 1). The two faces of the sheet are insulated, and the sheet is so thin that heat flow in it can be regarded as two-dimensional. The edges parallel to the x- axis are insulated, and the left-hand edge is maintained at the constant temperature O. If the temperature distribution u(1, y) f(y) is maintained along the right-hand edge, Set...
The heat that is conducted through a body must frequently be removed by other heat transfer...
The heat that is conducted through a body must frequently be removed by other heat transfer processes. For example, the heat generated in an electronic device must be dissipated to the surroundings through convection by means of fins. Consider the one-dimensional aluminum fin (thickness t 3.0 mm, width 20 cm, length L) shown in Figure 1, that is exposed to a surrounding fluid at a temperature T. The conductivity of the aluminum fin (k) and coefficient of heat convection of...
1) 2) 3) PROJECT #1 (2.5 Marks): The heat that is conducted through a body must...
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2)
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PROJECT #1 (2.5 Marks): The heat that is conducted through a body must frequently be removed by other heat transter processes. For example, the heat generated in an electronic device must be dissipated to the surroundings through convection by means of fins. Consider the one-dimensional aluminum fin (thickness t 3.0 mm, width Z 20 cm, length L) shown in Figure 1, that is exposed to a surrounding fluid at a temperature 1. The conductivity of the aluminum...
3/5 25 pts.J A slab of thickness L, made of material with constant thermal conductivity k, is undergoing a 1-D, steady heat transfer. Its boundary surface at x 0 is insulated while the boundary surface at x= 1 is kept at constant temperature T= oc. Heat energy is generated within the slab at a rate of 2. qx)o cos(rx/2L) is the energy generation rate per unit volume (Wm) at x= 0. where qo a. Develop an expression for the steady-state...
(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...
1. Consider the insulated heat equation up = cum, 0 <r<L, t > 0 u (0,t) = u (L, t) = 0, t > 0 u(x,0) = f(2). What is the steady-state solution? 2. Solve the two-dimensional wave equation (with c=1/) on the unit square (i.e., [0, 1] x [0,1) with homogeneous Dirichlet boundary conditions and initial conditions: (2, y,0) = sin(x) sin(y) (,y,0) = sin(x). 3. Solve the following PDE: Uzr + Uyy = 0, 0<<1,0 <y < 2...
Question 1 [Total 20 marks] (a) [5 marks] In a steady-state two-dimensional heat flow problem, the temperature, u, at any point in the domain (t, ) satisfies the differential equation u y(2-y) u= U0F With the given temperature boundary condition as follows: u(x, 0) = 0, u(x, 2) = x(4-x), 0 < x < 4 Calculate the temperature at the interior points a, b, and c using a mesh size h-1.
Question 1 [Total 20 marks] (a) [5 marks] In...
ry 82. Let f(x, y) - 0 if (x, y)-0 The graph of f is shown on page 813. a. Show that j x,0) - x for all x, and a0, y) y for all y. b. Show that (0, 0) ахау (0, 0) The three-dimensional Laplace equation dy dx is satisfied by steady-state temperature distributions T-f(x, y, z) in space, by gravitational potentials, and by electrostatic poten- tials. The two-dimensional Laplace equation ar'ду obtained by dropping the a2f/az2 term...
The two-dimensional heat equation reduces to Laplace's equation to = 0 if the temperature u is steady-state. u(x, y) is defined in 0<x<2 and 0 Sys2 and satisfy u(x,0) = u(x, 2) = u(0, y) = 0 and u(2, y) = 80 sin my. Answer the following questions. (1) Obtain two ODES (Ordinary Differential Equations) by the method of separation of variables. (2) Find u(x, y) satisfying the boundary condition. (3) Obtain the value of u(1,5).
6. a) For a thin conducting rod of length L = π, the temperature U(x, t) at a point 0 Sx S L at timet>0 is determined by the differential equation U, Uxx with boundary data U(x, 0) fx) and U(0,) UL, t)- 0 for all0. Show that for any positive integer k, the function U(x, t)- exp (-ak21) sin kx is a solution. It follows that Σ exp (-ak2 t) Bk sin kx is the general solution where Σ...
A sheet of metal coincides with the square in the xy-plane whose vertices (0, O), (1 0), (1, 1) and (0, 1). The two faces of the sheet are insulated, and the sheet is so thin that heat flow in it can be regarded as two-dimensional. The edges parallel to the x- axis are insulated, and the left-hand edge is maintained at the constant temperature O. If the temperature distribution u(1, y) f(y) is maintained along the right-hand edge, Set...
The heat that is conducted through a body must frequently be removed by other heat transfer processes. For example, the heat generated in an electronic device must be dissipated to the surroundings through convection by means of fins. Consider the one-dimensional aluminum fin (thickness t 3.0 mm, width 20 cm, length L) shown in Figure 1, that is exposed to a surrounding fluid at a temperature T. The conductivity of the aluminum fin (k) and coefficient of heat convection of...
1)
2)
3)
PROJECT #1 (2.5 Marks): The heat that is conducted through a body must frequently be removed by other heat transter processes. For example, the heat generated in an electronic device must be dissipated to the surroundings through convection by means of fins. Consider the one-dimensional aluminum fin (thickness t 3.0 mm, width Z 20 cm, length L) shown in Figure 1, that is exposed to a surrounding fluid at a temperature 1. The conductivity of the aluminum...
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