Estimate the steady state temperature on the plate a quarter
circle radius
r = 1 and f (θ) = θ2 - π shown in the
figure.
(a) Establish the equations to
be solved and their boundary conditions.
(b) Determine the coefficients of the series obtained.
(c) Give the solution to the problem.
Homework Answers
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Estimate the steady state temperature on the plate a quarter
circle radius
r = 1 and...
2. Find the steady-state temperature u(r,0) in a semicircular plate of radius r 2 if [10<0<...
2. Find the steady-state temperature u(r,0) in a semicircular plate of radius r 2 if [10<0< π/2 u(2,0) and the edges 0 = 0 and 0 = T are insulated. 0 /20 T пл sin 2 1 2 + cos(n0) Ans: u(r,0) п
Find the steady-state temperature u(r.0) in a circular plate of radius r = 1 if the temperature on the circumference is...
Find the steady-state temperature u(r.0) in a circular plate of radius r = 1 if the temperature on the circumference is as given (show all work!): 0 0 T 1) u(1,0) = 0, T<02T
Find the steady-state temperature u(r.0) in a circular plate of radius r = 1 if the temperature on the circumference is as given (show all work!): 0 0 T 1) u(1,0) = 0, T
Problem 2(30 points) Consider the steady-state temperature distribution in a square plate with dimensions 2 m...
Problem 2(30 points) Consider the steady-state temperature distribution in a square plate with dimensions 2 m x 2 m. There is a heat generation of ġ(x.y)=6x [W/my], and the thermal conductivity of k=1[W/(m-°C)]. The temperature on the top boundary is given by a piecewise function, f(x), which is defined below. x(4- x²)+10 0<x<1 | x(4- x?) + 20, 1<x<2 The bottom boundary is insulated. The temperatures on left-handed and right-handed boundary are maintained at constants 10[°C] and 20 [°C] as...
3. (20 points) Denote u(ar, y) the steady-state temperature in a rectangle area 0 z 10,...
3. (20 points) Denote u(ar, y) the steady-state temperature in a rectangle area 0 z 10, 0yS 1. Find the temperature in the rectangle if the temperature on the up side is kept at 0°, the lower side at 10° while the temperature on the left side is S0)= sin(y) and the right side is insulated. Answer the following questions. (a) (10 points) Write the Dirichlet problem including the Laplace's equation in two dimensions and the boundary conditions. (b) (10...
2.76 A plate in the form of a sector of a circle of radius a has...
2.76
A plate in the form of a sector of a circle of radius a has a
central angle beta, as shown in Fig. 2-30. If the circular part is
maintained at temperature f(0), 0 < theta < beta, while the
bounding radii are maintained at temperature zero, find the
steady-state temperature everywhere in the sector.
Answer:
2.76. A plate in the form of a sector of a circle of radius a has central angle B, as shown in Fig....
1- Consider waves propagating in a vibrating quarter-circular membrane: at2 The displacement u(r,...
1- Consider waves propagating in a vibrating quarter-circular membrane: at2 The displacement u(r, e t) is zero on the entire boundary at all times. a) Write down explicitly the three boundary conditions expressed above. b) Starting by the method of separation of variables, find the solution and show that it is given by ui (r, θ' t) = Σι Σ J1(A ct) sinde) [A, cos(JA Ct)+B, sin(vA ct)], where l is a positive even integer, and n is a positive...
Compute the steady-state temperature distribution in an infinitely long cylindri cal wedge of radius a and...
Compute the steady-state temperature distribution in an infinitely long cylindri cal wedge of radius a and angle B, whose cross-section is illustrated below. The two straight sides of the wedge are held at zero temperature, while the curved edge is at uniform temperature uo uo Here are a few points to consider in r solution to this problem (a) In polar coordinates, the steady-state temperature satisfies You are required to use the usual approach of separation of variables and to...
3. A circular plate of unit radius, whose faces are insulated, has half of its boundary...
3. A circular plate of unit radius, whose faces are insulated, has half of its boundary kept at constant temperature u, and the other half at temperature uz (see figure). Find the steady state temperature pf the plate. The steady state heat flow is written as oʻu 1du 1 8²u ar?'r or a dz = 0, with the boundary condition 0<¢ <T, luz, < < 21. u(1,0) = {u, After using the proper separation of variable method, we have u=dy...
An important concern in the study of heat transfer is to determine the steady-state temperature d...
An important concern in the study of heat transfer is to determine the steady-state temperature distribution of a plate when temperature around the boundary is known. Assume the plate shown in the figure represents a cross section of a metal beam, with negligible heat flow in the direction perpendicular to the plate. Let A, B, C, and D denote the temperatures at the four interior nodes of the mesh in the figure. The temperature at a node is approximately equal...
10. (21 pts]Find the steady-state temperature distribution in a solid cylinder with radius R and height...
10. (21 pts]Find the steady-state temperature distribution in a solid cylinder with radius R and height H, if the boundary temperatures are set as 0 on the bottom surface, ugra/R? on the top surface and Uz/H on the curved surface 1 [4 pts) Write the governing equation and boundary conditions 2) [17 pts]Solve the problem
2. Find the steady-state temperature u(r,0) in a semicircular plate of radius r 2 if [10<0< π/2 u(2,0) and the edges 0 = 0 and 0 = T are insulated. 0 /20 T пл sin 2 1 2 + cos(n0) Ans: u(r,0) п
Find the steady-state temperature u(r.0) in a circular plate of radius r = 1 if the temperature on the circumference is as given (show all work!): 0 0 T 1) u(1,0) = 0, T<02T
Find the steady-state temperature u(r.0) in a circular plate of radius r = 1 if the temperature on the circumference is as given (show all work!): 0 0 T 1) u(1,0) = 0, T
Problem 2(30 points) Consider the steady-state temperature distribution in a square plate with dimensions 2 m x 2 m. There is a heat generation of ġ(x.y)=6x [W/my], and the thermal conductivity of k=1[W/(m-°C)]. The temperature on the top boundary is given by a piecewise function, f(x), which is defined below. x(4- x²)+10 0<x<1 | x(4- x?) + 20, 1<x<2 The bottom boundary is insulated. The temperatures on left-handed and right-handed boundary are maintained at constants 10[°C] and 20 [°C] as...
3. (20 points) Denote u(ar, y) the steady-state temperature in a rectangle area 0 z 10, 0yS 1. Find the temperature in the rectangle if the temperature on the up side is kept at 0°, the lower side at 10° while the temperature on the left side is S0)= sin(y) and the right side is insulated. Answer the following questions. (a) (10 points) Write the Dirichlet problem including the Laplace's equation in two dimensions and the boundary conditions. (b) (10...
2.76
A plate in the form of a sector of a circle of radius a has a
central angle beta, as shown in Fig. 2-30. If the circular part is
maintained at temperature f(0), 0 < theta < beta, while the
bounding radii are maintained at temperature zero, find the
steady-state temperature everywhere in the sector.
Answer:
2.76. A plate in the form of a sector of a circle of radius a has central angle B, as shown in Fig....
1- Consider waves propagating in a vibrating quarter-circular membrane: at2 The displacement u(r, e t) is zero on the entire boundary at all times. a) Write down explicitly the three boundary conditions expressed above. b) Starting by the method of separation of variables, find the solution and show that it is given by ui (r, θ' t) = Σι Σ J1(A ct) sinde) [A, cos(JA Ct)+B, sin(vA ct)], where l is a positive even integer, and n is a positive...
Compute the steady-state temperature distribution in an infinitely long cylindri cal wedge of radius a and angle B, whose cross-section is illustrated below. The two straight sides of the wedge are held at zero temperature, while the curved edge is at uniform temperature uo uo Here are a few points to consider in r solution to this problem (a) In polar coordinates, the steady-state temperature satisfies You are required to use the usual approach of separation of variables and to...
3. A circular plate of unit radius, whose faces are insulated, has half of its boundary kept at constant temperature u, and the other half at temperature uz (see figure). Find the steady state temperature pf the plate. The steady state heat flow is written as oʻu 1du 1 8²u ar?'r or a dz = 0, with the boundary condition 0<¢ <T, luz, < < 21. u(1,0) = {u, After using the proper separation of variable method, we have u=dy...
An important concern in the study of heat transfer is to determine the steady-state temperature distribution of a plate when temperature around the boundary is known. Assume the plate shown in the figure represents a cross section of a metal beam, with negligible heat flow in the direction perpendicular to the plate. Let A, B, C, and D denote the temperatures at the four interior nodes of the mesh in the figure. The temperature at a node is approximately equal...
10. (21 pts]Find the steady-state temperature distribution in a solid cylinder with radius R and height H, if the boundary temperatures are set as 0 on the bottom surface, ugra/R? on the top surface and Uz/H on the curved surface 1 [4 pts) Write the governing equation and boundary conditions 2) [17 pts]Solve the problem
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