A metallic cylinder has radius, R and length, L. All surfaces of the cylinder are kept...

 

Transcribed Image Text:

A metallic cylinder has radius, R and length, L. All surfaces of the cylinder are kept at a temperature, Ts, except for the top surface which is kept at a temperature, To. Determine the steady-state temperature distribution in the cylinder. Assume that all physical properties of the metal are constant. a) State the appropriate boundary conditions and introduce a new variable so that three of the boundary conditions are homogeneous. b) Solve the PDE using separation of variables and apply the three homogeneous boundary conditions. (Notice from the graphs of Bessel functions that Bessel functions have infinitely many zeros. There are resources like MATLAB, and tables that can provide you with these values if you ever need them. The eigenvalues (2n) of this problem are related to zeros of a Bessel's function. So, you will simply state that the eigenvalues are solutions to a Bessel's function (you still need to specify which order and which kind of Bessel's function). c) Apply the final boundary conditions and the orthogonality condition. (Note that after separation of variables the ODE is a Sturm-Liouville problem and therefore the resulting eigenfunctions must be orthogonal). Some integrals that may be useful: √ Axª Ja-1₁ (Ax)dx = x²1 (λx) C fax" la-₁(λx) dx = x“I (Ax) X 1 to" (At)dt = x² (1²(2x) + J₁²(^x)) 2 A metallic cylinder has radius, R and length, L. All surfaces of the cylinder are kept at a temperature, Ts, except for the top surface which is kept at a temperature, To. Determine the steady-state temperature distribution in the cylinder. Assume that all physical properties of the metal are constant. a) State the appropriate boundary conditions and introduce a new variable so that three of the boundary conditions are homogeneous. b) Solve the PDE using separation of variables and apply the three homogeneous boundary conditions. (Notice from the graphs of Bessel functions that Bessel functions have infinitely many zeros. There are resources like MATLAB, and tables that can provide you with these values if you ever need them. The eigenvalues (2n) of this problem are related to zeros of a Bessel's function. So, you will simply state that the eigenvalues are solutions to a Bessel's function (you still need to specify which order and which kind of Bessel's function). c) Apply the final boundary conditions and the orthogonality condition. (Note that after separation of variables the ODE is a Sturm-Liouville problem and therefore the resulting eigenfunctions must be orthogonal). Some integrals that may be useful: √ Axª Ja-1₁ (Ax)dx = x²1 (λx) C fax" la-₁(λx) dx = x“I (Ax) X 1 to" (At)dt = x² (1²(2x) + J₁²(^x)) 2

Answer rating: 100% (QA)

ANSWER a at R R Q2RL ... View the full answer