problem4-05

# problem4-05 - PROBLEM 4.5 KNOWN Boundary conditions on four...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: PROBLEM 4.5 KNOWN: Boundary conditions on four sides of a rectangular plate. FIND: Temperature distribution. SCHEMATIC: y q′′ s W T1 T1 0 0 x L T1 ASSUMPTIONS: (1) Two-dimensional, steady-state conduction, (2) Constant properties. ANALYSIS: This problem differs from the one solved in Section 4.2 only in the boundary condition at the top surface. Defining θ = T – T∞, the differential equation and boundary conditions are ∂ 2θ ∂ 2θ + 2 =0 ∂x 2 ∂y θ(0, y) = 0 θ(L, y) = 0 θ(x,0) = 0 k ∂θ ∂y = q′′ s (1a,b,c,d) y=W The solution is identical to that in Section 4.2 through Equation (4.11), ∞ nπx nπy θ = ∑ Cn sin sinh L L n=1 (2) To determine Cn, we now apply the top surface boundary condition, Equation (1d). Differentiating Equation (2) yields Continued…. PROBLEM 4.5 (Cont.) ∂θ ∂y = ∞ ∑ Cn n=1 y=W nπ nπx nπW sin cosh L L L (3) Substituting this into Equation (1d) results in q′′ s = k ∞ ∑ A n sin n=1 nπx L (4) where An = Cn(nπ/L)cosh(nπW/L). The principles expressed in Equations (4.13) through (4.16) still apply, but now with reference to Equation (4) and Equation (4.14), we should choose nπx . Equation (4.16) then becomes f(x) = q ′′/k , g n (x) = sin s L L An = q′′ nπx s ∫ sin L dx k0 L ∫ sin 0 2 nπx dx L = q′′ 2 (-1) n+1 + 1 s kπ n Thus Cn = 2 q′′L (-1) n+1 + 1 s k n 2 π 2 cosh(nπW/L) The solution is given by Equation (2) with Cn defined by Equation (5). (5) ...
View Full Document

## This homework help was uploaded on 04/07/2008 for the course BENG 130, 103B, taught by Professor Gough during the Spring '08 term at UCSD.

### Page1 / 2

problem4-05 - PROBLEM 4.5 KNOWN Boundary conditions on four...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online