Page 1 PROBLEM 3.1 KNOWN: One-dimensional, plane wall ...

PROBLEM 3.34KNOWN: Hollow cylinder of thermal conductivity k, inner and outer radii, r i and r o ,respectively, and length L.FIND: Thermal resistance using the alternative conduction analysis method.SCHEMATIC:ASSUMPTIONS: (1) Steady-state conditions, (2) One-dimensional radial conduction, (3)No internal volumetric generation, (4) Constant properties.ANALYSIS: For the differential control volume, energy conservation requires that q r = q r+drfor steady-state, one-dimensional conditions with no heat generation. With Fourier’s law,dTdTqr=− kA =− k( 2 π rL)(1)drdrwhere A = 2πrL is the area normal to the direction of heat transfer. Since q r is constant, Eq.(1) may be separated and expressed in integral form,qrrodr2 π L∫=−rirToTi∫( )k T dT.Assuming k is constant, the heat rate is( i − o)( )2 π Lk T Tq r =.ln r o / riRemembering that the thermal resistance is defined asRt≡∆T/qit follows that for the hollow cylinder,ln ( r o / ri)R t = .