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

PROBLEM 3.108KNOWN: Rod (D, k, 2L) that is perfectly insulated over the portion of its length –L ≤ x ≤ 0 andexperiences convection (T ∞ , h) over the portion 0 ≤ x ≤ + L. One end is maintained at T 1 and theother is separated from a heat sink at T 3 with an interfacial thermal contact resistance R ′′ tc.FIND: (a) Sketch the temperature distribution T vs. x and identify key features; assume T 1 > T 3 >T 2 ; (b) Derive an expression for the mid-point temperature T 2 in terms of thermal and geometricparameters of the system, (c) Using, numerical values, calculate T 2 and plot the temperaturedistribution. Describe key features and compare to your sketch of part (a).SCHEMATIC:ASSUMPTIONS: (1) Steady-state conditions, (2) One-dimensional conduction in rod for –L ≤ x ≤ 0,(3) Rod behaves as one-dimensional extended surface for 0 ≤ x ≤ +L, (4) Constant properties.ANALYSIS: (a) The sketch for the temperature distribution is shown below. Over the insulatedportion of the rod, the temperature distribution is linear. A temperature drop occurs across thethermal contact resistance at x = +L. The distribution over the exposed portion of the rod is nonlinear.The minimum temperature of the system could occur in this portion of the rod.(b) To derive an expression for T 2 , begin with the general solution from the conduction analysis for afin of uniform cross-sectional area, Eq. 3.66.θmx mx( x) = C1e + C2e −0≤ x ≤+ L(1)where m = (hP/kA c ) 1/2 and θ = T(x) - T ∞ . The arbitrary constants are determined from the boundaryconditions.At x = 0, thermal resistance of roddθ⎞ θ1−θ( 0)qx( 0)=− kAc ⎟ = kAc θ1= T1−T∞dx ⎠x=0 L0 0 1mC0 01e − mC2e = ⎡ 1 ( C1e C2e)L θ − +⎢⎥⎤(2)⎣⎦Continued …..