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

PROBLEM 3.30KNOWN: Geometry and surface conditions of a truncated solid cone.FIND: (a) Temperature distribution, (b) Rate of heat transfer across the cone.SCHEMATIC:ASSUMPTIONS: (1) Steady-state conditions, (2) One-dimensional conduction in x, (3)Constant properties.PROPERTIES: Table A-1, Aluminum (333K): k = 238 W/m⋅K.2 2 3A= D /4 a /4 x ,ANALYSIS: (a) From Fourier’s law, Eq. (2.1), with π ( π )4qxdx=−kdT.π a2x3Hence, since q x is independent of x,4qxx dx T2∫=−k dTπ a x1 x3 ∫T1orx4qx⎡ 1 ⎤( 1)2 ⎢− k T T .π a 2x2⎥=− −⎣ ⎦ x 1Hence2q⎡x 1 1⎤T= T 1 + ⎢ − ⎥.π a2k ⎢x 2x2⎥⎣ 1 ⎦(b) From the foregoing expression, it also follows thatπ a2k T2 − Tq1x =2 ⎡1/x 2 221/ x ⎤⎢−⎣ 1⎥⎦π-1( 1m ) 238 W/m⋅K $( 20 −100)Cqx= ×2 ⎡ −2 −2 -2( 0.225) − ( 0.075)⎤ m⎢⎣⎥⎦= it follows thatqx= 189 W.