Conduction-Cylindrical Coordinates

CONDUCTION HEAT TRANSFER Heat Transfer The surface area (A) for transferring heat through the pipe (neglecting the pipe ends) is directly proportional to the radius (r) of the pipe and the length (L) of the pipe. A = 2prL As the radius increases from the inner wall to the outer wall, the heat transfer area increases. The development of an equation evaluating heat transfer through an object with cylindrical geometry begins with Fourier’s law Equation 2-5. Q k A^� � � � � � DT Dr From the discussion above, it is seen that no simple expression for area is accurate. Neither the area of the inner surface nor the area of the outer surface alone can be used in the equation. For a problem involving cylindrical geometry, it is necessary to define a log mean cross-sectional area (A_lm). (2-7) A_lm A_outer A_inner ln � � � � � � � � A_outer A_inner Substituting the expression 2prL for area in Equation 2-7 allows the log mean area to be calculated from the inner and outer radius without first calculating the inner and outer area. A_lm 2 p r_outer L 2 p r_inner L ln � � � � � � � � 2 p r_outer L 2 p r_inner L 2 p L � � � � � � � � � � � � r_outer r_inner ln r_outer r_inner This expression for log mean area can be inserted into Equation 2-5, allowing us to calculate the heat transfer rate for cylindrical geometries. HT-02 Page 12 Rev. 0