2.35 PROBLEM KNOWN: Three-dimensional system described by cylindrical coordinates (r,,z) experiences transient conduction and internal heat generation. FIND: Heat diffusion equation. SCHEMATIC: See also Fig. 2.9.
ASSUMPTIONS: (1) Homogeneous medium. ANALYSIS: Consider the differential control volume identified above having a volume given as V = drrddz. From the conservation of energy requirement, q r q r +dr + q q +d + q z q z+dz + E g = E st . The generation and storage terms, both representing volumetric phenomena, are E g = qV = q ( dr rd dz ) Eg = Vc T/ t = ( dr rd dz ) c T/ t. (2,3) (1)
Using a Taylor series expansion, we can write q r+dr = q r +
q d , ( q r ) dr, q +d = q + r
()
q z+dz = q z +
( q z ) dz. z
(4,5,6)
Using Fouriers law, the expressions for the conduction heat rates are
q r = kA r T/ r = k ( rd dz ) T/ r
q = kA T/r = k ( dr dz ) T/r
(7) (8) (9)
q z = kA z T/ z = k ( dr rd ) T/ z.
Note from the above, right that schematic the gradient in the -direction is T/r and not T/. Substituting Eqs. (2), (3) and (4), (5), (6) into Eq. (1),
T ( q r ) dr q d ( q z ) dz + q dr rd dz = ( dr rd dz ) c . r z t
()
(10)
Substituting Eqs. (7), (8) and (9) for the conduction rates, find
T k ( rd dz ) r dr r
T T k ( drdz ) r d z k ( dr rd ) z dz (11)
T . t Dividing Eq. (11) by the volume of the CV, Eq. 2.24 is obtained.
+ q dr rd dz = ( dr rd dz ) c
1 T 1 T T T kr r + 2 k + z k z + q = c t r r r
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