Abstract
General expressions have been derived for unsteady temperature distribution in a finite hollow cylinder under the influence of a time dependent volume heat source and prescribed heat fluxes at the boundaries. By introducing certain artificial additional heat source functions, corresponding Pseudo-steady solutions are defined and by means of which the temperature fields are expressed in the form of uniformly convergent series solutions. By the application of integral transform techniques the expressions for temperature distribution are obtained in various forms which may be applicable to various cases of technological importance.
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Abbreviations
- a :
-
thermal diffusivity
- K:
-
thermal conductivity
- 2l :
-
length of the cylinder
- R1 :
-
radius of the inner cylinder
- R2 :
-
radius of the outer cylinder
- r, ϕ, Z :
-
cylindrical coordinate
- t :
-
time in seconds
- T (r, ϕ, Z, t):
-
unsteady temperature distribution defined in equations (1), (2) and (3)
- T0 :
-
characteristic temperature
- T0j (r, ϕ, Z, t),j :
-
0, 1, 2, 3, 4; Pseudo-steady temperature distribution
- Q (r, ϕ, Z, t):
-
volume heat source function
- f j (r, ϕ, t),j :
-
1, 2; heat fluxes on the surfacesZ = − l, Z = l respectively
- f j (ϕ, Z, t),j :
-
3, 4; heat fluxes on the surfacesr = R1,r = R2 respectively
- N′ (r, ϕ, Z):
-
initial temperature distribution in cylinder
- \(x = \frac{r}{{R_1 }},\phi ,z\) :
-
\(\frac{Z}{{R_1 }}\); non-dimensional cylindrical coordinates
- 2b :
-
\(\frac{{2l}}{{R_1 }}\); non-dimensional length of the cylinder
- F0 :
-
\(\frac{{at}}{{R_1 ^2 }}\); Fourier number
- θ (x, ϕ, z, F0):
-
\(\frac{{T(r,\phi ,Z,t)}}{{T_0 }}\); non-dimensional temperature distribution defined by equations (4), (5) and (6)
- θ0j(x, ϕ, z, F0):
-
\(\frac{{T_{0j} (r,\phi ,Z,t)}}{{T_0 }},j = 0,1,2,3,4\); non-dimensional Pseudo-steady temperature distribution defined by equations (18) and (19)
- P0(x, ϕ, z, F0):
-
\(\frac{{R_1 ^2 }}{{KT_0 }}Q(r,\phi ,Z,t)\); Pomerantesev criterion
- \(K_{i_j } (x,\phi , F_0 )\) :
-
\(\frac{{R_1 }}{{KT_0 }}f_j (r,\phi ,t), j = 1,2\) Kirpichev criteria for the surfacesz = − b, z = b respectively
- \(K_{i_j } (\phi ,z, F_0 )\) :
-
\(\frac{{R_1 }}{{KT_0 }}f_j (\phi ,Z,t), j = 3,4\) Kirpichev criteria for the surfacesx = 1,x = R respectively
- I k (x):
-
modified Bessel function of the first kind of orderk and argumentx
- K k (x):
-
modified Bessel function of the second kind of orderk and argumentx
- δ ij :
-
Kronecker delta
- I k ′(λ n R):
-
\(\left[ {\frac{d}{{dx}}\{ I_k (\lambda _n x)\} } \right]_{\theta = R} \).
References
Olcer, Nurettin, Y... “On the theory of conductive heat transfer in finite regions,”Int. J. Heat Mass Transfer, 1964,7, 307–314.
—.. “On the theory of conductive heat transfer in finite regions with boundary conditions of the second kind,”Ibid. 1965.8, 529–556.
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(Communicated by Prof. R. S. Mishra,f.a.sc.)
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Rai, K.N., Kumar, S. & Lalmani Temperature fields in a hollow cylinder in presence of heat source under the boundary conditions of the second kind. Proc. Indian Acad. Sci. 77, 62–82 (1973). https://doi.org/10.1007/BF03049818
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DOI: https://doi.org/10.1007/BF03049818