Summary
An analytical solution is presented for nonhomogeneous, one-dimensional, transient heat conduction problems in composite regions, such as multilayer slabs, cylinders and spheres, with arbitrary convection boundary conditions on both outer surfaces. The method of solution is based on separation of variables and on orthogonal expansion of functions over multilayer regions. Similar analytical procedures are available in the literature for a variety of combinations of boundary conditions, but not including the ones considered here, although such cases are encountered in practice very often. The effect of layer thermal properties on density of eigenvalues and accuracy of solution is also examined.
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Abbreviations
- a i :
-
Thermal diffusivity of layeri
- c i :
-
Specific heat of layeri
- C in ,D in :
-
Constants in Eq. (24)
- D :
-
Determinant of the coefficients of the set of Eqs. (28)–(31)
- e :
-
Number of eigenvalues used in a solution
- f i (x):
-
Initial condition for dependent variableθ i (x,t)
- f n * :
-
Coefficient defined by Eq. (36)
- F i (x):
-
Initial temperature field of layeri
- h 1,h m :
-
Heat transfer coefficients
- I n *(t):
-
Function defined by Eq. (38)
- j(N) :
-
Number of eigenvalues lying from 0 toN
- k i :
-
Thermal conductivity of layeri
- N n :
-
Coefficient defined by Eq. (39)
- p :
-
Parameter in Laplace differential operator,p=0,1,2
- q i (x,t):
-
Function defined by Eqs. (9), (10), (11) or (58)
- t :
-
Time
- T i (x,t):
-
Temperature of layeri at depthx and timet
- T s1 ,T sm :
-
Surrounding temperatures
- V n *(t):
-
Function defined by Eq. (37)
- x :
-
Space coordinate
- X in (x):
-
Eigenfunction
- β n :
-
Eigenvalue
- Γ n (t):
-
Function given by Eqs. (48) or (52)
- θ i (x,t):
-
Dependent variable linked toT i (x,t) by Eq. (8)
- ϱ i :
-
Density of layeri
- Φ in (x),Ψ in (x):
-
Functions defined by Eqs. (25)–(27)
- i :
-
Number of layer
- m :
-
Total number of layers
- n :
-
Number of eigenvalue
References
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Antonopoulos, K.A., Tzivanidis, C. Analytical solution of boundary value problems of heat conduction in composite regions with arbitrary convection boundary conditions. Acta Mechanica 118, 65–78 (1996). https://doi.org/10.1007/BF01410508
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DOI: https://doi.org/10.1007/BF01410508