Abstract
Transient methods, such as those with pulse- or stepwise heating, have often been used to measure thermal diffusivities of various materials including layered materials. The objective of the present study is to derive an analytical solution of the temperature rise in a multilayered material, the front surface of which is subjected to pulse- or stepwise heating. The Laplace transformation has been used to obtain the analytical solution. This solution will enable us to establish the appropriate measurement method for thermophysical properties of the multilayered material. It is also shown that the present solution can be extended to functionally gradient materials (FGM), in which thermophysical properties as well as compositions change continuously.
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Abbreviations
- A :
-
Constant
- a :
-
Thermal diffusivity
- B :
-
Constant
- b :
-
Vector defined in Eq. (18)
- C :
-
Matrix defined in Eq. (18)
- c :
-
Specific heat capacity
- D :
-
Matrix defined in Eq. (18)
- E :
-
Element of matrix E
- E :
-
Matrix [ = DC −1]
- Fo:
-
Fourier number [=ar/L 2]
- L :
-
Thickness of the sample material
- l :
-
Thickness of the layer
- n :
-
Number of layers in the sample material
- p :
-
Numerator of the image function for the temperature response
- Q :
-
Heat input per unit area
- q :
-
Denominator of the image function for the temperature response
- R :
-
Residue
- S :
-
Function defined in Eq. (100)
- s :
-
Parameter in the Laplace transformation
- t :
-
Time
- U :
-
Laplace transform of the characteristic equation
- V :
-
Temperature ratio
- W :
-
Heat input function
- x :
-
Vector defined in Eq. (31)
- z :
-
Distance
- α :
-
±1
- γ :
-
Positive root of the characteristic equation
- δ :
-
Delta function
- η :
-
Thermal diffusion time [=l/√a]
- Θ :
-
Laplace transform of the non-dimensional temperature
- θ :
-
Nondimensional temperature
- Λ :
-
Heat-penetration coefficient [=λ/√a]
- λ :
-
Thermal conductivity
- ρ :
-
Density
- ϰ :
-
Parameter defined in Eq. (52)
- ω :
-
Parameter defined in Eq. (53)
- e :
-
Apparent
- i :
-
Value of the ith layer
- i/j :
-
(Quantity of the ith layer) divided by (quantity of the jth layer)
- m :
-
Mean
- ':
-
Differentiation
- *:
-
Inside the layer
References
W. J. Parker, R. J. Jenkins, C. P. Butler, and G. I. Abbott, J. Appl. Phys. 32:1679 (1961).
T. Y. R. Lee, Thermal Diffusivity of Dispersed and Layered Composites, Ph.D. thesis (Purdue University, Lafayette, IN, 1977).
N. Araki and K. Natsui, Trans. Japan Soc. Mech. Engr. (Ser. B) 49:1048 (1983) (in Japanese). [Translated into English in Heat Transfer Japan. Res. 14:36 (1985).]
N. Araki, A. Makino, and J. Mihara, Int. J. Thermophys. 13:331 (1992).
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Araki, N., Makino, A., Ishiguro, T. et al. An analytical solution of temperature response in multilayered materials for transient methods. Int J Thermophys 13, 515–538 (1992). https://doi.org/10.1007/BF00503887
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DOI: https://doi.org/10.1007/BF00503887