Abstract
The temperature response in functionally gradient materials (FGM), subjected to pulseor stepwise heating at the front surface, is evaluated. Applicability of the approximate solution for the temperature response is investigated by comparing it with an exact analytical solution for the FGM in which thermophysical properties have certain profiles. When the FGM is composed of conventional solid materials, appropriateness of the approximate solution for the FGM is demonstrated as far as the temperature response near the rear surface is concerned. The approximate solution is also compared with the solution for the multilayered material. It is shown that an eight-layered material can be regarded as an FGM, as far as the temperature response at the rear surface is concerned, and that the approximate solution can predict the temperature response within 6% error. Because of its simplicity and fair degree of agreement, the approximate solution is anticipated to be used not only for qualitative but also for quantitative prediction of the temperature response near the rear surface of the FGM in engineering applications.
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Abbreviations
- a :
-
Thermal diffusivity
- c :
-
Specific heat
- Fo:
-
Fourier number [=at/l 2]
- f :
-
Correction factor
- L :
-
Thickness of the sample material
- l :
-
Thickness of the layer
- n :
-
Number of layers in the sample material
- p :
-
Exponent in the profile function of a
- Q :
-
Heat input per unit area
- s :
-
Parameter in the Laplace transformation
- t :
-
Time
- V :
-
Normalized temperature response
- W :
-
Heat input function
- X :
-
Mixture ratio
- y :
-
Variable
- z :
-
Distance
- α :
-
Constant in the profile function of Λ
- α j,m :
-
±1
- β :
-
Constant in the profile function of Λ
- γ :
-
Positive root of the characteristic equation
- ɛ :
-
Perturbed term
- ζ :
-
Normalized thermal diffusion time defined in Eq. (14)
- η :
-
Thermal diffusion time \([ = {l \mathord{\left/ {\vphantom {l {\sqrt a }}} \right. \kern-\nulldelimiterspace} {\sqrt a }}]\)
- η L :
-
Total thermal diffusion time defined in Eq. (14)
- Θ :
-
Laplace transform of the temperature rise
- θ :
-
Temperature rise
- Λ :
-
Heat-penetration coefficient \([ = {\lambda \mathord{\left/ {\vphantom {\lambda {\sqrt a }}} \right. \kern-\nulldelimiterspace} {\sqrt a }}]\)
- λ :
-
Thermal conductivity
- ρ :
-
Density
- Φ :
-
Correction term
- χ :
-
Parameter defined in Eqs. (5) and (6)
- ω :
-
Parameter defined in Eqs. (5) and (6)
- F :
-
Front surface
- i :
-
Value of ith layer
- i/j :
-
(Quantity of the ith layer) divided by (quantity of the jth layer)
- P :
-
Pulsewise heating method
- R :
-
Rear surface
- S :
-
Stepwise heating method
- I:
-
Component I
- II:
-
Component II
- *:
-
Inside the layer
References
N. Araki, A. Makino, and J. Mihara, Int. J. Thermophys. 13:331 (1992).
N. Araki, A. Makino, T. Ishiguro, and J. Mihara, Int. J. Thermophys. 13:515 (1992).
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Ishiguro, T., Makino, A., Araki, N. et al. Transient temperature response in functionally gradient materials. Int J Thermophys 14, 101–121 (1993). https://doi.org/10.1007/BF00522665
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DOI: https://doi.org/10.1007/BF00522665