Abstract
We treat numerically in this paper, the transient analysis of a conjugated heat transfer process in the thermal entrance region of a circular tube with a fully developed laminar power-law fluid flow. We apply the quasi-steady approximation for the power-law fluid, identifying the suitable time scales of the process. Thus, the energy equation in the fluids is solved analytically using the well-known integral boundary layer technique. This solution is coupled to the transient energy equation for the solid where the transverse and longitudinal heat conduction effects are taken into account. The numerical results for the temporal evolution of the average temperature of the tube wall, θ av, is plotted for different nondimensional parameters such as conduction parameter, α, the aspect ratios of the tube, ɛ and ɛ 0 and the index of power-law fluid, n.
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Abbreviations
- c :
-
Specific heat of the power-law fluid
- Gz:
-
Graetz number defined by Gz = π RPe/L
- h :
-
Thickness of the tube wall
- L :
-
Length of the tube wall
- Pe:
-
Peclet number defined by \( {\text{Pe}} = \rho c\bar uR/\lambda \)
- R :
-
Internal radius of the tube
- t :
-
Physical time
- T :
-
Temperature
- T ∞ :
-
Entrance temperature of the power-law fluid
- \( \bar u \) :
-
Average velocity of the power-law fluid
- u :
-
Velocity
- r, z:
-
Cylindrical coordinates
- α:
-
Heat conduction or conjugated parameter, defined in Eq. 9
- δ:
-
Thermal thickness of the power-law fluid, defined in Eq. 2
- ɛ:
-
Aspect ratio of the wall, defined in Eq. 9
- ɛ 0:
-
Aspect ratio of the wall, defined in Eq. 9
- λ:
-
Thermal conductivity of the power-law fluid
- λ w:
-
Thermal conductivity of the wall
- ρ:
-
Density of the power-law fluid
- σ:
-
Nondimensional coordinate, defined in Eq. 15
- θ:
-
Nondimensional temperature of the power-law fluid, defined in Eq. 16
- θ w:
-
Nondimensional temperature of the wall, defined in Eq. 16
- η:
-
Nondimensional coordinate, defined in Eq. 15
- χ:
-
Nondimensional coordinate, defined in Eq. 15
- τ:
-
Nondimensional time, defined in Eq. 16
- ζ:
-
Nondimensional coordinate, defined in Eq. 15
- l:
-
Conditions at the upward end of the wall
- w:
-
Conditions at the wall
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Acknowledgements
This work has been supported by a research grant no. 43010-Y of Consejo Nacional de Ciencia y Tecnología at Mexico.
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Luna, N., Méndez, F. & Bautista, O. Numerical analysis of the transient conjugated heat transfer in a circular duct with a power-law fluid. Heat Mass Transfer 41, 659–666 (2005). https://doi.org/10.1007/s00231-004-0600-y
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DOI: https://doi.org/10.1007/s00231-004-0600-y