Abstract
A numerical solution using a combination of boundary vorticity method and line iterative relaxation method is presented for the hydrodynamically and thermally fully developed combined free and forced laminar convection with upward flow in inclined tubes subjected to the thermal boundary conditions of axially uniform wall heat flux and peripherally uniform wall temperature at any axial position. The numerical solution converges up to a reasonably high value of the characteristic parameter where an asymptotic behaviour for flow and heat transfer resultsfalready appears, and further results can be obtained by a linear extrapolation. The tube inclination angle or body-force orientation effects on flow and heat transfer characteristics are clarified for upward laminar flow configuration, and show that in high Rayleigh number regime the tube orientation effect has considerable influence on the results in the neighborhood of horizontal direction. Typical graphical results for flow and heat transfer are presented, mainly to illustrate the effects of Rayleigh number and tube inclination. The numerical results show that the perturbation method is invalid for the present problem.
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Abbreviations
- A :
-
axial pressure gradient in fluid, − (∂P/∂Z+ρwg sin α)
- a :
-
radius of tube
- C :
-
axial temperature gradient, ∂T/∂Z
- f :
-
friction factor, 2τ w/(ρxxx0057; 2), or a dummy variable
- g :
-
gravitational acceleration
- xxx0068; :
-
average heat transfer coefficient
- k :
-
thermal conductivity
- M :
-
number of divisions in R-direction
- N :
-
number of divisions in φ-direction
- Nu :
-
Nusselt number, xxx0068;(2a)/k
- P :
-
pressure
- Pr :
-
Prandtl number, ν/κ
- R, φ, Z :
-
cylindrical coordinates
- Ra :
-
Rayleigh number, βgCa 4/νκ
- Re :
-
Reynolds number Aa 3/4ρν 2
- r :
-
dimensionless radial coordinate, R/a
- T :
-
local temperature
- U, V, W :
-
velocity components in R, φ and Z directions
- u, v, w :
-
dimensionless velocity components in R, φ and Z directions
- α :
-
angle of tube inclination, see Fig. 1
- β :
-
coefficient of thermal expansion
- ε :
-
a presented error, see equation (12)
- θ :
-
dimensionless temperature difference, (T w−T)/ReCaPr
- κ :
-
thermal diffusivity, k/ρC p
- μ :
-
viscosity
- ν :
-
kinematic viscosity, μ/ρ
- ξ :
-
vorticity function defined by equation (7)
- ρ :
-
density
- τ :
-
shear stress
- ψ :
-
dimensionless stream function
- ▽2 :
-
dimensionless Laplacian operator, ∂2/∂r 2+∂/r ∂r+∂2/r 2 ∂φ 2
- b:
-
value at boundary
- i, j :
-
space subscripts of grid point in R and φ directions
- O:
-
condition for pure forced convection in horizontal tube
- w:
-
value at wall
- n :
-
nth iteration
- -:
-
average value
References
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Iqbal, M. and J. W. Stachiewicz, J. Heat Transfer, Trans. ASME 88 (1966) 109.
Hwang, G. J. and K. C. Cheng, Heat Transfer 1970 (Proc. Fourth International Heat Transfer Conference), 4, NC3.5, Elsevier Publishing Company, Amsterdam.
Newell, J. H. Jr., and A. E. Bergles, J. Heat Transfer, Trans. ASME 92 (1970) 83.
Hwang, G. J., Thermal instability and finite amplitude convection with secondary flow, Ph. D. thesis, Dept. Mechanical Engineering, University of Alberta, Edmonton, Alberta, CANADA (1970).
Hwang, G. J. and K. C. Cheng, Proc. 12th Midwestern Mechanics Conference, University of Notre Dame (1971).
Hallman, T. M., Combined forced and free convection in a vertical tube, Ph. D. thesis, Purdue University (1958).
Cheng, K. C., G. J. Hwang and M. Akiyama, Int. J. Heat Mass Transfer 15 (1972) 172.
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Cheng, K.C., Hong, S.W. Combined free and forced laminar convection in inclined tubes. Appl. Sci. Res. 27, 19–38 (1973). https://doi.org/10.1007/BF00382474
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DOI: https://doi.org/10.1007/BF00382474