Abstract
The thermal entrance region heat transfer problem (Graetz problem) for fully developed laminar flow in curved pipes with uniform wall heat flux is approached by an alternating direction implicit method for Dean numbers ranging from 0 to an order of 100. The effects of using several different finite-difference approximations for convective terms due to secondary flow in the energy equation on heat transfer result are studied. The effect of secondary flow on developing temperature field in the thermal entrance region is studied by considering the temperature profiles, isothermals and axial distributions of average wall temperature and bulk temperature. Heat transfer results are presented for Pr = 0.1, 0.7, 10 and 500.
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Abbreviations
- a :
-
radius of pipe
- C :
-
constant, (a 3/4νμ) (∂P 0/R c ∂Ω)
- Gz :
-
Graetz number, Pe(2a/R c Ω)
- h:
-
average heat transfer coefficient
- K :
-
Dean number, Re(a/R c)1/2
- k :
-
thermal conductivity
- M, N :
-
number of divisions in R and φ directions
- Nu :
-
local Nusselt number, h(2a)/k
- Pe :
-
Peclet number, RePr
- P 0 :
-
axial pressure and a function of R c Ω only
- Pr :
-
Prandtl number, ν/α
- q w :
-
uniform wall heat flux
- R, φ, R c Ω :
-
cylindrical coordinates
- R c :
-
radius of curvature of a curved pipe
- Re :
-
Reynolds number, (2a) W/ν
- r :
-
dimensionless radial coordinate, R/a
- r c :
-
dimensionless radius of curvature of a curved pipe, R c/a
- T :
-
local fluid temperature
- T 0 :
-
uniform fluid temperature at thermal entrance
- U, V, W :
-
velocity components in R, φ and R c Ω directions
- u, v, w :
-
dimensionless velocity components in r, φ and r c Ω directions, U/(νa), V/(νa), W/(Cν/a), respectively
- α :
-
thermal diffusivity
- β :
-
coefficient defined in equation (4)
- θ :
-
dimensionless temperature difference, (T−T 0)/(q w a/k)
- ν :
-
kinematic viscosity
- b:
-
bulk temperature
- i, j :
-
space subscripts of a grid point in R and φ directions
- w:
-
value at wall
- —:
-
average value
References
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Akiyama, M. and K. C. Cheng, Int. J. Heat Mass Transfer 15 (1972) 1426.
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Akiyama, M., Cheng, K.C. Graetz problem in curved pipes with uniform wall heat flux. Appl. Sci. Res. 29, 401–418 (1974). https://doi.org/10.1007/BF00384162
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DOI: https://doi.org/10.1007/BF00384162