Summary
For steady one-dimensional flow between parallel plates at unequal temperatures solutions are given for the Nusselt number in the entrance region, as well as for the fully developed flow with unequal temperatures and heat flux.
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Abbreviations
- C p :
-
constant pressure specific heat
- C ne :
-
even constant in Sturm-Liouville solution
- C no :
-
odd constant in Sturm-Liouville solution
- D :
-
equivalent diameter 4y 0
- f :
-
friction factor\(\tau _w /(\raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} \rho u^2 )\)
- G :
-
constant defined in text
- h :
-
heat transfer coefficientq/(t w −t b )
- n2,K:
-
constants in Deisslers equations
- k :
-
thermal conductivity
- q a :
-
heat input rate on one side of the duct
- q c :
-
heat input rate on the opposite side of the duct
- Q :
-
total heat input rate
- t :
-
fluid temperature atz
- t b :
-
bulk mixing temperature
- t i :
-
initial fluid temperature atx=0
- t wa :
-
wall temperature on one side
- t wc :
-
wall temperature on opposite side
- t wm :
-
arithmetic mean wall temperature
- u :
-
velocity of fluid at y
- U b :
-
bulk mean velocity
- u + :
-
dimensionless velocity
- x :
-
distance from the position of the temperature jump
- x + :
-
dimensionless distancex/D
- y :
-
distance from the wall
- y o :
-
distance from wall to centre line
- y + :
-
dimensionless distance from the wall
- z :
-
distance from the centre line
- z 0 :
-
distance from centre line to wall
- z + :
-
dimensionless distance from centre line =y 0 +−y +
- Z n :
-
eigenfunction
- Re :
-
Reynolds number 4u b y 0/v
- Pr :
-
Prandtl number
- Nu a :
-
Nusselt numberh a 4y o /k on one side
- Nu c :
-
Nusselt numberh c 4y o /k on opposite side
- α:
-
thermal diffusivityk/ ρ C p
- ε m :
-
eddy diffusivity for momentum
- ε h :
-
eddy diffusivity for heat
- γ:
-
defined as (v+ε)/v
- λ n :
-
eigenvalue of equation (24)
- v :
-
kinematic viscosity
- θ q :
-
dimensionless temperature for the uniform heat input case defined ast−t i )/(Q z 0/2k)
- ϑ:
-
dimensionless temperature for the uniform temperature case defined as (t−t i )/(t wa −t wm )
- θ 1 :
-
fully developed dimensionless temperature
- θ 2 :
-
entrance region dimensionless temperature
- σ:
-
fluid density
- τ:
-
shear stress at positiony
- τ w :
-
wall shear stress
References
Barrow, H., Int. J. Heat & Mass Transfer (1961) 306
Seban, R. A., Trans. AS ME,72 (1950) 789.
Stein, R. P., The Dependence of the Heat Transfer Coefficient on the Ratio of Heat Fluxes from the Walls of Parallel Plane Flow Channels, Reactor Heat Transfer Conference, U.S. Atomic Energy Commission, Nov. 1956, New York City.
Karman, T. von, Trans. A.S.M.E.61 (1939) 705.
Deissler, R. G., Analysis of Turbulent Heat Transfer, Mass Transfer and Friction in Smooth Tubes at High Prandtl and Schmidt Numbers. NACA report 1210, 1955.
Sparrow, E. M., T. M. Hallman and R. Seigel, Appl. sci. Res.A 7 (1957) 37.
Seigel, R. and E. M. Sparrow, A.S.M.E. (1960) 152.
Corcoran, W. H., F. Page Jr., W. G. Schlinger, B. H. Sage and D. K. Breaux, Ind. Eng. Chem.44 (1952) 410.
Knudsen, J. G. and D. L. Katz, Fluid Mechanics and Heat Transfer, McGraw-Hill, p. 206.
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Hatton, A.P. Heat transfer in the thermal entrance region with turbulent flow between parallel plates at unequal temperatures. Appl. sci. Res. 12, 249–266 (1963). https://doi.org/10.1007/BF03184976
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DOI: https://doi.org/10.1007/BF03184976