Abstract
A new model for the heat transfer in turbulent pipe flow is presented based on a modified form of the mixing length theory developed by Cebeci [1] for boundary layer flow problems. The model predicts the velocity and temperature distributions and the Nusselt number for fluids with low, medium and high Prandtl numbers (Pr=.02 to 15) and fits the available experimental data very accurately for values of Reynolds number exceeding 104. Expressions for the eddy conductivity and for the turbulent Prandtl number are presented and shown to be dependent upon the Reynolds number, the Prandtl number, and the distance from the tube wall.
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Abbreviations
- A + :
-
damping factor for eddy viscosity
- a :
-
tube radius, \(a^ + = a\sqrt {\tau _{\text{W}} /\rho /\upsilon }\)
- B + :
-
damping factor for eddy conductivity
- C p :
-
specific heat at constant pressure
- k :
-
thermal conductivity
- k m , k n :
-
mixing length constants for momentum and heat, respectively
- l :
-
mixing length, \(l^ + = l\sqrt {\tau _{\text{W}} /\rho /\upsilon }\)
- N u :
-
Nusselt number
- Pr, PR :
-
Prandtl number, ν/2
- Pe :
-
Péclet number, Pe=Re Pr
- p :
-
static pressure
- q :
-
heat flux
- Re, RE :
-
Reynolds number, u(2a)/ν
- r :
-
radial coordinate
- St :
-
Stanton number
- T :
-
temperature, \(T^ + = \frac{{(T_{\text{W}} - T)C_{p\tau {\text{w}}} }}{{q\sqrt {\tau _{\text{w}} /} \rho }}\)
- u :
-
axial velocity, \(u^ + = u/\sqrt {\tau _{\text{w}} /\rho }\)
- x :
-
axial coordinate
- y :
-
transverse coordinate normal to the wall, \(y^ + = y\sqrt {\tau _{\text{w}} /\rho /\upsilon }\)
- α :
-
thermal diffusivity
- μ :
-
viscosity
- ε m, ε h :
-
kinematic eddy viscosity and eddy conductivity, respectively
- ν :
-
kinematic viscosity
- ρ :
-
density
- τ :
-
shear stress
- b :
-
bulk
- h:
-
heat
- m:
-
mixed mean
- w:
-
at wall
References
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Na, T.Y., Habib, I.S. Heat transfer in turbulent pipe flow based on a new mixing length model. Appl. Sci. Res. 28, 302–314 (1973). https://doi.org/10.1007/BF00413075
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DOI: https://doi.org/10.1007/BF00413075