Page 431 - Bird R.B. Transport phenomena

§13.4 Temperature Distribution for Turbulent Flow in Tubes 413

Integrating this equation twice and then constructing the function в using Eq. 13.4-10,
we get

в = С<Г + C j = Щ i£ + Q * = ^ (13.4-12)
j
0 o п d£ + C 2
{t)
in which it is understood that a is a function of ^, and iQ) is shorthand for the integral
КЬ= {*<№<% (13.4-13)

The constant of integration Q is set equal to zero in order to satisfy B.C. 1. The constant
C is found by applying B.C. 2, which gives
o
C = I J ф Ш = [Id)]" 1 (13.4-14)
o
The remaining constant, C can, if desired, be obtained from Condition 4, but we shall
2/
not need it here (see Problem 13D.1).
We next get an expression for the dimensionless temperature difference © - ® , the
b
n
"driving force" for the heat transfer at the tube wall:

In the second line, the order of integration of the double integral has been reversed.
The inner integral in the second term on the right is just /(1) - l(g), and the portion
containing 7(1) exactly cancels the first term in Eq. 13.4-15. Hence when Eq. 13.4-14 is
used, we get

But the quantity 1(1) appearing in Eq. 13.4-16 has a simple interpretation:

1(1) = \\H^(\v rdr)^-- = \ ^ - (13.4-17)
z
Finally, we want to get the dimensionless wall heat flux,
q<P _ 2
(13.4-18)
k(T - T ) " ©o " ®ь
0 b
the reciprocal of which is 2
mm 2
(
(
D \ (v ) ) Jo Ш + " » / ) ( P / P " ) ] Ч ( }
z
To use this result, it is necessary to have an expression for the time-smoothed velocity
distribution v z (which appears in 1(0), the turbulent kinematic viscosity v {i) as a function
(0
of position, and a postulate for the turbulent Prandtl number Pr .

2
Equation 13.4-19 was first developed by R. N. Lyon, Chem. Eng. Prog., 47, 75-79 (1950) in a
paper on liquid-metal heat transfer. The left side of Eq. 13.4-19 is the reciprocal of the Nusselt
number, Nu = hD/k, which is a dimensionless heat transfer coefficient. This nomenclature is
discussed in the next chapter.