Page 434 - Bird R.B. Transport phenomena

416 Chapter 13 Temperature Distributions in Turbulent Flow

This equation can be integrated once to give
(/)
Pr F/=f J + C (13.5-10)

The constant of integration may be set equal to zero, since, according to Eq. 5.6-20, F = 0
at £ = 0. A second integration from 0 to f then gives

'o 1 +\{C& 2
2 2
= -Pr (/) ln(l +\(C& ) (13.5-11)
or
щ = (1 + \(С & У " (13.5-12)
2
2Рг
3
Finally, comparison of Eqs. 13.5-12 and 13.5-8 with Eq. 5.6-21 shows that the shapes of
the time-smoothed temperature and axial velocity profiles are closely related,
Pr " (
(13.5-13)
an equation attributed to Reichardt. This theory provides a moderately satisfactory ex-
2
planation for the shapes of the temperature profiles. 1 The turbulent Prandtl (or Schmidt)
number deduced from temperature (or concentration) measurements in circular jets is
about 0.7.
The quantity C appearing in Eq. 13.5-12 was given explicitly in Eq. 5.6-23 as C 3 =
3
(0
V3/1 бтЛ/ТТрО / ^ ), where J is the rate of momentum flow in the jet, defined in Eq. 5.6-
2. Similarly, an expression for the quantity/(0) in Eq. 13.5-12 can be found by equating
the energy in the incoming jet to the energy crossing any plane downstream:

wC (T 0 - T,) = [ V \ P C v (f - T )r dr dO (13.5-14)
p
p z
}
J Q J Q
Insertion of the expressions for the velocity and temperature profiles and integrating
then gives

Combining Eqs. 13.5-3,13.5-8, 5.6-23,13.5-12, and 13.5-15 then gives the complete expres-
sion for the temperature profiles T(r, z) in the circular turbulent jet, in terms of the total
momentum of the jet, the turbulent viscosity, the turbulent Prandtl number, and the
fluid density.

13,6 FOURIER ANALYSIS OF ENERGY TRANSPORT IN
TUBE FLOW AT LARGE PRANDTL NUMBERS
In the preceding two sections we analyzed energy transport in turbulent systems by use
of time-smoothed equations of change. Empirical expressions were then required to de-
scribe the turbulent fluxes in terms of time-smoothed profiles, using eddy transport coef-

2 H. Reichardt, Zeits. f. angew. Math. u. Mech., 24, 268-272 (1944).