Page 437 - Bird R.B. Transport phenomena

§13.6 Fourier Analysis of Energy Transport in Tube Flow at Large Prandtl Numbers 419

for the Fourier-transformed temperature @(Y, 0, z, v). The transformed boundary condi-
tions are

Inlet condition: at z = 0, 0(^0,2, v) = 0 for У > 0 (13.6-16)
Wall condition: at Y = 0, v) = forO < z < L (13.6-17)
Here again, the unit impulse function 8{v) appears as the Fourier transform of the func-
tion g(t) = 1 in the long-duration limit.
Two types of contributions appear in Eq. 13.6-15: real-valued zero-frequency im-
pulses 8{v) from functions and products independent of t, and complex-valued functions
of v from time-dependent product terms. We consider these two types of contributions
separately here, thus decoupling Eq. 13.6-15 into two equations.
We begin with the zero-frequency impulse terms. In addition to the explicit 8(v)
terms of Eq. 13.6-15, implicit impulses arise in the convolution terms from synchronous
oscillations of velocity and temperature, giving rise to the turbulent energy flux q (f) =
pC v'T' discussed in §13.2. The coefficients of all the impulse terms must be proportional
p
functions of a, in order that the dominant terms at each point remain balanced (i.e., of
comparable size) as a —» 0. Therefore, the coefficient к of the convective impulse terms,
including those from synchronous fluctuations, must be proportional to the coefficient
1/3
а/к 2 of the conductive impulse term, giving к ос а , or
к = Pr" 1/3 D (13.6-18)
for the dependence of the average thermal boundary layer thickness on the Prandtl number.
The remaining terms in Eq. 13.6-15 describe the turbulent temperature fluctuations.
They include the accumulation term 2mv®' and the remaining convection and conduc-
tion terms. The coefficients of all these terms (including liriv in the leading term) must
be proportional functions of a in order that these terms likewise remain balanced as a —»
0. This reasoning confirms Eq. 13.6-18 and gives the further relation v ос к, or

^ o c ! c P (13.6-19)
(v )
z
for the frequency bandwidth A^ of the temperature fluctuations. Consequently, the
1/:
stretched frequency Pr V and stretched time Pr~ £ are natural variables for reporting
1/3
Fourier analyses of turbulent forced convection. Shaw and Hanratty 4 reported turbu-
lence spectra for their mass transfer experiments analogously, in terms of a stretched fre-
quency variable proportional to Sc 1/3 v (here Sc = fx/p4t AB is the Schmidt number, the
mass transfer analog of the Prandtl number, which contains the binary diffusivity 4b AB, to
be introduced in Chapter 16).
Thus far we have considered only the leading term of a Taylor expansion in к for
each term in the energy equations. More accurate results are obtainable by continuing
1/3
the Taylor expansions to higher powers of к, and thus of Pr" D. The resulting formal
solution is a perturbation expansion
1/3
1/3
0 - © (Y, в, z, Pr i/) + K@W, 0, z, Pr ^) + • • • (13.6-20)
0
for the distribution of the fluctuating temperature over position and frequency in a given
velocity field. _
The expansion for T (the long-time average of the temperature) corresponding to Eq.
13.6-20 is obtained from the zero-frequency part of 0,
0 = 0 (У, в, z) + к©!(У, 0, z) + • • • (13.6-21)
О

D. A. Shaw and T. J. Hanratty, AIChE Journal, 23,160-169 (1977); D. A. Shaw and T. J. Hanratty,
4
AIChE Journal, 23, 28-37 (1977).