Page 690 - Bird R.B. Transport phenomena

670 Chapter 21 Concentration Distributions in Turbulent Flow

as shown in Fig. 21В.З(я). A small voltage is applied across the plates to produce a deposition
of Ag on the cathode (lower plate) and to polarize the circuit completely (that is, to maintain
the Ag + concentration at the cathode very nearly zero). Forced diffusion may be ignored, and
the Ag + may be considered to be moving to the cathode by ordinary (that is, Fickian) diffu-
sion and eddy diffusion only. Furthermore, this solution is sufficiently dilute that the effects
of the other ionic species on the diffusion of Ag + are negligible.
(a) Calculate the Ag + concentration profile, assuming that (i) the effective binary diffusivity
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2
of Ag + through water is 1.06 X 10~ cm /s; (ii) the truncated Lin, Moulton, and Putnam ex-
pression of Eq. 5.4-2 for the turbulent velocity distribution in round tubes is valid for "slit
flow" as well, if four times the hydraulic radius is substituted for the tube diameter; (iii) the
plates are 1.27 cm apart, and W /p is 11.4 cm/s.
0
(b) Estimate the rate of deposition of Ag on the cathode, neglecting all other electrode reactions.
(c) Does the method of calculation in part (a) predict a discontinuous slope for the concentra-
tion profile at the center plane of the system? Explain.
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Answers: (a) See Fig. 21B.3(W; (b) 6.7 X 10~ equiv/cm 2 • s
21B.5. Mixing-length expression for the velocity profile.
(a) Start with Eq. 5.5-3, and show that for steadily driven, fully developed turbulent flow in
a tube

(b) Next set т, 2 = т^ 0 + r^, where т^° is given by the cylindrical coordinate analog of Eq. 5.2-9,
and т*2 by Eq. 5.5-5. Show that Eq. 21B.5-1 then becomes

(c) Obtain Eq. 21.4-13 from Eq. 21B.5-2 by introducing the dimensionless symbols used in the
former equation.