from the Popovich and Hummel work. The Popovich and Hummel work shows a probability of laminar flow of 40% at y + = 5.05. Their paper states, "Adjacent to the wall there is a very thin layer of thickness yf = 1.6 f 0.5 in which the velocity gradient on all 147 photographs is essentially linear, but the slope of the gradient changes with time." This then is the laminar sublayer thickness, not y + = 5.05. Of the 147 photographs, 60 show some form of broadening of the trace in the laminar sublayer. I interpret these data to indicate that eddies from the fully turbulent region in the pipe approach within y + = 1.6 =k 0.4 of the pipe wall. However, small segments from these eddies penetrate the laminar layer to the wall, as shown by 41% of the photographs. The frequency of these penetrations or eddies was observed by Shaw and Hanratty. Thus there is an effective eddy frequency in this laminar sublayer. My work uses the Shaw and Hanratty frequency data and results in an effective y + of 0.45, which certainly appears in reasonable agreement with the over-all observed sublayer y + = 1.6 f 0.4.
Thomas states that my work is artificial. The purpose of my work was to show the result of numerical integration t o obtain the best available equations for momentum, heat, and mass transport with respect to experimental heat and mass transfer data. Thomas' criticism of the response of the model a t high Schmidt numbers is not valid because he used the wrong equation in his analysis. His criticism of the eddy approach distance is related to the wrong value of y + from the Popovich and Hummel observations. h criticism of artificiality is justified in regard to all empirical equations that are the basis for momentum, heat, and mass transport relationships like those presented in my paper. Unfortunately, knowledge of fluid mechanics is not available for a fundamental understanding of these relationships, so the empirical route has no alternative a t the present time.
G. A . Hughmark Ethyl Corp. Baton Rouge, La. 70821
Some Problems in Chemical Reactor Analysis with Stochastic Features SIR:Pell and Aris (1969) examined a system of stochastic differential equations as a model for a chemical reactor that is subject to random fluctuations in temperature. Such equations are currently receiving a great deal of attention as models for systems subject to random influences, particularly in connection with the important problem of the control of such systems. These problems are frequently met in chemical engineering applications, and there is no doubt that interest will develop quickly among chemical engineers as well. It is opportune, therefore, to draw attention to an important question of interpretation, which arises when stochastic differential equations are used as models for physical systems. Pell and Aris write the system of equations as dt = - ( a + .)y + 9
where a and q are Gaussian white-noise stochastic processes. However, the pathological properties of white noise render any precise definition of Equation 1 impossible, and this equation has meaning only as far as it is a symbolic representation of the I t 6 equation dy = -aydt
- ydw + dub
(2)
where w and wo are Wiener (Brownian motion) processes. It is misleading to regard a and cyo as derivatives of Wiener processes, since sample paths of such processes are, with probability 1, nondifferentiable almost everywhere (Doob, 1953, p. 394). Equations such as 2 have been given precise mathematical interpretations by Doob (1942), Ita (1951), and Gihman (1955). Pell and Aris have based their analysis on the associated Focker-Planck-Kolmogorov equation (3)
(We use the notation of the paper under discussion, except that subscripts are suppressed where appropriate. We restrict our discussion to the one-dimensional case, which brings out all the pertinent points of interest.) The link be190
l&EC FUNDAMENTALS
VOL. 9 NO. 1 FEBRUARY 1970
tween Equations 2 and 3 is through coefficients A and B , and it is just here that the important question of interpretation arises. A(yo)
69 = Y(t0
+ 6 4 - Y(t0)
=
Lim E[6y[y01/6t
=
to+*t
-
io
aydt
-
(4)
6t+O
vdw(t)
+
s,
to+*t
dwo(t)
(5)
The last two integrals are stochastic integrals (Doob, 1953, p. 426). From Equation 2,
(9)
The ambiguity in interpretation results from difficulty in defining the integrals in Equations 8 and 9. The limit of the usual Riemann sum is not unique in these cases. Consider a partition of the interval to, to 6t: to < t~ < . . . . . . t N = to 6t, max(tt+l - ti) = A.
+
+
i
I t 6 defines these integrals as N-1
I2 = lim A+O
i- 0
[w(ti)- w(to)l[w(tt+d - w(tJ1
[This is a loose version of Ita’s definition, but serves the purposes of the discussion. A precise definition is given by Dynkin (1965, Theorem 7.1).] N-1
[wo(tt)
and I4 = lim
-
w ~ ( t ~ ) l [ w ( t t+ ~w) ( t J 1 , and
i-0
A-0
Stratonovich (1968, p. 44) defines them as N-1 12
lim
=
I1/2[dtt+J
A d 0 i-0
N-1
{ 1/2[w(tt+d - w(tt)P
= lim i-0
A+O
+ w(ti11 - w ( G j {w(tt+l) - w(tt) 1 [w(tt)
+
- w(to)I[w(tt+1) - W ( t i ) I j
I n the same way N-1 14
C
= lim A+O
i-0
{ 1/2[~o(ti+1)- wo(ti)I [wo(tt+l) - w(UI
+
[wo(tt) - wo(to)I [w(ti+l) - N t t ) I
1
E [ I , ] = E[Z,] = 0
Berlin. 196.5. -~.~~~ I ~ - - -
E[I2] = 0
(It@ N-1 (t
