Generalization of Treatment
Equation
The treatment can be extended to cover the case of the introduction of a unit pulse a t stagef (feed) and the observation a t stage m (measuring) in a system of stages 1 . . . . . n, as discussed by Bell and Babb ( 7 ) . Symbols j and k are used for arbitrary (running) stagr: numbers. I n Equations I O , for i := 0 as applied to zeroth moments, the unit input is noiv to be moved from the first to the fth equation. The summation of all equations yields = 1 as before (see Equation 11). I am indebted to S. J. Vellenga for the observation that in order to find M i , j it is not necessary to work through the set of equations starting from M i , n . If Equations 10 are multiplied by factors 1, 1, . . . . . . . . . . .1, x , A’, . . . . . . . . xn-.’, the r,um of the left-hand members is M t , j . I t is convenient now to introduce a function FCy) with the properties
y
s 0 ; F(,)
y L 0;
=
0
=
y
I
(18 )
fir’ . 0 , 3. =
Ml,k =
In our case the Characteristic equation is in view of x
(1 a)
(204
4
=
and has roots p1 = 1 and y2 = x . I t is easy to prove that either root may now serve as the multiplier in an extension of AndrC’s method. I n general, both terminal unknowns M i , l and Mi,n would remain [ ( 2 ) , page 592, Equation 41, but because of the special values of their coefficients in the first and last equations, these being equal to one and the same root of the characteristic equation, one or the other of these unknowns cancels out, depending on which root is used as the multiplier. This is also the reason why with x F i r - j ) as the multiplier only M i , i remains. I n the usual theory of (difference equations the general solution is derived first and the boundary conditions are introduced later, when shortcuts such as the above are likely to be overlooked. Application of Equations 10 for i = 1 with the M0,,given by
CORRESPONDENCE
-
tanhmm
[’ - (’ -k
&A)
(21)
xF(’-~)
zT2
k = l
,yF(J--Ic) iF(/-1)
(22)
1 ‘ 1
This is a solution to Bell and Babb’s problem. For the special applications f = 1 andlor m = n functions F ( f 1 ) and/or F(k - m ) are identically equal to zero. While for low n these series expansions are very practical, for high n they must be rearranged, using summation formulas for geometric series. The case f = 1 ; m = n has been discussed (Van der Laan’s formula).
-
For f # I ; m = n one finds: FIRSTMOMENTS.For k 6 f =
-k
.xf-“(f
-
1)
+ (1 - x’)/(l (1
Fork 2 ( f MI,^ = 7[(k
- 1) -f -
1)
- x)
- xn--/+1 )/(I
+ - 211
(234
- Xn--k+l)/(I - 211
(23b)
+ (1 - x f ) / ( l - + x)
(1
SECONDMOMENT AT n. Summation of Equations 23 over k ( 1 5 k 6 n) and multiplication by 27 yields M 2 ”, whence by subtraction of the square of MI,,, (in accordance with Equation 23b) : 7’
M z , -~ Mi.n2 = -[n(l
- x)’ - x’f - 4{f(l
- x’)
- x’)
- 2x(1 - x ” ) -
- x)
fX ) X /
+ 4~ + 11
(24)
For f = 1 this reduces to Equation 1. The most general solution of Equations 21 and 22 (for f # 1 and m # n ) should be obtained by the same method. literature Cited
( 1 ) Bell, R. I>.,Babb, A. L., Chem.Eng.Sci. 20, 1001 (1965).
( 2 ) Jordan, C., “Calculus of Finite Differences,” 1st ed., Budapest, 1939; 2nd ed., Chelsea Publ. Co., New York, 194’7. ( 3 ) Molerus, O., Chem. Ingr. Tech. 38, 141 (1966). ( 4 ) Klinkenberg, A , , Trans. Inst. Chem. Engrs. 43, T141 (1965). (5) Overcashier, R. H., Todd, D. E., Olney, R. B., A . I. Ch. E . J . 5,54 (1959). (6) Retallick. 1%‘. E., IND. ENG. CHEM.FUNDAMENTALS 4, 88 (1965). (7) Van der Laan, E. T., Chem. Eng. Sct. 7,187 (1958).
ADRIAAN KLINKESBERG Bataafse Internationale Petroleum Maatschappz‘j N . V.
The Hague, Netherlands RECEIVED for review May 14, 1965 ACCEPTED January 10, 1966
Aris also pointed out that the transverse diffusion effect will always be important asymptotically as m + 0 3 . For this m-* rather than the & = m-l found from the onecase, & dimensional treatments. N
TRANSVERSE DIFFUSION IN CATALYST PORES SIR: I n a previously published communication with theabove title ( I ) , there is an error which has been kindly pointed out by R. Ark. T h e expansion for small values of (m/L), Equation 9 , lacks some terms of order (m/L)’; the correct expression is & =
-t F ( f - 7 )
I = n
k = n
M z , m=
f(1
+ 242’ + = 0 = a/(1 + y’ - (1 + x)y + x 0 -
xF(1-k)
Again, for i = 2:
(1
+ a)y’
z
j = 1
a = 0.
(1
7
,F,r--j)
(19) The above technique for obtaining each M i , j direct is related to Andrh’s method for solving first-order linear difference equations [ ( Z , pages 58’7, 5921. In this method the recurrent relationsp, - apn-l = V ( n ) are multiplied by successive powers of a and the products are added together. Constant a may be interpreted as the root of the “characteristic equation” y -
the same procedure:
] = n
M1.k
when the kth term of the above series is written as x ~ ( ~ - ’ ) For i = 0, only the f t h Equations 10 has a known term different from zero-i.e., unity-so that the result of the summation is simply:
19 now gives by
(Ty
+
(FYI
literature Cited
(1) Bischoff, K. B,, (1966).
cUEM, F~~~~~~~~~~~5 ,
135
Kenneth B. Bischof The Llniversity of Texas,Austin, T e x . VOL. 5
NO. 2
MAY 1966
285
