Exact uniqueness and multiplicity criteria for an nth order reaction in a CSTR

Shorter Communications

Exact uniqueness and multiplicity criteria for an nth order reaction in a CSTR (Rcceiued 21 October

1979; accepted 18 January

When a single nth order (n 2 0) exothermic irreversible chemical reaction occurs in a continuously stirred tank reactor (CSTR), the dimensionless steady-state temperature y satisfies the equation F(Y) b

(l+B-y)nexp[y(l-t)l=~ Y-l

lgYll+g

(1)

For the special case of n = 1, the cubic eqn (3) becomes a quadratic one, yielding the well known uniqueness criterion BY < 4(1+ P).

y = TIT,

YZ-y(Y-l)~O

H=Ua 4PG

Da = V&T,,&,“-‘lq

(2)

The use of T, as the reference temperature gives an identical equation for both the cooled and the adiabatic case [l]. In a recent communication Tsotsis and Schmitz[2] derived necessary and sufficient conditions for multiple solutions of eqn (1). The conditions are based on the requirement that the function F(y) is not monotonic in (I, I+ p). Thus, multiplicity occurs for some L&s,if and only if UY, “, R 8) = (n -l)Y~+(Y+l+P-n)Y2-y(2+B)Y+Y(l+/3) (3) has a negative value for some y in (1,l +B). Using the transformation (4) x = (y - 1)/S this condition may be expressed as M(x,y,B,n)=a*‘+bx~+c~+l
(5)

for some x in (0, I), where

b=B[u-2(1-n)+Bl~B(y-n)

(6)

c = pr2 + (n - o/s - rl A m

Icy~lt~.

(11)

and sutlicient condition for uniqueness for ri(ltB)*/a

for

r!?
Y<4

for

@21.

(12)

We prove here that the determination of the region in the parameter space for which multiplicity occurs for any n > 0 does not require consideration of the constraints placed by (8) and that for any ,!l uniqueness exists for all Do if and only if r=v:,

(13)

where y$ is the largest real zero of the cubic function d (defined by (7)). Very strong, explicit bounds on y,$ are given in [3] and may he used instead of tinding the exact roots of I$. It should be noted that either Fig. 3 in [2] or Pi. 2 in [3] which demarcate the multiplicity region for the adiabatic case, can be applied also for the case of a cooled CSTR by defining p and y as in eqn (2) here. Chang and Caio[4] derived two conditions similar to (7) and (8). which determine the region of multiplicity for an nth order reaction in a nonadiabatic reactor. An unfortunate choice of To instead of T, as the reference temperature introduced two additional parameters in the steady-state equation and prevented a compact presentation of the results. As will be shown here, the two conditions may be reduced to the single criterion (13).

d(y,8.n)4Ay’tBY2+Cy+D<0 and when the local miminum of M, xti, -b •t “‘:“.”

y=Y~[(n-I)Y+(l+B-n)l~G(Y

(y-1)(1+/!-y)

(7)

.

(14)

G has a unique minimum for y in (1,1 t 0). Thus,(14) has two solutions in (1, 1 + f3) for y > y*, one solution for y = y”, and no solution for 0 < y C y*. Hence, y* is the exact uniquenessmultiplicity boundary. Since

for all y in (l,i+@), n>O and p>O, y* must be a unique, continuous, monotonic decreasing function of /3. For any prescribed values of n and fi the cubic function 6 has either one real root (y3) or three real roots (y, < y* < y,). Since d is negative for suSiciently large y it follows that condition (7) can be satisfied if either

1,

where A,_@

B = 26[( 1- 2n)B2 + 2(2 - n)fl + I+ n]

-4/3(1-n)(3+2n)-(l--n)a

@ “)

’ ’

is such that

-3d <

C=-~‘-4(3-5n)~‘t2(4n2tlln-11)~2

The equation r = 0 may be rewritten as

- VI.

Tsotsis and Schmitz[2] have shown that this occurs if and only if the equation M = 0 has three real roots, i.e. whenever

D=4(1tB-#(l+/3).

Thus, the necessary all Do is [3]

for

decreasing

proof

a=(n-l&a

ocx&=

(10)

For a zeroth order reaction F(y) is a monotonic function in ( 1,l t p) if and only if

where T =T,+HT, In 1+fi

1980)

(9)

Y’Y3

(16)

or if YI c Y < Yz.

(17)

211

Shorter Communications However. it was shown that for any fi and n multiplicity exists for all Y larger than the critical value y*. Thus, multiplicity may exist only for (18)

Y > 71 k v:.

We shall now prove that condition (8) is satisfied for any y which satisfies (IS), so that condifion (18) is both necessary and suflicientformultiplicity for some Da. Consider first reactions with an order n > 1. Here, a, b, and 3a + 2b + c are positive. Thus, *

mm

-(3a+Zb+c) \‘(b*-3a~)+(b+3a)<~*

_,=~W3nc)-(b+3d= 3a

‘I

!19) x,,,~,, is positive if and only if c is negative, i.e. when

b

0

I x

According to (5), M(Kg,,8,n)~O

(21)

for any I in [O, I]. while M(r*) must vanish for some x in (0.1). Since

Fig. 1. Schematic graphs of M for y equal to gl, g2 and g,. Conditions (3X33) are satisfied if and only if either Y~g,~g2+3(l-n)B,

$+3Xlr-I)<0

forO
(22)

it follows from (21) and (22) that

Y>g,:2gz-g,+3(1-n)/3.

y’= Y:>gl.

Q.E.D.

(23)

Consider now a reaction with an order of 0
(35)

The functions g, and g, intersect only at &. Thus, for any 0 > PI2 either condition (34) or (35) are satisfied for any y and 0 0.

b=B(y-gJ>O

(34)

Or

(36)

(24) A schematic of M for this case is shown in Fig. I(b). Since M(y*) vanishes for some x in (0,1). it follows from (22). (31) and (36) that

and c=B(g,-Y)
(25) Y*= Yr>gd.

(37)

For a fixed n the monotonic graphs of g, and g2 intersect only at @=~,*=-n+V(l+n*-n)

(26)

We conclude that also in this case condition (8) is automatically satisfied by condition (IR), which is therefore necessary and sufficient for multiplicity for some l)o. Acknowlcdgcment-This work is part of a research project supported by the National Science Foundation (Grant ENG 7505336).

Thus, conditions (24,25) are satisfied if

It

7>g2

for

0~P~f-h

(23

Y’&?

for

f%
(28)

can be easily shown that for all x in [O,11 and O< n < 1 M(x,g,,B,n)>O

forBzB1z

M(x, 82. B, n) > 0 for 812 2 B > 0.

(2% (30)

A schematic of ti for these cases is shown in Fig. l(a). Since M(y*) must vanish for some x in (0, l), it follows from (2% (29) and (30) that 7* ’ sup(g,, gz)

Apartment of Chemical Engineering Universityof Houston Houston, TX 77004, U.S.A.

(31)

NOTATION

a CC LG E

According to (19), I,,,~”will be smaller than unity if either bt3azzO

(32)

or 3a+2b+c>O.

*Author to correspondence should be addressed.

(33)

G” H AH f M n i T

area through which heat transfer occurs concentration of reactant heat capacity Damkiihler number activation energy function defined by (1) function defined by (14) dimensionless heat exchange parameter heat of reaction reaction rate constant function d&ted by (5) reaction order volumetric flow rate universal gas constant temperature

T. M. LEIB D. LUSS*

212

Shorter Communications

U overall heat transfer coefficient V volume of reactor y dimensionless temperature

tn reference 0 feed

Gnek symbols t¶ dimensionless heat of reaction y dimensionless activation energy r function defined by (3) p density 6 function defined by (7)

RERERENCW

[I] Kauschus W., Demont J. and Hartmann K., Chem. Engng Sci. 1978 33 1283. [2]l~;otsis T. T. and Schmitz R. A., Chem. Engng Sci. 1979 34 [3] Van den Bosch B. and Luss D., Chem. Engng Sci. 1977 32

203. [4] Chang H. C. and Calo J. M., Chem. Engng Sci. 1979 34 285.

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Modified design procedure for a sparger reactor sequence (Received 24 December 1979; accepted 16 June 1980) In an earlier communication[l] we have indicated a general graphical design procedure for a sequence of sparger reactors in which a second order liquid phase reaction proceeds in a stagewise fashion. The prediction of the reactant concentration in each stage and hence the conversion depended on a search procedure initiated along a straight line representing the mass balance equation at the given stage and drawn from the known feed stage located on the abscissa in a E-IU-~~ diagram for the given system. The mass balance equation is given by:

The 1.h.s. of eqn (I) is a linear function of the unknown M and is drawnasastraightlinefromthefeedstate~withaslopeoequalto:

k,

60

v = - a.D,+zklA+ B

80

100

12c

M

The line intersects several E,(M) parametric curves and the correct estimate of M satisfies eqn (1). The trial and error procedure often involves interpolation between discrete E; curves and contributes to errors in prediction. This can be obviated by using the E-M relationship on Cartesian co-ordinates in which the parameter Q= M/(E;- I) rather than E; acts as the parameter. Q is independent of the reactant concentration and its value can be fixed straight away from known information for the system:

Fig. I. Modified parametric plot of enhancement factor relation and design of a sequence of sparger reactors.

cor-

(3) In Fig. I the modified design procedure is worked out for the same example cited in the earlier communication. The parameter Q has a value 3 and is represented parametrically on a line from the origin (1.0) in the PM figure. The feed state ti has a value of 75 and is locafed on the abscissa by point E The slope of the The point of intersection with the operating line is -l/6. parametric line P immediately fixes the exit state from reactor 1. Proceeding on similar lines, Sand U denote the exit states in the other two reactors. It is seen that the computed conversions in the reactors differ somewhat from those reported in rhe earlier communication. It is believed that the earlier values are in error since they depended on interpolation procedures. It is of interest to note that the parametric relationship E-M-Q is linear for all values of Q and is represented as a pencil of lines drawn from

Fig. 2. Dependence of slope of parametric plot of Fig. I on Q.