Optimum Fractional Conversion for Chemical Reactions at Equilibrium Michael R. Samuelsl and John D. Eliassen2 Department of Chemical Engineering, University of Delaware, Newark, D E 1971 1
Although it i s generally accepted that the fractional conversion of a given species in an equilibrium reaction can be increased b y adding an excess of the other reactanis, it i s shown that this i s not uniformly correct. In fact, it i s found that under some circumstances, adding a large excess of the other reactants can actually decrease the fractional conversion of the component it is wished io react. Equations are developed from which to determine the feed composition which leads to the maximum fractional conversion of a given species.
A l t h o u g h much attention has been given to det,ermining the effect' of reaction conditions (pressure arid t'emperature) on the equilibrium compositions of a reacting mixture, much less attention has been devoted to t'he question of preparing reactant feed streams which lead to optimum equilibrium conversions a t the specified reaction condit,ions. I n this paper we shall examine the effect' of reactant compositions on one form of optimum equilibrium conversions. I n particular we shall examine the effect of reactant feed ratios on the fractional conversion of any particular reactant species. We will show that the accepted method for achieving this goal is partially incorrect,, and will, in fact, be counterproductive in some circumstances. Let us consider t,he chemical reaction -aldl
-
Cuzilz . . . . =
,
...
ac-lAc-l
+ acd,
(1)
where
At
i t h reacting species stoichiomet'ric coefficient of A i ,positive for a product, negative for reactant. I n many reactions of industrial importance, one or more of the reactants is considerably more valuable t'han the remainder. In this case it may be desirable to charge the reactor in such a way that the great,est possible fractional conversion of these valuable components is achieved, eren if the fractional conversion of the remaining components is quite lowfor example, consider the methanol synthesis reaction = ai=
CO
+ 2H2 = CHsOH
(2)
Usually, hydrogen would be considerably more expensive than CO and thus (if we for the moment ignore the possibility of recycling t'he unreacted feed) one could easily picture a situation in which the optimum reaction feed mixture would be that which led to the maximum fractional conversion of HZ t'o CHsOH, more-or-less independent of the amount of CO required. The normal response in this situation would be to simply use a large excess of CO and thereby drive the reaction to near completion with respect to hydrogen. This approach has in fact been suggested by Denbigh (1966) who states "The effect of a n excess of one of the reactants is to decrease the yield relative to this reactant, and to increase the yield relative to the ot'her. For example, in the reaction To whom correspondence should be addressed.
* Present address, General Electric Co., Selkirk, SY
12158
+
discussed above (A 2B = C), let it be supposed that recirculation of uncoverted A and B is for some reason impractical. Then if d is a more valuable raw material than B, i t might be found advantageous to operate with an excess of B to raise the degree of conversion relative to A." However, as we shall show, this statement is not uniformly correct: For the methanol synthesis considered above, a large excess of CO results (surprisingly) in a low fractional conversion of hydrogen, rather than a high one. For a very large excess of CO, essentially no hydrogen will react. Theory
T e may show under what' conditions t'his unusual behavior will result by carefully examining the equilibrium composit,ions for the reacting species. We presume that the reaction conditions (pressure and temperature) are fixed and known. Thus the outlet compositions are related through the equilibrium conditions:
II ( y i p ; K'
=
yd(yi)-, cyI
-+
Equation 3 then becomes:
&I).
where
-
I n this case the total exponent on nIois a negative quantity, and the right-hand side of Equation 7 goes to zero as nIu m . Since the ai's and n,"s on the left-hand side of Equation 7 are fixed, the only way that the left-hand side of Equation 7 can vanish as nIo-+ m is if X 0. Thus we observe that the S vs. nIocurve passes through zero a t both nIu 0 and nIu a .Since we know the curve is positive for all intermediate nIo,the full curve must appear as shown in Figure 2, and passes through a maximum at' some intermediate nIo.As we will show, the maximum in X does not generally occur a t the stoichiometric feed ratio. Thus for the case where a < a I , n e observe that the optimum fract'ional conversion of species 1 occurs a t some intermediate value of nIo.If nIois increased above this value, S will actually decrease. Thus the statement quoted from Denbigh in the beginning of this discussion is clearly in error under these conditions. The feed compositions which correspond to the maximum fractioiial conversion of species 1 in these cases may be determined as follows: Equation 4 for the outlet mol fractions is substituted into Equation 3, the definition of K,. Taking the differential of the natural logarithm of K , then yields -+
-+
Son. let u;i esaniiiie the behavior of X as any one particular (reactant) nIo (1 # 1) is varied between zero and infinity,
while the remaining nio's are held constant. For nIo
=
0,
-Ymu.t :Ao equal zero; clearly if one reactant is missing, no renction c a n occur. A s nIois increased above zero, X begins to $0 t1i:it' Equation 5 is satisfied. However, a t no point can S exceed ( a l n i o / ~since i) the ith term in the denominator of Equation 5 would be negative. S o w let us examine the behavior of X for larger nIo.*is 7~1' appronclies infinity, the total number of moles in the equilibriuni mixture approaches nIo.(Remember, the amounts of the remaining feeds have been held constant'). Thus in the limit of nIo-+ a , the right-hand side of Equation 5 approaches K (nIo)a.Similarly, the I t h term in the denominator becomes ( n I U ) a I .Thus Equation 5 approaches:
I n the right-hand side of Equation 7 , the ( a I ) in the exponegative number, but the CY may be positive, negative, or zero depending on the st,oichiometry of the reaction under consideration. \Ye now consider three possible cases: Case 1. a > aI (either CY is posit,ive, or a smaller negative iiumber than a I ) . In t'liis case the total exponent on nIois a posit'ive quantity $0 that the right-hand side of Equation 7 grows without bound as nIO m . Since the numerator 011 the left-hand side of Equation 7 is bounded ab nIo m , one of the terms in the denominator must vanish as nIo-+ a . T h a t is, one reactant is completely depleted. Kliich reactant it is depends on the feed conipositioii uid stoichiometr\- of the reaction. However, the main point to lie observed here is that X nioiiotonically increases n-it,h nIO,until reaching the limit imposed by the feed condi-
-+
-it fixed temperature and pressure, K , = const, so that d In K , = 0. I n addition, if we seek the conditions where X is maximized, dX = 0. Thus, Equation 8 reduces to:
-+
-+
384
Ind. Eng. Chern. Process Des. Develop., Vol. 11, No. 3,
1972
where no =
n I othe total number of moles of reactants fed 2 = 1
to t h e reactor. Equation 9 is now expressed as a single summation.
X
Although there are several alternative conditions which might be imposed on this equation, only two will be examined here. I n the first case, we assume that not only is n10 fixed, but also that the inlet amounts of all but one reactant are fixed. Thus dnto = 0 for all i # I . We then ask what value of nIO will maximize X . Equation 10 tells us that we must select nIOsuch that
Figure 2. Variation of X with n; when a:
< cyI
and that ff1
CY1
(19)
or where R
which can be solved for nI by noting that
ffR =
ffi
i=l
R
+
Thus, summation of Equation 17 from 2 to R yields
nto = nFO nro
no = i=l
where R equals the number of reactants and nFOis the total number of moles of all reactants except component I fed to the reactor, and is considered here to be fixed. Thus,
no - 1 =
(o~R
- cy1)
no 01
or
(14) Equation 14 tells us precisely how much of component I to feed with a mixture containing nFomoles of other reactants to maximize the fractional conversion of component 1. Since aI < 0 because 1 is a reactant, then the condition stated earlier, CY - cyr < 0 is necessary if nIOis t o be positive. This result also correctly predicts that nIOmust become infinite when 01 - cyI = 0, but it predicts a negative nroif CY - aI > 0. The latter result is clearly of no physical significance. We have previously shown that when CY > 0, X will become increasingly large as nIO +. m . Apparently in this case X asymptotically approaches a maximum and the derivative, dX/dnro, is never truly zero. -Inother interesting case is that in which we fix n10 = I , but now allow all other reactant mole numbers to vary freely. Again we seek the feed conditions under which the fractional conversion of species 1, is maximized. Since dn10still equals zero, Equation 10 yields
\
ff1
If the dnio are to be allowed to vary freely, then Equation 15 will be satisfied only if
Thus we find that
which allows us to determine the feed rates of all reacting species such that the fractional conversion of species 1 is maximized. Discussion
We may summarize the results of the preliminary discussion as follows: If the stoichiometric coefficient of a reactant in a reaction is less than or equal to the sum of all the stoichiometric coefficients (both products and reactants), then increasing the amount of that component fed t o the reactor will increase the fractional conversion of all other reactants. However, if the stoichiometric coefficient of a reactant is greater than the sum of all coefficients, then the fractional conversion of the remaining reactants will increase for low values of nIO,but will go through a maximum and then decrease with further increases in nrO.For example, consider the ammonia synthesis reaction. If we consider hydrogen to be species 1, the reaction would be written: 3H2
+ Kz = 2XH3
(23)
where ff1
ff1
ffl
or solving for n t :
a!
(17)
Thus we conclude that each of the remaining reactants, 2 through R , should be in stoichiometric ratio to each other, though it is not clear what ratio these should bear to %lo. T o determine this ratio we note that
=
-3,
ff2
=
-1,
CY3
= 2,
cy
= -2
(234
I n this case, cyz > CY, and so increasing the ratio of nitrogen t o hydrogen in the feed a t first causes an increase in the fractional conversion of hydrogen until the maximum X is reached. Further increases in the nitrogen feed rate cause a decrease in the fractional conversion of hydrogen. On the other hand, if we wish to examine the effect of hydrogen on the fractional conversion of N2, the reaction would be written: Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 3, 1972
385
“OPTIMUM FEED” (FOR MAX. H p CONVERSION1
Ky=10,000
I
Ky=1000 Ky=100
-
Ky=lO
Ky=l.O
Ky.0.l
I
-1, a2
I
+ 3H2 = 2NH3
r\;Z
a1 =
I
-
=
-3,
=
+2, a
=
-2
(24) I n this case a2 < a , and so an increase in the hydrogen feed rate causes a monotonic increase in the fractional conversion of nitrogen. The nitrogen feed rate for maximum conversion of hydrogen may be obtained from Equation 14 where it is assumed that nFo = 1 mol hydrogen. The optimum nitrogen feed rate is then
nIo =
a3
(-1)(1 mol) 1 -2
+
=
Thus, the nitrogen feed rate should equal the hydrogen feed rate for maximum conversion of hydrogen. The effect of nitrogen feed rate on the fractional conversion of hydrogen is shown (quantitatively) in Figure 3 for various values of K,, the equilibrium factor. Several conclusions may be drawn from Figure 3: (1) The maximum in the hydrogen conversion does fall a t the equimolar feed rate predicted above; (2) the maximum is a rather broad one, with essentially optimum hydrogen conversions obtained for 0.5 < noN2< 2 mol of nitrogen/mol of hydrogen fed to the reactor; and (3) although a stoichiometric feed ratio produces hydrogen conversions not too far below the maximum, nitrogen feed rates below the stoichiometric result in hydrogen conversions well below the maximum. For the case where there are only two reactants as in the ammonia synthesis, Equations 22 and 14 yield the same result since there is only one free mole number in either case. However, if three or more reactants are present, Equation 22 gives the reactant feed rates which yield the true maximum fractional conversion of species 1, while equation 14 yields the maximum conversion obtainable when all of the reactant feed rates except one have been predetermined. literature Cited
Denbigh, K., “The Principles of Chemical Equilibrium,” 2nd ed.. 11 176. Cambridge Univ. Press. Cambridge. England (1966). ’ Prigogine, I., Defay, R., “Chemical Thermodynamics,” Longmans, Green, S e w York, KY (1954).
-
1 mol
Y
,
RECEIVED for review May 6, 1971 ACCEPTED January 7 , 1972
Optimal Process System Design Under Conditions of Risk Joel Weisman University of Cincinnati, Cincinnati, OH 46661
A. G. Holzman Cniversity of Pittsburgh, Pittsburgh, PA 15613
Chemical and nuclear systems usually contain a number of poorly known quantities which can b e characterized only in terms of probability distribution functions. Process design is, therefore, really carried out under risk. A mathematical programming procedure dealing with these risk elements is described. The procedure is extended to consideration of the effect of a nonlinear value for money. The results of the computations allow the selection of optimum design margins for use in detailed system design.
I t is now widely recognized that the overall design of a process system may be considered in terms of a formal optimization problem. It is generally desired to maximize profit or minimize costs 1% hile meeting any restrictions imposed by management, market conditions, or the physical relationships governing the process. The problem may therefore be stated as Minimize (or maximize) 386
F(z1,x2
xn)
Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 3, 1972
(1)
subject to the constraints
Gt(xi,xz
xn)
{E/
bt,i
= 1, 2,
m
(2)
The function to be minimized or maximized is referred to as the objective function and the variables, x3, which can be controlled by the designer, are called the decision variables.
