Ind. Eng. Chem. Fundam. 1980, 19, 130-131
130
Comments on: “Cubic Equations of State-Which?’’
Sir: In the comprehensive paper on the subject, “Cubic Equations of State-Which?” (Martin, 1979),the following was presented as the most general form of the volumecubic equation of state with pressure, P, a function of specific volume, V , and temperature, T 4T) + 6(T) p = -RT v (V + P)(V + 7) V ( V + P ) W + 7) (1) Here R is the universal gas constant, CY and 6 are functions of temperature, and and y are constants. (It was pdinted out, however, that the latter could also be taken as temperature functions.) The statement was made that, “By specializing the constants and by straightforward algebraic rearrangement including simple translation in volume, all forms of cubic equations can easily be obtained from eq 1.” The generality of eq 1 has been challenged (Kumar and Starling, 1980),so this discussion is given to show the background of the equation and the various forms in which it may be displayed. Some of these forms were not given in the foregoing paper because the paper already seemed too long with over one hundred equations. The reasoning which was employed in the previous paper was as follows. The virial equation to the third power in volume is
P)(V + y) so that a t a given T RT(V+P)(V+y)-aV+6 P= V ( V + P)(V + 7) R T [ V + (P + y ) V + By] - CYV+ 6
Now with each V one can associate a constant by simple linear addition, so that a(T) + 6(T) p = - -RT t ( V h)(V + 6) ( V + € ) ( V+ X)(V + 6) (3) which a t a given temperature has five independent constants. That this is a cubic equation can be seen by multiplying both sides by (V E)( V A)( V 6). One may ask why V + t was not used in the second term instead of either V + h or V + 6. This was done so that one could obtain two-term cubic equations if 6 were zero; for if the denominator of the second term were say ( V + t ) ( V + 6) with 6 = 0, the equation would be a quadratic. There is a temptation also to associate three different constants with the V’s in the third term rather than repeating 6 , A, and 8. Multiplying out shows quickly this would give a sixth-degree equation rather than a third degree. Thus, the most constants that can be added to the V‘s is three, making the total of five for the equation as a whole. If eq 3 is translated by e , as shown in the pioneer paper on volume translation (Martin, 1967), we get 4T) RT P= v t - t (V + x - t ) ( V + 6 - €)
P V / R T = (1 - ( 2 t - P - y + C Y / R T + ) ~[ t 2 - (P + y ) t + Pr + d / R T + S / R m p 2 ) / ( 1- (3t - P - y ) p + [3t2- 2 ( +~ Y ) t + ~ 7 - [t3 1 -~ (P+~y ) t 2+ ~ y t 1 ~(8)~ 1 Now let dl = -2t + P + y - a / R T , d2 = t2 - (P + y) + Py + d / R T + 6/RT, d3 = -3t + + y, d4 = 3t2- 2(P + 7) t + Py, and d, = -t3 + (6 + y) t2 - Pyt to obtain, with z = PVIRT 1 + dlp + d2p2 z = (9)
v+
+
+
+
+
+
+
d(T)
(V + t - t ) ( V + x - t ) ( V + 6 - t) ( 4 ) If the constant quantities, X - t and 6 - E, are taken as P and y, respectively, this becomes p = -RT 4T) 6(T) v (V + P)(V + y) V ( V + P)(V + y) which is just eq 1. Either eq 1 or eq 3 could have served as the starting point for the subject paper (Martin, 1979), but the former was chosen because of its being slightly simpler. T o proceed to other alternative forms of eq 1, multiply through the top and bottom of the right side by V ( V +
V + (P + r ) V + PrV
Then translate by t to get P= ( R T [ ( V- t ) 2+ (P + y ) ( V - t ) + 073 - CU(V - t)+ 6 ) / ( ( V- t ) 3+ (P + r ) ( V - t ) 2+ Pr(V - t ) )= ( R T V - [BRTt - RT(P + 7) CY] V+ RTt2 - RT(P + 7 ) t + RT& + at + S)/(V- [3t - (P + y)]V + [3t22(P + Y)t + 671 V - t3 + (P + 7)t2 - Pr t ) ( 6 ) Next, letting al = -2t + (P + y) - CYIRT, a2 = t2- (0 + y ) t by + a t / R T 6/RT, a3 = -3t /3 y, a4 = 3t2- 2(P + y ) t + by, and a5 = -t3 + (P + y ) t 2- Pyt, we obtain R T ( V alV + a2) p=(7) V + u3V + a4V + u5
+
+
+
+ +
+
which is the equation given by Abbott (1978) as suggested by Robinson (1977). Next in eq 6 multiply both sides by V I R T and then divide the numerator and denominator of the right side by V. Noting that density, p = 1 / V , the result is
1 + d3p
+ d4p2 + d5p3
This is the equation offered by Kumar and Starling (1980) with the statement, “there is no previously reported cubic equation of state from which eq 9 can be obtained.” They also state, “the equation presented by Martin is not the most general form of a volume-cubic equation of state.” Clearly, both of these propositions must .be rejected as not correct since eq 9 has been obtained from eq 1 by the procedures clearly stated in the comprehensive paper. A further statement should be made about the above alternative forms of the cubic equation of state. Although they may give a little additional insight into the equation, they really add nothing, particularly because the best cubic equation has been shown to be
or
+
0019-7874/80/10 19-0 130$01.OO/O
(5)
pR=
TR Z,VR - tf’,/RTC -
-
27(A - BTR) (11) 64(ZcV~+ - tP,/RTc)2
These are the generalized forms in terms of reduced temperature, pressure, and volume that follow directly from eq 1by the arguments previously given (Martin, 1979). To put them into ratios of volume or density expansions would give no advantage whatsoever. 0 1980 American Chemical Society
131
Ind. Eng. Chem. Fundam. 1980, 79, 131-132
Literature Cited Abbott, M. M., 176th National Meeting of the Chemical society, Miami Beach, Fla., Sept 1978. Kumar, K. H., Starling, K. E., Ind. Eng. Chem. Fundam., preceding correspondence in this issue, 1980. Martin, J. J., Ind. Eng. Chem. Fundam., 18, 81 (1979).
Martin, J. J., Ind. Eng. Chem., 59(12), 34 (1967). Robinson, R. L., Jr., personal communication to M. M. Abbott, Sept 1977.
Department Of Engineering The University of Michigan Ann Arbor, Michigan 48109
Joseph
J'
Experience with an All-Glass Internal Recycle Reactor
Sir: We would like to clarify several issues raised in the recent correspondence of Berty (1979) and the article by Fitzharris and Katzer (1978). We find that an all-glass internal recycle reactor has limited but very useful application in kinetic and poisoning studies of real catalysts-particularly monolithic catalysts. Our quartz internal recycle reactor is of the same design as that described by Fitzharris and Katzer (1978). We c o n f i i e d ideal-mixing behavior of the reactor by transient response tests. We were also able to reproduce methanation turnover numbers and activation energies on supported nickel in pellet, powder, and monolithic form within 20-25% of those obtained in tubular reactors and within a factor of 2 compared to data from the literature (see Table I). Berty criticized the all-glass reactor, commenting that a t low pressure the glass impeller may not develop sufficient pressure head to cause significant flow through a bed of catalyst particles. Based on pressure drop calculation, he predicted a flow through a single particle diameter deep bed in the glass reactor of about 0.2 m s-l (at 1 atm and 1000 rpm), approximately one-sixth the flow needed to produce incipient turbulence. We checked this experimentally by means of a hot-wire anemometer measuring air flows developed by the stirring unit at ambient pressure (86 kPa) through monolithic and particulate catalysts. We also calculated flows using Berty's equations for comparison with our experimental measurements of air flows. Table I1 shows the calculated and measured flows. In the case of the particulate catalysts, the measured flows are a factor of 10 less than calculated values. We believe the lower than expected flow can be attributed to a thin layer of quartz wool used to support the particulate samples. Comparison of the measured flow with the outlet flow indicates a recirculation ratio of unity for the particulates. However, for the monolithic catalyst the measured flow is only a factor of 3 lower' than the calculated value; this is anticipated since the length of the bed exceeds the monolithic channel diameter by about a factor of 3. Moreover, the observed recirculation ratio is 15 for the monolith. Thus a reasonable flow with considerable recycle can be developed through the monolith but not through the particulate samples. In actual operation, higher flows and recirculation ratios would be possible since the reactor can be pressurized to obtain at least a factor of 2 increase in pressure with a proportional increase in pressure head. Moreover, the depth of the monolithic bed could be decreased by at least a factor of 2-3. In other words, it is possible to obtain flows of 0.5-0.6 m s-l through a monolithic catalyst bed in the all-glass reactor. Berty suggested that a velocity greater than 1.4 m sdl is desirable to obtain turbulent flow in a catalyst bed. The implication is that turbulent flow is needed to avoid nonideal flow patterns which adversely affect the measurement of reaction rates. However, our experimental results in Table I suggest that this is not a serious problem for our thin beds and particular reaction conditions. 0019-7874/80/1019-0131$01 .OO/O
Table I turnover no. x
catalyst
All-Glass Internal Recycle Reactor 6% Ni on A1,0, powder 3.2 6% Ni on A1,0, pellets 3.1 3% Ni on A1,0, monolith 2.2 Tubular Fixed-Bed Reactors 6% Ni on A1,0, spheresa 5% Ni on A1,0, powderb
2.5 1.0
a Jarvi, G. A., Mayo, K. B., Bartholomew, C. H., t o be published in Chem. Eng. Commun., 1979. Vannice, M. A., J. Catal., 3 7 , 449 (1975). Calculated using his rate expression and activation energy. In molecules of CH, produced/site-s (at 500 K , 1 atm).
Table I1
catalyst
calcd flouP of air. mis
measd flow of air. m/s
0.20 0.20
0.02 0.02
1 1
0.33c
0.10
15
40 mesh Al,O,b 0.32 cm diameter A1,0, pellets cordierite monolith 31 o/cmZd
recirc ratio
Using the equation of Berty (1979) and 1000 rpm. Supported on a 1-2 mm layer of quartz wool. Using the pressure drop expressions of Hegedus, presented at the 166th National Meeting of the American Chemical Society, Chicago, Ill., Aug. 1973. 1 cm long X 2.5 cm diameter. a
Presumably if laminar flow patterns were affecting the rate measurements, a change in flow direction through the reador (particularly of the Fitzharris-Katzer design) would reveal this; however, upon reversing the inlet and outlet streams, the rates and conversions were found to be the same (within -+3-5%). Perhaps in thin catalyst beds, entrance effects and developing laminar flow serve to produce plug-flow-like patterns. Thus our experience confirms Berty's opinion that the all-glass recycle reactor is limited primarily to the study of films and thin catalyst beds a t low pressure. Nevertheless, we think it has broader application than he implied. I t is in our opinion a convenient, low-cost tool for obtaining accurate kinetic and poisoning data of practical as well as theoretical value for almost any real catalyst, because in principle any solid catalyst can be prepared in monolithic form (Deluca and Campbell, 1977). In addition, the quartz reactor enables sulfur poisoning studies to be conducted at low but industrially relevant poison concentrations (e.g., 0.001 to 1 ppm of H2S)where essentially all practical metal systems and even Pyrex introduce serious adsorption and/or contamination problems (Fitzharris and Katzer, 1979; Bartholomew, 1979). We feel that this development by Fitzharris and Katzer of a reactor system inert to contamination by sulfur a t such low concentrations is a remarkable contribution. 0 1980 American
Chemical Society
