I
WILLIAM F. STEVENS Northwestern Technological Institute, Evanston, 111.
Chemical Engineering Kinetics Use in the Scale-Up of Chemical Processes Attention has turned to the fundamental approach of first measuring the rate of reaction and developing procedures for direct design of the full-scale reactor. Analog and digital computers speed the complicated mathematical analysis often required
o m
of the most interesting and important phases of the work of the chemical engineer is the design and specification of commercial chemical reactors. T h e size required for the specified production rate is determined and optimum operating conditions are evaluated. Reactor design is usually based on laboratory and pilot plant investigations of the process, and scale-up is based on chemical engineering kinetics. At present, the fundamentals governing the rates of chemical reactions are not completely understood ; the chemical engineer must study each reaction in the laboratory to obtain rate data, after which he may apply scale-up procedures of reactor design. Chemical Kinetics
“Kinetics” refers to the science of systems in motion. Chemical kinetics deals with the rate of a chemical reaction and all the factors that influence it. A complete kinetic study usually results in a proposed formulation of natural laws to correlate and explain the effects of the variables on the reaction, which can be used as the basis for scale-up of the process. I t seems logical to consider the science of chemical engineering kinetics as made u p of two distinct parts: Investigation of the effects of the various process variables upon the reaction rate, resulting in the development of the rate equation (chemical kinetics) Application of the rate equation to the design of a commercial chemical reactor (reactor design) Such a n approach may be likened to the over-all approach used in the design of a heat exchanger. T h e first step is to determine the over-all heat transfer coefficient, and the second is to use the coefficient to size the exchanger. However, there are two major differences between heat exchanger design and re-
actor design: The general heat transfer equation, q = UAAT,, is applicable to all heat exchanger problems; the chemical reaction rate equation may take a different form for each situation, depending upon the reaction mechanism. Heat transfer coefficients may be calculated directly from physical properties and flow conditions to good accuracy; reaction rate constants must be experimentally determined for each reaction. Therefore, chemical engineers must be concerned with the experimental determination of reaction rate equations and rate constants, as well as with their use in the design of commercial reactors. Classification of Reactions. One method in common use emphasizes the phases present, referring to the reaction as homogeneous if only one phase is involved, or heterogeneous if more than one phase exists. Another procedure classifies reactions according to the method of operation to obtain rate data-Le., batch or flow. This may be extended to include the type of equipment to be used for the full-size reactor. Representative types include : batch, semibatch, and flow, either longitudinal or with complete mixing. The rate of a chemical reaction is a function of the concentrations of the components existing in the reaction mixture, temperature, pressure, and variables associated with the catalyst, if one is present. To evaluate these variables, it is desirable to carry out experiments in which a minimum number of variables are changing simultaneously. Carefully obtained experimental data may be used first to determine the form of the rate equation applicable for the reaction under study, after which the rate constant may be evaluated. The resulting equation and its constants may be used as a basis for scale-up to commercial scale. Smith (6) and other authors have emphasized that a careful distinction should be made between development
of a satisfactory rate equation and determination of the mechanism of the reaction. A satisfactory rate equation is one that may be used with confidence to design commercial-scale equipment for carrying out the reaction. The mechanism of a reaction refers to the exact sequence of steps involved in carrying out the reaction. Evaluation of such a mechanism is usually a much more complicated procedure than development of a satisfactory rate equation. I t is usually necessary, for economic reasons, to ‘design a commercial process before the mechanism of the reaction involved has been determined; the chemical engineer must generally employ empirically determine8 rate equations when applying kinetic principles to scale-up problems. Homogeneous Reactions
Homogeneous reactions are commonly studied as either batch or flow reactions. Early investigators in kinetics worked with batch reactions exclusively; the restraint of constant volume was usually satisfied, whether the reaction mass was gaseous or liquid. Later workers turned to flow reactions for laboratory investigation, because of their similarity to commercial procedures of continuous operation. I n flow reactions involving gaseous systems, constant pressure conditions prevail. Care must be taken to differentiate between these two modes of operation. The common method of classifying homogeneous reactions is by the “order of reaction.” If the reaction rate of a constant volume reaction is expressed by r = k.,CjC~
the order of reaction is the sum of a and b, the nymber and kind of molecules that actually simultaneously combine and not necessarily to the molecular proportions used in the stoichiometric equation representing the over-all equaVOL. 50, NO. 4
APRIL 1958
591
tion. The order of a reaction can be determined only from experimental data taken on the reaction itself. Such data are most easily obtained for a batch reaction operating a t constant volume. Constant temperature operation is also necessary, because the reaction rate constant, k , changes rapidly with temperature. The data taken are usually the concentration of a key reactant or product a t various time intervals during the course of the reaction. Corrigan (2) has summarized the data treatment procedures that may be used to determine the order. Values of k,, the reaction velocity constant, can be obtained simultaneously. Many industrial processes conduct reactions in a flow system a t essentially constant pressure. The volume occupied by a unit mass of reaction mixture may vary, thus affecting the concentrations of the reactants and products. For this reason, reactions in flow systems must be treated by using equations involving partial pressures rather than concentrations. I n a flow reaction, the reaction rate equation above should be expressed as: =
k,pjpj
Therefore, it is necessary to be able to calculate rate constant k,, if the value of rate constant k, is known from batch data. Hougen and Watson (3) have shown that, in general, where n is the order of the reaction (n = a b). I t is possible to design a commercial reactor directly, after the complete kinetic study has been made and the rate equation determined. However, in many cases pilot plant studies must be carried out to solve engineering problems that arise during the final reactor design calculations. Such investigations are usually made with equipment duplicating the commercial unit in every way except size. The resultant data are interpreted in terms of assumed rate equations, and the most likely form is determined. Then the best empirical rate equation can be used for design of the commercial reactor. Smith (6) describes many of the techniques used and summarizes their advantages and limitations.
+
Heterogeneous Reactions
Heterogeneous reactions are of great interest to chemical engineers, because many important commercial reactions involve more than one phase-reactions in which the reactants are actually in different phases, and the large group of catalytic reactions in which the catalyst is present in a phase different from the reactant mixture. I n this latter case, the simultaneous processes
592
of diffusion between phases and chemical reaction on the catalyst complicate the reaction rate equations and make determination of even an empirical rate equation a complex procedure. Most of the theoretical developments in the field of heterogeneous catalytic reactions have been limited to gas-solid systems, in which the reactants and products are in the gas phase and catalyst is a solid. For such systems, Hougen and Watson ( 4 ) have suggested that, when catalyzed by a solid, a gas-phase chemical reaction actually occurs on the surface of the catalyst and involves the reaction of molecules or atoms which are chemically absorbed on the active centers of the surface. If a reactant in the main fluid phase is to be converted catalytically to a product in the main fluid phase, the reactant must be transported to the fluid-solid interface, be chemically absorbed on the solid surface, and undergo reaction to form the chemically absorbed product. The product must then be desorbed and transferred from the interface to the main fluid phase. Simultaneous consideration of the rates of all these operations would give an over-all rate equation far too complex for use. Fortunately, however, in the great majority of cases of interest, one of the steps is slower than any of the others, and is controlling; a simplified equation will apply, under the assumption that all other steps are a t equilibrium. Hougen and Watson (3) have developed mechanism equations for many of these simpler situations. It is necessary only to carry out laboratory studies, using the resulting data to determine the most likely mechanism. Hummel (5) and Stevens (7) have made a rather complete study of the dehydrogenation of butane on a chromiaalumina catalyst. After the catalyst activity was stabilized, attention was turned to determination of the most likely mechanism for the reaction. Integral bed operation was used. By operating a t a constant temperature of 1050" F , complete integral curves of conversion us. reciprocal space velocity were obtained at several levels of constant pressure, 1.75, 3.0, 5.0, and 7.0 atm. absolute. Care was taken to obtain data a t relatively high space velocities, so that the integral curves could be accurately extrapolated to a zero value of reciprocal space velocity. The initial rates of dehydrogenation were then determined by graphical differentiation of each curve, and the resulting values of T O plotted us. x , the total pressure (Figure 1). Considerable information as to the mechanism of the dehydrogenation reaction was obtained merely by a study of these initial rates. As first suggested by Hougen and Watson (4) and later enlarged upon by Smith ( 6 ) , initial rates for a reaction of the type A G X S may be a simple
INDUSTRIAL AND ENGINEERING CHEMISTRY
+
0.065
5
0.055
I
/
3
PRESSURE
I I 5 (ATMOSPHERES)
I
I
7
Figure 1, Initial rate?data for butane dehydrogenation
function of pressure. If activated adsorption of the reactant were the controlling step, ri) should be directly proportional to pressure; if activated dcsorption of a product controls, ro should be constant with pressure. Figure 1 indicates that neither could have been controlling. Preliminary work by Hummel (5) had shown that mass transfer to the catalyst surface and diffusion within the catalyst pellet were of negligible importance. Thus, it seemed probable that butane dehydrogenation on chromia-alumina catalyst follows a mechanism in which the surface reaction controls. Two mechanisms may be postulated for a reversible surface reaction of the type A % R S, as suggested by Hougen and Il'atson ( 3 ) . An absorbed butane molecule might react to form a n absorbed butylene molecule (on the same site) and a molecule of hydrogen in the gas phase adjacent, or an absorbed butane molecule might react with a n adjacent vacant site to form two absorbed product molecules, one on each site. T h e complicated rate equations that hold for a mechanism of surface reaction controlling (3) may be written in simpler form for the initial rate a t zero conversion :
+
Eale&&
I
a
I
I
I
I
(ATMOSPHEBES)
Figure 2 . Check of dehydrogenation mechanism equation
Single-site
CKA,?~ *01
=
Each of these equations can be rearranged to a linear form:
Dual-site
Therefore, it is possible to choose the more likely of the two surface reaction mechanisms by computing ~ / r o and dqo for each pressure and plotting them against pressure. T h e one giving the best straight line will be the more probable mechanism form. This was done with the butane dehydrogenation data. From Figure 2 it can easily be seen that the dual-site mechanism gives the better fit. Additional information concerning the constants in the dual-site mechanism equation was obtainable from the graph of d q oversus T , Reference to the equation for this line shows that: Intercept = ___
,
Thus, it was possible to evaluate constants C and KA from the available values of slope and intercept. T h e results were: KA = 0.286, C = 0.255. Additional data enabled evaluation of the other constants in the complete equation, following the procedures of Hougen and Watson (5); the resulting mechanism equation was suitable for use as the basis for scale-up calculations. Modern Reactor Design Techniques
Reactor design has long involved tedious and expensive experimentation through successive stages of pilot plant reactions. Today, the direct approach is finding progressively wider acceptance. Chu ( 7 ) has described the modern technique and summarized its limitations. The key factor is the establishment of a reliable kinetic rate equation, using a careful experimental procedure similar to that described above. The kinetic equation can then be combined with the basic design equations for material balance, energy balance, heat transfer rate, and pressure drop, and the system solved simultaneously to give optimum operating conditions and reactor size. Solution of the equations is involved and time-consuming, by conventional techniques. However, electronic computers, both analog and digital, overcome the mathematical complexity.
I n effect, the new approach to reactor design transforms the emphasis of the scale-up problem from pilot plant experimentation to computer simulation and analysis. The new design procedure should result in the optimum design for the desired commercial reactor without additional pilot plant investigation. However, before such a design is accepted as final, the validity of all assumptions made in the mathematical derivation of the basic design equations must be checked. Certain possible causes for limitations of the applicability of the kinetic rate equations should be investigated.
Longitudinal Diffusion. Concentration gradients along the longitudinal direction of flow reactors tend to cause diffusion of both products and reactants. T h e magnitude of such diffusion must be kept negligible by operating with a high mass velocity, in both the experimental determination of the rate equation and the commercial reactor. Radial Gradient. I n the usual design equations for a flow reactor, no allowance is made for radial gradient of temperature or concentration. Modification of the heat transfer coefficient during scale-up is usually sufficient to correct for this. However, in highly exothermic reactions the effect is too great to ignore; scale-up by increasing the number of reactor tubes, while keeping the same tube size, will help to avoid difficulties. Wall Effects. The ratio of wall area to reactor volume varies with the size of the reactor. Existence of wall effects should be investigated during determination of the rate equation by operating a t several wall-to-volume ratios. I t is usually best to use the same material of construction for both the experimental and the commercial reactor. Impurity Accumulation. Often secondary reactions are not noticed during one-pass experimental studies. Such reactions may yield products which greatly affect the over-all reaction when present in significant amounts. Continuous recycle in a commercial process may accumulate such products in the system. Therefore, the experimentally determined rate equations should be checked a t several operating conditions under prolonged recycle before they are accepted as valid. Catalyst Life. In general, the activity of a catalyst declines continuously with use. Care must be taken to allow for this fact in the scale-up calculation. The magnitude of the activity fall-off must be determined by extensive experimentation, and the resulting experimental data must be corrected to a uniform catalyst activity before reactor design techniques are applied. Conclusions
Chemical engineering kinetics is a n important tool in the scale-up of chemical processes. Laboratory experiments, carefully run, can be interpreted in
terms of empirical reaction rate equations. These equations, in combination with design equations involving material balances, energy balances, heat transfer rates, and pressure drops, can then be used to determine the optimum design of commercial reactors. Additional experimentation may be needed to tie down the effects of mass and heat transfer on the empirical reaction rate equations. Computers have recently been utilized to carry out involved calculations. As their use develops, the direct reactor design technique should become the preferred procedure for applying the principles of chemical engineering kinetics to scale-up problems. Nomenclature
heat transfer area, sq. feet temperature-dependent overall rate factor CA, C, = concentrations of A or B, lb. moles/cu. foot KA = absorption equilibrium constant for A k, = reaction rate constant, in concentration units k, = reaction rate constant, in pressure units n = order of reaction p A , p B = partial pressures of A or B, atm. = over-all heat transfer rate, Q B.t.u./hr. r = reaction rate, lb. mole converted/hr. cu. foot ro = initial reaction rate, lb. mole converted/hr.-lb. catalyst R = gas constant, in consistent units T = temperature, in consistent units U = over-all heat transfer coefficient, B.t.u./hr.-sq. foot
A C
= =
-
-O
F.
AT,
= mean temperature difference,
n
=
O
F.
reactor pressure. atm.
literature Cited
(1) Chu, J. C., Chem. Eng., November and December 1956 and January 1957. (2) Corrigan, T. E., Ibid., July, August, September, and October 1954. (3) Hougen, 0. A., Watson, K . M., “Chemical Process Principles,” vol. 111, Wiley, New York, 1947. (4) Hougen, 0. A , , Watson, K. M., 1ND. ENG.CHEM. 35, 529 (1943). (5) Hummel, H. H., Ph.D. thesis, University of Wisconsin, 1948. (6) Smith, J. M., “Chemical Engineering Kinetics,” McGraw-Hill, New York, 1957. (7) Stevens, W. F., Ph.D. thesis, University of Wisconsin, June 1949. RECEIVED for review September 9, 1957 ACCEPTEDJanuary 6, 1958 Division of Industrial and Engineering Chemistry, Symposium on Collection of Engineering Data on a Small Scale, 132nd Meeting, ACS, New York, N. .Y., September 1957. VOL. 50, NO. 4
APRIL 1958
593
