Interaction of Chemical Reactions and Transport. 1. An Overview

Ind. Eng. Chem. Res. 2005, 44, 5199-5212

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Interaction of Chemical Reactions and Transport. 1. An Overview Stuart W. Churchill† Department of Chemical and Biomolecular Engineering, University of Pennsylvania, 311A Towne Building, 220 South 33rd Street, Philadelphia, Pennsylvania 19104

Chemical reactions, fluid mechanics, heat transfer, and mass transfer have both direct and indirect interactions. The direct ones, which are examined here, involve the effects of the fundamental mechanisms of reaction and transport on one another. The indirect ones, which arise from combined reactors, heat exchangers, and separators, are to be examined separately. Although most of the direct interactions have been recognized for decades, the investigations of such phenomena experimentally, theoretically, and numerically are very limited in number and scope. Furthermore, they are scarcely mentioned in textbooks on either chemical reaction engineering or transport. They could readily be incorporated in the software for computer-aided process design but have not, possibly because of the significant added complexity but more likely because of a lack of recognition of their importance or even their existence. In a supplementary investigation, selective numerical calculations have shown that, although the interactions between reaction and transport are of second order in magnitude in some applications, they are critical in others. Accordingly, in another supplementary investigation, new concepts, generalizations, and asymptotes have been devised to encompass these interactions and thereby abet their inclusion in future textbooks and software. Introduction The existence of a radial variation in the mixed-mean velocity in a tubular reactor for all rates of flow is not open to question, but the effects of the velocity distribution on the conversion are often ignored in design and analysis. The effects of the molecular diffusion of energy and species on the conversion have received only limited study and the effects of transport by turbulence almost none. The effects of interactions between reaction and transport, including that of the changes in the number of moles and in the temperature field on the velocity field, that of the heat of reaction on the heat-transfer coefficient, that of the change in composition on the mass-transfer coefficient, and that of the turbulent fluctuations on the rate constant, have been the subject of only a few analyses and experiments of limited scope or of none at all. In elementary textbooks on chemical reaction engineering, the primary effect of the laminar velocity profile is often mentioned cursorily, but in advanced as well as elementary textbooks on reaction engineering, the other effects and interactions go unmentioned. Accordingly, the common failure to account for them in process design and analysis is hardly surprising. Sophisticated mathematical techniques continue to be developed for various aspects of chemical reaction, as illustrated by the presentations at the MaCKie-2002 conference [see work by Constales et al.1 and, in particular with respect to tubular reactors, Balakotaiah and Chakraborty2]. However, these advances, although praiseworthy, contribute little to the task at hand, namely, undergraduate education and process design. The radial velocity distribution in tubular reactors, which is a consequence of momentum transfer, results in a radial variation in the rate of reaction and thereby † Tel.: (215) 898-5579. Fax: [email protected].

(215) 573-2093. E-mail:

in the composition in an Eulerian sense even for reactions that are negligibly energetic. Insofar as the reaction or reactions are energetic (the usual case), the consequent changes in the temperature field result not only in changes in the reaction rate constant but also, in a gas, in a change in the mixed-mean velocity as well as in the radial and longitudinal velocity distribution. A change in the velocity distribution results in a radial as well as a longitudinal variation in temperature and thereby results in a further variation in the rate of reaction, the temperature, and the composition in both a gas and a liquid. The velocity distribution plays an additional role in that it serves as a weighting function for the local concentration in the integration to determine the mixed-mean conversion. The radial and longitudinal variations in composition and temperature result in radial and longitudinal molecular diffusion because of the molecular motions and, if the flow is turbulent, because of the eddy motions as well. Chemical reactions modify the heat- and mass-transfer coefficients, as compared to those for no reaction, by virtue of changing the distributions in temperature and composition. These latter changes may be very important if heat and mass transfer are modeled in terms of the mixed-mean temperature and concentration, respectively, but not otherwise. In turbulent flow, the fluctuations in temperature and composition modify the rate of reaction if it is expressed in terms of the timeaveraged temperature and composition, as is usually the case. The design and the analysis of the operation of chemical reactors are more complicated and difficult than their counterparts for flow, heat exchange, and mass exchange because, in all but the most trivial cases, more than one species and more than one reaction mechanism must be considered, leading to a set of nonlinear algebraic equations that are coupled with the equations of conservation for energy and each independent chemical species in partial differential form. In

10.1021/ie049256u CCC: $30.25 © 2005 American Chemical Society Published on Web 02/22/2005

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Ind. Eng. Chem. Res., Vol. 44, No. 14, 2005

some chemical processes in tubular flow at high temperature, such as pyrolysis and combustion, which involve gaseous free radicals, the number of significant reaction mechanisms may exceed 100, each of which incorporates two or three parametric, numerical constants, most of them empirical and somewhat uncertain in value. Fortuitously, as long as a reasonably complete and accurate set of mechanisms is utilized, the overall numerical uncertainty in the prediction of the final composition is quite constrained by the multiple, somewhat self-compensating paths of reaction. Global models may yield fair predictions for the temperature and major species but are useless for predictions of minor species such NO and CO from combustion. The solution of these arrays of nonlinear algebraic equations is in itself often more difficult than the numerical integration of the differential equations of conservation. Reaction, heat transfer, and mass transfer are highly coupled, and the dependence of the individual rates of reaction on temperaature is highly nonlinear. Momentum transfer may also be coupled, although less critically, by virtue of the dependence of the viscosity and density on temperature and composition. It is the totality of these very real complications that has led to the widespread utilization of grossly oversimplified models such as “plug flow”, global models, and sometimes isothermality in reactor design. Logically, a very general differential and algebraic model for this complicated behavior might simply be solved numerically for the purposes of process design, and indeed with the present state of fluid mechanics, transport, chemical kinetics, computer hardware, computer software, and numerical algorithms, this approach is perhaps feasible for any given set of conditions. A large library of reaction rate mechanisms has been developed, and the principal uncertainty in the modeling of tubular chemical reactors arises from the effects of turbulence, if it is present, rather than from the chemical mechanisms. Unfortunately, the aforementioned gross simplifications are now so deeply and comfortably ingrained in reactor engineering that their use has persisted even though advances in computer hardware and software have largely eliminated their justification in practice. On the other hand, simple models are essential in a first course in chemical reactor engineering in order to provide students with a conceptual understanding before they are introduced to and confused by all of the details and complexities. At the present time, the primary simple model for tubular reactors consists of A f B, rate mechanisms, and equilibrium expressed in terms of concentrations rather than fugacities, invariant physical properties, constant pressure, and, most seriously, “plug flow”, which together with its counterparts of space velocity and space time, is usually taken to imply a uniform radial temperature and composition. The challenge is to describe the real behavior in tubular reactors in such a way that the simplifications and idealizations made in the interests of a conceptual understanding are not embedded beyond recall in the student’s mind in school and thereby afterward in practice. Responding to this challenge, rather than improving the practice of process design itself, has been the primary objective of the present investigation. The treatment of heat transfer in elementary textbooks provides a pattern for this task. Insofar as physical property variations and viscous dissipation are negligible, convective heat transfer in channels may be

expressed in terms of simple algebraic expressions for Nu as a function of Re with only one parameter, namely, Pr. (See, for example, work by Churchill and Zajic.3) The first step in the current investigation has been to try to identify and/or derive asymptotes and analogies for tubular chemical reactors analogous to those used so successfully by Churchill and Zajic and others for forced convection in channels and to try to use these elements to develop a comparable generalized structure. The nonlinear dependence of the rate of chemical reactions on composition and temperature and the coupling of the differential balances for energy and each independent species suggest that greater difficulty may be expected in deriving asymptotes in closed form than for convective heat and mass transfer, that some idealizations must be made, and that generalization may be impeded because of the large number of parameters associated with the mechanisms of chemical reaction. Some success in devising simple but realistic models has been achieved by Churchill4 and Yu et al.5 by considering various aspects of the behavior of tubular chemical reactors in isolation, that is, of one aspect at a time. These segregated analyses have an obvious shortcoming, namely, that the results cannot be expected to be completely general, but that seems to be a less serious impediment than expected. The calculations of Yu et al. suggest that an even more successful generalization may be achieved in terms of finitedifference models and methodologies for their solution rather than in terms of correlative equations. The restriction of the analysis herein to homogeneous liquid- and gas-phase reactions in round tubes has been to keep the manuscript to a reasonable length. Some of the corresponding overlooked effects that occur in other types of chemical reactors and for other conditions, and in particular for combined reactors and exchangers, have been examined by Churchill.6 Nonenergetic Reactions Asymptotic Formulations for a First-Order Irreversible Reaction in Laminar Flow. The first and most important step in developing improved representations for chemical reactions in tubular flow would appear to be to eliminate outright the ubiquitous postulate of “plug flow”, which occurs in the real world only for semisolid materials such as ice cream pushed through a tube by a wooden rod. On the other hand, the many solutions in the literature based on that postulate need not necessarily be discarded in that they can be re-interpreted as asymptotic solutions for perfect radial mixing of the composition and thereby for radial uniformity in any flow. The rate of a first-order reaction Af B is usually expressed as

rˆ ) kCA

(1)

The correlation of the rate of reaction in terms of concentration rather than fugacity is questionable on theoretical grounds but, nevertheless, is an almost universal practice. The rate of reaction according to eq 1 may be equated to the resulting rate of change in composition in fully developed flow with perfect radial mixing of composition to obtain

(

um -

)

dCA ) kCA dL

(2)

Ind. Eng. Chem. Res., Vol. 44, No. 14, 2005 5201

For subsequent simplicity and convenience, eq 2 may be rewritten in terms of the conversion of species A, namely, Z ≡ 1 - CA/CA0, as follows:

um

dZ ) k(1 - Z) dL

(3)

In the absence of heat exchange with the surroundings and for the flow of a liquid or of a gas with a negligible pressure drop, k and um are effectively invariant, and eq 2 may be integrated from Z ) 0 at L ) 0 and the result rearranged as

Zp ) 1 - e-kL/um

(4)

The subscript p designates the asymptote for the physically conceivable case of perfect radial mixing of the composition rather than, as might be inferred, the solution for the hypothetical case of “plug flow”. The radial transfer of momentum in a round tube gives rise to a radial variation in the velocity, which, in turn, gives rise to a radial variation in composition because the fluid at the centerline moves more rapidly and has a shorter residence time for reaction in any segment of length. In the asymptotic extreme of no radial mixing (negligible diffusion) and fully developed flow, an expression for the conversion Zr in any annular element at any radius r may be adapted from eq 3 simply by substituting Zr for Zp and u for um. In the laminar regime of flow, the fully developed velocity distribution is given by

r2 u )21um a

[ ( )]

(5)

The local conversion Zr at each radius within the reactor is then given as a function of length by

Zr ) 1 - e-kL/2um[1-(r/a) ] 2

(6)

The corresponding mixed-mean conversion is equal to the integral of Zr, weighted by the local velocity distribution, over the cross section, that is, by

Zm )

( ) ()

∫01Zr uum

d

r a

2

(7)

Substituting for ur/um from eq 5 in eq 7 then results in the following general expression for Zm in fully developed laminar flow:

∫01Zr[1 - (ar ) ] d(ar ) 2

Zm ) 2

2

(8)

Substituting Zr from eq 6 in eq 8 results, in turn, in the following expression for the mixed-mean conversion due to a first-order reaction:

Zm ) 2

∫01( 1 - e-kL/2u [1-(r/a) ])[1 - (ar ) ] d(ar ) m

2

2

2

(9)

Examination of eq 9 reveals that the mixed-mean conversion has a double dependence on the velocity distribution, on the one hand, because of the radial variation in the conversion and, on the other hand, because of its weighting by the flow, and that these two effects are somewhat compensatory. The postulate of “plug flow” disregards both. The integral of eq 9 may

be expressed in nominally closed form as

Zm ) 1 - E3{kL/2um}

(10)

Here E3{x} is a tabulated function called the exponential integral of order 3. Equation 10 constitutes the classical solution for a first-order, equimolar, irreversible, nonenergetic reaction in fully developed laminar flow with negligible pressure drop, negligible radial transport of species, and no heat exchange with the surroundings. This solution is inconvenient in two respects. The first inconvenience is that values of E3{x} must be looked up in a handbook such as that by Abramowitz and Stegun7 or obtained from a software package, although the following simple asymptote may be used to evaluate E3{x} for values of the argument larger than about 50:

E3{x} f e-x/(3 + x)

(11)

The second inconvenience is that neither the general solution nor the asymptotic one may readily be inverted to give kL/um as an explicit function of Zm. To avoid the inconvenience of looking up values of E3{x}, values of Zm may be determined by evaluation of the integral in eq 9 by quadrature. An even better alternative, at least educationally, is to exercise and reinforce the existing skills of the students in numerical analysis by utilizing stepwise integration of finite-difference equations corresponding to eqs 3 and 8 or 9. Such a procedure has the merit of being applicable for other reaction mechanisms and conditions for which an analytical solution is not possible. Other than its usefulness in calculating values for E3 for very large values of kL/um, eq 11 has the merit of allowing the derivation of the exact value of 2 for Llam/ Lp, that is, the ratio of the required lengths of a tubular reactor in laminar flow with negligible radial mixing to that for perfect radial mixing, in the limit as complete conversion is approached (see work by Churchill4 for a detailed derivation). This limiting value of the ratio of lengths is a measure of the maximum error due to the postulate of “plug flow”. On the other hand, the maximum error, if expressed in terms of the conversion for a fixed length, is only about 12%. This is a warning concerning the critical dependence of a conclusion on the arbitrary choice of a criterion. Asymptotic Formulations for a First-Order Irreversible Reaction in Turbulent Flow. Hill8 presented a very elegant review and analysis of chemical reactions in homogeneous turbulence, but the behavior that is described appears to be of more intrinsic than practical interest because the major changes in conversion in a shear flow are expected to occur near the wall, where the turbulence is anything but homogeneous. Fox9 has written an entire book on computational models for reacting flows but without illustrative calculations even for fully developed tubular flow and at too sophisticated a level for use in a first course in reaction engineering. For fully developed turbulent flow in a round tube, the same idealizations as those for laminar flow are implied plus three more, namely, negligible transport of chemical species and energy by the turbulent fluctuations, a negligible effect of the turbulent fluctuations on the rate of reaction itself, and negligible viscous dissipation. The differential formulations for u and um in fully developed turbulent flow in a round tube may

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be expressed, per Churchill,10 in the following generalized forms:

( (

) )

du+ ) 1 -

y+ [1 - (u′v′)++] dy+ + a

(12)

dum+ ) 1 -

y+ 3 [1 - (u′v′)++] dy+ + a

(13) um

Here y ) a - r, u+ ) u(F/τw)1/2, y+ ) y(Fτw)1/2/µ, a+ ) a(Fτw)1/2, and (u′v′)++ ) -Fu′v′/τ. To implement the numerical solution of eqs 12 and 13, Churchill11 devised the following correlating equation for the dimensionless local turbulent shear stress:

(u′v′)++ )

| { exp

([ ( ) ] 0.7

y+ 10

}

3 -8/7

dZ (dL ) ) 2kC Zp )

A0(1

- Z)2

1

(17) (18)

um 1+ 2kCA0L

It follows that in fully developed laminar flow the local conversion is given by

+

(

)| )

1 6.95y+ -1 1+ + + 0.436y 0.436a a+

-8/7 -7/8

(14)

Integration of eqs 12 and 13, with (u′v′)++ from eq 14, is not possible in closed form. Evaluation of the corresponding integral formulations by quadrature is feasible, but stepwise integration from y+ ) 0 to y+ ) a+ of finite-difference representations for eqs 12 and 13 for any value of a+ > 150 (Re > 4000) is straightforward and more efficient computationally. This latter process results in predictions of the local velocity within 0.4% and of the mixed-mean velocity within 0.1% of the best experimental data (see work by Churchill et al.12). As an example of the application of eqs 12-14 to a tubular reactor, the following expression for the conversion due to a first-order reaction in a longitudinal filament of fluid in turbulent flow at a dimensionless distance y+ from the wall may be inferred from eq 4 and ur/um ) u+/um+: +/u+)

Zr ) 1 - e-(kL/um)(um

(15)

Equation 7 remains applicable, and the finite-difference formulation for the mixed-mean concentration in turbulent flow in terms of the dimensionless variables of eqs 12-14 is thereby

dZm )

formulations for a few such reaction mechanisms follow then illustrative numerical results. a. Second-Order Reaction. The expressions for a second-order, irreversible, equimolar reaction, 2A f B + C, in flow in a round tube with perfect radial mixing, corresponding to eqs 3 and 4 for a first-order reaction, are

( )( )

-2 y+ u+ (kL/um)(um+/u+) (1 e ) 1 dy+ (16) a+ um+ a+

Stepwise integration with respect to y+ from 0 to a+, of a finite-difference representation for eq 16, together with ones for eqs 12 and 13, while again using eq 14 for (u′v′)++, was utilized by Yu et al.5 to calculate the first accurate numerical results for a first-order reaction in fully developed turbulent flow. Such calculations are within the capabilities of current students and most practicing engineers, although they may not be acquainted with eqs 12-14. Asymptotic Formulations for Other Reaction Mechanisms. The methodology utilized to formulate expressions for a first-order reaction in the asymptotic limits of perfect radial mixing and negligible radial mixing is readily adapted for other reaction mechanisms, including reversible, nonequimolar, and multiple ones. Stepwise integration of finite-difference representations is also directly applicable, although care must be taken to account for the interactions. Illustrative

Zr )

1 um[1 - (r/a)2] 1+ kCA0L

(19)

and the mixed-mean conversion by

[

(

Zm ) 2φ 1 - φ ln 1 +

1 φ

)]

(20)

Here, φ ) kCA0L/um. Surprisingly, the solution for the mixed-mean conversion for this second-order reaction in fully developed laminar flow with negligible radial mixing of species and energy is simpler than that for a first-order reaction in the sense that a tabulated function does not appear. The exact asymptotic value of 4/3 for Llam/Lp in the limit as complete conversion may be derived from eq 20 (again see work by Churchill4). The methodology for the calculations for a first-order reaction in fully developed turbulent flow is readily adapted for this second-order one. b. Reversible Reactions. For a first-order reversible reaction, A f B, it can be shown that all of the elements of the solution for an irreversible reaction, namely, eqs 4, 5, 10, 12, 17, and 18, are directly applicable if L and Z are both multiplied by (1 + K)/K, where K ) k/k′ is the equilibrium constant and k′ is the reaction rate constant for the reverse reaction. c. Consecutive Reactions. Perhaps the simplest scheme of coupled, multiple reactions is A f Bf C. The two expressions corresponding to eq 2 are

( (

) )

um -

dCA ) kACA dL

(21)

um -

dCB ) kBCB dL

(22)

while the three corresponding to eq 4, that is, for perfect radial mixing, are

CA/CA0 ) e-kAL/um

(23)

kA CB ) (e-kAL/um - e-kBL/um) CA0 kA + kB

(24)

CC ) CA0 - CA - CB

(25)

Expressions for the corresponding local values for fully


developed laminar flow with no radial mixing may be obtained simply by substituting 2um[1 - (r/a)2] for um in eqs 22 and 23 and then substituting these expressions in eq 25. In turn, expressions may be derived for the mixed-mean concentrations by integrating these expressions for CA and for CB weighted by u/um ) 2[1 - (r/a)2] over the cross section. These integrations may be carried out analytically in terms of integral functions or numerically by quadrature or a stepwise procedure. For turbulent flow, the entire process is necessarily carried out by stepwise integration. For more complex sets of equimolar reactions, the combined process, including the determination of the concentrations for perfect radial mixing, is most conveniently carried out by stepwise integration for each independent chemical species. It may be noted that, for the reacting system A f Bf C, only two chemical species are independent. In the development herein, A and B were arbitrarily chosen as the independent species and C was chosen as the dependent one. Some consideration of this choice is appropriate in more complex systems because some formulations may be simpler, and the stepwise calculations may then converge more rapidly. d. Nonequimolar Reaction. For a nonequimolar, irreversible, liquid-phase, first-order reaction A f 2B, eqs 2-4, 9, 10, 17, and 18 are applicable without modification because the change in density is negligible. On the other hand, for an ideal-gas-phase reaction, the formulation must be modified significantly to account for the change in the mixed-mean velocity due the change in the number of moles. For perfect radial mixing of energy and the chemical species, um may simply be replaced by um0(1 + Zp), where um0 is the rate of flow at the inlet. Equation 3 is then replaced by

um0

)

)

kL 1 ) 2 ln - Zp um0 1 - Zp

(27)

An alternative formulation may be attained by utilizing the theorem of the mean (see, for example, work by Churchill,13 p 408), which results in

(

)

kL 1 ) (1 + Zp)M ln um0 1 - Zp

(28)

Here, the subscript M designates the integrated-mean value. Equation 28 is nominally exact, but an error may be, and usually is, introduced by the approximation utilized for the mean value. Three representative arbitrary approximations are the arithmetic, logarithmic, and geometric average of the limiting values of 1 + Zp, namely, 1 and 1 + Zp. Churchill4 tested the accuracy of these approximations with the exact values obtained from eq 27 and concluded that the geometric mean was the best. The first complexity associated with one or more nonequimolar gas-phase reactions is apparent from the algebraic structure of eq 27 in that it cannot be inverted analytically to give Zp as an explicit function of kL/um0. The second and more serious complexity is that Zp, as given by eq 27, is not directly adaptable for Zr in laminar and turbulent flow because the expansion due to the

(29)

The other is the pseudo-stationary-state flow, namely

u ) (u/um)S(1 + Zm)um0

(30)

where (ur/um)S is the velocity distribution for fully developed flow. The procedure based on this latter expression is more complicated because Zm, the mixedmean value, is not known in advance, but the approximation of the flow is intuitively more realistic. The resulting differential expression is

2um0[1 - (r/a)2](1 + Zm)

dZ ) k(1 - Zr) dL

(31)

with Zm given by eq 8. A stepwise method of solution may be devised by expressing eq 31 in a finite-difference form such as

∆

(

)

k∆L/um0 1 ) 1 - Zr 2(1 + Zm)[1 - (r/a2)]

(32)

and eq 8 as a summation such as r)a

∑

r)0

(26)

and eq 4 by

(

u ) u0(1 + Zr)

Zm ) 2

dZ 1-Z )k dL 1+Z

(

reaction increases as the wall is approached, resulting in a two-dimensional field of flow. An exact expression for the resulting perturbation of the flow has apparently never been derived or calculated, although the latter is presumably possible by means of two-dimensional computational fluid dynamics. As a stopgap, two limiting conditions are considered here. One is the perturbed velocity profile corresponding to no radial flow, namely

[ ( )] ( )

Zr 1 -

r

a

2

∆

r

2

(33)

a

Then, starting from Zr ) 0 and Zm ) 0 at L ) 0, ∆Zr, and thereby Zr at k∆L/um0, is calculated from eq 32 for a chosen step size and a series of values of r/a, using Zm ) 0. Next, Zm is calculated from eq 33, using these values of Zr, and that value is used in the second step in k∆L/um0, etc. The rate of convergence of this procedure with step size and the accuracy of the solution may be improved by means of a more sophisticated finitedifference formulation, such as using a mean value of Zm over ∆L in eq 32 rather than the value at the beginning of the step. A similar procedure may be utilized for turbulent flow with y+)a+

Zm ) 2

∑ Zr

y+)0

( )[ ( )] ( ) u+

um+

1-

y+

a+

∆

y+

a+

(34)

Illustrative Numerical Results. The potential error due to the common postulate of perfect radial mixing (or “plug flow”) may be illustrated by comparisons of the numerical values computed by Yu et al.,5 using the above solutions and formulations for nonenergetic reactions in laminar and turbulent flow with negligible radial mixing and with perfect radial mixing. Such comparisons depend critically upon the choice of the dependent variable. For example, as illustrated in Figure 1, the ratio of the required length of a reactor in laminar flow with negligible radial mixing of species to that for perfect radial mixing (plug flow), as computed by a finite-difference method, approaches the limiting

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Figure 1. Ratio of the length of a tubular reactor for equimolar, irreversible, nonenergetic reactions in fully developed laminar flow with negligible radial mixing to that for perfect radial mixing (“plug flow”) [from Yu et al.5].

Figure 2. Ratio of the length of a tubular reactor for a first-order, equimolar, irreversible, nonenergetic reaction in fully developed turbulent flow with negligible radial mixing to that for perfect radial mixing (“plug flow”) as a function of a+ ) Re(f/8)1/2 [from Yu et al.5].

theoretical value of 2 as the conversion approaches unity for a first-order reaction and approaches the corresponding limiting value of 4/3 for a second-order reaction. On the other hand, the fractional increase in the conversion due to perfect mixing at any fixed reactor length is, as mentioned before, no more than 12% for a first-order reaction and no more than 10.4% for a second-order reaction. For turbulent flow, the conversion depends on the Reynolds number as well. The ratio of the required lengths is plotted in Figure 2 for a first-order reaction and in Figure 3 for a second-order one as a function of a+ ) Re(f/2)1/2. By virtue of the following semitheoretical correlating equation of Churchill11

um+ ) (2/f)1/2 ) 3.29 -

( )

227 50 + + + a a

2

+

1 ln(a+) 0.436 (35)

a one-to-one relation exists between a+, u+, f, and Re ) 2aumF/µ ) 2a+um+. According to Churchill et al.,12 eq 35, which is based on eq 14, represents the best experimental data for f within 0.1%. On the basis of this expression, the values of Re ) 4015, 17 100, 37 818, 487 839, and 5 938 704 correspond to the values of a+ ) 150, 500, 1000, 10 000, and 100 000. The ratios of reactor length are seen to be less than those in laminar flow, but they still differ significantly

Figure 3. Ratio of the length of a tubular reactor for a secondorder, equimolar, irreversible, nonenergetic reaction in fully developed turbulent flow with negligible radial mixing to that for perfect radial mixing (“plug flow”) as a function of a+ ) Re(f/8)1/2 [from Yu et al.5].

from unity for moderate values of Re. Although the plotted values extend to Re = 106, Churchill and Pfefferle14 have asserted that turbulent flow is attained in tubular reactors only for combustion or detonation or for thermal cracking in tubes of very large diameter. Hence, the curves for a+ ) 500 and greater probably do not represent realistic conditions. Generalizations. Generalized expressions for Zp and Zm for a few limited classes of single reactions have been formulated by Churchill.4 On the other hand, the common finite-difference methodology utilized for the illustrative calculations by Yu et al.5 is an alternative and much broader form of generalization. Idealizations. The above formulations, solutions, and numerical results for nonenergetic reactions incorporate several idealizations that were noted only in passing. These as well as others will now be identified and quantified one by one insofar as possible. a. Pressure Drop due to Friction and Inertia. The pressure drop due to friction in fully developed laminar flow in a round tube, as given by the theoretical expression

f ) 12/Re

(36)

or in fully developed turbulent flow as represented by eq 35, along with, in each case, the pressure change due to inertial effects, may readily be taken into account in finite-difference formulations and their numerical solutions for reacting mixtures if the change in the velocity due to nonequimolar reactions and energetic effects is accounted for. However, illustrative calculations for representative conditions indicate that the perturbation of the conversion in tubular reactors due to both of these sources of pressure drop is ordinarily negligible in both the laminar and turbulent regimes of flow. b. Deviations from Fully Developed Flow near the Inlet. The fluid motion in the inlet region of a tubular reactor depends on the configuration at the entrance, is ordinarily two-dimensional, and may require a significant distance to approach the onedimensional velocity distribution corresponding to fully developed flow. A measure of the order of magnitude of the effect of the development of the velocity on the conversion is provided by the difference of the solutions for the conversion in “plug”, laminar, and turbulent flow. On the basis of the aforementioned illustrative calcula-


tions, it may be inferred that entrance effects have a negligibly small effect on the small conversions that are attained near the inlet. c. Finite Radial Diffusion. Cleland and Wilhelm15 carried out numerical calculations for the effect of molecular diffusion on the conversion for a first-order, equimolar, irreversible reaction at constant density and temperature for a series of values of Df/ka2 ) 2Sc(L/a)(um/kL)/Re ranging from 0 to ∞ and of kL/2um ranging from 0 to 2.0 and concluded that diffusion was significant (which they defined as a decrease of greater than 1% in the conversion) for Df/ka2(kL/2um) ) 4Sc(L/a)/Re < 2 × 10-3. They further concluded from this result that radial diffusion is ordinarily negligible for liquids but may not be for gases in tubes of small diameter. Additionally, they carried out experimental work for a pseudo-first-order liquid-phase reaction (the hydrolysis of acetic hydride in an aqueous solution) and measured conversions corresponding to slightly higher rates of diffusion than those predicted. The discrepancies that they observed were attributed primarily to natural convection generated by temperature differences of up to 0.21 K, which stemmed from the exothermic heat of reaction and a cooling jacket. Finlayson and Rosendoll16 recently devised a computer code called CRDT (Chemical Reactor Design Tool) for student use in carrying out numerical calculations for tubular reactors and presented some illustrative results based on an unnecessarily crude model for the effects of turbulence. Ekambara and Joshi17 recently reviewed previous work and developed a new model for the effects of axial mixing in a reactor in fully developed laminar flow, but their illustrative calculations are for nonreacting flows. The models of Fox9 also include molecular and turbulent diffusion. Despite these contributions, this is a topic that needs further study. d. Reaction Mechanisms in Terms of Concentrations. The error due to expressing the reaction mechanisms in terms of concentrations rather than in terms of fugacities cannot be evaluated without specifying the reaction, and thereby the chemical species, as well as the temperature and pressure. Furthermore, the rate constants in the literature are almost universally expressed in terms of concentration and cannot be reinterpreted in terms of fugacities without knowing the conditions under which they were measured. One obvious shortcoming of expressing rates of reaction in terms of concentration is that the equilibrium constant based on concentrations is pressure-dependent, whereas that based on fugacities is not. e. Pseudo Stationary State for the Concentrations of Free Radicals. The concept of Christiansen18 and others of a pseudo stationary state for the concentrations of free radicals is one of the greatest advances of all time in reaction kinetics in that it explains the previously puzzling observations of fractional and bilinear mechanisms of reaction. However, as shown by Pfefferle,19 this concept results in a grossly unsatisfactory approximation for the prediction of the formation of pollutants such as NO and CO from combustion at high temperature because of an insufficient number of collisions between the molecules and free radicals in the flamefront. Fortunately, the present state of computational software and hardware, as well as of the database for free-radical reactions, suggests that this approximation is no longer necessary or appropriate.

Energetic Effects Most gas-phase reactions are significantly energetic, resulting in longitudinal temperature gradients and, except in the extreme case of perfect mixing, in radial gradients as well. Furthermore, most reaction mechanisms are strongly dependent on temperature, thereby resulting in the use of external heat exchange for control and safety. The prediction or analysis of tubular reactors for energetic reactions thereby invokes the solution of a partial differential energy balance coupled to differential species balances and possibly coupled to the differential momentum balances as well. Because the overall objective of the work reported herein is generalization, the identification or derivation of asymptotes is even more critical than that of nonenergetic reactions. Energetic effects on the velocity field, and by that means on the conversion, are generally less than those due to nonequimolar gas-phase reactions because the tolerable fractional change in the absolute temperature for adiabatic conditions is generally less than that in the longitudinal molar flux, which may double, as in the example of A f B + C. The only other energetic effects to be examined herein are that of the turbulent fluctuations in temperature on the reaction rate constant and that of the heat of reaction on the heat-transfer coefficient for exchange with the surroundings. Effect of the Temperature on the Rate of Reaction. The effect of the temperature on the rate of most single-reaction mechanisms can be represented reasonably well by the Arrhenius equation

k ) k∞e-E/RT

(37)

Here k is the rate constant, k∞ the frequency factor, E the energy of activation, R the universal gas constant, and T the absolute temperature. Sometimes a dependence on the absolute temperature to a small power is included as a factor. That nuance is unimportant in what follows directly but will be considered in a subsequent context. Dependence of the Temperature on Conversion. For fully developed laminar or turbulent flow in the extreme limit of vanishingly small radial and longitudinal mixing of energy and species and no exchange of energy with the surroundings (adiabatic conditions), the following one-to-one relationship exists between the absolute temperature and the chemical conversion Z:

T ) T0 +

QRZ c

(38)

It is convenient in most applications to rearrange eq 38 in the following dimensionless form:

( )

QR T )1+Z T0 cT0

(38A)

Here QR is the exothermic heat of reaction per mole of A reacted, c the mean capacity of the mixture, and T0 the absolute temperature at the inlet. For simplicity, QR and c are postulated to be invariant. Equation 38A is applicable for the mixed-mean temperature Tm as a function of the mixed-mean conversion Zm for all conditions but, in terms of Tr and Zr for an annular filament of fluid, only for the limiting cases of complete and negligible radial mixing.

5206


Dependence of the Rate Constant on Conversion. Equations 37 and 38A may be combined as follows to eliminate T in favor of Z: -(E/RT0)(T0/T)

k ) k∞e

) k ∞e

(-E/RT0)/[1+Z(QR/cT0)]

(39)

For QR/RT0 , 1, which applies to most reactions carried out adiabatically in a round tube, eq 39 may be approximated as follows: 2

k = k∞e(-E/RT0)[1-Z(QR/cT0)] ) k0eZ(EQR/RcT0 )

(40)

The quantity

k0 ≡ k∞e-E/RT0

(41)

may be recognized as the rate constant at the inlet, whereas k∞ is that for an asymptotically large temperature. It should be noted that for QR/RT0 , 1, the three parameters, namely, k∞L/um, E/RT0, and QR/cT0, are replaced by two, namely, k0L/um and EQR/RcT02. This reduction, which may not have been proposed previously, constitutes a significant gain in terms of correlation and the required amount of computation, insofar as QR/RT0 , 1. The dimensionless quantity EQR/RcT02 that characterizes the effect of the heat of reaction in this regime is hereafter designated as the asymptotic thermal parameter and symbolized by ψ. For an exothermic reaction, QR and ψ are positive and k increases with Z, whereas for an endothermic reaction, they are negative and k decreases. It should be noted that, because E/RT0 is always positive, QR/cT0 and EQR/RcT02 necessarily have the same sign. Small absolute values of QR/cT0 do not necessarily imply small absolute values of ψ. The mathematical range of validity of eq 40 and thereby the characterization of the behavior by ψ has been examined by Yu et al.,5 who found the practical range considerably greater than might have been expected. Illustrative Formulations and Solutions for Liquid-Phase Reactions. Formulations and solutions illustrating energetic effects for several simple reactive mechanisms and conditions in the adiabatic flow of a liquid in a round tube follow for perfect radial mixing and for fully developed laminar and turbulent flow with negligible radial mixing. a. Liquid-Phase Reaction with Perfect Radial Mixing. The equation for the conservation of species A for a liquid-phase (constant-density), first-order irreversible reaction A f B in the limiting case of perfect radial mixing of energy and species may, by virtue of eqs 3, 37, and 38A, be expressed as

dZp ) k∞(1 - Zp)e(-E/RT0)/[1+Zp(QR/cT0)] um dL

(42)

Equation 42 may be integrated formally from Zp ) 0 at L ) 0 to obtain

k∞L ) um

-E/RT0[1+Zp(QR/cT0)]

Zpe

∫0

1 - Zp

dZp

(43)

The integration posed by eq 43 may be carried out in closed form in terms of exponential functions, but the determination of numerical values then requires “looking up” tabulated values. Alternatively, the integral may be evaluated numerically by quadrature, but stepwise

numerical integration of a finite-difference expression such as

[

] [ ]

k∞∆L E/RT0 1 ) exp ∆ ln um 1 - Zp 1 + Zp(QR/cT0)

(44)

is more efficient computationally. Equation 44 is implicit in Zp for a specified value of k∞L/um. One resolution of that problem is to solve this expression for several values of Zp and then determine the desired value of Zp by interpolation (or extrapolation) of calculated values of k∞L/um. Another resolution is to solve eq 42 stepwise for a specified value of k∞L/up using an initial or mean value for the exponential term over each interval. A third alternative is the following approximate algebraic relationship between Zp and k∞L/u, as obtained by formal integration of eq 43 with an integrated-mean value for the exponential term:

{ [

E/RT0 k ∞L ) exp um 1 + Zp(QR/cT0)

]} { } ln

M

1 1 - Zp

(45)

For a specified value of kpL/um, it is convenient to invert eq 45 to obtain

{( ){ [

k∞ L -E/RT0 exp um 1 + Zp(QR/cT0)

Zp ) 1 - exp

]} }

(45A)

M

For the asymptotic condition of Zr(QR/cT0) , 1, the rate constant may be represented by eq 40, resulting in the approximation of eq 42 by

um

dZp = k0(1 - Zp)eZpψ dL

(46)

If Zpψ , 1 as well, eq 46 may be approximated, in turn, by

um

dZp = k0(1 - Zp)(1 + Zpψ) dL

(47)

Equation 47 may be integrated analytically to obtain

[

]

1 + Zpψ k0L 1 ) ln um 1+ψ 1 - Zp

(48)

which may be expressed explicitly in terms of k0L/u as

Zp )

1 - e-k0(1+ψ)L/um

(48A)

1 + ψe-k0(1+ψ)L/um

Because eq 48A is in closed form and fully explicit in Zp for a specified value of k0L/u, it is the most convenient of all of the solutions in this section for this quantity, but it is applicable only for Zpψ , 1. It follows from eq 48 that insofar as ψ , 1,

[

]

ln(1 + Zpψ) L 1 ) 1LT0 1 + ψ ln( 1 - Zp)

(49)

Here, LT0 is the required length for a nonenergetic firstorder, equimolar, irreversible reaction with perfect radial mixing and negligible heat exchange with the surroundings. An alternative approximate solution for Zpψ , 1 may be devised by utilizing an integrated-mean value for the


exponential term in eq 47 to obtain

(

k0L 1 1 ) ln um 1 - Zp (1 + Zpψ)M

)

(50)

For all of the previously mentioned choices of a mean value, including the logarithmic mean, which is the most logical choice here, eq 50 reduces, for a sufficiently small value of Zpψ, to

k0L ) um

(

)

1 1 ln Zpψ 1 - Zp 1+ 2

(51)

Relative to eq 48A, eq 51 is less accurate and is implicit rather than explicit in Zp. However, its relative simplicity proves to be a significant advantage. It follows from eq 50 that in the regime of Zpψ , 1 the effect of the heat of reaction on the required length of the reactor for any specified conversion in the limit of perfect radial mixing may be represented by

1 L ) = LT0 (1 + Zpψ)M

Zpψ 1 =1Zpψ 2 1+ 2

(52)

Equation 52 is less accurate than eq 49 but has advantages in simplicity and conceptuality. These several approximate solutions are evaluated quantitatively by Yu et al.5 b. Liquid-Phase Reaction in Fully Developed Laminar or Turbulent Flow. The local conversion, Zr, in fully developed laminar flow may be obtained from eq 43, 44, 45A, 48A, 50, or 51 simply by substituting 2um[1 - (r/a)2] for um. The mixed-mean conversion may then be determined from eq 8 or eq 33. Insofar as a mean value of (1 + Zpψ)M with respect to r/a may be postulated, the following expression for the mixed-mean value corresponding to eq 50 may be adapted from eq 10:

[

Zm ) 1 - E3

kL 2um(1 + Zpψ)MM

]

phase flow are very similar to that above for a firstorder reaction in laminar flow and therefore, in the interest of brevity, are not presented here. However, a surprising generality may be noted, namely, that eq 52 remains applicable for Zpψ , 1. However, it should be noted that, because of the different effective weighting functions in the integrals, the exact integrated-mean values of (1 + Zpψ)M and (1 + Zpψ)MM are not identical with those for a first-order reaction. Even so, this generality proves to be useful as an approximation for other reactions. The energy balance for multiple reactions invokes further coupling, but the stepwise numerical integrations remain straightforward. Formulations and Solutions for Energetic GasPhase Reactions. The molar concentrations change with temperature in a gaseous mixture, and for an ideal gas, the following relationship exists between the mixedmean velocity and the mixed-mean concentration in flow through a round tube:

( )

um QR T ) ) 1 + Zm um0 T0 cT0

Equation 54 is applicable for perfect radial mixing of momentum, energy, and species. However, the changing mixed-mean velocity results in a two-dimensional velocity in the laminar regime and in the time-mean velocity in the turbulent regime just like for a nonequimolar gasphase reaction. Again, as a reasonable approximation for both the laminar and turbulent regimes, the velocity distribution, u/um, is herein postulated to be unaffected by the temperature-induced variation in um, and thereby eq 30 is presumed to remain applicable. a. First-Order Ideal-Gas Reaction with Perfect Radial Mixing. For a first-order irreversible equimolar reaction A f B, with perfect radial mixing of energy and composition, the longitudinal variation of the conversion may be represented on the basis of eq 54 by

[ ( )] QR cT0

um0 1 + Zp

dZp ) dL

[

k∞(1 - Zp) exp

(53)

The applicability of eq 53 is limited to Zpψ , 1, and even then it is very uncertain because of the double averaging. Its principal value is in identifying the firstorder effect of the heat of reaction in liquid-phase flow. For quantitative predictions, the mixed-mean conversion should be calculated by numerical integration of one of the expressions for Zr per eq 7 or summation per eq 33. With all of these expressions except for eq 48A and for eq 45A with a second averaging, Zp, and thereby Zr, must be calculated iteratively for a specified value of k∞L/um or k0L/um. A better methodology computationally is to carry out all of the calculations stepwise and simultaneously. The corresponding determinations of Zr and Zm for turbulent flow, based on eqs 43, 44, 45A, 48A, 50, and 51 for Zp, are readily adapted from those already described for a nonenergetic first-order reaction. c. Other Reactions in Liquid Flows. Algebraic complexities may preclude analytical solutions, but the above methodology for stepwise numerical solutions is readily adaptable to most other single-reaction mechanisms in liquid flows. The formulations and calculations for an energetic second-order reaction in liquid-

(54)

-E/RT0

1 + Zp(QR/cT0)

]

(55)

Here um0 is the mixed-mean velocity at the inlet. Equation 55 can be integrated formally, numerically, or approximately to determine Zp as a function of k∞L/ um0 for a series of values of E/RT0 and QR/cT0. The formal integration from Zp ) 0 at L ) 0 may be expressed as

k ∞L ) um0

[ ( )] [ ∫ Zp

1 + Zp

0

QR cT0

E/RT0

exp

1 + Zp(QR/cT0) 1 - Zp

]

dZp (56)

The integral of eq 56 may be expressed in closed form, but the result invokes one or more integral functions and hence is inconvenient for numerical evaluations. The solution of eq 55 may also be expressed in terms of the integrated-mean value of the exponential as follows:

{ [

E/RT0 k ∞L ) exp um0 1 + Zp(QR/cT0)

]} [ ln

M

]

1 + Zp(QR/cT0) 1 - Zp

(57)

5208


For the purpose of numerical stepwise solution, eq 54 can be expressed in a finite-difference form such as

k∞∆L ) um0

[ ( )] [ 1 + Zp

QR cT0

exp

E/RT0

1 + Zp(QR/cT0)

] ( ) ∆ ln

1 (58) 1 - Zp

The rate of convergence of the stepwise solution may be improved by including 1 + Zp(QR/cT0) in the finite difference on the right-hand side, resulting in

[

]

E/RT0 k∞∆L ) exp ∆Ω um0 1 + Zp(QR/cT0)

(59)

where

[ ( )] ( ) ( ) QR cT0

Ω ) 1 + Zp

ln

QR 1 - Zp 1 - Zp cT0

(60)

For a specified value of k∞L/um0, a solution of eq 60 may be carried out for several trial values of Zp and interpolation may be used to determine the one that gives the specified value of k∞L/u0. If a mean value is taken for the exponential term, eq 56 can be integrated formally to obtain

{ [

Ω ) exp

E/RT0

]}

k∞ L 1 + Zp(QR/RT0) M um0

(61)

For a specified value of k∞L/um0, eq 60 must be solved iteratively to determine the value of Zp corresponding to the approximate value of Ω determined from eq 61. For Zpψ , 1, eq 55 may be reduced to

( )]

[

um0 1 + Zp

QR cT0

[ ( )]

dZp EQR ) k0(1 - Zp) 1 + Zp dL RcT02

(62) which may be integrated analytically to obtain

{(

) (

)

k 0L 1 1 E ) 1+ ln + um0 1 + (EQ /RcT 2) RT0 1 - Zp R 0 EQR E QR - 1 ln 1 + Zp (63) RT0 cT0 RcT 2

(

) [

( )]} 0

As contrasted with eq 48, its analogue for a liquid-phase (constant-density) reaction, eq 63 cannot be rearranged to be explicit for a specified value of k0L/u0, and hence this approximate solution for Zp(QR/cT0) , 1 and Zp(EQR/RcT02) , 1 has somewhat limited utility. However, an explicit but slightly more approximate alternative to eq 63 may be derived from eq 55 by utilizing the arithmetic-mean value for 1 + Zp(QR/cT0), namely, 1 + (Zp/2)(QR/cT0), for its integrated-mean, leading to

Zp )

[1 + (Zp/2)(QR/cT0)]M(1 - e-k0L/um0) 1 + ψe-k0L/um0

(64)

b. First-Order Ideal-Gas Reaction in Fully Developed Laminar and Turbulent Flow. Equations 54-64 for Zp are applicable for Zr in fully developed laminar and turbulent flow if 2um0[1 - (r/a)2] and (u+/

um+)um0, respectively, are simply substituted for um0. Equation 33 remains directly applicable for Zm for laminar flow and eq 34 for turbulent flow. c. Other Energetic Reactions. Expressions for other energetic gas-phase reactions with perfect radial mixing and those for laminar and turbulent flow with negligible radial mixing are readily adapted from the preceding formulations for nonenergetic reactions and those immediately preceding for a first-order energetic gas-phase reaction, although they will generally be more complex. Solutions in closed form are not to be expected. Stepwise, iterative, or approximate integration is necessary. Numerical Results for Energetic Reactions in Laminar and Turbulent Flow. Yu et al.5 also carried out illustrative calculations for the effect of the heat of reaction on equimolar irreversible liquid-phase reactions. As representative values, they choose E/R of 1.96 × 104 and 4.9 × 103 K for first- and second-order reactions, respectively, a value for T0 of 700 K, and a range of absolute values of Q/c (the change of temperature due to complete reaction) from 0 to 100 K, resulting in a range of absolute values of Q/cT0 from 0 to 1/7, and a range of absolute values of ψ from 0 to 4 and from 0 to 1 for first- and second-order reactions, respectively. The stepwise numerical integrations produced the somewhat surprising result that L/LT0 is primarily a function of ψ, varies only slightly with the order of the reaction, and is effectly independent of the rate and mode of flow. Equation 52 was found to provide a very good prediction for laminar and turbulent flow as well as for perfect radial mixing for both first- and secondorder reactions for values of ψ , 0.125, but as might have been expected, the predictions became increasingly inaccurate as the absolute value of ψ increased. Effect of Turbulence on the Reaction Rate Constant. The turbulent fluctuations directly affect the rate of a reaction because of the nonlinear dependence of the rate on temperature and, except for first-order reactions, on the composition. For example, insofar as the Arrhenius equation is applicable, the increase in the rate produced by an upward fluctuation in temperature is greater than the decrease produced by the corresponding downward fluctuation. It follows that the timeaveraged temperature and concentration of the reactant are not necessarily applicable in expressions for the rate of reaction. The following analysis of these effects is adapted from that of Glassman,20 which starts with a global mechanism for r, the rate of combustion, namely

rˆ ) kCAn ) k∞CAnT me-E/RT

(65)

The dependence on T m is included in eq 65 in compensation for the representation of many free-radical reaction mechanisms by a global model. Equation 67 may be differentiated and divided by itself to obtain

dcA E dT drˆ )n + m+ rˆ cA RT T

(

)

(66)

Glassman suggests n ) 2, E/R ) 20 000, and T ) 2000 as typical values for the combustion of hydrocarbons. If, as a first approximation, the fractional fluctuations in CA and T are assumed to be equal and m to be the order of unity, temperature effects may be inferred to dominate because m + E/RT . n.


If the influence of the turbulence on T m is neglected, the transient rate constant may be expressed in terms of its value at the time-averaged temperature, T h , as follows:

k{T} T h E 1) exp RT h T k{T h}

[ (

)]

(67)

If the fluctuating temperature is expressed as T ) T h (1 + Rφ{t}), where R is the half-range of the turbulent fluctuations in temperature and -1 e φ{t} e 1, where the integral of φ{t} over time is zero, describes the fluctuations, it follows that 1 - T h /T ) Rφ{t}/(1 + Rφ{t}) ≈ Rφ{t}. Substituting this expression in eq 66, expanding the result in series, and integrating to obtain k h {T h }, the time-averaged value, lead to the following expression for δk, the maximum fractional change in the rate constant in terms of β ) ER/RT h:

δk )

k{T} - k{T h} β2 e eβ - (1 + β) ≈ 2 k{T h}

(68)

If a sinusoidal fluctuation is arbitrarily postulated, its integral over time reduces δk by a factor of 2. On the basis of the arbitrary postulates of a 10% fluctuation in temperature and k ) 0.01 s-1, δk is approximately equal to 0.25. Although this estimate of a 25% increase in the rate constant due to the turbulent fluctuations is subject to considerable uncertainty, it suggests that the effect of the turbulent fluctuations may be significant. Vulis21 carried out test calculations for combustion similar to those of Glassman and also concluded that the fluctuations in temperature increased the rate of reaction significantly and that those in composition had a negligible effect. Chinitz et al.22 examined the effect on the rate of combustion reactions of three different probability distributions for the fluctuations in temperature. They found that the fractional amplification of the rate of reaction decreases as the time-averaged temperature increases but increases as the amplitude of the fluctuations and the energy of reaction increase. The latter two effects are consistent with the simple model of Glassman. Additional calculations to define and generalize the effects of turbulent fluctuations on multiple as well as single reactions for a wide range of conditions are needed before any quantitative generalizations can be formulated, but these several analyses suggest that predictions that ignore these effects, particularly at extreme conditions, should be accepted only with caution. Effect of the Heat of Reaction on the HeatTransfer Coefficient. Insofar as a detailed twodimensional model is utilized, heat transfer through the wall is accounted for simply by means of the thermal boundary condition, for example, by a uniform wall temperature or a uniform heat flux density, and a heattransfer coefficient need not be introduced. Indeed, its use would appear to be inconsistent with chemical kinetic calculations based on the local temperature, composition, and velocity. On the other hand, the effect of the reaction on such a coefficient may be of intrinsic interest insofar as it provides understanding of otherwise inexplicable results and perhaps guidance in choosing optimal conditions. The earliest and one of the few solutions to account for the effects of a chemical reaction on heat transfer in laminar flow is that of Rothenberg and Smith.23 They

postulated a first-order irreversible but not necessarily equimolar reaction in laminar flow with invariant physical properties, including density, and a uniform temperature at the wall equal in value to that at the inlet. Thus, the fluid is heated by the wall if the reaction is endothermic and cooled by the wall if it is exothermic. They solved a finite-difference representation of this model numerically for a variety of presumably representative conditions. The heat-transfer coefficient based on the mixed-mean temperature was found to increase over that for inert flow for both endothermic and exothermic reactions and by as much as a factor of 4 for a strongly endothermic one. They attributed this enhancement primarily to the radial diffusion of the reactant. To provide a better understanding of the process of enhancement of the heat- and mass-transfer coefficients due to a chemical reaction, a very idealized solution was derived in the current investigation for a spatially uniform reaction and a uniform heat flux density from the wall in fully developed laminar flow in a tube. Small but finite perturbations in the temperature of the fluid due to reaction and to the imposed heat flux were postulated, thereby allowing the rate of reaction and the density and viscosity of the fluid to be postulated to be essentially uniform. These conditions presumably lead to fully developed convection. The resulting solution may be expressed as

Nu )

(

48

)

3QE 11 1 + 11

(69)

Here Nu ) 2jwa/λ(Tw - Tm) represents the Nusselt number, QE ) aqR/2jw is the ratio of the input of energy by the reaction to that at the wall, jw is the heat flux density at the wall, a is the radius of the tube, λ is the thermal conductivity of the fluid, Tw and Tm are the wall and mixed-mean temperatures, and qR is the uniform heat of reaction per unit volume. As QE f 0, Nu f 48/11, which is the well-known solution for no reaction. For positive values of QE corresponding to the combination of an exothermic reaction and heating from the wall or the converse of both, Nu decreases slowly as QE becomes large relative to 3/11. For negative values of QE, corresponding to the combination of an endothermic reaction and heating from the wall or the converse, Nu increases and becomes unbounded as 3QE/11 f -1. Despite the gross idealization of a uniform rate of reaction, the predicted effects for positive and negative values of QE are given credence by virtue of their analogy to the wellknown ones for the effect of viscous dissipation on the heat-transfer coefficient in the laminar regime. It should be noted that the effect of the heat of reaction could alternatively be accounted for in this instance without any change in the value of Nu by means of redefinition of the heat-transfer coefficient in terms of an enthalpy difference rather than a temperature difference, as well as by numerical solution of the equations of conservation. A far greater number of analyses appear to have been carried out to define the effects of homogeneous reactions on the heat- and mass-transfer coefficients in turbulent flow in a round tube than in laminar flow. This is surprising not only because of the greater complexity and uncertainty in the modeling but also because, as previously mentioned, fully developed tur-

5210


bulent flow is rarely attained in tubular reactors. Because of this restriction and because all of the early models for transport by the turbulent fluctuations are of questionable accuracy, only representative analyses are mentioned here. Brian and Reid24 and Brian25 carried out analytical and finite-difference solutions, respectively, for asymptotic conditions (chemical equilibrium in the bulk of the fluid and a vanishingly small temperature difference). They generalized their results to some extent by expressing their model in terms of partial derivatives of the kinetic expression. They predict enhancements of the heat-transfer coefficient by as much as a factor of 10. The work of Edwards and Furgason26 is somewhat unique in that they carried out both experiments and finite-difference computations to determine the effect of the exothermic gas-phase decomposition of ozone on the heat-transfer coefficient. They found reductions of up to 27%. Ooms et al.27 carried out finite-difference solutions for first-order, irreversible, endothermic reactions in general, using penetration theory, which is of questionable accuracy as a model of the turbulent transport, and thereby determined the enhancement factor for the heat-transfer coefficient as a function of the Reynolds number Re ) 2aumF/µ, the Prandtl number Pr ) cµ/λ, and the Schmidt number Sc ) µ/FDf as well as of three parameters representing the imposed heat flux density at the wall, the rate of reaction, and the heat of reaction. Hanna et al.28 utilized an eddy diffusivity model of reasonable validity, postulated the analogy between heat and mass transfer, and made a number of numerically justified idealizations to derive an implicit algebraic solution for the enhancement of mass transfer by a first-order irreversible isothermal chemical reaction in the asymptotic limit of large Sc as a function of a+Sc1/3 and (kµ/τw)Sc1/3 only. Their expression predicts enhancements by as high as a factor of 4. They demonstrate good agreement of their algebraic expression with values obtained by numerical integration of their differential model, which is a necessary but not a sufficient test of its functional and numerical accuracy. Mitrovic and Papavassiliou29 used Lagrangian direct numerical simulation (DNS) to compute the effect of a first-order irreversible isothermal reaction on the masstransfer coefficient in turbulent flow between parallel plates of unlimited breadth with an imposed uniform flux of the reacting species at the wall. Their predicted values of the enhancement ratio ζ increased with L/b, and the fully developed values, which ranged up to 9.87 / ) 50 and Sc ) 50 000, were correlated closely by for t1/2 the expressions

[ ( ) ] 5.98

ζ) 1+ and

[ (

ζ) 1+

5.35 1/5.35

/ 0.262 t1/2

)]

4.39Sc0.223 / 0.400 t1/2

for Sc e 10

(70)

3 1/3

for Sc g 100 (71)

/ Here t1/2 ) t1/2τw/µ and t1/2 is the half-life for the reaction. Considering the differences in the modeling, the predicted values of ζ by Hanna et al.,28 when plotted as ζ versus kt1/2, were found to agree remarkably well with the essentially exact values of Mitrovic and Papavassiliou. Equation 70 is presumed to be applicable

for heat transfer with uniform heating at the wall and an endothermic reaction or the converse if the residence time required for the difference between the wall and mixed-mean temperature to attain half of its value at the inlet is taken as the half-life. Equation 71 is not relevant for heat transfer because Pr (the analogue of Sc) does not exceed 100 for ordinary liquids. Equation 69 suggests rearrangement and approximation of eq 70 / 0.262 / . 5.98 or t1/2 . 922 in order to obtain the for t1/2 following expression for uniform heating at the wall and for an exothermic reaction: / 0.262 5.35 / -1.4 ζ ) 1 + [5.98/t1/2 ] /5.35 ) 1 + 2673t1/2 (72)

Except for the most recent one, these several analyses for the effect of chemical reactions on heat and mass transfer are based on outdated models for turbulent transport. In addition, all of them, including the most recent, are for specific or very idealized mechanisms of reaction. Despite these defects and limitations, the results are consistent qualitatively with one another and with those for laminar flow. Bernstein and Churchill,30 in an investigation of the combustion of premixed propane and air in a refractory tube, provided an indirect confirmation of the gross effect of the heat of reaction on the convective heattransfer coefficient. They found it necessary to increase the convective coefficient by an order of magnitude over that predicted by the conventional correlating equations in order to obtain numerically stable solutions in good accord with their experimental observations. Without any hard evidence, they inferred that the apparent enhancement of the heat-transfer coefficient was due to the intensely energetic reaction. Summary and Conclusions The almost universal postulate of “plug flow” in tubular reactors is conceptually misleading and may lead to significant quantitative errors. The use of the space velocity or the space time as a variable implies “plug flow” and thereby should be avoided in tubular reactors. The primary source of error in the solutions for “plug flow” is the neglect of the radial variation of the conversion associated with the velocity distribution, but errors also arise from the neglect of the distributions of temperature and concentration arising from the velocity distribution as well as from the neglect of the change in the fluid motion resulting from both nonequimolar reactions and the heat of reaction in gas-phase reactions and, in turbulent flow, from the neglect of eddy diffusion and of the effect of the fluctuations in temperature and composition on the rate of reaction as well. Exact solutions in closed form are more difficult to derive for fully developed isothermal laminar flow and impossible for fully developed turbulent flow and/or for energetic reactions with or without heat transfer from the surroundings. However, differential models encompassing these aspects of behavior in both laminar and turbulent flow and for energetic gas-phase reactions are readily formulated and are readily solved numerically with the computer hardware, software, and finitedifference algorithms currently available to students as well as to practicing engineers. Because the students are well versed in this technology, it is a shame, if not a crime, to fail to enlist this capability in chemical reactor engineering. One generation of students with


experience in such modeling will inculcate this more general approach into those pockets of practice where it is not now utilized. Many of the effects identified herein are in need of further investigation. Examples are the effects of molecular and turbulent diffusion of energy and species and the effects of the heat of reaction and nonequimolar reactions on the field of flow and thereby on the mixedmean rate of reaction. Although a few of the concepts, models, derivations, and solutions presented herein appear in current textbooks on chemical reaction engineering, the majority do not. Examples of those that are usually missing include (1) the interpretation of the classical solutions for a hypothetical “plug flow” as those for the flow of a real fluid in the asymptotic limit of perfect radial mixing, (2) the interpretation of the classical solutions for laminar flow as those for the complementary asymptotic limit of negligible radial mixing, (3) the derivations of the exact limiting solutions of Llam/Lp ) 2 for a firstorder reaction and of Llam/Lp ) 4/3 for a second-order reaction, although the latter value but not the former one may be inferred by extrapolation of finite-difference solutions, (4) the formulations and numerical results for turbulent flow, (5) eq 30 or an alternative approximation for the radial velocity in nonequimolar and/or energetic gas-phase flows, (6) the use of integrated-mean terms of constrained variation in order to allow integration of the remaining integrand in closed form, thereby obtaining simpler expressions that are more readily interpreted and utilized, (7) the use of a doubly integratedmean term in order to identify the first-order dependence of the mixed-mean conversion on the heat of reaction, (8) the development of generalized solutions for nthorder irreversible, reversible, nonequimolar, and energetic reactions in both liquid- and gas-phase flows as illustrated by Churchill,4 (9) the use of partially integrated variables in finite-difference equations, such as in eq 32 and twice over in eq 59, thereby improving the rate of convergence, (10) the representation of the asymptotic temperature dependence of energetic reactions for which Q/RT0 , 1 in terms of two variables, namely, k0 and EQ/RcT02, rather than in terms of three, namely, k∞, Q/RT0, and E/RT0, (11) the derivation of the asymptotic solutions for EQ/RcT02 , 1 for first-order energetic reactions, (12) the effect of the turbulent fluctuations on the rate of reaction itself as approximated by eq 68 (from Glassman17), and (13) the dependence of the heat of reaction on the convective heattransfer coefficient as provided semiquantitatively by eq 69 and quantitatively by eq 72 (which is based on the DNSs of Mitrovic and Papavassilou29). The incorporation of most of the concepts, if not all of the details, in the undergraduate courses in reaction engineering and transport would appear to be essential in terms of understanding. Because the allotment of space and time in the curriculum as a whole and for reaction engineering and transport in particular is a zero-sum game, some current material must be omitted, which is a hard choice. Notation A, B, C ) reacting species a ) radius a+ ) a(τwF)1/2/µ b ) half-spacing between parallel plates Ci ) molar concentration of species i c ) specific heat capacity at constant pressure

Df ) diffusivity based on the molecular concentration E ) energy of activation E3{x} ) exponential integral of order 3 and argument x f ) Fanning friction factor ) 2τw/Fum2 jw ) heat flux density k ) reaction rate constant k′ ) rate constant for reverse reaction k∞ ) frequency factor K ) equilibrium constant based on concentrations ) k/k′ L ) length of the tubular reactor m ) exponent of T n ) order of reaction Nu ) Nusselt number ) 2jwa/λ(Tw - Tm) Pr ) Prandtl number ) cµ/λ QE ) ratio of the heat of reaction to the heat flux from the wall ) aqR/2jw QR ) exothermic heat of reaction per mole reacted qR ) uniform heat of reaction per unit volume R ) universal gas law constant Re ) Reynolds number ) 2aumF/µ r ) radial coordinate rˆ ) homogeneous rate of reaction Sc ) Schmidt number ) µ/FDf T ) absolute temperature t1/2 ) half-life of the reaction t/1/2 ) dimensionless half-life ) t1/2τw/µ u ) velocity in the axial direction u+ ) u(τwF)1/2/µ u′ ) fluctuating component of u v′ ) fluctuating component of velocity normal to the wall (u′v′)++ ) -Fu′v′/τ x ) arbitrary independent variable y ) distance from the wall y+ ) y(τwF)1/2/µ Z ) conversion ) 1 - CA/CA0 Greek Symbols R ) half-range of turbulent fluctuations β ) RE/RT h ∆ ) finite difference of δk ) maximum fractional change in the rate constant due to turbulent fluctuations λ ) thermal conductivity of the fluid µ ) dynamic viscosity ζ ) enhancement factor of heat- and mass-transfer coefficients due to the heat of reaction F ) specific density τ ) shear stress φ ) kCA0L/um φ{t} ) time variation of turbulent fluctuations in temperature ψ ) EQR/RcT02 Subscripts A, B, C ) species A-C lam ) in fully developed turbulent flow M ) integrated mean MM ) double integrated mean m ) mixed mean p ) for perfect radial mixing (“plug flow”) r ) at radial coordinate r S ) pseudo fully developed flow w ) at the wall 0 ) at the entrance

5212 -


) time-averaged

Literature Cited (1) Constales, D.; Marin, G. B.; Van Keer, R.; Nicolis, G.; Yablonsky, G. S. Mathematics in Chemical Kinetics and Engineering: An Introduction to the Special Issue Dedicated to Papers Presented at the MaCKiE-2002 Conference. Chem. Eng. Sci. 2003, 58, 4748. (2) Balakotaiah, V.; Chakroborty, S. Averaging Theory and Low-Dimensional Models for Chemical Reactors and Reacting Flows. Chem. Eng. Sci. 2003, 58, 4769. (3) Churchill, S. W.; Zajic, S. C. Prediction of Fully Developed Convection with Minimal Explicit Empiricism. AIChE J. 2002, 48, 927. (4) Churchill, S. W. The Interaction of Reactions and Transport. III. Asymptotes, Approximations, and Generalizations for Tubular Reactors. In review. (5) Yu, B.; Churchill, S. W.; Kawaguchi, Y. The Interaction of Reactions and Transport. IV. The First-order Effects of Flow and the Heat of Reaction in Tubular Reactors. In review. (6) Churchill, S. W. The Interaction of Reactions and Transport. II. Combined Reactors and Heat/Mass Exchangers. In review. (7) Abramowitz, M., Stegun, I. A., Eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables; NBS Applied Mathematics Series 55; U.S. Government Printing Office: Washington, DC, 1964. (8) Hill, J. C. Homogeneous Turbulent Mixing with Chemical Reaction. 1976, 8, 135. (9) Fox, R. O. Computatioal Models for Turbulent Reacting Flows; Cambridge University Press: Cambridge, U.K., 2003. (10) Churchill, S. W. New Simplified Models and Formulations for Turbulent Flow and Convection. AIChE J. 1997, 42, 1125. (11) Churchill, S. W. The Prediction of Turbulent Flow and Convection in a Round Tube. Adv. Heat Transfer 2001, 34, 255. (12) Churchill, S. W.; Shinoda, M.; Arai, N. An Appraisal of Experimental, Predictive and Correlative Contributions to Fully Developed Turbulent Flow in a Round Tube. Therm. Sci. Technol. 2002, 10 (No. 2), 1. (13) Churchill, S. W. The Intepretation and Use of Rate Data. The Rate Process Concept; McGraw-Hill Scripta: Washington, DC, 1974. (14) Churchill, S. W.; Pfefferle, L. D. The Refractory Tube Burner as an Ideal Stationary Chemical Reactor. AIChE Symp. Ser. 1984, No. 87, 279. (15) Cleland, F. A.; Wilhelm, R. H. Diffusion and Reaction in Viscous-flow Reactor. AIChE J. 1956, 2, 489. (16) Finlayson, B. A.; Rosendoll, B. M. Reactor/Transport Models for Design: How to Teach Students and Practitioners to

Use the Computer Wisely. In CACHE 2000; Malone, M. F., Trainham, J. A., Eds.; AIChE: New York, 2000; pp 176-191. (17) Ekambara, K.: Joshi, J. B. Axial Mixing in Laminar Pipe Flows. Chem. Eng. Sci. 2004, 3929. (18) Christiansen, J. A. On the Reaction between Hydrogen and Bromine. Mat.-Fys. Medd.sK. Dan. Vidensk. Selsk. 1919, 1, 1. (19) Pfefferle, L. D. Stability, Ignition, and Pollutant Formation Characteristics in a Thermally Stabilized Plug-Flow Reactor. Ph.D. Thesis, University of Pennsylvania, Philadelphia, PA, 1984. (20) Glassman, I. Combustion, 3rd ed.; Academic Press: San Diego, CA, 1996. (21) Vulis, L. A. The Rate of Turbulent Combustion. Fiz. Goreniya Vzryva (in Russian); English transl. Int. Chem. Eng. 1973, 13, 394. (22) Chinitz, W.; Antaki, P. J.; Kassar, G. M. The Effect of Temperature Fluctuations of Reaction Rate Constants in Turbulent Reacting Flows. Fluids Engineering Conference, Boulder, CO, 1981; pp 207-216. (23) Rothenberg, R. I.; Smith, J. M. Heat Transfer and Reaction in Laminar Flow Tube. AIChE J. 1966, 12, 213. (24) Brian, P. L. T.; Reid, R. C. Heat Transfer with Simultaneous Chemical Reaction: Film Theory for a Finite Reaction Rate. AIChE J. 1962, 8, 322. (25) Brian, P. L. T. Turbulent Flow Heat Transfer with a Simultaneous Chemical Reaction of Finite Rate. AIChE J. 1963, 9, 831. (26) Edwards, L. L.; Furgason, R. R. Heat Transfer in Thermally Decomposing Ozone. Ind. Eng. Chem. Fundam. 1968, 7, 330. (27) Ooms, G.; Groen, G.; de Graag, D. P.; Ballintijn, J. F. On Turbulent Pipe Flow with Heat Transfer and Chemical Reaction. Proceedings of the Sixth International Heat Transfer Conference; Hemisphere Publishing Corp.: Washington, DC, 1978; Vol. 5, pp 383-388. (28) Hanna, O. T.; Sandall, O. C.; Wilson, C. L. Mass Transfer Accompanied by First-Order Chemical Reaction for Turbulent Duct Flow. Ind. Eng. Chem. Res. 1987, 26, 2286. (29) Mitrovic, B. M.; Papavassiliou, D. V. Effects of a FirstOrder Chemical Reaction on Turbulent Mass Transfer. Int. J. Heat Mass Transfer 2004, 47, 43. (30) Bernstein, M. H. Pollutant Formation and Multiple Stationary States in Radiantly Stabilized Flames. Ph.D. Thesis, University of Pennsylvania, Philadelphia, PA, 1976.

Received for review August 16, 2004 Revised manuscript received November 19, 2004 Accepted November 24, 2004 IE049256U

Interaction of Chemical Reactions and Transport. 1. An Overview

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