5132
Ind. Eng. Chem. Res. 2008, 47, 5132–5145
Interaction of Reactions and Transport: Homogeneous Reactions in a Round Tube. IV. The First-Order Effects of Flow and Heat of Reaction Stuart W. Churchill,*,† Bo YU,‡ and Yasuo Kawaguchi§ Department of Chemical and Biomolecular Engineering, UniVersity of PennsylVania, 311A Towne Building, 220 South 33rd Street, Philadelphia, PennsylVania 19104; Beijing Key Laboratory of Urban Oil and Gas Distribution Technology, China UniVersity of Petroleum (Beijing), Beijing 10224, China; and Department of Mechanical Engineering, Tokyo UniVersity of Science, Hongo, Bunkyo-Ku, Tokyo 113-8656, Japan
The conversion due to first- and second-order, irreversible, equimolar reactions in fully developed isothermal laminar flow in a round tube was computed exactly for perfect, finite, and negligible molecular diffusion of species and energy. The analogous computations for fully developed turbulent flow were based on an essentially exact velocity distribution but an empirical model for the eddy diffusion of species and energy. The effect of a moderate heat of reaction was computed numerically for constant density, constant viscosity, and adiabatic conditions. The identification and implementation of a new dimensionless variable led to the discovery of several near-invariances and, thereby, to very simple but reasonably accurate algebraic representations for the mixed-mean conversion and the mixed-mean temperature for moderately energetic reactions. I. Introduction In most elementary textbooks on chemical reaction engineering, including recent ones, the treatment of homogeneous reactions in continuous flow through a tubular channel is limited primarily to the idealized case of plug flow, invariant density, and uniform temperature. The use of either the space-velocity or the space-time as the independent variable implies these same idealizations. Turbulent flow is often cited as a justification for the postulate of plug flow, although Churchill and Pfefferle1 assert that, except for combustion and thermal cracking, fully turbulent flow is rarely established in tubular reactors in either the laboratory or in practice. An expression for the conversion due to a second-order irreversible equimolar reaction in isothermal laminar flow without diffusion, as contrasted with that for plug flow, is often presented, but more or less as an afterthought. The consequence of these extreme simplifications is that most undergraduate students in chemical engineering complete a course in reaction engineering with the impression that plug flow and isothermality may safely be postulated in the design of a tubular reactor. Courses in process design and in advanced reactor engineering do little to undo these misimpressions. It is perhaps worth noting by way of contrast that textbooks and handbooks on convective heat transfer and/or mass transfer do not present expressions for Nu and Sh in terms of the space-velocity or space-time, and that expressions for heat and mass transfer in plug flow, if presented, are carefully identified as meaningful only in the sense of lower bounds. The state of computational fluid dynamics (CFD) and its counterparts for heat transfer and mass transfer, as well as of the available computer hardware and software, is currently such that a numerical solution for a tubular reactor can be carried out for any single reaction and any thermal boundary condition with virtually no idealizations (see, for example, the paper by Finlayson and Rosendoll2). On the other hand, the minimal number of reaction mechanisms to describe adequately some * To whom correspondence should be addressed. Tel.: (215) 8985579. Fax: (215) 573-2093. E-mail:
[email protected]. † University of Pennsylvania. ‡ China University of Petroleum (Beijing). § Tokyo University of Science.
chemical conversions such as pyrolysis and combustion is very large, perhaps exceeding 50 (see, for example, the paper by Pfefferle and Churchill3), and each mechanism introduces at least one empirical coefficient. Energetic reactions introduce one or more additional empirical coefficients and/or exponents for each reaction mechanism by virtue of the dependence of the rate constant on temperature. In addition, one or more empirical coefficients and exponents may be required to account for internal and external heat transfer. Mass transfer introduces several additional, if less critical, parameters. These coefficients and exponents, although empirical, are generally known with sufficient accuracy for all practical purposes. However, a difficulty arises from their number. Although a numerical solution is feasible for any given set of reaction mechanisms and for some simple thermal boundary conditions, it is virtually impossible to carry out calculations for a sufficiently broad set of mechanisms and conditions to support the development of correlating equations comparable to those for forced convective heat transfer, which can be generalized in terms of expressions for Nu as a function of only Re, Pr, and geometry (see, for example, the paper by Churchill and Zajic4)sthe one exception being the dependence of physical properties on temperature. This state of affairs impedes understanding and is the root cause of the persistence of such idealized concepts as plug flow, isothermality, space-velocity, and space-time in the textbooks on reactor engineering, and even in the computer simulation packages for process design. The objective of the overall investigation, of which these numerical calculations for idealized and limited conditions are a part, is primarily to improve the understanding of the process of chemical conversions in tubular flow and secondarily to provide the basis for improved predictions of this behavior. The numerical results presented here have five distinct roles in support of that objective: (i) to demonstrate that such calculations are within the capability of current undergraduate students in chemical engineering as well as of recent engineers in practice, (ii) to identify the significance and magnitude of the errors associated with the postulate of plug flow, (iii) to provide a database to test the range of validity of the asymptotes and approximations presented by Churchill,5 (iv) to serve as a database for the development of additional asymptotes and
10.1021/ie071229r CCC: $40.75 2008 American Chemical Society Published on Web 06/18/2008
Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 5133
approximations, and (v) to encourage and guide the development of more general models and methods of solution for chemical reactors in the computer packages used in chemical process design. To fulfill this objective, consideration was given to three undertakings: (i) the development and testing of an all-purpose computer program, (ii) the execution of numerical calculations with minimal idealizations for a narrow set of conditions, thereby reducing the task of correlation, and (iii) the execution of numerical calculations for bounding and/or asymptotic conditions in the hope of providing some insight and some guidance in correlation. The first of these procedures was eliminated from consideration because it is more directly the task of computer-aided design than of reaction engineering, and a compromise was made with respect to the second and third. The illustrative numerical calculations herein are primarily for single, first- or second-order, irreversible, equimolar homogeneous reactions in the fully developed, tubular flow of a fluid of constant density and viscosity with no heat exchange with the surroundings (adiabatic conditions) and either negligible or perfect radial transport of energy and species by molecular and eddy diffusion. A few supplemental calculations were carried out for finite molecular and eddy diffusion of species. The effects of heat exchange have already been examined and reported elsewhere. The reasons for the particular choices of asymptotic and/or idealized conditions are as follows: (1) Homogeneous reactions in tubular flow: This choice not only narrows the analysis but also leads to simpler calculations than would be required if stirred reactors, or if surface-coated tubular, packed-bed, or fluidized-bed catalytic reactors, were encompassed. Equivalent, separate investigations of each of these other reactors would appear to be worthwhile and feasible. (2) Single first-order and second-order irreversible reactions: This restriction is only in the interest of simplicity in the calculations and in their interpretation. The results are presumed to be representative qualitatively for other single-reaction mechanisms, reversible reactions, and even multiple reactions. Multiple reactions may pose computational difficulties such as stiffness but have been encompassed in our prior work. (3) Equimolar reaction: This choice was also made in the interest of simplicity and the possible revelation of generalities. The consideration of nonequimolar reactions does not pose any serious chemical-kinetic difficulty, but in the gas phase, it results in two-dimensional flow. (4) Fully developed flow: This idealization, which is often utilized without specific justification or even specific mention, is of great convenience because it is necessary if the fluidmechanical modeling is to be restricted to one dimension. It introduces error in the computed conversion due to the development of a boundary layer at the entrance, in liquid-phase flow due to changes in viscosity, and in gas-phase flow due to changes in concentration associated with nonequimolarity, temperature, and pressure. (5) Invariant density: This idealization avoids consideration of the continuous development of the flow associated with changes in density due to temperature and composition, which would require two-dimensional modeling and result in a loss of the very generality that is the objective of this analysis. It is perhaps the most serious departure from physical reality made in this analysis. (6) Invariant viscosity: The variation of the viscosity with temperature, and to a lesser extent with composition, introduces two complexities. First, it results in a continuous change in the
velocity distribution; second, it requires partial-differential modeling; and third and most serious, it prevents generalization if the viscosity is strongly composition-dependent, as is the usual case. This said, the postulate of an invariant viscosity, which is throughout this analysis, is perhaps the most serious departure from physical reality (7) Adiabatic conditions: This idealization was made in the interest of generality. In practice, energetic reactions in tubular reactors generally require heat exchange with the surroundings to prevent self-quenching if they are endothermic and to prevent thermal runaways if they are exothermic. (8) Perfect or negligible radial mixing of energy and species: The postulate of Pr and Sc approaching zero by virtue of the thermal conductivity and the molecular diffusivity, respectively, approaching very large values. The postulate of negligible radial mixing is also physically conceivable as an asymptotic condition for both Pr and Sc approaching an infinite value by virtue of the thermal conductivity and the molecular diffusivity respectively, approaching very small values. Fortunately, these two idealizations result in solutions that bound the effects of radial mixing for adiabatic conditions. It was, thereby, feasible to limit in number the computations for finite radial diffusion of species by molecular motion and/or turbulent fluctuations, both of which invoke partial-differential modeling and introduce additional parameters. (9) Expression of the rate mechanisms in terms of concentration rather than in terms of the chemical potential or fugacity: Significant deviations from ideality in the gas phase are to be expected only for high pressures, but those in the liquid phase may be the rule rather than the exception. The weak justifications for the acceptance here of the product of concentrations as a driving force for reaction are the lack of experimental or theoretical data for the rate of reaction in terms of activities and the universality of the usage of concentrations in both chemical kinetics and chemical reaction engineering. (10) Neglect of the effect of the turbulent fluctuations on the rate of reaction: This effect has been examined by Glassman6 and Churchill,7 both of whom concluded it to be significant only for the extremes of temperature encountered in combustion, detonation, and plasmas. As indicated, most of the itemized idealizations and restrictions were made primarily in the interests of simplification and illustration. The avoidance of all of these idealizations for any single condition is technically feasible in terms of current computational resources and the current preparation of undergraduate students and recent graduates in chemical engineering, but at the price of a loss of simplicity, understanding, and generality. As a counterbalance to these restrictions, the illustrative calculations encompass one effect that has received almost no attention in a generalized sense, namely, that of the heat of reaction, and one effect that has been neglected or approximated crudely in most prior modeling of reactors, namely, that of turbulent flow. Furthermore, the calculations herein for turbulent flow encompass a complete range of Re and, as demonstrated by Churchill et al.,8 utilize an essentially exact representation for the velocity distribution. It should be noted that the errors associated with the postulate of plug flow and other idealizations have been the subject of many numerical and experimental studies. The literature on this particular topic has been reviewed and summarized by Cutler et al.9 Because of the existence of this review, prior results are mentioned only insofar as they are directly relevant to the primary objective of this investigation, namely, to identify
5134 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 Table 1. Exact Solutions for the Range and Ratio of Conversion Due to Radial Diffusion First-Order Reaction, A f B, in Isothermal Laminar Flow kL/um
Zp
Zmlamn
1 - Zmlamn/Zp
Zp/Zmlamn
0 0.02 0.10 0.30 0.50 0.70 0.80 0.90 0.92 0.94 0.96 0.98 1.00 1.10 1.40 1.50 1.80 2.00 4.00 ∞
0 0.01980 0.09516 0.25918 0.39347 0.50341 0.55067 0.59343 0.60148 0.60937 0.61711 0.62469 0.63212 0.66713 0.75340 0.77687 0.83470 0.86466 0.98168 1.00000
0 0.01946 0.09018 0.23546 0.35064 0.44468 0.48544 0.52668 0.52964 0.53668 0.54350 0.55022 0.55680 0.58812 0.66788 0.69048 0.74860 0.78062 0.93974 1.00000
0 0.01724 0.05236 0.09153 0.10885 0.11667 0.11846 0.11922 0.11927 0.11929 0.11928 0.11921 0.11916 0.11843 0.11352 0.11120 0.10315 0.09712 0.04273 0
1.0000 1.0175 1.0553 1.1007 1.1221 1.1320 1.1344 1.1354 1.1356 1.1354 1.1354 1.1353 1.1353 1.1343 1.1281 1.1251 1.1150 1.1077 1.0445 1.0000
Equations 3 and 4 are identical to those for plug flow, but perfect radial mixing of energy and species is physically conceivable, at least in an asymptotic sense, whereas plug flow is purely hypothetical for any fluid or condition. On the other hand, insofar as radial mixing of species is neglected, eq 4 is applicable for any differential annular filament of fluid at radius r, in fully developed laminar flow if the appropriate local velocity, namely, 2um[1 - (r/a)2], is substituted for um. The following expression is thereby obtained for the conversion at any radius:
{
Zrlamn ) 1 - exp
(
)
(1)
dZp ) k(1 - Zp) dL
(2)
um -
dCAp ) kCAp dL
or as um
( )
Here, Z ) 1- CA/CA0 is the fractional conversion of the component A, and the subscript p designates perfect radial mixing. Equation 2 may readily be integrated with the condition Zp ) 0 at L ) 0, to obtain
(
1 kL ) ln um 1 - Zp
)
(3)
Equation 3 may be inverted to result in Zp ) 1 - e-kL⁄um
(4)
}
(5)
Here, the subscript rlamn designates the local conversion at radius r in fully developed laminar flow with negligible radial diffusion. Integration of the local conversion, as given by eq 5 and weighted by the velocity ratio, ur/um ) 2[1 - (r/a)2], over the cross section of the tube, results in the following expression for the mixed-mean conversion in fully developed laminar flow with no radial mixing: 1
generalities and, thereby, as already stated, to improve understanding and to provide the basis for improved predictions. II. Isothermal Reactions. Isothermal reaction in tubular flow implies nonenergetic reactions and no heat transfer through the wall. Such an idealized condition is approached for some liquidphase reactions, but the primary purpose of its examination here is to allow the development of a structure and of generalizations that carry over, at least in part, to the energetic reactions that are considered in Section III, and to the reactors with heat exchange that have been examined by Churchill and Yu10 and Yu and Churchill.11 II.1. Analytical Solutions for Perfect Radial Mixing. The analytical solutions for perfect radial mixing that follow in this section, although not new, are presented for two purposes. First, they serve as a standard against which the accuracy of the numerical solutions can be tested. Second, in some instances, they serve as components of new correlative and predictive models. They are, as indicated previously, for irreversible nonenergetic equimolar reactions in fully developed flow and, hence, are postulated implicitly to occur at constant pressure as well as temperature. II.1.1. First-Order Reaction. The change in composition due to the reaction A f B in tubular flow with perfect radial mixing may be expressed as
-kL 2um[1 - (r ⁄ a)2]
Zmlamn ) 2
∫ (1 - e
[
r -kL⁄2um 1-( )2 a
0
])[1 - ( ar ) ]d( ar ) ) 1 - 2E 2
2
3
{ } kL 2um
(6)
Here, E3{x} is the exponential integral of third order, a tabulated function (see, for example, the paper by Abramowitz and Stegun12). Their tabulation extends only to an argument of 100, but the following asymptote is applicable for larger ones and proves useful herein: E3{x} f
e-x 3+x
(7)
Extended tabulations may also be found in computer packages. However, the most expeditious method of determining values of Zmlamn for any value of kL/um may simply be the stepwise integration of the finite-difference equivalent of eq 6, namely,
(
{
∆Zmlamn ) 2 1 - exp
kL ⁄ 2um
})[
1-
( ar ) ]∆( ar ) 2
2
(8) 1 - (r ⁄ a) Test integrations of eq 8 produced values of Zmlamn that differed only in the sixth significant figure from those calculated using tabulated values of E3{x}. Equations 4 and 6 constitute upper and lower bounds, respectively, for the conversion. Their fractional difference 2
Zp - Zmlamn 2E3{kL ⁄ 2um} - exp{-kL ⁄ um} ) Zp 1 - exp{-kL ⁄ um}
(9)
represents the possible range of conversion that can be produced by the radial diffusion of momentum and species. Illustrative values of Zp, Zmlamn, and (Zp - Zmlamn)/Zp, with the latter calculated from eq 9, are presented in Table 1. An irregular set values of kL/um was chosen because the objective was to illustrate the behavior rather to provide a data bank. The fractional difference in conversion is observed to rise rapidly from zero, to be relatively invariant for 0.30 < kL/um < 1.5, to attain a maximum value of 0.11929 at kL/um ) 0.94, and then to decrease back to zero as kL/um further increases. The maximum fractional difference of 0.11929 is consistent with prior, less-accurate findings in the range of 0.11-0.12. The limited range of this quantity has often been cited in textbooks as a justification for the postulate of plug flow. The fractional
Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 5135 Table 2. Exact Solutions for Reactor Lengths for Perfect and Negligible Radial Mixing First-Order Reaction, A f B, in Isothermal Laminar Flow kLmlamn/um
Z
kLp/um
Lmlamn/Lp
0 0.10 0.50 1 2 4 8 16 32 ∞
0 0.09018 0.35064 0.55580 0.78062 0.93974 0.99477 0.99993 0.99999 1.00000
0 0.09451 0.43177 0.81373 1.5169 2.8091 5.2528 9.7047 18.3514 ∞
1.0000 1.0581 1.1580 1.2289 1.3184 1.4240 1.5230 1.6487 1.7437 2.0000
Table 3. Exact Solutions for the Range and Ratio of Conversion Due to Radial Diffusion Second-Order Reaction, 2A f B + C, in Isothermal Laminar Flow kL/um
Zp
Zmlamn
1 - Zmlamn/Zp
Zp/Zmlamn
0 0.02 0.10 0.20 0.28 0.29 0.30 0.40 0.50 0.60 0.70 0.90 1.00 2.00 4.00 8.00 16.00 ∞
0 0.03846 0.16667 0.28571 0.35897 0.36709 0.37500 0.44444 0.50000 0.54545 0.58333 0.64286 0.66667 0.80000 0.88888 0.94118 0.96970 1.00000
0 0.03685 0.15204 0.25666 0.32169 0.32896 0.33606 0.39912 0.45069 0.49830 0.53044 0.58951 0.61371 0.75628 0.85941 0.92377 0.96019 1.00000
0 0.04178 0.08775 0.10169 0.10386 0.10387 0.10384 0.10199 0.09861 0.09469 0.09067 0.08298 0.07944 0.05465 0.03317 0.01849 0.00980 0
1.0000 1.0436 1.0962 1.1132 1.1159 1.1159 1.1159 1.1136 1.1094 1.1045 1.0997 1.0904 1.0863 1.0578 1.0343 1.0188 1.0862 1.0000
increase of Zp over Zmlamn, which is seen in Table 1 to attain a maximum value of 0.1356, has also been cited in textbooks. However, an alternative criterion should also be noted, namely the ratio of the required lengths of reactor for a specified conversion. Equation 3 provides kL/um as an explicit function of Zp for perfect radial mixing, but eq 6 cannot be rearranged to give kL/2um as an explicit function of Zmlamn. When faced with this difficulty, Churchill5 devised the following procedure. First, Lp is expressed in terms of Z per eq 3, then Z is replaced by E3{kL/2um} per eq 6. The subscripts on Z in the resulting expression can be dropped because both eqs 3 and 6 are being applied for the same value of this variable, leading to kLmlamn ⁄ um Lmlamn kLmlamn ⁄ um kLmlamn ⁄ um ) ) ) Lp kLp ⁄ um 1 kLmlamn ln -ln 2E3 1-Z 2u
{
{ {
}
m
}} (10)
Equation 10 can be solved for either Lmlamn/Lp or Z for a chosen value of kLmlamn/um. A few illustrative values obtained by this procedure are presented in Table 2. The limiting value of Lmlamn/Lp for kLmlamn/um f ∞ can be determined by utilizing eq 7 for E3{kLmlamn/2um} in eq 10, thereby obtaining Lmlamn ) Lp )
kLmlamn ⁄ um
{
-ln{2e-kLmlamn⁄2um} + ln 3 +
kLmlamnn 2um
}
1 um kLmlamn -ln{2} 1 + + ln 3 + kLmlamn ⁄ um 2 kLmlamn 2um
{
} (11)
From inspection, the limiting value of Lmlamn/Lp for kLmlamn/ um f ∞ can be seen to be 2, which amounts to an increase of 100%. The numerically computed values of Lmlamn/Lp in Table 2 indicate that, although this ratio increases rapidly with kLmlamn/ um and becomes significant for fractional conversions >0.50, it does not closely approach the limiting value of 2 for the degrees of conversion of practical interest. Even so, a value of Lmlamn/ Lp greater than 1.4 for Z ) 0.94 is eye-opening and constitutes an alternative interpretation and measure of the possible error due to the postulate of plug flow. What is the explanation and significance of the greater effect displayed in Table 2 as compared with that in Table 1? Quite simply, those in Table 1 are in the context of operation (conversion for a specified length of reactor), while those in Table 2 are in the context of design (length of reactor for a specified conversion). II.1.2. Second-Order Reaction. The corresponding key expressions for a second-order, equimolar reaction, 2A f B + C, are
(
um -
( )
um
(12)
dZp ) 2kCA0(1 - Zp)2 dL 1
Zp )
Zrlamn )
)
dCA ) 2kCA2 dL
um 1+ 2kCA0L
(13)
1
)
1+
(14)
1 2φ
1 1 ) um[1 - (r ⁄ a)2] 1 - (r ⁄ a)2 1+ 1+ φ kCA0L
(15)
and 1
Zmlamn ) 2
a) ∫ 1 +[1u -[1(r-⁄ (ra)⁄ a)] d(r] ⁄ ⁄2kC 2
2
2
0
m
(
{
) 2φ 1 - φ ln 1 +
1 φ
A0L
})
(16)
Here φ ) kCA0L/um. It follows that
{
Zp - Zmlamn 1 ) 1 - (1 + 2φ) 1 - φ ln 1 + Zp φ
(
}) ) φ[(1 + 2φ) ln {1 + φ1 } - 2] (17)
and, using the same procedure as for eq 10, Lmlamn φmlamn 1 ) ) 2φmlamn - 1 Lp φp Z 1 ) - 2φmlamn 1 1 - φmlamn ln 1 + φmlamn
(
{
)
}
(18)
The limiting value of Lmlamn/Lp for φmlamn f ∞ as determined by numerical evaluations of eq 18 or by expansion of the logarithm is 4/3, which is much less than that for a first-order reaction but still very significant. Illustrative values of the fractional difference in conversion for a second-order reaction as calculated from eqs 14, 16, and 17 are shown in Table 3. The values of the fraction conversion due to radial mixing, (Zp - Zmlamn)/Zp, may be observed to rise rapidly from zero, to attain a maximum value of 0.1039 at kCA0L/ um ) 0.29, to be relatively invariant for 0.10 < kCA0L/um < 0.70,
5136 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008
( )
Table 4. Exact Solutions for the Ratio of Reactor Lengths Due to Radial Mixing Second-Order Reaction, 2A f B + C, in Isothermal Laminar Flow kLmlamn/um
Z
kLp/um
Lmlamn/Lp
0 0.25 0.5 1 2 4 8 16 ∞
0 0.29882 0.45069 0.61371 0.75628 0.85941 0.92377 0.96019 1.00000
0 0.21308 0.41024 0.79435 1.5515 2.0563 6.0592 12.0608 ∞
1.0000 1.1732 1.2188 1.2589 1.2891 1.3088 1.3203 1.3266 1.3333
(
)( )
(21)
( )
y+ 3[ 1 - (u ′ V′)++] dy+ (22) a+ Equations 21 and 22 are exact, but an empirical expression for (uj′V′)+2 ≡ Fu′V′ /τ′ is necessary for their numerical integration. For that purpose, the following correlating equation based on theoretical asymptotes with theoretical and empirical coefficients was devised
(19)
was solved stepwise and simultaneously with the corresponding finite-difference expression for a reaction in an annular element, which is kL 1 1 - Zp ∆ (20) 2 1 - (r ⁄ a)2 um The resulting values of Zrlamb, weighted by the local velocity ratio, that is, by 2[1 - (r/a)2], were then integrated numerically over the cross section to obtain Zmlamn. A Runge-Kutta method was used for the finite-difference calculations, and Simpson’s rule was used for the evaluation of integrals. An equivalent procedure was utilized for a second-order reaction. II.2.2. Fully Developed Turbulent Flow. The same idealizations as for laminar flow are implied here 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. For fully developed turbulent flow in a round tube, the differential formulations for u and um may be expressed, per Churchill,13 in the following generalized forms ∆Zrlamb )
and
y+ [ 1 - (u ′ V′)++] dy+ a+
du+ m) 1-
and thereafter to decrease back to zero. The maximum fractional increase of Zp over Zmlamn, which also occurs at kCA0L/um ) 0.29, is 0.1116. Thus, the possible error due to the postulate of plug flow for a second-order reaction is only slightly less than that for a first-order reaction. On the other hand, as illustrated in Table 4, the ratio of reactor lengths for a second-order reaction approaches its limiting value much more rapidly than for a first-order reaction and is greater than 1.3 even for a fractional conversion of 0.85. Figure 1 reveals the surprising result that the ratio Lmlamn/Lp, which is the length of a reactor in laminar flow with negligible diffusion of species to that for a reactor with perfect radial mixing, is larger for a second-order reaction than for a firstorder reaction for fractional mixed-mean conversions less than 0.65 despite the lesser limiting value of 4/3 as compared to 2. This behavior is not so evident from a comparison of the values in Tables 2 and 4 because the latter are tabulated in terms of regular values of kLmlamn/um, whereas the values are plotted versus Zm. The near-equality of the values of this ratio for fractional conversions less than 0.80 has possible utility as a generalization. II.2. One-Dimensional Finite-Difference Formulations. II.2.1. Fully Developed Laminar Flow. Although complete analytical solutions are given above for first- and second-order reactions in isothermal laminar flow with perfect and negligible radial mixing, numerical integrations of finite-difference formulations were carried out as a test of the accuracy of this methodology before its necessary application for nonisothermal conditions and turbulent flow. For example, for a first-order reaction, a finite-difference formulation for eq 2, such as ∆Zp ) (1 - Zp)∆(kL ⁄ um)
du+ ) 1 -
|
([ ( ) ] y+ 10
3 -8⁄7
{
}
(
-1 1 1+ 0.436y+ 0.436a+ 6.95y+ -8⁄7 -8⁄7 (23) a+ Equations 21–23 predict the local velocity in fully developed turbulent flow within 0.4% and the mixed-mean velocity within 0.1% of the best experimental data for a+ g 1000 (Re g 4000) (see paper by Churchill et al.8), but they are not applicable for developing flow. For a first-order reaction in fully developed turbulent flow, and the indicated idealizations, the following expression for the conversion in an annular filament of fluid at a dimensionless distance y+ from the wall can be inferred from eq 4: (u ′ V′)++ )
0.7
+ exp
)| )
{ ( )}
Zy+turbn ) 1 - exp -
+ kL um um u+
(24)
The corresponding formulation for the mixed-mean conversion is 1
Zy+turb ) 2
∫Z
y+
0
( )( ) ( ) u+ y+ y+ 1 d u+ a+ a+ m
(25)
Numerical values for Zy+turbn from a finite-difference solution rather than from eq 24 can be utilized in the integrand of eq 25. The formulation for a second-order reaction in fully developed turbulent flow follows simply from the replacement in eq 25 of Zy+turbn from eq 24 with that corresponding to eq 14, namely,
[
]
+ u+ m⁄u (26) 2φ II.3. Numerical Results from Finite-Difference Computations. Stepwise numerical calculations of the conversion were carried out for both first- and second-order equimolar reactions at 300 K in laminar isothermal flow with no radial n-1 mixing for 50 values of kCA0 L/um ranging from 0.001 to 500, and for turbulent flow with no radial mixing for 11 values of + a+ ranging from 150 to 5 × 105 (Re ) 2a+um ranging from 4 × 103 to 3.33 × 107). In the interest of brevity, only representative values are displayed herein. In order to test the accuracy of a simple finite-difference algorithm in this asymptotic case (fully developed laminar flow with no radial mixing), numerically calculated values of Zmlamn for a first-order reaction, as identified by the heading FD, are compared in Table 5 with exact values calculated from eq 6, using tabulated values of E3{kL/2um}. The agreement, with minor exceptions, is seen to be within 2 in the fifth significant figure. The primary value of the stepwise integrations for this case, for which an exact analytical solution exists, is to establish the
Zy+turbn ) 1 ⁄ 1 +
Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 5137
Figure 1. Ratio of length of reactor for first- and second-order reactions in laminar flow with negligible and perfect radial mixing. Table 5. Validation of Finite-Difference Computations First-Order Reaction, A f B, in Isothermal Laminar Flow with no Radial Diffusion kL/um
E3{kL/2um}
Zmlamn (1 - 2E3)
Zmlamn FD
0 0.10 0.25 0.40 0.60 0.80 1.10 1.50 2.20 3.00 4.00
0.50000 0.45491 0.39879 0.35194 0.30004 0.25728 0.20594 0.15476 0.09588 0.05673 0.03013
0 0.09018 0.20242 0.29612 0.39992 0.48544 0.58812 0.69048 0.80824 0.88654 0.93974
0 0.09016 0.20245 0.29611 0.39992 0.48543 0.58810 0.69047 0.80824 0.88652 0.93973
Table 7. Ratio of Reactor Lengths for a First-Order Reaction, A f B, in the Limits of Negligible and Perfect Radial Mixing
flow laminar turbulent
kL/um
Zp
Zmturbn
1 - Zmturbn/Zp
Zp/Zmturbn
0.00 0.20 0.40 0.60 0.80 1.00 2.00 3.00 4.00 5.00 ∞
0.00000 0.18127 0.32968 0.45119 0.55067 0.63212 0.86466 0.95021 0.98168 0.99326 1.00000
0.00000 0.17996 0.32631 0.44585 0.54370 0.62391 0.85535 0.94352 0.97768 0.99110 1.00000
0.00000 0.00723 0.01184 0.01184 0.01266 0.01299 0.01077 0.00901 0.00407 0.00217 0.00000
1.0000 1.0073 1.0120 1.0120 1.0128 1.0132 1.0109 1.0091 1.0041 1.0022 1.0000
validity of this numerical methodology for three other cases: first for turbulent flow; second for reaction mechanisms or a combination of reaction mechanisms for which analytical solutions are not available, possible, or convenient; and third for the nonisothermal conditions that are examined subsequently herein. The numerically computed values of (Zp - Zmturbn)/Zp for a+ ) 1000 (Re ) 37640) and k0a/um in Table 6 reveal that the conversion departs negligibly (
