Discussion
The objective function defined by eq 1 uses absolute errors. An objective function using relative errors can be constructed by setting
Nomenclature
ut = u1*/212
and
v,
any three of the four element balances are satisfied, the fourth one will also be satisfied. Hence any one of the columns of the element matrix may be eliminated to obtain an independent set of element balance equations.
= vi*/$+=
Since the flow rates of the various species may differ by several orders of magnitude, normally relative errors should be minimized. The problem as formulated here, does not force the solution to be all positive. If corrections are such that a corrected value is negative, the problem can be resolved by assigning higher weighting to that experimental flow rate. The number of independent element balance equations for any system is equal to the rank, R, of the element matrix, E, which must be equal to or less than the number of chemical elements, L , in the system. A system is said to be degenerate when R is less than L. The rank of matrix P is equal to that of E and hence P will be singular in degenerate cases; but the set of equations given by (12) will be compatible. Degenerate systems can be handled by the same computational procedure after eliminating (L- R) columns of E so that the remaining columns form an independent set. For example, the system consisting of HF, HCl, CHCl3, CHClzF, CHClF2, and CHFI is degenerate since there are four chemical elements in the system and the rank of the element matrix is 3. I n this case, if
d, = discrepancy in the j t h element balance (eq 13) etj = number of atoms of j t h element in a molecule of the ith species E = element matrix I = identity matrix L = number of elements N = number of chemical species o = null matrix Pi, = elements of matrix P , defined by eq 14 Q5 = solution to set of eq 15 objective function ut = weighting factor for inlet flow rate of the ith species vi = weighting factor for exit flow rate of the ith species Zt = inlet flow rate of the ith species Yi = exit flow rate of the ith species zj = Lagrange multiplier for the j t h element conservation equation
s =
SUPERSCRIPTS -
*
= =
experimentally measured value modified value or function
literature Cited
Kuehn, D. R., Davidson, H., Chem. Eng. Progr., 57 (6), 44 (1961). Vaclavek, V., Collect. Czech. Chem. Commun., 34, 2662 (1969). RECEIVED for review March 23, 1972 ACCEPTEDJanuary 29, 1973
Effect of Interphase Mass Transfer on Product Selectivity in Liquid-Phase Paraffin Chlorination Michael P. Ramage’ and Roger E. Eckert” School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907
Product selectivity in the liquid-phase chlorination of n-dodecane in a semibatch reactor was found to be a complicated function of temperature, agitation rate, and chlorine flow rate. These experimental results combined with a surface renewal theory for product selectivity, also developed in this study, are the basis for the following conclusions regarding the liquid-phase chlorination process. Interphase mass transfer limitations in the reaction between gaseous chlorine and liquid n-paraffins alter the yield ratio of secondary/primary monochlorides and cause the ratio of the selectivity to total monochlorides to decrease below the intrinsic value.
T h e monochlorides of the heavier saturated paraffins, such as ndodecane, are used as intermediates in several chemical processes. One specific use for the monochlorides of n-dodecane is in the manufacture of biodegradable detergents. The simplest way to chlorinate these heavier hydrocarbons is by treating liquid hydrocarbon with gaseous chlorine (“liquidphase chlorination”). This eliminates the dehydrohalogenation and pyrolytic reactions which would be associated with 1 Present address, Mobil Research and Development Corporation, Paulsboro, N. J. 08066.
248 Ind.
Eng. Chern. Process Des. Develop., Vol. 1 2 , No. 3, 1973
the vapor-phase chlorination of high boiling point hydrocarbons. In liquid-phase chlorination the effect of interphase mass transfer limitations on not only the rate of reaction but also on the selectivity has to be considered. Product selectivity is described here for fixed conversion by such responses as yield ratio of secondary/primary monochlorides and total yield of monochloride. Only one previous investigator (van de Vusse, 1966) has considered the effect of mass transfer on product distribution in liquid-phase saturated paraffin chlorination. The data of
other investigations also point to the possible significance of mass transfer. Hatch (1937) and Blouri, et al. (1963), determined the yield ratio of secondary/primary monochlorides as a function of temperature for the liquid-phase chlorination of n-heptane. Hatch chlorinated over a temperature range from -15 to 100' in a semibatch reactor. Blouri used a vertical cocurrent two-phase flow reactor and studied the chlorination from 15 to 80' ; neither investigator considered mass transfer. In Figure 1 the results of these two investigations are plotted in terms of log yield ratio us. reciprocal temperature. The observed activation energy (slope in Figure 1) for Blouri's data is twice that for Hatch's data. Under conditions where mass transfer resistance can affect product distribution, the ratio of the observed activation energy to the intrinsic value will differ from unity (Danckwerts, 1970). The exact value depends upon the degree of mass transfer control as will be discussed in the mass transfer section. Blouri also presented data for the effect of pressure on the yield ratio of secondary/primary monochloroheptane. He found that by increasing the reactor pressure from 1 to 4 atm a t 80', the yield ratio of secondary/primary monochlorides increased by 10%. Hass, et al. (1937), reported a pressure effect on the yield ratio of secondary/primary monochlorides in the liquid-phase chlorination of n-heptane. Since the mass transfer characteristics of a gas-liquid reaction system are a function of pressure and the intrinsic kinetic parameters of the liquid-phase reaction are probably not, it can be concluded that mass transfer resistance is a reasonable explanation for this pressure effect. van deVusse used a semibatch reactor to study the effect of interphase mass transfer on the selectivity to total monochlorides over polychlorides in the liquid-phase chlorination of n-decane a t 100". Monochloride yields under stirred conditions were compared to those under nonstirred conditions. Stirred conditions using both n-decane and solutions of ndecane in dichlorobenzene (10-20 mol % n-decane) and nonstirred conditions using pure n-decane gave the same selectivity to total monochlorides. But when dilute n-decane solutions were chlorinated under nonstirred conditions, a decrease in the selectivity to monochlorides was observed which van de Vusse attributed to mass transfer limitations. It has been shown that the works of Blouri, Hatch, and Hass indicate a mass transfer effect on product distribution in the liquid-phase chlorination of n-heptane. Further, van de Vusse has reported a mass transfer effect on the selectivity to total monochlorides over polychlorides in the liquid-phase chlorination of n-decane. Intrinsic Chemical Kinetics
The overall reaction paths for the yields of primary and secondary monochlorides in the liquid-phase chlorination of ndodecane are kdp
kP
+
C I Z H Z ~C12 + HC1 C12H26
+ Cl2
k,
+ n-CizH26Cl-
Clz
HC1
+ C1zHz4Clz
(1)
kds
HCI
+ sec-ClzHz6C1+ HC1 + ClZHz4Cl2(2) Clz
The term secondary monochloride refers to the total of the five potential isomers (2-chloro through 6-chloro) and these will be treated as a group in contrast to the single compound primary (1-chloro) throughout this paper. For the yield of total monochlorides, the above path is simplified to Ci2H26
+ Clz
km
+ CizH2rCl-
kd
HC1
Clz
HC1
+ CizHz4Clz
(3)
2. 123.
97.
72. 50. 30. TEMPERATURE ,*C
13.
-3.
-16.
Figure 1. Previous investigators have obtained different valves of y and different activation energies for liquidphase chlorination of C7H16
The free-radical chain mechanism (Dewar, 1949; Roberts and Caserio, 1965) for the reaction is (a) Clz + 2C1. (b) C1* R H + Re (c) R . Clz+ RC1 (d) C1. C1. + Clz (e) R . C1. + RCl (f) R . R . + Rz
+ + + + +
+ HC1 + C1.
(4)
where R H = paraffin or alkyl chloride. Initiation in step a can be accomplished either thermally a t temperatures above 100' (Roberts and Caserio, 1965) or by ultraviolet irradiation in the spectra range from 2700 to 4200 A (Gibson and Bayliss, 1933). Initiation and termination can also occur a t the wall of a glass reactor vessel (Bernstein, et al., 1969; Bratolyubov, 1961). But due to the high density of the liquid-phase system and the high reactivity of the radicals, these wall reactions are probably negligible compared to the number of radicals reacting in the bulk of the liquid (Calvert and Pitts, 1966). Pyrolytic and dehydrohalogenation reactions which can occur in vapor-phase chlorination are negligible due to the low temperatures used in this liquid-phase process. It has been shown both theoretically (Dewar, 1949) and experimentally (Stauff, 1942) that even though the intrinsic kinetics of liquid-phase chlorination are quite complex, the rate of formation of an alkyl chloride product can be represented by a power law model of the following form T, = IC,(C1~)'(RH)"(I)h (5) where a = m, d, p, s, dp, or ds. Since a power law model of this type only represents an asymptotic solution to the true kinetic model, the numerical values of IC,, j, m, and h in eq 5 can be a function of the experimental conditions. For the liquid-phase chlorination of ndecane, Barilli, et al. (1970), found the reaction orders in the power law models for monochloride and dichloride formation to be the same, only the rate constants differed. Using power law models which differ only in the rate constants to represent the intrinsic kinetics of ndodecane chlorination, a relationship between the final dodecane conversion CVD and the yield ratio of secondary/primary monochlorododecane, y, in the presence of dichlorination can be obtained by integrating the following two equations (Ramage, 1971)
ys =
(sec-CIzHzrC1) -- 1 (CnHzsCl) ( C ~ Z H Z ~ C ~(CVD) Z)
+
(sec-ClZHZ6C1) (CizHzs)o(l - 2) Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 3, 1973
249
Table I. Estimated Rate Constant Ratios for n-Decane and n-Dodecane Chlorination as Calculated by Ramage (1971) from Results of Fredericks and Tedder (1960, 1961) nDecane
looo
n-Dodecane ~
35O
780
1460
1670
215’
0.071 0.077 0.083 0.085 0.087 0.929 0.923 0.917 0.915 0.913 12.0 11.1 10.8 13.1 10.5 0.88 0.91 0.91 0.91 0.91 0.91 0.93 0.93 0.94 0.93 0.90 0.90 0.90 0.90
The intrinsic yield of monochlorides in the presence of dichlorination varies with conversion as follows
Ramage (1971) showed theoretically that 0 5 j 5 1,O 5 m 5 I, and 0 5 h 5 0.5; the exact value of each order depends upon which steps of free-radical mechanism are controlling. Stauff (1942) found experimentally that j = m = h = 0.5 for the photoinitiated chlorination of liquid hexadecane in solutions of high carbon tetrachloride concentration. For the thermal chlorination of n-decane in the absence of diluent, van de Vusse assumed that 3 = m = 1, while Barilli later reported 3 = Oandm = 1. The intrinsic rate constant ratios k p / k m , ke/km, kd/k,, kdp/km, and kd,/km are functions of the energetics of the reacting molecules and probably do not depend upon which steps in the free-radical mechanism are controlling. An estimate of these ratios can be obtained from the work of Fredericks and Tedder (1960, 1961). At 35, 78, and 146O, these investigators determined experimentally the degree to which one chloride atom a t various positions on a monochlorobutane molecule reduced the reactivity of the other hydrogen atoms in the molecule with respect to further chlorination. They found that the chloride atom reduced only the reactivity of the hydrogen atoms attached to the carbons which are a, p, and y to the chlorine atom. The effect on the y carbon is small. Using the results of Fredericks and Tedder, the rate constant ratios for n-dodecane chlorination were estimated (Table I). Values a t 167 and 215’ were obtained by extrapolation on an Arrhenius plot. For n-decane chlorination, the ratio k,&, was also estimated and agrees with the experimental value of 0.88 found by Barilli. From the above discussion, it can be seen that even in the absence of mass transfer limitations, selectivity is complex. In establishing the effect of mass transfer on y and ym, one must be careful that the controlling steps of the kinetic mechanism do not change over the range of experimental variables studied. Integration of eq 6, 7, and 8 using the rate constant ratios given in Table I shows that, in a semibatch reactor, y can increase by as much as 6501, and ym can decrease by as much as IS’% if m changes from 0.5 (as Stauff reported) to 250
Ind. Eng. Chem. Process Des. Develop., Vol.
12, No. 3, 1973
1.0 (as Barilli reported). Also if experiments are run in the presence of an aromatic solvent containing electron-donor substituents, the energetics of the free radicals will be altered and therefore the rate constant ratios will change (Russell, 1957, 1958). Finally it should be noted that even in the absence of mass transfer limitations, semibatch, CSTR, and tubular reactors can yield different values of y and ym since each reactor has a different concentration versus conversion history. Effect of Mass Transfer on Product Distribution
In this section mass transfer effects on several types of selectivity will be discussed. For some consecutive reactions in gas-liquid systems, analytical solutions for film theory (van de Vusse, 1966; Teramoto, et al., 1969; and Hashimoto, et al., 1968) and penetration theory (Szekely and Bridgewater, 1967; Harriott, 1970) models have been developed. These apply to simple reaction orders but the kinetics of ndodecane chlorination are not clearly established. Therefore, since the purpose of this paper is to show that mass transfer can affect selectivity, only a qualitative treatment will be presented. The effect of mass transfer on the yields of the several products is determined by the relative magnitude of certain parameters. The three regimes presented by Astarita (1967) for the effect of mass transfer on the rate of a single isothermal gasliquid reaction will be used. The gas-liquid interface is assumed to be composed of small stagnant liquid elements, which are continuously brought to the interface from a perfectly mixed bulk liquid by the motion of the bulk liquid itself. When an element is first brought to the interface, the concentrations of CIS,C12H26, and C12H2&1 in the element will be the same as those in the bulk liquid. While at the interface, absorption of chlorine into the element takes place by unsteady molecular diffusion. The time which an element is exposed to the interface is determined by the hydrodynamics of the bulk liquid. The equivalent diffusion time ( t D ) is a time average of the exposure times for all liquid elements and is defined as (9) Note that t D = (?r/4)8 and t~ = l/S where 8 is Higbie’s (1935) exposure time and S is Danckwerts (1951) fractional rate of surface renewal. The reaction time (t?) is defined as tD
= Dc/[kL0I2
The denominator of eq 10 is the maximum possible reaction rate of chlorine. In order for appreciable reaction to take place inside a surface element it must remain a t the interface for the same order of time as t,. If the chlorine does not react in the element, it reacts in the bulk liquid. Whereas t, is the required lifetime, t~ is the time actually available for reaction and diffusion in the elements. For both chemical reaction and diffusion to be important in the surface elements, t~ must be a t least the same magnitude as t,. Slow Reaction Regime. If the following condition is fulfilled tD
