Liquid Phase Reactor: Sucrose Inversion

Compared to continuous stirred tank reactors (CSTRs), plug flow reactors (PFRs) are usually better suited for fast reactions and large heat effects.² However, pressure drop and the development of "hot spots" can be problematic. Therefore, catalyst particle sizes can't be too small and careful startup procedures must be followed.

A PFR is mathematically equivalent to a large number of small, equally-sized CSTRs in series whose total volume or catalyst weight matches that of the system. When mixing occurs in the axial direction, the number of tanks, N, needed to describe reactor operation decreases. The model is called the "Tanks in Series" model. The parameters N and τ (space time) of the model can sometimes be obtained from the mean and variance of the residence time distribution (E-curve) of the reactor. For a PFR, the mean can be calculated exactly, and the variance is zero. For a real reactor, τ is usually estimated, and N is regressed from the model (Equation 2):

(2)

where "i" is the reactor number, C_Ao is the feed concentration of the limiting reactant (in this case, sucrose), Δf_Ai is the change in fractional conversion of "A" in the i the stirred tank, and r_Ai is the rate of reaction evaluated at the tank exit concentrations. This rate must be positive. Solving the mass balance for equal-sized CSTRs in series can also be used to determine the order of reaction for "A" using data from a real reactor and assuming the temperature can be kept reasonably constant and that N is known.

The forward rate equation for a catalytic reaction is almost always 1^st order in catalyst concentration and some positive order ≤2 in any reactant concentration. Products can sometimes inhibit the catalyst, causing the reactant's order to appear less than it really is. Even reactants can inhibit the catalyst, resulting in orders for a reactant closer to zero. For these reasons, catalytic reactions are often expressed by a "power-law" model:²

(3)

where  is the concentration of limiting reactant, k' is the apparent rate constant, and is the apparent reaction order. This model presupposes thatthe catalyst concentration is constant (it is absorbed into the true rate constant to give k'). In real life, catalysts often deactivate, i.e., C_cat [in mmol acid sites/gcat] decreases due to accumulation of catalyst poisons. For this reason one must either account for deactivation (express C_cat as a function of time onstream) or (preferably) collect data over a time period where the catalyst is relatively stable.

The kinetics of sucrose inversion are usually found to be first order in sucrose concentration and first order in concentration of catalyst sites. Lifshutz and Dranoff report a second-order rate constant k of ~0.029 mL/(mmol acid sites • min) for a catalyst similar to the one used here at 60 °C with an activation energy of 77 kJ/mol.⁴ Gilliland et al. report ~1.21 mL/(mmol acid sites • min) with an activation energy of 84 kJ/mol for a similar catalyst at these conditions.⁴ The significant differences in k can arise from several factors: (a) effects of heat and mass transfer on the kinetics; (b) poor flow distribution; (c) poor temperature control; and (d) varying states of catalyst activation.

The space time (analogous to residence time) for a catalytic reactor is usually expressed as  = W/Q, where W is the weight of catalyst and Q is the volumetric flowrate of feed. The units on the rate constant are adjusted to account for the units on space time (i.e., for a 1^st order reaction the units on k' would be the same as for 1/). Figure 1 illustrates the behavior of reactions of various kinetics orders in both a PFR and in a tanks-in-series model consisting of two tanks. Note that for positive orders, the PFR is always superior.

Figure 1. Computed fractional conversions (vs. space time) of sucrose using rate constants derived from the data for several sucrose feeds at 60°C.

To determine the amount of sugar, a polarimeter is used, which measures the degree of rotation of the light polarized by the analyte compound. Sugars are examples of compounds with enantiomers that can be differentiated by their optical activity, the ability to rotate light. A polarimeter is especially suited to measurement of concentrations in this experiment because the reactant sucrose rotates light to the right (positive optical rotation), whereas the products glucose and fructose rotate to the left (negative optical rotation).

The polarimeter determines the fractional conversions of sucrose after reaction in a packed bed reactor. A previous polarimeter calibration for a three different sucrose feeds is shown in Figure 3.

Figure 3. Relationship between degree of rotation and fractional conversion of sucrose for different feed concentrations.

Sample data are presented in Figure 4 for the reaction at 60 °C at varying sucrose feed concentrations. Fractional conversions were calculated directly from the polarimeter calibration curve using the following equation, where D is the degrees of rotation from the polarimeter:

    (4)

Figure 4. Sucrose inversion reaction at 60°C, 100 mL/min feed rate.

For both 0^th and 1^st order reactions, the conversion in a PFR is independent of feed concentration.² Additionally, k' should be invariant for 1^st order kinetics. Assuming the reactor to be a PFR, the 2^nd order rate constant, k₂ (mL/mmol sites • min), was determined by accounting for the 1^st order dependence the catalyst, and the pseudo-1^st order rate constant k' (mL/g_cat • min) was determined by ignoring the 1^st order dependence of the catalyst. The results of the pseudo-k' calculations are plotted in Figure 4. And the value of k₂ was found by dividing k' by the concentration of catalyst (mmol acid sites/g_cat) given previously.

  (5)

To determine whether the reaction kinetics were closer to 0, 0.5, 1, 1.5 or 2^nd order in sucrose, nonlinear regression of the mass balance was used, and the sum of squared errors was minimized for all three runs. In order to use nonlinear regression, an objective function was formulated based on the integrated PFR mass balance and the respective reaction order. For example, the following is the objective function for a 1.5 kinetic order in sucrose concentration:

(6)

Other objective functions can be formulated from the standard PFR mass balance solutions, which can be found in all kinetics textbooks.² The experimental data in Figure 4 were fit to the integrated PFR mass balances for 1, 1.5 and 2 orders with respect to sucrose. The sum of squared errors for the three reaction orders were determined to be 0.39, 0.16 and 1.3, respectively. Therefore, the best fit was found to be n = 1.5 order. This leads to a k' value of 35 (mL/g_cat • min).

It was initially thought that the kinetics were 1^st order with respect to sucrose.^2-3 Using this assumption, one can determine the number of equal volume CSTRs, N, in series that are required to model this reactor. Again, the sum of squared errors in the mass balance for all three runs were minimized to determine both N and k'. The data were fit to the tanks-in-series model for 1^st order reactions:

(7)

It was found that N = 2.1 "tanks" and k' = 0.62 mL/g_cat • min. This is not a great fit because the reaction order is not exactly 1. The data suggest a sucrose order > 1. The relative standard deviations of f_A were at most 2%, which is easily accounted for by the variation in temperature (as high as 9 °C). There was no evidence of catalyst deactivation. The fractional conversions for both a PFR and for two CSTR tanks in series was computed using the k's from nonlinear regression and plotted in Figure 1. For zeroth order, there was no difference between a PFR and CSTRs in series because the rate is independent of sucrose concentration. If the curves for 6 or greater CSTRs had been plotted, they would have coincided closely with the PFR curves. The predicted fractional conversions for two CSTR tanks in series is slower than a PFR for all reaction orders. The experimental data for 15 wt% sucrose is actually closer to a first-order reaction in PFR.

The error in k' can be estimated by comparing the differences in computed k' values at the average temperature deviation (4.5 °C) to the temperature of the reaction, 60 °C, using the Arrhenius equation and averaging the two literature activation energies. The estimated k' for 1.5 order kinetics at 64.5 °C is 52 (mL/mol)^0.5 mL • gcat^-1 • min^-1, which is almost 50% higher than the regressed value of 35 (mL/mol)^0.5 mL • gcat^-1 • min^-1. Slight variations in temperature can affect the k' greatly. 

The reaction does not behave exactly as expected because the apparent order n is > 1. Of all the phenomena that can cause such deviations in real reactors, deviations from ideal PFR behavior caused by axial mixing are suggested by the fact that fitting to the tanks-in-series model gives only a small number of tanks - for a perfect PFR, N should be at least 6. Such deviations are often found in relatively short beds, especially if the flow is multiphase (some water is vaporized in the reactor). However, another cause of the deviation is less apparent but probably even more important. The reaction is highly exothermic, and as mentioned, the temperature oscillated by as much as 9°C (mostly above the set point). The more sucrose in the feed, the more heat that will be generated. As might be expected, the oscillations were most significant with the 20 wt% feed. This suggests another reason for an apparent order n > 1: the more heat generated at a higher concentration of feed increases the reactor temperature more, which in turn increases the reaction rate resulting in a derived apparent order > the actual order. If the temperature is inadequately controlled, the reactor temperature might increase to the adiabatic limit. Deviations from ideal PFR behavior in both flow and temperature can affect the apparent kinetics derived from actual reactors, putting a premium on careful reactor scale-up to duplicate pilot-plant conditions of fluid flow and heat transfer.

Packed bed reactors have many uses in the chemical industry. Sulfuric acid, a chemical used to make hundreds of different products, is commonly manufactured in part using packed bed chemical reactors in series. Over 200 million tons are produced annually.In this reaction, sulfur dioxide and air are passed through fixed bed reactors in series (with intermediate heat exchangers for heat removal) containing a supported vanadium oxide catalyst at high temperatures.⁴ The SO₂ is oxidized to SO₃, which, when absorbed in water, makes sulfuric acid.

A more recent use for packed bed reactors is in production of biodiesel by transesterifying triglycerides, or esterifying fatty acids, with methanol. While biodiesel is produced in different ways, packed bed reactors can be advantageous for continuous production. Biodiesel is considered a renewable energy source because it is produced from algae or waste foodstuffs, and because it is biodegradable and non-toxic. Regardless of the catalyst used, it must be thoroughly purged from the product after the reaction, because even small amounts can render the fuel unusable.⁵

APPENDIX A – USING THE POLARIMETER
Polarimetry measures the extent to which a substance interacts with plane polarized light (light which consists of waves that vibrate only in one plane). It can rotate the polarized light to the left, to the right, or not at all. If it rotates polarized light to the left or to the right, it is “optically active”. If a compound does not have a chiral center, it will not rotate polarized light. The number of degrees and the direction of rotation are measured to give the observed rotation. The observed rotation is corrected for the length of the cell used and the solution concentration, using the following equation:
  (A1)
where: a = specific rotation (degrees) (literature value), l = path length (dm), and c = concentration (g/mL).
Comparing the corrected observed rotation to literature values can aid in the identification of an unknown compound. However, if the compounds are known, it is more common to prepare calibration standards of the unknowns and correlate the observed rotation to concentration.