Modeling of a Steady-State Tubular Reactor With Occurrence of a Suspected 1st-Order Reaction
Part (A) Assumption of an Ideal Plug-Flow Reactor (PFR)
An aqueous stream containing ethylacetate (denoted by A) at a concentration CAo (= 0.50 mol/L) is fed at a volumetric flow rate vo to the left of a cylindrical tubular reactor shown schematically in the figure below. The reactor is packed with uniform resin catalyst beads to a height H (= 40.5 cm) and 2 o its cross-sectional area is So (= 10.75 cm ). Upon entering the reactor, which is maintained at 40 C, ethylacetate undergoes the hydrolysis reaction, CH3COOC2H5 + H2O → CH3COOH + C2H5OH, with a suspected first-order rate expression, –rA = kCA (I)
where k is the rate constant to be determined and CA is the ethylacetate concentration in the aqueous stream that is dependent on the axial distance z from the reactor entrance. The aqueous stream flows through the packed bed with a uniform velocity everywhere (i.e., plug flow) and the reactor operates at a steady state (such that CA(z) is independent of time).
(a) By performing a steady-state mole balance on the volume element shown in the figure above (defined by the z and z+dz sections), show that the molar concentration of A in the tubular reactor is
dC A S o governed by the following differential equation: k C A (II) dz vo with the boundary condition: at z = 0, CA = CAo.
(b) Solve the above differential equation and show that the concentration of A in the reactor effluent
So H V is given by lnC Ae lnC Ao k( ) lnC Ao k (III) o vo where V ≡ SoH = 435.4 mL.
(c) A series of runs were carried out in which the feed solution of a constant concentration CAo is fed at a varying volumetric flow rate vo to the tubular reactor and the concentration of A in the reactor effluent was measured at each flow rate. The following set of data was obtained.
vo, mL/min ∞ (feed) 120 80 50 30 20 14 10
CA e, mol/L 0.500 0.374 0.324 0.249 0.157 0.088 0.042 0.015
Use the above data to construct an appropriate plot such that you can (i, 1%) validate Eqn (III) and thereby confirm the suspected first-order rate expression, Eqn (I), and (ii, 7%) determine the linear least-square value for the rate constant k. SHOW your work!
(d) Determine the 95% confidence region around your model regression line by calculating of values of the upper and lower limits of the model result that correspond to each of the eight vo values.
(e) Determine the 95% confidence interval for the rate constant k.
Part (B) Assumption of a Non-Ideal Plug-Flow Reactor
A more detailed analysis of the packed reactor and the data in Part A suggests a need to modify the reactor model to account for some back-mixing (due to the presence of void space caused by uneven packing) in the mid section of the reactor. The revised model below has been proposed that consists of three ideal reactors in series. In the revised model, the mid-section of the packed reactor that is suspected of having a void space that gives rise back-mixing, is modeled as a mixed-flow reactor with a volume Vm in which the reaction occurs with the solution being well mixed at all time. The beginning and final sections of the packed reactor that flank the mid-section are
modeled as ideal tubular reactors each with a volume V1. Previously the entire reactor was assumed 3 to be an ideal tubular reactor with a volume VTotal (= AoH = 435.4 cm ). Their performance is described by the analogs of Eqn (II) and (III) in Part A, see below.
As shown schematically above, the aqueous stream containing ethylacetate at a molar concentration CAo enters the first tubular reactor with a volumetric flow rate vo. The effluent from the first tubular reactor contains unreacted A at a concentration CA1 and enters the mixed-flow reactor with a volumetric flow rate vo. The effluent from the mixed-flow reactor contains unreacted A at a concentration CA2 and enters the second tubular reactor with a volumetric flow rate vo. The effluent from the latter contains unreacted A at a concentration CA3, which should ideally reflect CAe expt.
(a) Using a total volume balance on the three reactors in series and a steady-state mole balance on A in each of the three reactors, show that the following equations apply:
Volume balance: V1 Vm V1 2V1 Vm VTotal Ao H (IV) V kV lnC lnC k 1 C C exp( 1 ) For the first tubular reactor, A1 Ao or A1 A0 (V) vo vo C C A1 For the mixed-flow reactor, A2 kV (VI) 1 m vo V kV lnC lnC k 1 C C exp( 1 ) For the second tubular reactor, A3 A2 or A3 A2 (VII) vo vo C 2kV C Ao exp( 1 ) For the three reactors in series, Eqn (V) through (VII) give, A3 V v (VIII) 1 k m o vo (b) Re-analyze the data in Part A, but this time use Eqn (IV) and (VIII) in conjunction with a non- linear least-square regression to determine the parameters k and Vm that best fit the revised reactor model to the data. SHOW your work in arriving at your best-fit parameters. Show the values of SSD—the sum of squares of differences between CAe expt and CA 3—of the non-linear regression analysis as k and Vm are being varied. Hint: Consider the cases of Vm/V = 0, 0.05, 0.10, 0.15, 0.20, 0.25.
Part A Solution n8 n8 n8 2 1 2 1 V V 2 1 V 2 S x (xi x ) ( ] [ 16.09 min] n 1 i1 7 i1 vo i vo 7 i1 vo i
n8 n8 [y y ]2 [lnC ln C ]2 exp t i mod el i Ae expt i Ae mod el i S i1 i1 y,x n 2 6 V V ( )2 2 1 (x x ) 1 vo i vo y ln(C Ae ) (t , 0.025 )S y,x 1 2 (t 6, 0.025 )S y,x 1 2 n (n 1)S x 8 7S x t 6, 0.025 2.447
S y,x S y,x 0.0259 0.0016 k (t , 0.025 ) (t 6, 0.025 ) 2.447 S x n 1 S x 7 15.1 7 min min
95% confidence interval for k is given by k = ( 0.0745 ± 0.0016)/min or 0.0729/min < k < 0.0761/min
Part B Solution (a) Total volume of the three reactors in series should equal the volume of the actual tubular reactor being modeled, i.e., V1 + Vm + V1 = SoH = 435.4 mL or 2V1 + Vm = SoH = 435.4 mL (IV)
Eqn (V) & (VII) are the equivalent analogs of Eqn (III) in Part A kV kV C C exp( 1 ) C C exp( 1 ) A1 A0 (V) A3 A2 (VII) vo vo A steady-state mole balance on reactant A around the mixed-flow reactor gives, N 0 A (vC ) (vC ) V (r ) v C v C V (kC ) t A in A out A gen o A1 o A2 m A2 C C A1 v C (v kV )C or A2 V (VI) o A1 o m A2 1 k m vo C 2kV C Ao exp( 1 ) Eqn (V)−(VII) give A3 V v (VIII) 1 k m o vo