23-Chem-A4 Chemical Reactor Engineering · May 2013
Question 4 of 5: Acetone Pyrolysis in a Plug-Flow Reactor — First-Order Kinetics
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exams — May 2013 — 04-Chem-A4 Chemical Reactor Engineering. Three-hour, open-book exam; any non-communicating calculator permitted, Fogler’s Elements of Chemical Reaction Engineering allowed. Format: five questions, each 20 marks; any four constitute a complete paper (80 marks). All five are solved below for completeness. Per the paper’s instructions, all data are treated as exact and answers are given to three significant figures.
Reference texts: H. S. Fogler, Elements of Chemical Reaction Engineering (4th ed., Prentice Hall) — rate laws, batch/PFR/CSTR design equations, Arrhenius temperature dependence, integral & differential data analysis; O. Levenspiel, Chemical Reaction Engineering (3rd ed., Wiley) — reactor comparison and the tanks-in-series model; supporting thermochemical and property data from Perry’s Chemical Engineers’ Handbook (9th ed.).
Question 4: Acetone Pyrolysis in a Plug-Flow Reactor — First-Order Kinetics (20 marks)
Given. Isothermal PFR at 520 °C (793.15 K), 1.00 atm; pure acetone feed so the gas-phase expansion factor is $\varepsilon=1$ ($A\rightarrow2$ products). Cylindrical reactor 3.3 cm ID × 80 cm.
Quantity
Value
Temperature / pressure
520 °C (793.15 K) / 1.00 atm
Reactor
3.3 cm ID × 80 cm ⇒ $V=0.684$ L
$C_{A0}=P/RT$
0.0154 mol/L
Feed (g/h) → $X_A$
126→0.05, 46→0.13, 21→0.24, 12→0.35
Expansion factor $\varepsilon$
1
Find. (a) evidence of first-order kinetics and the rate constant $k$; (b) an independent differential-reactor estimate of $k$.
Figure 5 — Isothermal plug-flow reactor for acetone pyrolysis (A → ketene + CH₄, $\varepsilon=1$). Conversion is measured at four feed rates to test the first-order model.
Approach. Apply the first-order PFR design equation with the $(1+\varepsilon X)$ expansion term at each feed rate—constancy of $k$ proves first order—then verify with a low-conversion differential-reactor balance.
Set up the first-order PFR design equation with volume change. Acetone
pyrolysis $\text{CH}_3\text{COCH}_3\rightarrow\text{CH}_2\text{CO}+\text{CH}_4$ doubles the
moles, so for pure feed the expansion factor is $\varepsilon=y_{A0}\delta=1(2-1)=1$. For a
first-order reaction $-r_A=kC_A=kC_{A0}(1-X)/(1+\varepsilon X)$ the PFR design equation
integrates to
$$k = \frac{F_{A0}}{C_{A0}V}\Big[(1+\varepsilon)\ln\tfrac{1}{1-X}-\varepsilon X\Big].$$
Feed concentration and reactor volume. At 520 °C (793.15 K),
1 atm, pure acetone:
$$C_{A0}=\frac{P}{RT}=\frac{1}{(0.08206)(793.15)}=0.0154\ \text{mol/L},\qquad
V=\frac{\pi}{4}(3.3)^2(80)=684\ \text{cm}^3=0.684\ \text{L}.$$
Evaluate $k$ at each feed rate. With $F_{A0}=(\text{g/h})/58$ and
$\varepsilon=1$ so the bracket is $2\ln\frac{1}{1-X}-X$:
$$\begin{array}{ccc}
\text{Feed (g/h)} & X_A & k\ (\text{h}^{-1})\\\hline
126 & 0.05 & 10.9\\
46 & 0.13 & 11.2\\
21 & 0.24 & 10.6\\
12 & 0.35 & 10.1
\end{array}$$
The rate constant is essentially constant over a ten-fold change in feed rate, which is the
signature of first-order kinetics. The mean is
$$\boxed{k \approx 10.7\ \text{h}^{-1}\quad(\text{first order confirmed}).}$$
(b) Differential-reactor check. For the lowest conversion $(X=0.05<0.10)$
volume change is negligible, and the whole reactor is treated as differential:
$$-r_A=\frac{F_{A0}X}{V}=\frac{(2.172)(0.05)}{0.684}=0.159\ \text{mol/(L}\cdot\text{h)},\qquad
k=\frac{-r_A}{C_{A0}}=\frac{0.159}{0.0154}=\boxed{10.3\ \text{h}^{-1}}.$$
This matches the integral value (10.7 h$^{-1}$) within a few percent, confirming the result.
