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Chemical Reaction Engineering · Virtual lab

Reactor design, kinetics & conversion

Six self-contained experiments, each solved live in your browser with hand-rolled numerics — no libraries. The rate ODEs are integrated with RK4, the design equations are solved on the fly, and every plot is drawn on a canvas. Pick an experiment on the left, then move through Aim, Theory, Procedure, the live Simulation, a graded Self-assessment, and References.

1 · Batch reactor kinetics

Integrating dCA/dt = -k CA^n for zero, first and second order in a stirred tank
To study a constant-volume batch reactor by numerically integrating the species mole balance for an irreversible reaction A → products, for reaction orders 0, 1 and 2; to plot concentration and fractional conversion against time; and to confirm the analytic first-order result CA = CA0 e^(-k t) and half-life t½ = ln2 / k.

In a well-mixed, constant-volume batch reactor there is no inflow or outflow. The mole balance on reactant A reduces to the rate of change of concentration equalling the reaction rate:

dCA/dt = rA = −k · CA^n

where k is the rate constant, n the reaction order and CA the concentration of A. The fractional conversion is defined relative to the initial charge:

X = (CA0 − CA) / CA0  ⇒  CA = CA0 (1 − X)

Closed-form solutions exist for the integer orders, and the simulator's RK4 integrator must reproduce them exactly:

order 0:  CA = CA0 − k t  (until CA hits 0)
order 1:  CA = CA0 e^(−k t),  t½ = ln2 / k
order 2:  1/CA = 1/CA0 + k t,  t½ = 1 / (k CA0)

Note that only the first-order half-life is independent of the starting concentration — a fingerprint used to identify reaction order experimentally.

  1. Open the Simulation tab.
  2. Choose a reaction order (0, 1 or 2) with the segmented control.
  3. Set the rate constant k and initial concentration CA0 with the sliders.
  4. Press Run and watch the stirred tank fade as A is consumed; the live readouts show CA, X and the instantaneous rate.
  5. For order 1, read the half-life and check that it equals ln2 / k and does not change when you raise CA0.
  6. Switch the curve toggle between concentration and conversion and compare the shapes.

Stirred batch reactor idle

Batch reactor: the tank colour tracks the remaining concentration of A; the lower plot is the live integrated curve. RK4 with dt is used — toggle to overlay the analytic solution and verify they coincide.

Kinetics

0
1
2
0.35
2.0
1.0x
CA(t)
X(t)
t (min)
0
CA
2.000
Conversion X
0%
Rate −rA
--
Half-life t½
--
RK4 vs exact
--

The math

Integrator: RK4 on dCA/dt = -k CA^n. Half-life: order 1 gives ln2/k (independent of CA0); order 2 gives 1/(k CA0). The error stat is the maximum gap between the numeric and analytic curve.
  • Fogler, H. S. — Elements of Chemical Reaction Engineering, Ch. 1-5 (Mole Balances, Batch). Pearson.
  • Levenspiel, O. — Chemical Reaction Engineering, Ch. 3 (Interpretation of Batch Reactor Data). Wiley.
  • Virtual Labs (IIT) — Chemical Engineering: Batch Reactor, vlabs.ac.in.

2 · CSTR steady-state design

The design equation V = FA0 X / (−rA) for a continuous stirred-tank reactor
To apply the steady-state CSTR design equation to find the exit conversion of a single continuous stirred-tank reactor for a given volume, volumetric flow and reaction kinetics, and to visualise the inlet and outlet streams and the perfectly-mixed contents at the exit concentration.

A CSTR is assumed perfectly mixed, so the exit stream has exactly the contents' composition. A steady-state mole balance on A (in − out + generation = 0) gives the classic design equation:

V = FA0 · X / (−rA)|exit

where FA0 = CA0 · v0 is the molar feed rate and the rate is evaluated at the exit conditions CA = CA0(1−X). Defining the space time τ = V / v0, for an n-th order reaction:

τ · k · CA0^(n−1) (1−X)^n = X

For first order this rearranges to a clean closed form; for second order it is a quadratic in X. The simulator solves the implicit equation by bisection so any positive order works:

order 1:  X = kτ / (1 + kτ)

Unlike a batch or PFR, the whole CSTR operates at the (low) exit rate, so it generally needs the largest volume for positive-order kinetics.

  1. Open the Simulation tab.
  2. Pick the order, then set k, feed concentration CA0 and volumetric flow v0.
  3. Adjust the reactor volume V and watch the solver return the steady-state exit conversion.
  4. Confirm the first-order case against X = kτ/(1+kτ) using the displayed space time τ.
  5. Increase v0 (shorter τ) and note conversion falls; increase V and note it rises toward 1.

CSTR apparatus steady

CSTR: feed enters top-left at CA0, the tank is uniform at the exit concentration, product leaves bottom-right. The bar shows how exit conversion responds to volume.

Design inputs

0
1
2
0.50
2.0
1.0
4.0
Space time τ
--
Damkohler Da
--
Exit conversion
--
Exit CA
--
FA0
--
−rA exit
--

The math

Solver: bisection on tau*k*CA0^(n-1)*(1-X)^n - X = 0 over X in [0,1). Da = k*tau*CA0^(n-1). The whole tank reacts at the slow exit rate — that is why CSTRs are volume-hungry.
  • Fogler, H. S. — Elements of Chemical Reaction Engineering, Ch. 2 & 5 (Conversion and Reactor Sizing, CSTR). Pearson.
  • Levenspiel, O. — Chemical Reaction Engineering, Ch. 5 (Ideal Reactors for a Single Reaction). Wiley.
  • Virtual Labs (IIT) — Chemical Engineering: Continuous Stirred Tank Reactor, vlabs.ac.in.

3 · PFR — plug-flow reactor

Integrating dX/dV = (−rA)/FA0 down a tubular reactor
To integrate the plug-flow design equation along the length of a tubular reactor, obtaining conversion as a function of cumulative reactor volume, and to confirm the first-order analytic result X = 1 − e^(−kτ) where τ = V/v0.

In an ideal PFR the fluid moves as discrete plugs with no axial mixing; composition varies continuously down the tube. A differential mole balance over a slice dV gives the design equation:

dX/dV = (−rA) / FA0  with  −rA = k CA0^n (1−X)^n

Because FA0 = CA0 v0, dividing through introduces the space time and the integral becomes purely a function of kτ. Integrating from the inlet (X = 0) to volume V:

order 1:  X = 1 − e^(−kτ)
order 2:  X = kτ CA0 / (1 + kτ CA0)

The simulator marches X forward in V with RK4, so the curve is exact even for non-integer orders. Compare it to the CSTR result of the same τ: the PFR always converts more for positive order because it exploits the high inlet rate.

  1. Open the Simulation tab.
  2. Set order, k, CA0 and the volumetric flow v0.
  3. Drag the reactor volume slider; the marker sweeps down the tube and the conversion profile fills in.
  4. Note the tube colour gradient — dark feed at the inlet fading to converted product at the outlet.
  5. For order 1, verify the exit conversion equals 1 - exp(-k*tau) shown in the readout.
  6. Switch the overlay to compare the PFR profile with the single-CSTR exit at the same τ.

Tubular plug-flow reactor flowing

PFR: the top schematic shows the concentration gradient along the tube; the lower plot is conversion X versus cumulative volume. The dashed line marks the equivalent single-CSTR exit conversion.

Design inputs

0
1
2
0.50
2.0
1.0
6.0
PFR only
vs CSTR
Space time τ
--
Exit conversion
--
Exit CA
--
CSTR X (same τ)
--

The math

Integrator: RK4 on dX/dV = k*CA0^(n-1)*(1-X)^n / v0. The result depends only on k*tau. For order 1 the exact exit conversion is 1 - exp(-k*tau).
  • Fogler, H. S. — Elements of Chemical Reaction Engineering, Ch. 2 & 5 (Tubular Reactors). Pearson.
  • Levenspiel, O. — Chemical Reaction Engineering, Ch. 5 (Plug Flow Reactor). Wiley.
  • Virtual Labs (IIT) — Chemical Engineering: Plug Flow Reactor, vlabs.ac.in.

4 · CSTR vs PFR — Levenspiel plot

Comparing required volumes via the area under 1/(−rA) versus X
To compare the reactor volume required by a single CSTR and a single PFR to reach the same target conversion with identical kinetics, using the Levenspiel plot of 1/(−rA) against X, and to see graphically why the PFR is smaller for positive-order reactions.

Both ideal reactor volumes are integrals (or areas) on the same axes. Plotting the reciprocal rate 1/(−rA) against conversion X — the Levenspiel plot — turns sizing into reading an area:

V_PFR = FA0 · ∫₀^X dX / (−rA)  (area under the curve)
V_CSTR = FA0 · X / (−rA)|X  (rectangle to the exit point)

For positive-order kinetics the rate falls (so 1/(−rA) rises) as X grows. The PFR uses the small area under the rising curve; the CSTR is forced to use the full-height rectangle evaluated at the slow exit rate. Hence:

V_PFR < V_CSTR  (positive order)

The ratio V_CSTR / V_PFR grows with target conversion and with reaction order. For autocatalytic or negative-order behaviour the inequality can reverse — the geometry tells you instantly.

  1. Open the Simulation tab.
  2. Choose order, k and feed CA0.
  3. Slide the target conversion X and watch both shaded regions grow: the area (PFR) and the rectangle (CSTR).
  4. Read the two required volumes and their ratio; confirm PFR < CSTR for orders 1 and 2.
  5. Push X toward 1 and observe both areas blow up — the last few percent of conversion are the most expensive.

Levenspiel plot PFR smaller

Levenspiel: the orange shaded area is V_PFR / FA0; the cyan dashed rectangle is V_CSTR / FA0. Both share the same 1/(−rA) curve.

Kinetics and target

0
1
2
0.50
2.0
0.80
V_PFR / FA0
--
V_CSTR / FA0
--
Volume ratio
--
Smaller reactor
--

The math

PFR: composite-Simpson integral of 1/(-rA) from 0 to X. CSTR: rectangle X / (-rA)|X. Ratio = V_CSTR / V_PFR. For positive order the curve rises, so the rectangle beats the area — PFR wins.
  • Levenspiel, O. — Chemical Reaction Engineering, Ch. 5-6 (Single & Multiple Reactor Systems). Wiley.
  • Fogler, H. S. — Elements of Chemical Reaction Engineering, Ch. 2 (Reactor Sizing, Levenspiel plots). Pearson.
  • Virtual Labs (IIT) — Chemical Engineering: Comparison of Reactors, vlabs.ac.in.

5 · Arrhenius temperature law

k = A exp(−Ea/RT): the ln k versus 1/T straight line
To study how the rate constant depends on temperature through the Arrhenius equation, to construct the linearised ln k versus 1/T plot, to read the activation energy Ea from its slope and the pre-exponential factor A from its intercept, and to see how temperature changes shift reactor conversion.

The Arrhenius equation captures the strong temperature sensitivity of most reaction rates:

k = A · exp(−Ea / (R T))

where A is the pre-exponential (frequency) factor, Ea the activation energy, R = 8.314 J/mol·K and T the absolute temperature. Taking logarithms linearises it:

ln k = ln A − (Ea / R) · (1/T)

So a plot of ln k against 1/T is a straight line of slope −Ea/R and intercept ln A. Measuring k at two temperatures is enough:

ln(k2/k1) = −(Ea/R)(1/T2 − 1/T1)

A rough rule of thumb: a 10 K rise near room temperature roughly doubles many rates — the simulator lets you check this for any Ea you set.

  1. Open the Simulation tab.
  2. Set the activation energy Ea and pre-exponential factor A with the sliders.
  3. Drag the temperature marker along the curve; the dot moves along the Arrhenius and ln-plot lines together.
  4. Read the recovered slope (−Ea/R) and intercept (ln A) from the fitted line and confirm they match the inputs.
  5. Use the k(T2)/k(T1) readout to test the 10 K doubling rule for different Ea values.

Arrhenius plots linear in 1/T

Arrhenius: top is k versus T (curved); bottom is ln k versus 1/T (straight). The slope is −Ea/R; the intercept is ln A. Both dots track the temperature slider.

Parameters

60
18
350
k at T
--
ln k
--
Slope (−Ea/R)
--
Intercept (ln A)
--
k(T+10)/k(T)
--
Recovered Ea
--

The math

A least-squares line is fit to sampled (1/T, ln k) points; its slope m gives Ea = -m*R and intercept gives ln A. Recovered Ea should equal your input to within rounding.
  • Fogler, H. S. — Elements of Chemical Reaction Engineering, Ch. 3 (Rate Laws, Arrhenius). Pearson.
  • Levenspiel, O. — Chemical Reaction Engineering, Ch. 2 (Kinetics of Homogeneous Reactions). Wiley.
  • Atkins, P. & de Paula, J. — Physical Chemistry, Ch. on Reaction Kinetics. Oxford.

6 · Reaction order determination

Integral method: which linearisation of concentration-time data is straight
To determine the order of a reaction from synthetic batch concentration-time data using the integral method — testing the zero, first and second-order linearisations, identifying which one is a straight line by its correlation coefficient, and recovering the rate constant from the slope.

The integral method assumes an order, integrates the batch rate law and checks whether the data fall on a straight line. Each order has its own linear coordinate:

order 0:  CA  vs t  (slope = −k)
order 1:  ln CA  vs t  (slope = −k)
order 2:  1/CA  vs t  (slope = +k)

The correct order is the one whose transformed plot is straight — quantified by the coefficient of determination R² closest to 1. The rate constant follows from the slope of that best-fit line.

R² = 1 − SS_res / SS_tot

The simulator generates noisy data from a hidden true order and asks you to identify it — just as a real kinetic study would, where measurement scatter makes the wrong orders look almost-but-not-quite linear.

  1. Open the Simulation tab.
  2. Press New data set to generate noisy CA(t) measurements from a hidden true order.
  3. Toggle between the three linearisations and read each one's R².
  4. Pick the order whose plot is straightest (highest R²) and read off the recovered k.
  5. Press Reveal to check your call against the hidden true order, then add or remove noise and repeat.

Integral-method fit fit a line

Order determination: the points are the (transformed) data for the chosen trial order; the line is its least-squares fit. The straight one — highest R² — reveals the true order.

Analysis

CA
ln CA
1/CA
0.020
R² order 0
--
R² order 1
--
R² order 2
--
Best order
--
Recovered k
--
True order
hidden

The math

Each trial transforms CA, fits a line by least squares, and reports R-squared. The order with the highest R-squared is the answer; its slope magnitude is the rate constant k.
  • Levenspiel, O. — Chemical Reaction Engineering, Ch. 3 (Interpretation of Batch Reactor Data, integral & differential methods). Wiley.
  • Fogler, H. S. — Elements of Chemical Reaction Engineering, Ch. 7 (Collection and Analysis of Rate Data). Pearson.
  • Virtual Labs (IIT) — Chemical Engineering: Determination of Order of Reaction, vlabs.ac.in.