CSTR vs PFR vs Batch: Which Reactor, and Why

Every reaction engineering course hits the same fork in the road. You’ve got a reaction, you know its kinetics, and now someone asks you to pick a reactor. Batch? A CSTR? A PFR? The three get introduced as if they’re just different shapes of tank — and the exam question is always some version of “which one, and how big.”

Here’s the thing most lectures bury under the algebra: the choice between these three reactors comes down to a single variable behaving three different ways. That variable is concentration. Get how concentration lives inside each reactor and you’ll never have to memorize which one “wins” — you’ll be able to reason it out from scratch.

Three reactors, one real difference

Strip away the diagrams and here’s what actually separates them:

• Batch. You charge everything in, seal it, and let it run. Nothing flows in or out. Composition changes with time — you watch reactant concentration fall from start to finish on a clock.
• CSTR (continuous stirred-tank, a.k.a. mixed-flow). Feed pours in continuously, product pours out continuously, and the tank is stirred so well that it’s uniform everywhere. The catch: “uniform everywhere” means uniform at the exit composition. The moment fresh feed enters, it’s instantly diluted down to the final, lowest concentration.
• PFR (plug-flow tube). Fluid marches through a tube in lockstep, like a column of soldiers, with no mixing front-to-back. Composition changes with position. The inlet is nearly pure feed; the outlet is nearly product; every point in between is somewhere on the journey.

Same reaction, three different concentration stories. The CSTR runs entirely at its faded exit concentration; the PFR carries the full gradient from dark inlet to faded outlet.

A CSTR always operates at the lowest concentration in the system — its own exit. A PFR experiences the entire journey, from rich feed down to lean product. Batch does the same journey as the PFR, just spread over time instead of distance. That one distinction drives almost every reactor decision you’ll ever make.

Why the concentration story decides everything

For almost every reaction you’ll meet, rate increases with reactant concentration — a rate law like $-r_A = k C_A^n$ with $n > 0$. Higher $C_A$, faster reaction. Hold that thought against the picture above.

The CSTR is stuck running its whole volume at the stingy exit concentration. Every cubic meter of that tank is reacting slowly, because everywhere in it, $C_A$ is already knocked down to the final value. Low rate everywhere means you need a lot of volume to get the job done.

The PFR gets to cash in on the high concentrations near the inlet, where the reaction is ripping along, and only slows down as the fluid approaches the exit. Its average rate across the whole reactor is much higher — so it needs less volume for the same conversion. Batch works the same way through time: fast at the start when concentration is high, tapering as it depletes.

The design equations (and the one that surprises people)

All three come from the same starting point: a mole balance on reactant $A$. I’ll skip the full derivation here (that’s its own article), but the results are worth having side by side. Using conversion $X_A$ and feed concentration $C_{A0}$:

Batch (constant volume) — how long you hold it:

CSTR — note there’s no integral. The rate is a single value, evaluated at the exit conditions:

PFR — the rate changes down the tube, so you integrate:

Here $\tau = V/v_0$ is the space time — the volume divided by the volumetric feed rate, i.e. how long an average parcel of fluid spends inside. Multiply $\tau$ by $v_0$ and you get the reactor volume.

Now look again at the batch and PFR equations. They’re the same integral. That’s not a coincidence:

A PFR is a batch reactor on a conveyor belt. The clock time a batch reactor spends reacting is exactly the residence time a fluid parcel spends travelling down a plug-flow tube. Distance in a PFR plays the role that time plays in a batch reactor.

That single realization collapses two-thirds of your reactor toolkit into one idea. Solve a batch problem and you’ve basically solved the matching PFR problem.

The CSTR equation is the odd one out — and that’s the whole point. Because the tank is uniform at exit conditions, there’s nothing to integrate; you just evaluate the rate once, at the exit. Algebraically easier, but physically it’s exactly why the CSTR is the volume hog.

The Levenspiel plot: seeing the volume difference

There’s a graphical way to see all of this at once, and it’s worth burning into memory because exams lean on it hard. Plot $1/(-r_A)$ on the vertical axis against conversion $X_A$ on the horizontal. Since rate falls as conversion rises, $1/(-r_A)$ climbs as you move right.

On this plot, reactor volume is literally an area:

• The PFR space time is the area under the curve from $0$ to your target conversion — that’s what the integral means geometrically.
• The CSTR space time is a rectangle: width equal to the target conversion, height equal to $1/(-r_A)$ at the exit — the single value the whole tank runs at.

For normal kinetics the curve rises, so the CSTR rectangle swallows extra area above the curve. That gap — rectangle minus area-under-curve — is the wasted volume you pay for perfect mixing.

The rectangle always contains the area under a rising curve, so the CSTR is always the bigger reactor here. And you can see when the gap gets ugly: the steeper the curve shoots up near your target conversion (high orders, high conversions), the more volume a CSTR wastes relative to a PFR.

A worked example: how big is the penalty?

Let’s put real numbers on it. Say you’re running a liquid-phase, second-order reaction $A \rightarrow \text{products}$, with:

• Rate law $-r_A = k C_A^2$, with $k = 0.5\ \text{L}\,\text{mol}^{-1}\text{min}^{-1}$
• Feed concentration $C_{A0} = 2\ \text{mol/L}$, feed rate $v_0 = 10\ \text{L/min}$
• Target conversion $X_A = 0.80$

At 80% conversion the exit concentration is $C_A = C_{A0}(1 - X_A) = 2(0.2) = 0.4\ \text{mol/L}$, so the exit rate is $-r_A = 0.5(0.4)^2 = 0.08\ \text{mol}\,\text{L}^{-1}\text{min}^{-1}$.

CSTR — evaluate at the exit and you’re done:

PFR — integrate. For second order the integral has a clean closed form, $\int_0^{X}\frac{dX}{(1-X)^2} = \frac{X}{1-X}$:

Push the target to 90% and the penalty gets worse, because the second-order curve steepens. That’s the Levenspiel plot warning you in advance.

So when is a CSTR ever the right call?

If PFRs are so volume-efficient, why does anyone build a stirred tank? Because volume isn’t the only thing you’re optimizing — and this is exactly where exam questions like to flip on you. A CSTR earns its keep when low, uniform concentration is a feature, not a bug:

• Autocatalytic reactions, where a product speeds up the reaction, so the rate is actually slow at high reactant concentration and peaks somewhere in the middle. A CSTR can park right at that peak-rate composition; a PFR is forced to crawl through the slow early region. Here the CSTR can genuinely win on volume.
• Selectivity, when your desired reaction is lower-order than the side reaction. Running at low concentration (CSTR) starves the high-order byproduct pathway more than it starves the product you want. Concentration level is a selectivity knob, and the CSTR sets it low.
• Controlling a strong exotherm. A CSTR’s big, well-mixed, diluted contents and large heat-transfer area make runaway far easier to tame than a PFR with a hot spot building somewhere down the tube.
• Practical, unglamorous reasons: handling slurries and solids, keeping catalyst or microbial cultures suspended and alive, easy sampling, robust steady-state operation, cheap to run around the clock.

And you rarely have to choose just one — a CSTR followed by a PFR is a classic combo: the CSTR handles the messy, fast, exothermic front end at safe low concentration, then the PFR polishes off the last few percent of conversion efficiently.

Where batch fits

Batch skips the “volume efficiency” conversation almost entirely, because it’s chosen for different reasons: small production volumes, high-value products, and flexibility. Pharmaceuticals, specialty chemicals, and anything made in modest quantities lean batch because you can run different recipes in the same vessel and you don’t need the capital and control complexity of a continuous line. The cost you pay is downtime — charging, heating, cooling, discharging, and cleaning between runs — which is dead time that a continuous reactor never suffers. Once your production is large and steady, continuous almost always wins on cost per kilogram.

The decision, in one table

If you care most about…	Reach for	Because
Smallest volume, normal kinetics	PFR (or batch)	Exploits high inlet concentrations → higher average rate
Low concentration for selectivity or safety	CSTR	Whole tank sits at the lean exit concentration
Autocatalytic / microbial growth	CSTR (or CSTR→PFR)	Can operate at the peak-rate composition
Taming a strong exotherm	CSTR or cooled PFR	Dilution + heat-transfer area prevent runaway
Low volume, high value, flexibility	Batch	One vessel, many recipes; capital-light

Notice that not one of these rows is something you’d memorize as a fact. Each one falls straight out of the concentration picture from the top of this article. That’s the whole game: know how concentration lives in each reactor, and every “which reactor” question answers itself.