Reactor Cascade Startup & Disturbance Response

Reactor Cascade

Tanks in series N 4

Residence time per tank τ 1.5 s

Disturbance type

Disturbance amplitude 0.60

Live Readouts

Outlet conc. C_N(t)

1.00

Deviation from base

0.0%

Total residence time N·τ

6.0 s

How it works

Each tank is modelled as an ideal continuous stirred-tank reactor (CSTR): perfectly mixed, constant volume, constant volumetric flow Q. A mass balance on tank i gives a first-order ODE driven by the tank upstream of it:

dC_i/dt = (1/τ)·(C_(i-1) − C_i),   τ = V/Q  (per-tank residence time)
C_0(t) = feed concentration, shaped by the disturbance you trigger

For a unit step change in the feed, the exact closed-form response of the N-th (last) tank is the classic "tanks-in-series" curve:

C_N(t)/ΔC_in = 1 − e^(−t/τ)·Σ_(k=0)^(N−1) (t/τ)^k / k!

N (tanks in series) — with N = 1 the outlet jumps and relaxes exponentially (classic CSTR lag). As N grows, the response sigmoid-delays and sharpens: this is the numbering-up-vs-size-up trade-off from continuous-manufacturing scale-up — many small well-mixed stages start to approximate an ideal plug-flow reactor (PFR), which has zero back-mixing and a pure time delay.
τ per tank — sets how fast each stage relaxes toward its inlet; the total plant residence time is N·τ.
Disturbance type — Step models a sustained feed-composition upset (e.g. a raw-material lot change); Pulse models a transient slug (e.g. a valve hiccup) that clears after roughly one tank residence time; Sine models a periodic feed-pump ripple, testing whether the cascade damps or amplifies an oscillating input.
Deviation from base — the live gap between outlet and steady baseline is exactly what a real plant's inline monitoring watches to decide whether a batch is in-spec or must be diverted — the practical reason continuous lines need robust startup and deviation-handling strategy before hazardous or high-value chemistry is run through them.

Below baseline Near baseline Above baseline

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