ENV-07

CSTR first-order conversion

C = C0 /(1 + k τ) in an ideal CSTR.

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ReactorsFogler

Governing equation

C=\dfrac{C_0}{1+k\tau},\quad X=\dfrac{k\tau}{1+k\tau}

where

C_0: Inlet concentration (mg/L)
k: Rate constant (1/h)
\tau: Residence time (h)
C: Outlet concentration (mg/L)
X: Conversion (—)

Lecture brief

Historical brief

Streeter–Phelps (1925) oxygen sag, settling theory and Guldberg–Waage kinetics made water and air quality a rate problem. The lab computes sag, overflow and a snapshot of reactor mass balance. This sheet (ENV-07 — CSTR first-order conversion) is the form associated with Fogler. Working symbols: $C_0$ , $k$ , $\tau$ $\rightarrow$ $C$ , $X$ . A steady mass balance C0 − C + r τ = 0 with r = −k C solves immediately for the outlet of a continuous stirred tank.

Purpose

Purpose: compute

C

X

from

C_0

k

\tau

in Environmental via

C=\dfrac{C_0}{1+k\tau},\quad X=\dfrac{k\tau}{1+k\tau}

C = C0 /(1 + k τ) in an ideal CSTR. Use it when a real environmental question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

C_0 = 100.000\,\mathrm{mg/L}

k = 0.500\,\mathrm{1/h}

\tau = 4.000\,\mathrm{h}

, the governing relation

C=\dfrac{C_0}{1+k\tau},\quad X=\dfrac{k\tau}{1+k\tau}

yields

C = 33.333\,\mathrm{mg/L}

X = 0.667\,\mathrm{—}

. Ideal mixing, constant density, isothermal. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Inlet concentration C_0100 mg/LRate constant k0.5 1/hResidence time \tau4 h

Outputs

Outlet concentration C33.333 mg/L
Conversion X0.667 —

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Narration of this film

Ideal mixing, constant density, isothermal.

A steady mass balance C0 − C + r τ = 0 with r = −k C solves immediately for the outlet of a continuous stirred tank.

Reading speed

Watch on YouTube