CHE-02

PFR first-order space time

τ = (1/k) ln(1/(1−X)) for an isothermal PFR.

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ReactorsLevenspiel

Governing equation

\tau=\dfrac{1}{k}\ln\dfrac{1}{1-X}

$X \in \left[0.05, 0.99\right]$

where

k: Rate constant (1/s)
X: Conversion (—)
\tau: Space time (s)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

\tau=\dfrac{1}{k}\ln\dfrac{1}{1-X}

$X \in \left[0.05, 0.99\right]$

$X = 0.05$ $\tau = 1.026$

Along the curve

②

Domain of the variable — blue

X_{\min} = 0.05,\quad X_{\max} = 0.99

$X \in \left[0.05, 0.99\right]$

$X = 0.05$ $X = 0.05$

Along the curve

③

Response curve — green

\tau=\dfrac{1}{k}\ln\dfrac{1}{1-X}

$X \in \left[0.05, 0.99\right]$

$X = 0.05$ $\tau = 1.026$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\tau}{\mathrm{d}X}

$X \in \left[0.05, 0.99\right]$

$X = 0.056$ $\tau^{\prime} = 21.18$

Along the curve

Lecture brief

Historical brief

From CSTR/PFR mole balances and Arrhenius rates to McCabe–Thiele stages and NTU exchangers, chemical engineering is conservation plus equilibrium. The lab is that design arithmetic. This sheet (CHE-02 — PFR first-order space time) is the form associated with Levenspiel. Working symbols: $k$ , $X$ $\rightarrow$ $\tau$ . The PFR mole balance dX/dV = −r/FA0 integrates to −ln(1−X)/k for first-order constant-density flow.

Purpose

Purpose: compute

\tau

from

k

X

in Chemical engineering via

\tau=\dfrac{1}{k}\ln\dfrac{1}{1-X}

τ = (1/k) ln(1/(1−X)) for an isothermal PFR. Use it when a real chemical engineering question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

k = 0.050\,\mathrm{1/s}

X = 0.800\,\mathrm{—}

, the governing relation

\tau=\dfrac{1}{k}\ln\dfrac{1}{1-X}

yields

\tau = 32.189\,\mathrm{s}

. Plug flow, no axial dispersion, irreversible first-order. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Rate constant k0.05 1/sConversion X0.8 —

Outputs

Space time \tau32.189 s

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Narration of this film

Plug flow, no axial dispersion, irreversible first-order.

The PFR mole balance dX/dV = −r/FA0 integrates to −ln(1−X)/k for first-order constant-density flow.

Reading speed

Watch on YouTube