CHE-08

Damköhler number I

Da = k τ, reaction rate over convective rate.

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DimensionlessDamköhler 1936

Governing equation

\mathrm{Da}_I = k\tau

$\tau \in \left[0.1, 100\right]$

where

k: Rate constant (1/s)
\tau: Residence time (s)
\mathrm{Da}_I: Damköhler I (—)

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

\mathrm{Da}_I = k\tau

$\tau \in \left[0.1, 100\right]$

$\tau = 0.1$ $\mathrm{Da}_I = 0.02$

Along the curve

②

Domain of the variable — blue

\tau_{\min} = 0.1,\quad \tau_{\max} = 100

$\tau \in \left[0.1, 100\right]$

$\tau = 0.1$ $\tau = 0.1$

Along the curve

③

Response curve — green

\mathrm{Da}_I = k\tau

$\tau \in \left[0.1, 100\right]$

$\tau = 0.1$ $\mathrm{Da}_I = 0.02$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\mathrm{Da}_I}{\mathrm{d}\tau}

$\tau \in \left[0.1, 100\right]$

$\tau = 0.728$ $\mathrm{Da}_I^{\prime} = 0.2$

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-08 — Damköhler number I) is the form associated with Damköhler 1936. Working symbols: $k$ , $\tau$ $\rightarrow$ $\mathrm{Da}_I$ . Damköhler compared the chemical time 1/k to the residence time τ. Da ≫ 1 means equilibrium, Da ≪ 1 means frozen flow.

Purpose

Purpose: compute

\mathrm{Da}_I

from

k

\tau

in Chemical engineering via

\mathrm{Da}_I = k\tau

Da = k τ, reaction rate over convective rate. 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 = 0.200\,\mathrm{1/s}

\tau = 5.000\,\mathrm{s}

, the governing relation

\mathrm{Da}_I = k\tau

yields

\mathrm{Da}_I = 1.000\,\mathrm{—}

. First-order, homogeneous, single reaction. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Rate constant k0.2 1/sResidence time \tau5 s

Outputs

Damköhler I \mathrm{Da}_I1.000 —

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Narration of this film

First-order, homogeneous, single reaction.

Damköhler compared the chemical time 1/k to the residence time τ. Da ≫ 1 means equilibrium, Da ≪ 1 means frozen flow.

Reading speed

Watch on YouTube