CHE-24

Counterflow NTU effectiveness

ε = (1−e^{−NTU(1−Cr)})/(1−Cr e^{−NTU(1−Cr)}).

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Heat exchangersKays–London

Governing equation

\varepsilon=\dfrac{1-e^{-\mathrm{NTU}(1-C_r)}}{1-C_r e^{-\mathrm{NTU}(1-C_r)}}

$NTU \in \left[0.1, 8\right]$

where

NTU: Number of transfer units (—)
C_r: Capacity ratio (—)
\varepsilon: Effectiveness (—)

The figure, part by part

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①

Governing law — red

\varepsilon=\dfrac{1-e^{-\mathrm{NTU}(1-C_r)}}{1-C_r e^{-\mathrm{NTU}(1-C_r)}}

$NTU \in \left[0.1, 8\right]$

$NTU = 0.1$ $\varepsilon = 0.093$

Along the curve

②

Domain of the variable — blue

NTU_{\min} = 0.1,\quad NTU_{\max} = 8

$NTU \in \left[0.1, 8\right]$

$NTU = 0.1$ $NTU = 0.1$

Along the curve

③

Response curve — green

\varepsilon=\dfrac{1-e^{-\mathrm{NTU}(1-C_r)}}{1-C_r e^{-\mathrm{NTU}(1-C_r)}}

$NTU \in \left[0.1, 8\right]$

$NTU = 0.1$ $\varepsilon = 0.093$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\varepsilon}{\mathrm{d}NTU}

$NTU \in \left[0.1, 8\right]$

$NTU = 0.15$ $\varepsilon^{\prime} = 0.808$

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-24 — Counterflow NTU effectiveness) is the form associated with Kays–London. Working symbols: $NTU$ , $C_r$ $\rightarrow$ $\varepsilon$ . The ε–NTU method integrates the local energy balance of a counterflow exchanger, with Cr = Cmin/Cmax.

Purpose

Purpose: compute

\varepsilon

from

NTU

C_r

in Chemical engineering via

\varepsilon=\dfrac{1-e^{-\mathrm{NTU}(1-C_r)}}{1-C_r e^{-\mathrm{NTU}(1-C_r)}}

ε = (1−e^{−NTU(1−Cr)})/(1−Cr e^{−NTU(1−Cr)}). 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

NTU = 2.000\,\mathrm{—}

C_r = 0.500\,\mathrm{—}

, the governing relation

\varepsilon=\dfrac{1-e^{-\mathrm{NTU}(1-C_r)}}{1-C_r e^{-\mathrm{NTU}(1-C_r)}}

yields

\varepsilon = 0.7746\,\mathrm{—}

. Two streams, an NTU bar, an ε tick. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Number of transfer units NTU2 —Capacity ratio C_r0.5 —

Outputs

Effectiveness \varepsilon0.7746 —

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Narration of this film

Two streams, an NTU bar, an ε tick.

The ε–NTU method integrates the local energy balance of a counterflow exchanger, with Cr = Cmin/Cmax.

Reading speed

Watch on YouTube