Table 11.4 - Heat Exchanger NTU Relations

Flow Arrangement	Relation
Parallel flow	(11.28b) $NTU = - \frac{\ln [1 - ε (1 + C_{r})]}{1 + C_{r}}$
Counterflow	(11.29b) $\begin{matrix} NTU & = & \frac{1}{C_{r} - 1} \ln (\frac{ε - 1}{ε C_{r} - 1}) & (C_{r} < 1) \\ NTU & = & \frac{ε}{1 - ε} & (C_{r} = 1) \end{matrix}$
Shell-and-tube
One shell pass (2, 4, . . . tube passes)	(11.30b) ${(NTU)}_{1} = - {(1 + C_{r}^{2})}^{- 1 / 2} \ln (\frac{E - 1}{E + 1})$ (11.30c) $E = \frac{2 / ε_{1} - (1 + C_{r})}{{(1 + C_{r}^{2})}^{1 / 2}}$
n shell passes (2n, 4n, . . . tube passes)	Use Equations 11.30b and 11.30c with (11.31b, c, d) $\begin{matrix} ε_{1} = \frac{F - 1}{F - C_{r}} & F = {(\frac{ε C_{r} - 1}{ε - 1})}^{1 / n} & NTU = n {(NTU)}_{1} \end{matrix}$
Cross-flow (single pass)
C_max (mixed), C_min (unmixed)	(11.33b) $NTU = - \ln [1 + (\frac{1}{C_{r}}) \ln (1 - ε C_{r})]$
C_min (mixed), C_max (unmixed)	(11.34b) $NTU = - (\frac{1}{C_{r}}) \ln [C_{r} \ln (1 - ε) + 1]$
All exchangers (C_r = 0)	(11.35b) $NTU = - \ln (1 - ε)$