Table 11.3 - Heat Exchanger Effectiveness Relations

Flow Arrangement	Relation
Parallel flow	(11.28a) $ε = \frac{1 - \exp [- NTU (1 + C_{r})]}{1 + C_{r}}$
Counterflow	(11.29a) $\begin{matrix} ε & = & \frac{1 - \exp [- NTU (1 - C_{r})]}{1 - C_{r} \exp [- NTU (1 - C_{r})]} & (C_{r} < 1) \\ ε & = & \frac{NTU}{1 + NTU} & (C_{r} = 1) \end{matrix}$
Shell-and-tube
One shell pass (2, 4, . . . tube passes)	(11.30a) $ε_{1} = 2 {1 + C_{r} + {(1 + C_{r}^{2})}^{1 / 2} \times \frac{1 + \exp [- {(NTU)}_{1} {(1 + C_{r}^{2})}^{1 / 2}]}{1 - \exp [- {(NTU)}_{1} {(1 + C_{r}^{2})}^{1 / 2}]}}^{- 1}$
n shell passes (2n, 4n, . . . tube passes)	(11.31a) $ε = [{(\frac{1 - ε_{1} C_{r}}{1 - ε_{1}})}^{n} - 1] {[{(\frac{1 - ε_{1} C_{r}}{1 - ε_{1}})}^{n} - C_{r}]}^{- 1}$
Cross-flow (single pass)
Both fluids unmixed	(11.32) $ε = 1 - \exp [(\frac{1}{C_{r}}) {(NTU)}^{0.22} {\exp [- C_{r} {(NTU)}^{0.78}] - 1}]$
C_max (mixed), C_min (unmixed)	(11.33a) $ε = (\frac{1}{C_{r}}) (1 - \exp {- C_{r} [1 - \exp (- NTU)]})$
C_min (mixed), C_max (unmixed)	(11.34a) $ε = 1 - \exp (- C_{r}^{- 1} {1 - \exp [- C_{r} (NTU)]})$
All exchangers (C_r = 0)	(11.35a) $ε = 1 - \exp (- NTU)$