**3.2 Effectiveness-number of transfer units (ε-NTU)**

When more than one of the inlet and outlet temperatures of the heat exchanger is unknown, LMTD may be calculated by trial and errors solution. Another approach to calculating the rate of heat transfer is the effectiveness number of transfer units (*ε*-NTU) method. The *ε*-NTU can be expressed according to Eq. (26) where *Cpmin* is the minimum value between the heat capacity of cold fluid (*Cp,C*) and hot fluid (*Cp,H*). The effectiveness (*ε*) can be defined as the ratio of the actual heat transfer rate (*q*) and the maximum possible heat transfer rate (*qmax*) according to Eq. (27).

$$NTU = \frac{UA\_m}{\mathbf{C}p\_{\text{min}}} \tag{26}$$

$$
\varepsilon = \frac{q}{q\_{\text{max}}} \tag{27}
$$

**Figure 4.**

*Correction factor for common shell and tube heat exchangers [3]. (a) One-shell pass and 2, 4, 6, etc. (any multiple of 2), tube pass. (b) Two-shell pass and 4, 8, 12, etc. (any multiple of 4), tube pass. (c) Singlepass cross-flow with both fluids unmixed. (d) Single-pass cross-flow with one fluid unmixed and other unmixed.*

Where:

$$\mathbf{q} = \mathbf{C}p\_{,H}(T\_{H,in} - T\_{H,out}) = \mathbf{C}p\_{,\mathcal{C}}(T\_{C,out} - T\_{C,in}) \tag{28}$$

$$q\_{\text{max}} = \mathbf{C}\_{p \text{ min}} \left( T\_{H, \text{in}} - T\_{C, \text{in}} \right) \tag{29}$$

The heat transfer rate using the *ε*-NTU method can express as [14]:

$$Q = \varepsilon C\_{P\min} \left( T\_{H,in} - T\_{C,in} \right) \tag{30}$$

**Tables 2** and **3** show the effectiveness and NTU relations for heat exchangers, respectively. It should be noted that *Cr* is the capacity and it can be defined as follows:

$$\mathbf{C}\_r = \frac{\mathbf{C}\_{P\text{ min}}}{\mathbf{C}\_{P\text{ max}}} \tag{31}$$


#### **Table 2.**

*Effectiveness relation for heat exchangers [15].*


#### **Table 3.**

*NTU relation for heat exchangers [15].*
