# Fabrizio Sarghini » 25.Heat Exchangers - II Part

### Heat Exchangers

If the heat exchanger has a complex structure, for example multiple passage of tubes in multiple shells, the method requires altering LMDT for an appropriate correction factor F $\Delta T_{eff}=F\cdot \Delta T_{LMDT}$

which is computed as a function of the 2 parameters $P=\frac{t_2-t_1}{T_1-t_1}$ $R=\frac{T_1-T_2}{t_2-t_1}$

### Heat Exchangers – NTU Effectiveness Method

The Number of Transfer Units (NTU) method is used to calculate the rate of heat transfer in heat exchangers (especially counter current exchangers) when there is insufficient information to calculate the Log-Mean Temperature Difference (LMTD).

When the fluid inlet and outlet temperatures are specified or can be determined by simple mass balance, then the LMTD method can be used, whereas when this information is not available the NTU or the effectiveness method is used.

### Heat Exchangers – NTU Effectiveness Method

The maximum possible heat transfer which can be hypothetically achieved in a counter flow heat exchanger of infinite length is given by the product of the maximum possible temperature difference $\Delta T_{max}=T_{HI}-T_{CI}$

multiplied by the smaller heat capacity rate $C_{\min}=\min(c_{PH}\cdot m_h, c_{PC}\cdot m_C)=\min(C_H, C_c)$

### Heat Exchangers – NTU Effectiveness Method

The quantity $W_{\max}=C_{\min}(T_{HI}-T_{CI})$

is the maximum heat which could be transferred between the fluids.
The effectiveness ε $\epsilon=\frac W{W_{\max}},\epsilon \in [0,1]$

is the ratio between the actual heat transfer rate and the maximum possible heat transfer rate where $W=C_H(T_{HI}-T_{HO})=C_C(T_{CO}-T_{CI})$

### Heat Exchangers

The knowledge of e for a particular heat exchanger together with the inlet conditions of the enables us to calculate the amount of heat transferred between the fluids by using $W=\epsilon \cdot W_{\max}$

For any heat exchanger it can be shown that $\epsilon =f\Biggl(NTU, \frac {C_{\min}}{C_{\max}}\Biggr)$

For given geometries, e can be computed using correlations in terms of the heat capacity ratio $C=\frac{C_{\min}}{C_{\max}}$

and the number of transfer units, $NTU=\frac{UA}{C_{\min}}$

where U is the overall heat transfer coefficient and A   is the heat transfer area.

### Heat Exchangers – NTU Effectiveness Method

The specific equation for the effectiveness of a parallel flow heat exchanger can be written as $\epsilon =\frac{1-e^{[-NTU(1+C)]}}{1+C}$

while in the special case of condensation or vaporisation C=0 and therefore the effectiveness is given by $\epsilon =1-e^{-NTU}$

### Shell and Tube Heat Exchangers

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