NTU 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). In heat exchanger analysis the fluid inlet and outlet temperatures are specified or can be determined by simple mass balance the LMTD method can be used ,but when these information are not available The NTU or The Effectiveness method is used.

To define the effectiveness of a heat exchanger we need to find the maximum possible heat transfer which can be hypothetically achieved in a counter flow heat exchanger of infinite length.Therefore one fluid will experience the maximum possible temperature difference which is the difference of $\ T_{h,i}-\ T_{c,i}$ (The temperature difference between the inlet temperature of the hot stream and the inlet temperature of the cold stream).The method proceeds by calculating the heat capacity rates (i.e. mass flow rate multiplied by specific heat) $\ C_{h}$ and $\ C_{c}$ for the hot and cold fluids respectively, and denoting the smaller one as $\ C_{min}$.

A quantity

$q_{max}\ =C_{min}(T_{h,i}-T_{c,i})$

is then found,where $\ q_{max}$ is the maximum heat which could be transferred between the fluids. According to the above equation to experience the maximum heat could be transferred the heat capacity should be minimized since we are using the maximum possible temperature difference.This justifies the reason of using $\ C_{min}$ in the equation.

The effectiveness(E),the ratio between the actual heat transfer rate and the maximum possible heat transfer rate

$E\ ={\frac {q}{q_{max}}}$

where

$q\ =C_{h}(T_{h,i}-T_{h,o})\ =C_{c}(T_{c,o}-T_{c,i})$ and $q_{max}\ =C_{min}(T_{h,i}-T_{c,i})$

Effectiveness is dimensionless quantity between 0 and 1 inclusive. Now therefore if we know E for a particular heat exchanger with the inlet conditions of the flows we can calculate the amount of heat is being transferred between the fluids by

$q\ =EC_{min}(T_{h,i}-T_{c,i})$

For any heat exchanger it can be shown that

$\ E=f(NTU,{\frac {C_{min}}{C_{max}}})$

For given geometries, $\ E$ can be calculated using correlations in terms of the 'heat capacity ratio'

$C_{r}\ ={\frac {C_{min}}{C_{max}}}$

and the number of transfer units, $\ NTU$

$NTU\ ={\frac {UA}{C_{min}}}$

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

For an example we can obtain a specific equation for the effectiveness of a parallel flow heat exchanger as

$E\ ={\frac {1-exp[-NTU(1+C_{r})]}{1+C_{r}}}$

Similar effectiveness relationships can derived for Concentric tube heat exchangers and Shell and tube heat exchangers as well. Such relationships differ from each other depending on the type of the flow(counter current,concurrent,cross flow),number of passes(in shell and tube exchangers) and mixed or unmixed.

Note that the $C_{r}\ =0$ is a special case scenario where in the heat exchanger condensation or vaporisation is occurring.Hence in this special case heat exchanger behavior is independent of the flow arrangement.Therefore the effectiveness is given by

$E\ =1-exp[-NTU]$

References

[1] heat- exchanger performance by the LMTD and -NTU methods.

2. F. P. Incropera & D. P. DeWitt 1990 Fundamentals of Heat and Mass Transfer, 3rd edition, pp. 658–660. Wiley, New York

3. F. P. Incropera , D. P. DeWitt , T. L. Bergman & A. S. Lavine 2006 Fundamentals of Heat and Mass Transfer ,6 th edition , pp 686–688. John Wiley & Sons US