NTU method

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$\ C_{\mathrm {min} }=\mathrm {min} [{\dot {m}}_{c}c_{p,c},{\dot {m}}_{h}c_{p,h}]$

A quantity:

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

is then found, where $\ q_{\mathrm {max} }$ is the maximum heat that could be transferred between the fluids per unit time. $\ C_{\mathrm {min} }$ must be used as it is the fluid with the lowest heat capacity rate that would, in this hypothetical infinite length exchanger, actually undergo the maximum possible temperature change. The other fluid would change temperature more slowly along the heat exchanger length. The method, at this point, is concerned only with the fluid undergoing the maximum temperature change.

The effectiveness ($\epsilon$), is the ratio between the actual heat transfer rate and the maximum possible heat transfer rate:

$\epsilon \ ={\frac {q}{q_{\mathrm {max} }}}$

where:

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

Effectiveness is a dimensionless quantity between 0 and 1. If we know $\epsilon$ for a particular heat exchanger, and we know the inlet conditions of the two flow streams we can calculate the amount of heat being transferred between the fluids by:

$q\ =\epsilon C_{\mathrm {min} }(T_{h,i}-T_{c,i})$

For any heat exchanger it can be shown that:

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

For a given geometry, $\epsilon$ can be calculated using correlations in terms of the "heat capacity ratio"

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

and the number of transfer units, $\ NTU$

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

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

For example, the effectiveness of a parallel flow heat exchanger is calculated with:

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

Or the effectiveness of a counter-current flow heat exchanger is calculated with:

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

For counter-current flow heat exchanger with $C_{r}\ =1$:

$\epsilon \ ={\frac {NTU}{1+NTU}}$

The effectiveness-NTU relationships for crossflow heat exchangers and various types of shell and tube heat exchangers can be derived only numerically by solving a set of partial differential equations. So, there is no analytical formula for their effectiveness, but just a table of numbers or a diagram. These relationships are differentiated from one another depending (in shell and tube exchangers) on the type of the overall flow scheme (counter-current, concurrent, or cross flow, and the number of passes) and (for the crossflow type) whether any or both flow streams are mixed or unmixed perpendicular to their flow directions.

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

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

References

• F. P. Incropera & D. P. DeWitt 1990 Fundamentals of Heat and Mass Transfer, 3rd edition, pp. 658–660. Wiley, New York
• F. P. Incropera, D. P. DeWitt, T. L. Bergman & A. S. Lavine 2006 Fundamentals of Heat and Mass Transfer ,6th edition, pp 686–688. John Wiley & Sons US