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, if the fluid inlet and outlet temperatures are specified or can be determined by simple energy balance, the LMTD method can be used; but when these temperatures are not available either the NTU or the effectiveness NTU method is used.

The effectiveness-NTU method is very useful for all the flow arrangements (besides parallel flow and counterflow ones) because the effectiveness of all other types must be obtained by a numerical solution of the partial differential equations and there is no analytical equation for LMTD or effectiveness, but as a function of two variables the effectiveness for each type can be presented in a single diagram.

To define the effectiveness of a heat exchanger we need to find the maximum possible heat transfer that 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 ${\displaystyle \ T_{h,i}-\ T_{c,i}}$ (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) ${\displaystyle \ C_{h}}$ and ${\displaystyle \ C_{c}}$ for the hot and cold fluids respectively, and denoting the smaller one as ${\displaystyle \ C_{\mathrm {min} }}$:

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

Where ${\displaystyle {\dot {m}}}$ is the mass flow rate and ${\displaystyle c_{p}}$ is the fluid's specific heat capacity at constant pressure.

A quantity:

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

is then found, where ${\displaystyle \ q_{\mathrm {max} }}$ is the maximum heat that could be transferred between the fluids per unit time. ${\displaystyle \ C_{\mathrm {min} }}$ 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 (${\displaystyle \epsilon }$), is the ratio between the actual heat transfer rate and the maximum possible heat transfer rate:

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

where:

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

Effectiveness is a dimensionless quantity between 0 and 1. If we know ${\displaystyle \epsilon }$ 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:

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

For any heat exchanger it can be shown that:

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

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

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

and the number of transfer units, ${\displaystyle \ NTU}$

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

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

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

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

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

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

For a balanced counter-current flow heat exchanger (balanced meaning ${\displaystyle C_{r}\ =1}$, which is a scenario desirable to reduce entropy):

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

A single-stream heat exchanger is a special case in which ${\displaystyle C_{r}\ =0}$. This occurs when ${\displaystyle C_{\mathrm {min} }=0}$ or ${\displaystyle C_{\mathrm {max} }=\infty }$ and may represent a situation in which a phase change (condensation or evaporation) is occurring in one of the heat exchanger fluids or when one of the heat exchanger fluids is being held at a fixed temperature. In this special case the heat exchanger behavior is independent of the flow arrangement and the effectiveness is given by:[1]

${\displaystyle \epsilon \ =1-e^{-NTU}}$

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.