The number of transfer units (NTU) method is used to calculate the rate of heat transfer in heat exchangers (especially parallel flow, counter current, and cross-flow exchangers) when there is insufficient information to calculate the log mean temperature difference (LMTD).
Alternatively, this method is useful for determining the expected heat exchanger effectiveness from the known geometry.
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.
First, you must know the specific heat capacity of your two fluid streams, denoted as
This information can usually be found in a thermodynamics textbook,[1] or by using various software packages.
is the maximum rate of heat that could be transferred between the fluids per unit time.
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.
), is the ratio between the actual heat transfer rate and the maximum possible heat transfer rate: where the real heat transfer rate can be determined either from the cold fluid or the hot fluid (they must provide equivalent results): Effectiveness is a dimensionless quantity between 0 and 1.
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: Then, having determined the actual heat transfer from the effectiveness and inlet temperatures, the outlet temperatures can be determined from the equation above.
For any heat exchanger it can be shown that the effectiveness of the heat exchanger is related to a non-dimensional term called the "number of transfer units" or NTU: For a given geometry,
From this energy balance, it is clear that NTU relates the temperature change of the flow with the minimum heat capacitance rate to the log mean temperature difference (
Starting from the differential equations that describe heat transfer, several "simple" correlations between effectiveness and NTU can be made.
, which is a scenario desirable to enable irreversible entropy production to be reduced given sufficient heat transfer area): A single-stream heat exchanger is a special case in which
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:[3] For a crossflow heat exchanger with both fluid unmixed, the effectiveness is: where
corresponds to the unmixed fluid, the solution is All these formulas for crossflow heat exchangers are also valid for
Additional effectiveness-NTU analytical relationships have been derived for other flow arrangements, including shell-and-tube heat exchangers with multiple passes and different shell types, and plate heat exchangers.
[2] However, a mass transfer-analogous definition of the effectiveness-NTU method requires some additional terms.
One common misconception is that gaseous mass transfer is driven by concentration gradients, however, in reality it is the partial pressure of the given gas that drive mass transfer.
In the same way that the heat transfer definition includes the specific heat capacity of the fluid, which describes the change in enthalpy of the fluid with respect to change in temperature and is defined as:
This specific mass capacity should describe the change in concentration of the transferring gas relative to the partial pressure difference driving the mass transfer.
With this information, the NTU for gaseous mass transfer of gas 'x' can be defined as follows:
is the overall mass transfer coefficient, which could be determined by empirical correlations,
is the surface area for mass transfer (particularly relevant in membrane-based separations), and
One particularly useful application for the above described effectiveness-NTU framework is membrane-based air dehumidification.
From here, all of the previously described equations can be used to determine the effectiveness of the mass exchanger.
It is very common, especially in dehumidification applications, to define the mass transfer driving force as the concentration difference.
When deriving effectiveness-NTU correlations for membrane-based gas separations, this is valid only if the total pressures are approximately equal on both sides of the membrane (e.g., an energy recovery ventilator for a building).