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. The effectiveness-NTU method is very useful for all the flow arrangements (besides parallel flow, cross flow, and counterflow ones) but 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.
Defining and using heat exchanger effectiveness To define the effectiveness of a heat exchanger we need to find the maximum possible heat transfer that can be hypothetically achieved in an ideal counter-flow heat exchanger of infinite length. Therefore one of the fluids would experience in that conditions the maximum possible temperature change, which is the difference of T h , i − T c , i {\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 mass flowrates ( m ˙ {\displaystyle {\dot {m}}} ) of the two streams exchanging heat must be known (here, the cold stream is denoted with subscripts 'c' and the hot stream is denoted with subscripts 'h'). The method proceeds by calculating the heat capacity rates (i.e. mass flow rate multiplied by specific heat capacity) for the hot and cold fluids respectively
C h = m ˙ h c p , h {\displaystyle \ C_{h}={\dot {m}}_{h}c_{p,h}} and C c = m ˙ c c p , c {\displaystyle \ C_{c}={\dot {m}}_{c}c_{p,c}}
Here, c p {\displaystyle c_{p}} is the fluid specific heat capacity at constant pressure. Note that in this formulation the specific heat capacities of the fluids are considered constant. Since the specific heat capacity is by definition the derivative of enthalpy with respect to temperature: c p = ∂ h ∂ T {\displaystyle c_{p}={\frac {\partial h}{\partial T}}} the products m ˙ c p {\displaystyle {\dot {m}}c_{p}} represent the capacity of enthalpy transport of each flow, per unit of temperature change. By conservation of energy, the total enthalpy change of both fluids must be the same (in absolute value) when they pass through the ideal heat exchanger. Therefore, the fluid with the smaller heat capacity rate will be the one that experience the maximum temperature change, whereas the other fluid would change temperature more slowly along the heat exchanger length. Therefore, the maximum possible heat transfer rate between the fluids is determined by the following expression:
Q ˙ m a x = C m i n ( T h , i − T c , i ) {\displaystyle {\dot {Q}}_{\mathrm {max} }\ =C_{\mathrm {min} }(T_{h,i}-T_{c,i})}
where
C m i n = m i n ( C h , C c ) {\displaystyle \ C_{\mathrm {min} }=\mathrm {min} (C_{h},C_{c})}
Then, the effectiveness of the heat exchanger ( ϵ {\displaystyle \epsilon } ), is defined as the ratio between the actual heat transfer rate and this maximum theoretically possible heat transfer rate:
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