NTU method

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 ${\displaystyle T_{h,i}-T_{c,i}}$ (the temperature difference between the inlet temperature of the hot stream and the inlet temperature of the cold stream). The mass flowrates (${\displaystyle \dot{m}}$) 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

${\displaystyle C_h={\dot{m}}_h{c}_{p,h}}$ and ${\displaystyle C_c={\dot{m}}_c{c}_{p,c}}$

Here, ${\displaystyle c_p}$ 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:

${\displaystyle c_p=\frac{\mathrm{\partial }h}{\mathrm{\partial }T}}$

the products ${\displaystyle \dot{m}{c}_p}$ 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:

${\displaystyle {\dot{Q}}_{\mathrm{m}\mathrm{a}\mathrm{x}}=C_{\mathrm{m}\mathrm{i}\mathrm{n}}(T_{h,i}-T_{c,i})}$

where

${\displaystyle C_{\mathrm{m}\mathrm{i}\mathrm{n}}=\mathrm{m}\mathrm{i}\mathrm{n}(C_h,C_c)}$

Then, the effectiveness of the heat exchanger (${\displaystyle ϵ}$), is defined as the ratio between the actual heat transfer rate and this maximum theoretically possible heat transfer rate:

${\displaystyle ϵ=\frac{\dot{Q}}{{\dot{Q}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}}$

where the real heat transfer rate can be determined either from the cold fluid or the hot fluid (they must provide equivalent results):

${\displaystyle \dot{Q}=C_h(T_{h,i}-T_{h,o})=C_c(T_{c,o}-T_{c,i})}$

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

${\displaystyle \dot{Q}=ϵC_{\mathrm{m}\mathrm{i}\mathrm{n}}(T_{h,i}-T_{c,i})}$

Then, having determined the actual heat transfer from the effectiveness and inlet temperatures, the outlet temperatures can be determined from the equation above.

Relating Effectiveness to the Number of Transfer Units (NTU)

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:

${\displaystyle ϵ=f(NTU,\frac{C_{\mathrm{m}\mathrm{i}\mathrm{n}}}{C_{\mathrm{m}\mathrm{a}\mathrm{x}}})}$

For a given geometry, ${\displaystyle ϵ}$ can be calculated using correlations in terms of the "heat capacity ratio," or ${\displaystyle C_r}$ and NTU:

${\displaystyle C_r=\frac{C_{\mathrm{m}\mathrm{i}\mathrm{n}}}{C_{\mathrm{m}\mathrm{a}\mathrm{x}}}}$

${\displaystyle NTU}$ describes heat transfer across a surface

${\displaystyle NTU=\frac{UA}{C_{\mathrm{m}\mathrm{i}\mathrm{n}}}}$

Here, ${\displaystyle U}$ is the overall heat transfer coefficient, ${\displaystyle A}$ is the total heat transfer area, and ${\displaystyle C_{min}}$ is the minimum heat capacity rate. To better understand where this definition of NTU comes from, consider the following heat transfer energy balance, which is an extension of the energy balance above:

${\displaystyle \dot{Q}=C_{min}(T_o-T_i{)}_{min}=UA\mathrm{\Delta }{T}_{LM}}$

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 (${\displaystyle \mathrm{\Delta }{T}_{LM}}$). Starting from the differential equations that describe heat transfer, several "simple" correlations between effectiveness and NTU can be made. For brevity, below summarizes the Effectiveness-NTU correlations for some of the most common flow configurations:

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

${\displaystyle ϵ=\frac{1-\exp [-NTU(1+C_r)]}{1+C_r}}$

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

${\displaystyle ϵ=\frac{1-\exp [-NTU(1-C_r)]}{1-C_r\exp [-NTU(1-C_r)]}}$

For a balanced counter-current flow heat exchanger (balanced meaning ${\displaystyle C_r=1}$, which is a scenario desirable to enable irreversible entropy production to be reduced given sufficient heat transfer area):

${\displaystyle ϵ=\frac{NTU}{1+NTU}}$

A single-stream heat exchanger is a special case in which ${\displaystyle C_r=0}$. This occurs when ${\displaystyle C_{\mathrm{m}\mathrm{i}\mathrm{n}}=0}$ or ${\displaystyle C_{\mathrm{m}\mathrm{a}\mathrm{x}}=\mathrm{\infty }}$ 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:

${\displaystyle ϵ=1-e^{-NTU}}$

For a crossflow heat exchanger with both fluid unmixed, the effectiveness is:

${\displaystyle ϵ=1-\exp (-NTU)-\exp [-(1+C_r)NTU]\sum _{n=1}^{\mathrm{\infty }}{C}_r^n{P}_n(NTU)}$

where ${\displaystyle P_n}$ is the polynomial function

${\displaystyle P_n(x)=\frac{1}{(n+1)!}\sum _{j=1}^n\frac{n+1-j}{j!}{x}^{n+j}}$

If both fluids are mixed in the crossflow heat exchanger, then

${\displaystyle ϵ={\left[\frac{1}{1-\exp (-NTU)}+\frac{C_r}{1-\exp (-NTU\cdot C_r)}-\frac{1}{NTU}\right]}^{-1}}$

If one of the fluids in the crossflow heat exchanger is mixed and the other is unmixed, the result depends on which one has the minimum heat capacity rate. If ${\displaystyle C_{\mathrm{m}\mathrm{i}\mathrm{n}}}$ corresponds to the mixed fluid, the result is

${\displaystyle ϵ=1-\exp \left(-\frac{1-\exp (-NTU\cdot C_r)}{C_r}\right)}$

whereas if ${\displaystyle C_{\mathrm{m}\mathrm{i}\mathrm{n}}}$ corresponds to the unmixed fluid, the solution is

${\displaystyle ϵ=\frac{1}{C_r}(1-\exp \{-C_r[1-\exp (-NTU)]\})}$

All these formulas for crossflow heat exchangers are also valid for ${\displaystyle C_r=1}$.

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.