NTU method

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.

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. \ C_h = \dot m_h c_{p,h} and \ C_c = \dot m_c c_{p,c} Here, 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=\frac{\partial h}{\partial T} the products \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: \dot Q_\mathrm{max}\ = C_\mathrm{min} (T_{h,i}-T_{c,i}) where \ C_\mathrm{min}=\mathrm{min}(C_h, C_c) Then, the effectiveness of the heat exchanger ( \epsilon ), is defined as the ratio between the actual heat transfer rate and this maximum theoretically possible heat transfer rate: \epsilon \ = \frac{\dot Q}{\dot Q _\mathrm{max}} where the real heat transfer rate can be determined either from the cold fluid or the hot fluid (they must provide equivalent results): \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 \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: \dot Q \ = \epsilon C_\mathrm{min} (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. \ \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," or C_r and NTU: C_r \ = \frac{C_\mathrm{min}}{C_\mathrm{max}} \ NTU describes heat transfer across a surface NTU \ = \frac{U A}{C_\mathrm{min}} Here, U is the overall heat transfer coefficient, A is the total heat transfer area, and 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: \dot Q= C_{min} (T_{o} -T_{i})_{min} = UA \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 ( \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: \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 a balanced counter-current flow heat exchanger (balanced meaning C_r \ = 1 , which is a scenario desirable to enable irreversible entropy production to be reduced given sufficient heat transfer area): \epsilon\ = \frac{NTU}{1+NTU} A single-stream heat exchanger is a special case in which C_r \ = 0 . This occurs when C_\mathrm{min}=0 or 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: \epsilon \ = 1 - e^{-NTU} For a crossflow heat exchanger with both fluid unmixed, the effectiveness is: \epsilon \ = 1 - \exp(-NTU) - \exp[-(1+C_r)NTU] \sum_{n=1}^\infty C_r^n P_n(NTU) where P_n is the polynomial function 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 \epsilon \ = \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 C_{\mathrm{min}} corresponds to the mixed fluid, the result is \epsilon \ = 1 - \exp\left( -\frac{1-\exp(-NTU\cdot C_r)}{C_r} \right) whereas if C_{\mathrm{min}} corresponds to the unmixed fluid, the solution is \epsilon \ = \frac{1}{C_r} (1 - \exp\{-C_r[1 - \exp(-NTU)] \}) All these formulas for crossflow heat exchangers are also valid for 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. It is common in the field of mass transfer system design and modeling to draw analogies between heat transfer and mass transfer. 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, a mass transfer-analogous specific mass capacity is required. This specific mass capacity should describe the change in concentration of the transferring gas relative to the partial pressure difference driving the mass transfer. This results in a definition for specific mass capacity as follows: c_{p-x}= \frac{d \omega_x}{dP_x} Here, \omega_x represents the mass ratio of gas 'x' (meaning mass of gas 'x' relative to the mass of all other non-'x' gas mass) and P_x is the partial pressure of gas 'x'. Using the ideal gas formulation for the mass ratio gives the following definition for the specific mass capacity: c_{p-x}= \frac {M_x/M_{other}} {P_{other}} Here, M_x is the molecular weight of gas 'x' and M _{other} is the average molecular weight of all other gas constituents. With this information, the NTU for gaseous mass transfer of gas 'x' can be defined as follows: NTU_x \ = \frac{U_mA_m}{\dot{m}c_{p-x}} Here, U_m is the overall mass transfer coefficient, which could be determined by empirical correlations, A_m is the surface area for mass transfer (particularly relevant in membrane-based separations), and \dot{m} is the mass flowrate of bulk fluid (e.g., mass flowrate of air in an application where water vapor is being separated from the air mixture). At this point, all of the same heat transfer effectiveness-NTU correlations will accurately predict the mass transfer performance, as long as the heat transfer terms in the definition of NTU have been replaced by the mass transfer terms, as shown above. Similarly, it follows that the definition of C_r becomes: C_r \ = \frac{(\dot{m}c_{p-x})_{min}}{(\dot{m}c_{p-x})_{max}} One particularly useful application for the above described effectiveness-NTU framework is membrane-based air dehumidification. In this case, the definition of specific mass capacity can be defined for humid air and is termed the "specific humidity capacity." c_{p-h}=\frac {M_{wv}/M_{air}} {P_{air}}=\frac{0.62198}{P_{air}}=\frac{0.62198}{P_{total}-P_{wv,inlet}} Here, M_{wv} is the molecular weight of water (vapor), M_{air} is the average molecular weight of air, P_{air} is the partial pressure of air (not including the partial pressure of water vapor in an air mixture) and can be approximated by knowing the partial pressure of water vapor at the inlet, before dehumidification occurs, P_{wv,inlet} . From here, all of the previously described equations can be used to determine the effectiveness of the mass exchanger.

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