Figure 1 shows a generalized Heat Exchanger in which heat is transferred between two streams (stream 1 and stream 2).

The rate of heat transfer,
, between the streams may be expressed as a function of the area available for the transfer of heat, A, the overall heat transfer coefficient, U, and a mean temperature difference, ΔT_{m}, such that

Estimating the rate of heat transfer for a given design of heat exchanger requires techniques for estimating the Overall Heat Transfer Coefficient and techniques for estimating the mean temperature difference.

Techniques for estimating mean temperature difference are based upon the following assumptions:

the heat exchangers have only two streams;

heat exchange with the surroundings is negligible;

there is a linear relationship between specific enthalpy and temperature for both streams (i.e., constant specific heat capacities);

the overall heat transfer coefficient between the stream is constant throughout the heat exchanger;

where a heat exchanger consists of multiple parallel paths, the flowrates and heat transfer areas in each path are identical.

The above assumptions are most likely to be met when both streams are single-phase fluids (i.e. all liquid or all gas) and where the temperature changes are small such that the specific heat capacities and other properties of the fluids stay constant throughout the heat exchanger. The approach could be applied to heat exchangers involving boiling or condensing but only under circumstances where there are no significant changes in overall heat transfer coefficient. Heat exchangers involving the onset of boiling or condensation or the dryout transition are therefore not suitable for treatment using the traditional mean temperature difference approach. Such heat exchangers will need to be analyzed using techniques which make allowance for changes in heat transfer coefficient.

With the above assumptions, the rate of heat transfer in any geometry of heat exchanger can, in principle, be calculated. The simplest case is pure countercurrent flow. Here the mean temperature difference can be expressed in terms of the inlet and outlet temperatures of each stream.

For a heat exchanger with countercurrent flow, the mean temperature difference is known as the log mean temperature difference, ΔT_{LM}. The log mean temperature difference is the maximum mean temperature difference that can be achieved in any geometry of heat exchanger for any given set of inlet and outlet temperatures. For any other type of heat exchanger, the mean temperature difference can be expressed as

where F is always less than or equal to 1. Estimating the mean temperature difference in a heat exchanger by calculating the log mean temperature difference and estimating F is known as the *F factor method*.

*F* varies with geometry and thermal conditions. The thermal conditions are defined by parameters such as the overall heat transfer coefficient, U, the area available for heat transfer, A, the mass flow rates of the two steams
and
, the specific heat capacities of the two streams c_{1} and c_{2}, and the temperature change in each stream (T_{1,in} - T_{1,out}) and (T_{2,in} - T_{2,out}).

For any given geometry, F is often presented as a function of two nondimensional parameters, R, the ratio of the thermal capacities of the two streams and P (sometimes known as effectiveness, E), the ratio of the achieved heat transfer rate to the maximum possible heat transfer rate. These parameters are typically defined as

and

where

Unfortunately, there are no standard definitions of R and P. Users of the F method should always check the definitions used by any supplier of F value information and apply identical definitions when using this information in heat transfer calculations. This variation of definition can lead to problems when comparing data from different authors.

Values of F are frequently presented as graphs showing the relationship between F and P for a range of values of R. Figure 2 shows a typical relationship. The figure shows that F always tends to 1 as the amount of heat transferred reduces to zero. The figure also shows that for any given value of R, there is typically a maximum achievable value of P. The value of F changes rapidly as P approaches its maximum value. Because of this sensitivity, heat exchanger designs are rarely developed near the maximum value of P and are typically restricted to conditions which give values of F greater than 0.8.

**Figure 2. Typical relationship between F and P for various values of R (Based on Single E-shell with any even number of tube passes).**

The F factor can be used for both design and rating calculations. For design calculations, the mass flowrates, specific heat capacities and required temperature changes will be specified. R and P can therefore be calculated directly. The design engineer will need to check that the required value of P is less than the maximum value of P for the specified value of R. The value of F can then be found from the graph. The combination of the log mean temperature difference and F gives the mean temperature difference and with an estimate of the overall heat transfer coefficient, the required heat transfer area can be established. If the required value of P is greater than the maximum value for the specified value of R, a different type of heat exchanger must be considered until a feasible design is found. A simple countercurrent heat exchanger will always be able to achieve any design requirement but may be physically impractical.

For rating calculations, the geometry of the heat exchanger and its heat transfer area, the mass flowrates and the specific heat capacities of the streams and hence the overall heat transfer coefficient and the inlet temperatures will be defined. It will therefore be possible to calculate R but not P. The rating is estimated using an iteration which may start by guessing a heat transfer rate and calculating the exit temperatures, the log mean temperature difference and P and then using the figure to estimate F. The resulting heat transfer rate is then calculated from F, the log mean temperature difference, the overall heat transfer coefficient and the heat transfer area. The guessed and calculated heat transfer rates are compared and the guessed value is adjusted until convergence is achieved.

Published relationships for F in graphical form are available for most geometries of shell and tube heat exchanger and a range of geometries of crossflow heat exchanger (see guide to references at the end of this section). The size of the graphical presentations rarely allows values of F to be estimated to better than two significant figures. This accuracy of estimation is consistent with the overall accuracy of the mean temperature difference approach and the lack of compliance with the underlying assumptions. Attempts to improve the accuracy in the estimation of F values are therefore unlikely to produce significant benefits for heat exchanger designers.

An alternative method of presenting mean temperature difference information is known as the *effectiveness—N _{TU}* method. This method is based upon exactly the same initial assumptions. The heat transfer behavior are presented as a relationship between effectiveness, E (defined in a similar way to P), the ratio of the thermal capacities of the streams, R, and the number of heat transfer units, N

_{TU}which as calculated from the expression

where (
)_{smaller} is the smaller of (
)_{1} and (
)_{2}.
Unfortunately, the parameters E, R and N_{TU} are again not consistently defined and any user should check the definition used by the supplier of data and apply those definitions when using the data.

Effectiveness-N_{TU} information is typically presented graphically as the relationship of effectiveness against N_{TU} for various values of R. Figure 2 shows a typical relationship. This shows that the effectiveness tends to zero as the N_{TU} tends to zero and the effectiveness tends to a maximum value as N_{TU} becomes large.

**Figure 3. Typical relationship between E and NTU for various values of R (Based on Single E-shell with any even number of tube passes).**

Users of the effectiveness-N_{TU} technique are not required to calculate the log mean temperature difference when carrying out design or rating calculations. For design calculations, E and R can be calculated from the mass flowrates, specific heat capacities and inlet and outlet temperatures. The value of N_{TU} can be read from the graph for the chosen design of heat exchanger and used to calculate the required surface area. In rating calculations, the surface area, mass flowrates and specific heat capacities can be used to calculate R and N_{TU}. The value of E can be read from the graph and used to calculate the rate of heat transfer in the heat exchanger. It can therefore be argued that the effectiveness-N_{TU} method can be used for rating calculations without the need for an iteration. In reality, since heat transfer coefficient will change with temperature, it is still likely that an iterative calculation will be required.

Effectiveness-N_{TU} relationships are generally presented graphically and may be used to produced estimates to two significant figures. Again, because of deviations from the underlying assumptions, this level of precision in the calculations is consistent with the precision of the overall method. An attempt has been made to produce a single algebraic expression with a number of variable coefficients to represent a range of common heat exchangers, *ESDU, 93012*. Approximately 90 geometries were represented by an expression with 14 variable coefficients. The curve fitting approach matches the exact relationships to better than 2%. By incorporating the algebraic expression into a computer program, a range of geometries and designs can be readily accessed. The accuracy of the computer calculations does not, however, bring any increase in the accuracy of the overall method.

F values or effectiveness-N_{TU} relationships are obtained by making simplifying assumptions about the geometry of a heat exchanger and then carrying out a process of integration. The integration can either be carried out algebraically (most designs of shell and tube heat exchanger) or using finite element methods (most designs of crossflow heat exchanger). In all but the simplest of geometries, the process of integration is complex. For shell and tube exchangers, the resulting algebraic expressions require care in application to avoid error. For example, for the single E-shell with any even number of passes, the expression linking N_{TU} with E is

where

and

For crossflow heat exchangers, particularly with more than one pass, the finite element integration often involves iteration as well. Users of mean temperature difference techniques are therefore advised to use the graphical presentations of these integrations or curve fits to the data.

Table 1 gives references to sources of graphical data for various types of heat exchanger.

#### REFERENCES

ESDU, 85042. Effectiveness-N_{TU} relationships for the design and performance rating of two-stream heat exchangers, ESDU, Data Item, 85042, December 1985.

ESDU, 86018. Effectiveness-N_{TU} relationships for the design and performance rating of two stream heat exchangers, ESDU, Data Item, 86018, July 1986, Amended July 1991.

ESDU, 87020. Effectiveness-N_{TU} relationships for the design and performance evaluation of multi-pass crossftow heat exchangers, ESDU, Data Item 87020, October 1987, Amended November 1991.

ESDU, 88021. Effectiveness-N_{TU} relationships for the design and performance evaluation of additional shell and tube heat exchangers, ESDU, Data Item 88021, November 1988, Amended July 1991.

ESDU, 91036. Algebraic representations of effectiveness-N_{TU} relationships, ESDU, Data Item 91036, November 1991.

Kays, W and London, A. L. (1984) *Compact Heat Exchangers. Third Edition.* McGraw-Hill.

Kern, D. Q (1950) *Process Heat Transfer, First Edn.*, McGraw-Hill.

Perry (1973) *Chemical Engineers' Handbook*, McGraw-Hill.

Pignotti, A. (1984), Matrix formalism for complex heat exchangers, Trans ASME, *Journal of Heat Transfer*, Vol. 106, pp. 352-360.

Taborek, J. (1983) *Heat Exchanger Design Handbook, Section 1.5*, Hemisphere Publishing Corporation.

TEMA (1978) *Standards of Tubular Exchanger Manufacturers Association, Sixth edn.*