Tube banks are employed in a wide range of heat exchangers. The average heat flux,
, is expressed as a linear function of the mean temperature difference between the bulk of the fluid in the tube bank and the wall, according to a rate equation

where
is the rate of heat transfer,
is an overall heat transfer coefficient, and ΔT_{M} is some representative temperature difference between the tube wall and the bulk of the fluid. A is the total heat transfer area, which for plain tubes is given by

where N is the number of tubes of outer diameter D and length L. The overall heat transfer coefficient,
is nondimensionalized in terms of an average Nusselt Number,
, according to

where λ is the fluid thermal conductivity, or alternatively as an average Stanton Number,
,

where c_{p} is the fluid specific heat.

In tube banks,
is frequently correlated according to

where Reynolds number, Re, is defined by

u_{max} is the bulk velocity in the minimum cross section, ρ and η are the fluid density and viscosity respectively. The Prandtl number, Pr, is just,

where c_{p} is the specific heat, and λ is thermal conductivity. The rate of heat transfer may also be expressed in terms of an energy balance, which for single-phase heat transfer may be written,

where T_{in} and T_{out} are the bulk temperatures at the inlet and outlet of the bank, and
is the thermal capacitance of the fluid, presumed constant over the range of interest

is the total mass flow rate through all of the passages in the tube bank, and c_{p} is averaged over the range of interest.

A local heat transfer coefficient may be defined in a similar fashion to Equation 1. The local Nu is a function of a number of parameters such bank type and geometry, flow Re, pressure gradient, location within bank etc. Figure 1 shows local Nu distributions in the interior of tube banks as a function of the angle φ measured from the front of the cylinder. It can be seen that the staggered-geometry Nu-distribution is similar to the single cylinder case, with a maximum occurring at φ = 0°. For the in-line case, Nu rises to a maximum at the reattachment point around 45°. It can also be seen that heat transfer is somewhat higher for both in-line and staggered tube banks, than for single cylinders. This is due to increased free-stream turbulence (as a result of proceeding rows) and shear, due to constriction of the flow passages. For most applications the engineer is not concerned with the details of local heat transfer in tube banks and the reader is referred to Žukauskas and Ulinskas (1988) for further information on the subject.

### Experimental Data and Empirical Correlations

Two major sources of data on both fluid flow and heat transfer are the Delaware group in the USA and the group in the Republic of Lithuania. The work at the University of Delaware on overall heat transfer in tube banks has formed the basis for heat exchanger calculations for many years while the recent book by Žukauskas et al. (1988) is a comprehensive reference on heat transfer in tube banks based on many years of research. It contains extensive information on local and overall heat transfer. It appears that the Delaware data was gathered at nominally constant wall temperature, T_{w}, conditions and that of the Lithuanian group at constant wall heat flux,
. Numerous other data also exist, in addition to these two main sources.

A number of empirical correlations may also be found. The Engineering Science and Data Unit correlate
vs. Re according to Equation (5) [ESDU (1978)]. Values of the coefficients c and m are given in Table 1 Re is based on u_{max} (see Tube Banks, Cross Flow Over.) The correlation of Žukauskas and Ulinskas (1988) agrees well with experimental data, but exhibits poor continuity across certain Re ranges. Several other correlations may be found in the literature. Agreement among heat transfer correlations is generally better than for pressure drop correlations. In most correlations,
is treated as being primarily a function of Re and configuration (i.e., the pitch-to-diameter ratios are often treated as being relatively unimportant over a wide range). Similarly, the effect of thermal boundary conditions (constant T_{w} vs. constant
), though often significant, is usually simply ignored.

**Table 1. Coefficients used to calculate overall heat transfer in tube banks using Eq. (5) with n = 0.34, p = 0.26. From ESDU International plc (1973)**

Tradition is to plot
or
in the form of log-log plots with both the Pr-dependence and the property-variation being removed by defining

The heat-transfer factors j, k (unprimed) are Pr-independent. Primed values j', k' indicate that the influence of *property variations* across the boundary layer has also been taken into consideration. The Delaware group chose n = 1/3, i.e., j is the so-called *Colburn heat transfer factor*. ESDU (1973) and Žukauskas and Ulinskas (1988) propose similar, but slightly higher, values n = 0.34 and 0.36, respectively (over most of the range). Figure 2 shows j and k as a function of Re for Pr = 1. It can be seen that heat transfer is a little higher for staggered than in-line geometries at lower Re.

The influence of fluid properties is accounted for in Equation 5 by the term (Pr_{w}/Pr)^{p}, where Pr is evaluated at the mean bulk temperature, and Pr_{w} at Tw_{w}. The Delaware group use (η_{w}/η)^{p}, the matter is largely one of preference (but their value of p = 0.14 was based on the *Seider-Tate correlation* for flow inside tubes). ESDU (1973) suggest p = 0.26 regardless of whether the fluid is being heated or cooled.

The choice of reference temperature is a matter for concern since in most situations, the engineer must use a mean value for thermal design/analysis. For constant T_{w} the appropriate temperature to use in Eq. (1) is the *log-mean temperature difference*,

where ΔT_{in} = T_{in} – T_{w} and ΔT_{out} = T_{out} − T_{w} are the differences between the fluid and wall temperatures at the inlet and outlet, respectively (see also Mean Temperature Difference.) The reference temperature, T_{M}, for ennumerating property values should be T_{w} + ΔT_{LM}. For constant
the arithmetic-mean temperature should be used,

For most applications it is immaterial whether properties, ρ, η etc., are evaluated at the log-mean or the arithmetic-mean temperature, and the latter is often employed for convenience. For large numbers of banks and low values of the product Re.Pr the choice of T_{M} is quite important.

Empirical correlations are obtained from data for idealized tube banks. In practice, Nu is a function of a number of other parameters which are often accounted for by writing,

where
is the idealized-correlation-based Nusselt number, and the k-coefficients account for deviations from this situation.

One example, already mentioned, is the factor (Pr_{w}/Pr)^{p}. Other factors could include the effects of finite numbers of rows, angle of attack, or other considerations. These are discussed below.

Idealized correlations, described above, are for deep tube banks, where the number of rows, N_{row}, is large. For each of the first few rows in a tube bank, heat transfer may be substantially different (usually less) than occurs deep in the bank. Figure 3 shows the correction factor k_{1}, as a function of N_{row}.

In many situations, the flow is not one of pure crossflow, i.e., β < 90°. Heat transfer is reduced from by a factor k_{2}. Figure 4 shows k_{2} vs. β.

Two commonly-employed methods of enhancing heat transfer in tube banks are the use of rough surfaces and finned tubes (see Augmentation of Heat Transfer.)

*Rough tubes* are used to increase the ratio of heat transfer to drag by increasing turbulence. The effects appear to be more pronounced for staggered than in-line geometries. Žukauskas and Ulinskas (1988) propose the following prescription for staggered geometries:

where k is the roughness height.

External fins may be employed with gas crossflow, in order to increase the heat-transfer area (internal fins are also used sometimes). The book by Stasiulevičius and Skrinska (1988) is devoted to the subject, as is a chapter in Žukauskas et al. (1988). When using *finned tubes* in a design, the engineer must use Nu-correlations appropriate to finned tubes, and properly account for temperature variations throughout the fin (see also Extended Surface Heat Transfer.)

*Nu correlations for finned-tube banks*

A number of general purpose correlations for finned tube-banks may be found:

Gnielinski et al. in the *Heat Exchanger Design Handbook* (1983), ESDU (1984, 1986),

Stasiulevičius and Skrinska (1988), and Žukauskas and Ulinskas (1988)

These typically assume forms similar to Equation (5), but with additional parameters to account for the fin geometry. The engineer should obtain Nu correlations specific to the particular geometrical configuration under consideration, whenever possible.

When variation in temperature across the fins is significant, the rate equation, Equation (1), is rewritten as

A is the total area for heat transfer, A = A_{f} + A_{w}, where A_{f} is the total area of the fins and A_{w} that of the exposed tube-wall. E_{f} is a surface effectiveness factor,

and η_{f} is a *fin efficiency*, i.e., the ratio of the actual rate of heat transfer through the fins to that which would occur if all fin material were at constant T_{w}.

Figure 5 is a plot of hf vs. nondimensional height h*, at various diameter ratios,
. λ_{f} is the thermal conductivity of the fin, δ the fin thickness and
the average heat transfer coefficient. Primed values,
h'
and
D'
, indicate that they are adjusted for heat transfer through the fin-tip, as indicated in Figure 5. Charts of η_{f} vs. h* have been devised for a variety of geometries; the article by Gnielinski et al. in the *Heat Exchanger Design Handbook* (1983) contains a selection. These should be used in preference to Figure 5, which was obtained from the one dimensional fin equation solution,

where
h'
is calculated according to Schmidt [Stasiulevičlus and Skrinska (1988)].

Bypass streams are relatively ineffective in transferring heat, and are usually treated as adiabatic, i.e., only the main flow stream is used to compute heat transfer. Since the temperature and hence the viscosity differs from that in the main flow lanes, this must be taken into consideration when computing the bypass and main mass-flow rates.

### Methods for Calculating the Thermal Performance of Tube Banks

How to compute the performance of a tube bank will depend to a larger extent on which of the parameters,
,
, A, T_{in}, T_{out},
Ċ
etc., are prescribed or required, whether fluid properties are constant, and the precision required. It is common to differentiate between "rating" and "sizing" of heat exchangers. This section is intended to provide some guidelines for the reader, not a single general-purpose procedure. Two cases will be considered in detail: plain tubes at constant T_{w} and non-isothermal plain or finned tubes:

### Plain tubes, constant T_{w}

The log-mean temperature-difference (LMTD) approach is based on the fundamental assumption that the heat flux is directly proportional to the temperature difference between the bulk of the fluid at some interior location and the wall, i.e., the rate equation, Equation 1, with ΔT_{M} prescribed according to Equation (12).

Unless all temperatures are known, a priori, a trial-and-error method based on successive applications of the rate equation combined with a heat balance is required. An alternative approach, due to Nusselt, is to postulate the rate of heat transfer to be proportional to the difference between the inlet and wall temperatures,

The *effectiveness*, E, represents the ratio of the actual to the maximum possible heat transfer. For constant T_{w}

where the quantity A
/C is known as the **Number of Transfer Units**, NTU. The advantage of T_{in} as reference (if known) is that no iteration is required provided fluid properties are constant and upstream values can be used. Even if these vary, the E-NTU-method is often preferable, as T_{M} affects the calculation of
via property variations only. A typical procedure is as follows:

Given T_{in} guess a value for T_{out} (unless known).

Calculate Re and Pr based on property values evaluated at the arithmetic-mean bulk temperature, and Pr_{w} at T_{w}.

Compute the value of
for an ideal bank using Equation (5) applying corrections, k1, k2 etc. as discussed above, to obtain
and hence
for the actual bank, Equation (14).

Compute the overall rate of heat transfer,
, using one of two methods: either calculate ΔT_{LM} from Equation (12), and use the rate equation (1) to obtain
, or calculate E from
using Equation (20) and obtain
using Equation (19).

Compute the exit bulk temperature, T_{out} from
, by means of a heat balance, Equation (8).

Reiterate steps 2–6 based on the new value of T_{out} (unless prescribed) until satisfactory convergence is obtained.

If changes in temperature are large, break the bank up into an individual or groups of rows, and proceed in a sequential fashion from the inlet to the exit.

### Plain or finned tubes, general case

In many practical engineering applications, the tube-side fluid also undergoes significant changes in temperature. Under these circumstances, Equation (1) is no longer appropriate, and the rate of heat transfer is

where U, the Overall Heat Transfer Coefficient, based on the outer surface A (including fins) is given by,

where the subscript "bank" refers to the external bank-side crossflow, and the subscript "tube" refers to the internal tube-side longitudinal flow. The terms on the right-hand side of Equation (22) may be regarded as resistances to the flow of heat in the presence of a temperature difference. A_{i} is the total area for heat transfer on the inner tube surface, E_{f bank} and E_{f tube} are fin-surface-effectiveness correction factors for externally and internally-finned tube surfaces (equal to unity for plain tubes), and R_{f bank} and R_{f tube} are fouling factors which may be estimated using the recommendations in the Standards of the Tubular Exchanger Manufacturers Association, TEMA (1988). R_{w} is the tube-wall resistance which for plain tubes, of length L, is given by,

where D_{i} is the inner diameter, and λ_{w} is the wall thermal conductivity. For finned tubes, R_{w} is configuration-dependent.

Commonly-employed heat exchanger design methods are the LMTD-based F-correction-factor method, the *P-NTU* method and the θ method. For the F-correction method, AT_{LM}, is defined by Equation (12), but with ΔT_{in} and ΔT_{out} the temperature differences between the two working fluids at the inlet and outlet. Heat transfer is reduced by a factor F, which is a function of the capacitance ratio, R, and the thermal effectiveness, P, as defined in Table 2, which also gives with the modified rate equation used to calculate
. The *P-NTU* method presumes the thermal effectiveness, P (similar to E), to be a function of R, as well as the number of transfer units, *NTU* = AU/C_{cold}, of the cold fluid. For the third method, θ = θ(R, NTU_{cold}) but a modified energy balance, instead of a modified rate equation (see Table 2) is used to compute
. Charts of F, P, and θ may be found in standard references on heat exchangers, often in combined form [see Taborek, *Heat Exchanger Design Handbook* (1983)]. Regardless of which method is used, two
correlations are needed to calculate U: one, as per Equation (5), to calculate the bank-side crossflow heat transfer coefficient,
, the other to calculate the internal tube-side heat transfer coefficient,
(See Tubes Single-Phase Heat Transfer in). The temperatures at the solid-fluid interfaces, T_{w bank} and T_{w tube}, are obtained as a linear combination of T_{M bank} and T_{M tube}, using the ratio of the resistances in Equation (22) as weighting factors. T_{w} may be taken as being the temperature between the fluid and the fouling resistance. A typical calculation would then proceed as follows:

Given T_{in bank} and T_{in tube}, guess values for T_{om bank}, T_{out tube}, T_{w bank}, T_{w tube} (unless known).

Calculate Re and Pr for tube-side and bank-side fluids at the arithmetic-mean bulk temperatures, and values of Pr_{w} for the two fluids based on T_{w bank} and T_{w tube}.

Compute
and
using the appropriate correlations applying corrections k_{1}, k_{2} as necessary, to obtain
and
. Calculate the overall heat transfer coefficient, U, Equation (22).

Compute the overall rate of heat transfer,
: Either calculate P and R based on the guessed values and obtain F(P,R) or compute R and NTU_{c} to obtain P(R,NTU_{c}) or θ(R,NTU_{c}) from a chart. Calculate
from the formulae in Table II.

Compute T_{out bank} and T_{om tube} from
, by means of heat balances applied to both tube-side and bank-side fluids Equation 8. Hence calculate T_{w bank} and T_{w tube} (a single value, T_{w}, will often suffice).

Re-iterate steps 2–5 based on the new values of T_{out bank}, T_{out tube} (if necessary) until satisfactory convergence is obtained.

**Table 2. Design methods commonly used to calculate thermal performance in heat exchangers**

The choice of whether to use the F-correction, *P-NTU*, or θ methods will depend on the application and may result in substantial simplifications to the general procedure detailed above. If all temperatures are known, the F-correction method can be used with properties enumerated at T_{M}. Conversely, if fluid property variations are negligible and upstream values may be used, the *P-NTU* and θ methods can be used advantageously. If property variations are significant, but small enough to be considered perturbations about a mean, all the above lumped-parameter-type schemes will necessarily be iterative. For large variations in T_{M}, a numerical scheme is preferred.

While traditional methods are meritorious, the majority of heat exchangers are now designed using computer-software. Many approaches are possible; for example the *E-NTU* method may be performed over a discrete number of cells corresponding to one or more cylinders. An alternative procedure is to start from first principles, and solve pairs of equations of the form

on a cell-by-cell basis. If an upwind-difference scheme is used,

and upstream values are used to ennumerate properties, no iteration is required. Better accuracy can be achieved with a central-difference scheme

Some iteration is now necessary, though by proceeding as indicated in Figure 6, a simple Jacobi point-by-point method will converge rapidly. It is now possible to account for variations of U due to changes in T_{b} on a cell-by-cell basis. T_{w} may also be obtained, at an intermediate calculation in each cell to enumerate the dependence of U on Pr_{w}. Variations in other properties may be dealt with on a cell-by-cell basis as necessary, depending on the nature of the problem.

The rate equations defined, Equation (24), may be considered like a set of simple enthalpy-conservation equation (convection-source-term equations). It is possible to devise much more sophisticated two-phase enthalpy equations, which account for bank-side mixing, 3D phenomena etc., in addition to bulk convection. It is also possible to generate momentum equations to account for flow-related effects, bypassing, finite-number of rows, etc. For complex flows, the definition of U may be of limited use. Numerical methods differ from traditional methods in that there is no need to rely on the premise that heat transfer is governed by a rate equation. The reader is referred to the articles by D. B. Spalding in the *Heat Exchanger Design Handbook* (1983), and elsewhere. The use of spreadsheet programs, specialized heat-exchanger-design programs and general-purpose computational fluid dynamics software should be considered prior to writing source code.

Simple lumped-parameter type schemes once formed the basis for all practical heat exchanger design. The situation is changing, and the application of computer software is not only increasing in use and range of application, but is also enhancing our understanding of the physics of flow within the passages of heat exchangers.

#### REFERENCES

Engineering Sciences Data Unit (1973) Convective Heat Transfer During Crossflow of Fluids Over Plain Tube Banks, ESDU Data Item No. 7303J, London, November.

Engineering Sciences Data Unit (1984) Low-fin Staggered Tube Banks: Heat Transfer and Pressure Drop for Turbulent Single Phase Cross Flow, ESDU Data Item No. 84016, London.

Engineering Sciences Data Unit (1986) High-fin Staggered Tube Banks: Heat Transfer and Pressure Drop for Turbulent Single Phase Gas Flow, ESDU Data Item No. 86022, London, 1986.

*Heat Exchanger Design Handbook* (1983) Vol. 1–4, Hemisphere, New York.

Standards of the Tubular Exchanger Manufacturers Association, 7th edn. (1988) New York.

Stasiuleviĉius, J. and Skrinska, A. (1988) *Heat Transfer of Finned Tube Bundles in Crossflow*, 1988. A. A. Žukauskas and G. F. Hewitt (Eds.), Hemisphere, New York. DOI: 10.1016/0142-727X(88)90051-3

Žukauskas, A. A. and Ulinskas, R. V. (1988) *Heat Transfer in Tube Banks in Crossflow*, Hemisphere, New York.