Tube banks are commonly-employed design elements in heat exchangers. Both plain and finned tube banks are widely found. Tube bundles are a sub-component in shell-and-tube heat exchangers, where the flow resembles crossflow at some places, and longitudinal flow elsewhere. (The term tube *bank* is often used, in the literature, to denote a crossflow situation and *bundle* to indicate longitudinal flow, however this convention is far from universal.) The flow may be *single-phase* or *multiphase*: boilers and condensers containing tube banks find a wide range of applications in industry, in addition there may be *combustion*, e.g., in a furnace heat exchanger.

Figure 1 shows the two basic tube-bank patterns involving either a rectangular or a rhombic primitive unit. These are referred to as *in-line tube banks* and *staggered tube banks* respectively. These are characterized by crosswise and streamwise pitch-to-diameter ratio's, a and b,

where D is the cylinder diameter, s_{T} the crosswise (transverse) pitch, and s_{L} the streamwise pitch. Most commonly encountered tube banks are the in-line square (a = b), rotated square (a = 2b) and equilateral triangle (a = 2b/√3). Tube banks with the product axb <1.25^{2} are referred to as compact, while those with axb> 4 are considered widely-spaced.

**Figure 1. Schematic of (a) an in-line and (b) a staggered tube bank illustrating nomenclature, and showing location of minimum cross section.**

Analytical expressions for the flow of an ideal fluid in in-line and staggered tube banks have been derived in the form of power series [Beale (1993)]. For widely spaced banks, the pressure coefficients are similar to the sinusoidal distribution observed in single cylinders. For compact banks there are significant differences in both in-line and staggered tube banks, in the latter case two additional pressure extrema can occur at φ = ±45°, where φ is the angle measured from the front of the cylinder.

The flow of a real fluid in the body of a tube bank also resembles flow past a single cylinder, though with significant differences. (See Tubes, Crossflow over.) The flow Reynolds Number, Re, is defined by

where ρ is the fluid density, ū_{max} is the maximum bulk velocity i.e. the bulk velocity in the minimum cross section (see Figure 1), D is the cylinder diameter, and η is the fluid viscosity. At low Re, the flow is Laminar with *separation* occurring at around ρ = 90°, resulting in stable Vortices forming behind each cylinder. For staggered banks, the upstream flow is typically a maximum between the preceding tubes, so the impinging flow bifurcates at the front-leading edge of each cylinder. For in-line banks, cylinders are in a comparatively dead zone downstream of the preceding cylinder's wake, and there are two re-attachment points at around φ = ±45°.

As Re increases, vortices are shed from each cylinder in an alternate fashion. (See Vortex Shedding.) Due to the overall pressure gradient, transient motion tends to occur at higher Re than for single tubes, particularly for compact banks. The number of vortices present and the motion of the streams depends on a and b; neighboring streams may be in-phase, out-of-phase, or uncorrelated. Wake-switching is observed in staggered tube banks, while in in-line banks there is an instability in the shear-layer between the *wake* and main flow with some of the detached vortices being entrained by the free-stream flow.

As Re further increases, the wake becomes turbulent, and there is substantial free-stream turbulence, due to the influence of the preceding upstream rows (see Turbulence.) The Boundary Layer region remains laminar up to the critical Re of around 2 × 10^{5}, at which point transition to turbulence occurs and the separation point moves downstream. The engineer should be aware that both vortex-shedding and turbulent buffeting can induce vibrations in tube banks. (See Vibration in Heat Exchangers.)

The most widely adopted reference bulk velocity for use in Eq. (2) is the bulk interstitial velocity which occurs in the minimum cross section, i.e., the maximum bulk velocity, ū_{max} [Bergelin et al (1950, 1958), Žukauskas et al. (1988), ESDU (1974)]. Some authors, e.g., Kays and London (1984), base Re on a hydraulic radius. The Engineering Sciences Data Unit, ESDU (1979), base Re on the so-called mean superficial velocity, ū_{mean}, i. e., the average velocity that would occur if the tubes were removed, see Figure 1. The mean superficial velocities is useful when simulating large-scale flow in heat exchangers. It can readily be shown that

for all in-line banks, and staggered banks with a < 2b^{2} – 1/2. For some compact staggered banks, where a > 2b^{2} – 1/2, the minimum cross section occurs across the diagonal and so

The hydrodynamic parameter of most interest to the heat exchanger designer is the overall pressure loss coefficient, often expressed in terms of an Euler Number Eu,

where
is the mean pressure drop across a single row. Numerous alternative definitions abound in the literature, for instance, Bergelin et al. (1950, 1958) define a Friction Factor f = 4Eu. Others simply use the symbol f to denote Eu. The friction factor of Kays and London (1984) is such that f = (a – 1)Eu/π for large banks. ESDU (1979) use a different definition for pressure loss coefficient, again, based on Ū_{mean}, i.e. there has been little effort to-date to standardize parameters.

Many experimental studies have been conducted on flow in tube banks, starting in the early part of this century. Comprehensive data were gathered by workers at the University of Delaware in the 40s and 50s. Their data were primarily in the low and intermediate range of Re. These were summarized in two reports, Bergelin et al. (1950, 1958), containing original data, and several papers [see ESDU (1974), for a concise bibliography of experimental data].

A group at the Institute of Physical and Technical Problems of Energetics of the Lithuanian Academy of Sciences, have also published a large number of papers on both flow and heat transfer in tube banks over a wide Re range. The book by Žukauskas et al. (1988) contains detailed discussions on numerous aspects of the subject, while the article by Žukauskas (1987) is a substantially shorter, but comprehensive review of fluid flow and heat transfer in tube banks. Charts of Eu vs. Re are provided for in-line and staggered banks at various pitch ratios. Achenbach has conducted research on tube banks in the high Re turbulent flow regime [for references, see Žukauskas et al. (1987)].

Various empirical correlations of pressure drop data have been devised over the years. The Eu vs. Re correlations of the Lithuanian group are commendable; they have been reconciled with numerous sources of externally-gathered experimental data, in addition to data gathered by the authors themselves. They are included here. Others such as those based on the Delaware groups work, could have been reproduced equally well, having formed an integral part of the thermal design of shell-and-tube exchangers, in the West for many years; see the articles by Taborek in the Heat Exchanger Design Handbook (1983) and Mueller in the *Handbook of Heat Transfer* [Rohsenhow and Hartnett (1972)].

Figures 2 and 3 show Eu vs. Re (based on ū_{max}) for in-line square and equilateral triangle tube banks. The *Heat Exchanger Design Handbook* (1983) contains analytical expressions approximating these curves. These take the form of a power series,

for all in-line and rotated square banks, except in-line banks with b = 2.5, for which

Values of the coefficients c_{j} for in-line and staggered tube banks are given in Table 1 and Table 2, respectively. The reader is cautioned that these equations render poor continuity across certain ranges of application. Other mathematical correlations also exist, for example, ESDU (1979). Agreement between ESDU/Delaware, and the Lithuanian group's curves is not particularly good, especially in the low-intermediate Re range [Beale (1993)].

**Figure 2. Pressure drop coefficient vs. Reynolds number for in-line tube banks. From Heat Exchanger Design Handbook (1983).**

**Figure 3. Pressure drop coefficient vs. Reynolds number for staggered tube banks. From Heat Exchanger Design Handbook (1983).**

**Table 1. Coefficients, c _{i}, for use in Equations (7) and (8) to generate pressure drop coefficients for in-line square banks. From Heat Exchanger Design Handbook (1983)**

**Table 2. Coefficients, c _{i}, for use in Eq. (7) to generate pressure drop coefficients for equilateral triangle banks. From Heat Exchanger Design Handbook (1983)**

In most practical tube banks it is necessary to modify Eu for several effects. This usually done as follows:

where Eu' is the value calculated from the correlation for an ideal bank, and the k_{i} are correction factors. Corrections are typically required to account for the geometry, size, and location of the tube bank, deviations from normal incidence of the working fluid, and variations in the fluid properties due to temperature and pressure changes. These are detailed below.

This may be accounted for by multiplying Eu' by k_{i} as is shown in the insets to Figures 2 and 3. Mathematical expressions for k1 may be found in the *Heat Exchanger Design Handbook* (1983).

For nonisothermal conditions it is necessary to account for variations in viscosity across the boundary layer. Many authors achieve this by setting

where η_{w} is the viscosity at T_{w} and η is the viscosity at the mean bulk temperature, T_{M}, of the tube bank. Žukauskas and Ulinskas [*Heat Exchanger Design Handbook* (1983)] recommend

For many applications, the temperature effects on ρ, μ etc. are negligible, and it is sufficient to calculate these at the arithmetic mean of the inlet and exit bulk temperatures (if known). However, if the temperature change in the bank is such as to affect the fluid properties significantly, it is necessary to compute Re and Eu on a row-by-row basis.

If the number of rows, N_{row}, in the streamwise direction are sufficiently small, it is necessary to modify Eu for *entrance-length effects*. Figure 4 shows the entrance-length factor k_{3} as a function
of N_{row}.

The flow may deviate from pure crossflow in either of two ways:

The angle of attack, α, may not be 0°, i.e., the bank may be rotated at some arbitrary angle to the flow.

There may be a component in the azimuthal (longitudinal) z-direction, b ≠ 90°, sometimes referred to as inclined crossflow.

Butterworth (1978) used the analogy of fluid flow in porous media and noted the mean pressure drop to be nominally the same for *in-line* and *rotated tube banks* with a/b ≈ 1, concluding the overall pressure gradient to be independent of a. Thus the mean pressure drop may be calculated using the normal incidence correlations. The physical significance of the reference velocity ū_{max} is vague; however, it may still be calculated from the mean superficial velocity Ū_{mean} using Equation (4) or (5).
Ū_{mean} should be regarded as a local volume-averaged quantity (see Figure 5) where the volume V includes the space occupied by the cylinders

Since Re and Eu are based on velocity and pressure gradients in the α-direction, the normal component of the pressure drop is reduced by a factor

It being understood that there is also a pressure gradient in the crosswise-direction. For a/b ≠ 1 anisotropy may be a problem, but the α-direction flow-resistance can still be estimated from the normal-incidence, α = 0°, 90°, values.

For this case the flow resistance is anisotropic, i.e.,
is not in the β-direction, due to the axial (longitudinal) drag being less than the crosswise component. Figure 6 shows k_{5} vs. β, where k_{5} is the ratio of the normalized streamwise component of the pressure drop to the value if the same mass flow had been in pure crossflow.

The use of roughened and other enhanced surfaces to increase heat transfer is widespread. In addition to being deliberately employed, surface roughness tends to naturally increase when smooth tubes become fouled. When the mean height of the roughness elements, k, is sufficient, turbulence is enhanced. It is generally maintained that the influence of surface roughness is more significant in staggered banks than in-line banks. The *Heat Exchanger Design Handbook* (1983) contains some guidelines for calculating k_{5} as a function of k/D, for staggered tube banks.

Banks employing *finned tubes* are often found. Figure 7 is a schematic showing some of the more common arrangements, with examples of circular, spiral, axial and plate-fins. It is common to distinguish between so-called low-finned and high-finned tubes, according to the fin-height to diameter ratio. Fins may be straight or tapered. Correlations for Eu vs. Re, or equivalent, may be found in *The Heat Exchanger Design Handbook* (1983), Žukauskas et. al (1988), Stasiulevičius and Skrinska (1988), ESDU (1984, 1986) and elsewhere.

Tube banks seldom extend to the walls of the containing vessel. Also in staggered tube banks, alternate cylinders are missing near a wall. (Experimentalists may employ dummy half-tubes or corbels to eliminate this effect.) Near the wall the flow is faster, due to decreased resistance. An iterative procedure, similar to Wills and Johnson’s method, in shell-and-tube heat exchanger analysis [Hewitt et al. (1994)] is recommended. Let the subscripts "bank" and "bypass" refer to the main and bypass flow lanes, as illustrated in Figure 8, and suppose that

The fractional mass flow through the bank, F_{bank} is,

where

The procedure is iterative because Eu_{bank} and Eu_{bypass} are functions of
and
. The pressure drop may be calculated from either n_{bank}, n_{bypass}, or the average value,
,

Eu_{bank} is obtained using Equation (9), Figures 2 or 3, in the usual fashion. Bypass pressure drop correlations are uncommon, and often proprietary [but see ESDU (1974)]. As a rough approximation, Eu_{bypass} may be estimated from Figure 2 assuming effective values of a and b, namely, a = 2s_{bypass}/D, b = s_{L} (in-line bank) or a = 2s_{bypass}/D, b = 2s_{L}/D (staggered bank). In the latter case the effective number of rows is N_{row}/2.

Equation 14 is based on the premise that pressure variations across the inlet and outlet are insignificant, something which may or may not be the case. For this reason, the engineer should attempt to use data incorporating bypass effects, if available. The lack of such data suggest that more research is needed on this important subject.

The procedure for calculating the overall pressure drop in tube banks is as follows:

If either variations in fluid properties due to changes in temperature obtained from a heat transfer calculation (see Tube Banks, Single-phase Heat Transfer in), or bypass effects are significant, the calculation procedure are iterative.

Numerical studies have gained popularity in recent years. It is convenient to differentiate between detailed calculations of flow within the passages of tube banks and overall performance calculations.

Many results have now been obtained for tube banks using both finite-volume and finite-element methods. Numerical methods have been used to simulate laminar transient flow, large-eddy turbulent flow, inclined flow β ≠ 0°, and also 3D secondary flow effects in finned tubes. Beale (1993) contains a review of recent work.

Most detailed numerical simulations have been conducted for either laminar or high-speed turbulent regimes. Far less numerical data are available in the intermediate Re range, in which most heat exchangers operate, where the influence of free-stream turbulence is of paramount importance. Because of the complex nature of flow within the passages of tube banks, and the inadequacies of existing turbulence models, numerical experiments have not yet, and probably never will, render laboratory work obselete. The two activities are not, however, mutually exclusive, and many of the problems described above are readily amenable to the methods of computational fluid dynamics.

These methods range from simple automatations of earlier methods to detailed three-dimensional flow calculations using the techniques of Computational Fluid Dynamics. Overall performance predictions are still based on empirically-based correlations of pressure drop (Equations (7) and (8)) and heat transfer. However, these are embedded in a computer code which is used to predict the overall performance of the heat exchanger as a whole. Important effects such as entrance phenomena, bypassing, variable properties etc. can thus readily be accommodated. In modern heat-exchanger design, computer-based methods have already supplanted hand-calculation techniques to some extent, a trend which will doubtless continue in the future. The challenge is to incorporate the physics and engineering experience into future heat exchanger design software.

#### REFERENCES

Beale, S. B. (1993) Fluid Flow and Heat Transfer in Tube Banks, Ph.D. Thesis, University of London.

Butterworth, D. (1978) The development of a model for three-dimensional flow in tube bundles, Int. *J. Heat Mass Trans.*, 21, 253–256.

Bergelin, O. P., Colburn, A. P., and Hull, H. L. (1950) Heat transfer and pressure drop during viscous flow across unbaffled tube banks, *Engineering Experiment Station Bulletin No. 2*, University of Delaware.

Bergelin, O. P., Leighton, M. D., Lafferty, W. L., and Pigford, R. L. (1958) Heat transfer and pressure drop during viscous and turbulent flow across baffled and unbaffled tube banks, *Engineering Experiment Station Bulletin No. 4*, University of Delaware.

Engineering Sciences Data Unit (1979) Crossflow Pressure Loss over Banks of Plain Tubes in Square and Triangular Arrays including Effects of Flow Direction, ESDU Data Item No. 79034, London.

Engineering Sciences Data Unit (1980) Pressure Loss during Crossflow of Fluids with Heat Transfer over Plain Tube Banks without Baffles, ESDU Data Item No. 74040, London, 1980.

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.

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

Hewitt. G. F., Shires, G. L., and Bott, T. R. (1994) *Process Heat Transfer*, CRC Press, Boca Raton, FL.

Kays, W. M. and London, A. L. (1984) *Compact Heat Exchangers*, McGraw-Hill, New York.

Rohsenhow, W and Hartnett, J. P. (Eds.), (1973) *Handbook of Heat Transfer*, McGraw-Hill, New York. DOI: 10.1016/0017-9310(75)90148-9

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

Žukauskas, A. A. (1987) Heat transfer from tubes in crossflow, *Advances in Heat Transfer*, 18., 87–159, Academic Press.

Žukauskas, A. A., Ulinskas, R. V. and Katinas, V. (1988) *Fluid Dynamics and Flow-induced Vibrations of Tube Banks*, English-edition Editor, J. Karni, Hemisphere, New York.