In this situation motion of the condensate film, and hence the local film thickness and heat-transfer rate, are influenced by the surface shear stress due to the vapor flow, as well as by gravity. For a pure vapor with laminar flow of the condensate film and with vapor flow normal to the tube axis, this problem is now quite well understood. The theoretical approach for condensation on a plane surface with flow of vapor parallel to the condensing surface is relatively straightforward. The complication which arises in the case of the tube in crossflow is separation of the vapor boundary layer at some position around the tube, the point of separation being affected by the "suction" effect due to condensation.

The simplest model is that of Shekriladze and Gomelauri (1966), who used the asymptotic expression for the vapor Shear Stress at the condensate surface

where
is the local condensation mass flux, U_{δ} is the tangential velocity of the condensate surface and U'
is the tangential velocity at the outer 'edge' of the vapor boundary layer. It is sometimes implied in the literature that the shear stress on the condensate surface is attributable to two components: "dry shear" and "momentum transfer by condensation", the latter being given in Equation (1). Since the velocity of the vapor Immediately Adjacent to the condensate surface is Equal to that of the surface, no momentum is "transferred by condensation". The correct interface condition is continuity of shear stress from vapor to liquid. Equation (1) is valid for relatively high condensation rates. The Shekriladze–Gomelauri model assumes potential flow outside the vapor boundary layer, so that

where U_{∞} is the vapor approach velocity and φ is the angle measured from the forward stagnation point. Equations (1) and (2) indicate that τ_{δ} is positive for all φ, so that boundary layer separation does not occur in this simple approach.

Shekriladze and Gomelauri (1966) used Equations (1) (with the additional approximation
U' << u_{δ}
) and Equation (2), otherwise treating the condensate film as in Nusselt (1916). Numerical results for the dependence of film thickness, and hence heat-transfer rate, on position around the tube were found. As indicated by Rose (1988), for vertical vapor downflow, the mean Nusselt number for the tube given by the Shekriladze–Gomelauri model can be approximated to within 0.4% by

where Nu is Nusselt Number, Re is Reynolds Number defined as U_{∞}ρd/η, d is the diameter of the tube and ρ and η are the density and viscosity of the condensate, respectively.

The quantity F measures the relative importance of gravity and vapor shear stress and may be written

where Gr_{c} is ρΔρd^{3}/η^{2}, Δρ is the difference between condensate and vapor density, (ρ – ρ_{v}), h_{lg} is specific enthalpy of phase change and λ is thermal conductivity of condensate. (F is commonly written as Pr/Fr·H, where Pr is Prandtl Number, Fr is Froude Number and H is c_{p} ΔT/h_{1g} sometimes known as Jakob Number or phase change number and P_{p} is isobaric specific heat capacity of condensate. This is somewhat misleading since c_{p} and Pr play no part when, as in this model, the condensate flow is laminar and convection terms are neglected.)

Equation (1) underestimates the shear stress on the condensate film up to the separation point and overestimates this thereafter. From the viewpoint of calculating the mean heat transfer for the whole tube these effects tend to compensate for each other. Equation (4) is in broad agreement with experimental data (see Figure 1).

Figure 1 also shows that, for values of F greater than around 10, the vapor velocity effect is small. Notice that for F → ∞ Eq. 4 gives

i.e. the Nusselt (1916) result. It is also seen that for F less than around 0.1, gravity effects are unimportant and Equation (4) approximates to its form when the condensate flow and heat transfer are dominated by the vapor shear stress (F = 0)

Several more complete and detailed approaches have been used by various investigators and are discussed by Rose (1988). In particular, consideration has been given to:

accurate calculation of the vapor shear stress and vapor boundary layer separation by matching the shear stress on either side of the vapor-liquid interface,

direction of vapor flow in relation to that of gravity,

pressure variation in the condensate film resulting from vapor flow around the cylindrical surface.

The general conclusion, broadly supported by experimental data, is that, for the purposes of calculating the mean heat-transfer coefficient for the tube, Equation (4) seems to be generally satisfactory. There is, however, evidence that, at higher vapor velocities, Equation (4) might overestimate (but not excessively) the mean heat-transfer coefficient for steam, while underestimating (possibly more significantly) values for refrigerants when F < about 1. In the former case this may be due, in part, to the variation of pressure, and hence saturation temperature, around the tube, while the latter has been attributed to onset of turbulence in the condensate film.

#### REFERENCES

Michael, A. G., Rose, J. W. and Daniels, L. C. (1989) Forced convection condensation on a horizontal tube — experiments with vertical downflow of steam, *J. Heat Transfer*, 111, 792–797.

Nusselt, W. (1916) Die Oberflachenkondensation des Wasserdampfes, *Z. Vereines Deutsch. Ing.*, 60, 541–546, 569–575.

Rose, J. W. (1988) Fundamentals of condensation heat transfer: Laminar film condensation, *JSME Int. Journal*, Series II, 31, 3, 357–375.

Shekriladze, I. G. and Gomelauri, V. I. (1966) Theoretical study of laminar film condensation of flowing vapor, *Int. J. Heat Mass Trans.*, 9, 581–591.