Boiling is most often understood as a phase transition from a liquid to a vapor state involving the appearance of vapor bubbles on a hot surface. In this respect, forced convection boiling and pool boiling have much in common. However, forced convection imparts a number of specific features to the conditions of bubble production and breakaway into the bulk of the liquid. The structures of vapor-liquid mixtures resulting from boiling and mixing of liquid and vapor phases also differ appreciably from each other.

Forced convection intensifies these processes compared to free motion accompanying pool boiling. Figure 1 shows variation of the bulk flow temperature
and the wall temperature T_{w} in a heated channel with boiling heat transfer when a subcooled liquid with
, where T_{s} is the saturation temperature, is supplied to the inlet. It also shows the change of the basic flow regimes along vapor-generation channel.

**Figure 1. Regions of heat transfer in forced convective bailing (vertical tube shown horizontally for diagramatic purposes).**

T_{w} is lower than T_{s} in the zone AB. Therefore, an ordinary convective heat transfer occurs between the wall and the liquid in this zone. The same is observed at the section BB', where the wall superheat ΔT = T_{w} – T_{s} is insufficient to activate nucleation centers. The first vapor bubbles appear on the wall at the point B'. The degree of the wall overheating needed for incipience of boiling depends on local values of heat flux density
, mass velocity of liquid in
and its subcooling
. We note that despite overheating of the liquid layers near the hot wall, the bulk flow temperature
at the point B' remains lower that T_{s}. As a result the so-called "surface boiling" or "subcooled boiling" is observed.

The subcooled boiling zone extends up to point C where
becomes equal to T_{s}, and the vapor quality of the flow x = 0. Here, x = (h – h_{L})/h_{LG}, where h is the flow enthalpy, h_{L} the saturated liquid enthalpy on the saturation line, and h_{LG} the latent heat of vaporization. The zone of saturated liquid boiling follows, where
and x > 0. Initially, the vapor bubbles in subcooled boiling (B'C) do not break away from the wall or slip along it. At the condition of net vapor generation, bubbles leave the wall and are condensed in the flow of subcooled liquid; after this point, an ever increasing quantity of vapor is accumulated in the flow core. Eventually (for x > 0) the condensation process ceases.

In the neighborhood of point D, where the fraction of the channel cross section occupied by vapor is fairly large, an annular flow arises with the liquid film flowing by the channel wall and a vapor core being in the center (see Annular Flow). Within this regime, heat transfer occurs directly from the wall to the interface and, eventually, this process becomes so efficient that the wail temperature is insufficient to sustain nucleate boiling which is then suppressed. As the vapor-liquid mixture flows to still higher vapor quality region the quantity of liquid in the flow decreases and at a certain boundary vapor quality x_{h} (point E) wall dryout sets in, i.e., there is no longer any liquid-to-heat-transfer-surface contact, and the wall temperature rises. A transition occurs to dispersed, or the fog-type, flow of the mixture (zone EG).

The description of change of regimes with the growth of vapor quality x set forth above is slightly simplified. Actually the region B'E covers the bubbly (1) slug (2) churn (3) and annular-dispersed (4) flow regimes in a vertical channel (depicted in Figure 2b) and bubbly (1), plug (2), stratified (3), wave (4), slug (5), and annular-dispersed (6) flows in horizontal channels (Figure 2a). The wider variety of regimes in horizontal channels is due to gravity accounting for flow stratification.

For the most part, the above flow regimes are also observed in the channels of more complex geometry, such as annular and curvilinear channels and assemblies of fuel rods. Determination of the most probable flow regime can be conveniently done using the so-called regime charts (see Gas-Liquid Flow). The most commonly encountered ones are the Baker and Taitel-Dukler diagrams for horizontal flows and the Hewitt-Roberts and Oshinowo-Charles diagrams for upward and downward vertical flows, respectively. It should be noted that the regime can be appreciably affected by the conditions of mixture injection to the channel and the presence and intensity of heat input on the wall. Therefore, the diagrams cannot be considered as universal and can be used only tentatively.

Heat transfer in forced flow boiling is determined by both transfer of heat accumulated by vapor in the bubbles being broken off and by liquid convection.

Universal design formulas fitting all the regimes are not available for heat transfer. Commonly, individual relations are used for each region. Thus, for subcooled boiling the heat flux removed from the hot wall is represented as a sum of two components, viz., a convective flux and that of boiling with convective motion

The component
, where α_{L}, the liquid heat transfer coefficient, is calculated by the formulas for single-phase convection heat transfer (see Forced Convection). The component
, where α_{cb} is the heat transfer coefficient that is used to determine the nucleate boiling flux
. It approaches the heat transfer coefficient in pool boiling α_{pb} calculated with respect to
rather than the total heat flux q. It is commonly assumed that α_{cb} = (0.7 – 0.8)α_{pb}, where α_{pb} = Aq_{cb}p^{m} (see Pool Boiling).

In the subcooled boiling region (T_{b} < T_{s}) the combined effect of nucleate boiling and forced convection is as illustrated in Figure 3 The dependence of heat transfer coefficient on the velocity of the liquid U_{L} with a single-phase convection without boiling α = α_{L} is plotted as straight line 3. As the heat flux density grows,
heat transfer enhances and the dependences for a shift upward. It is also obvious that at low mixture velocities U_{L} has only a small effect on heat transfer, and curves 1 and 2 run horizontally with α close to the appropriate heat transfer coefficients in pool boiling α_{pb}. Conversely, at high U_{L} its effect on α turns out to be determining and curves 1 and 2 approach straight line 3.

**Figure 3. Variation of heat transfer coefficient with velocity and heat flux in the subcooled boiling region.**

Quantitatively the heat transfer coefficient α_{cb} in the region of joint effect of nucleate boiling and forced convection is well described by Kutateladze's formula

where, as before, α_{cb} = (0.7 – 0.8) α_{pb}.

In the developed boiling region use is often made of the approach formulated by Rohsenow and extended by Chen

where α_{mic} = α_{pb}S_{c} determines the contribution of microconvection or nucleate boiling and α_{mac} = α_{L}F_{c} determines the contribution made by macroconvection or forced convection.

Chen presents the curves for S_{c} and F_{c} which best describe experimental data of a great many researchers for water and some organic liquids. These curves are presented in Figure 4, where Fc is graphed as a function of Martinelli's parameter X

where (dp/dz)_{L} and (dp/dz)_{G} are the frictional pressure gradients for the liquid and gas phases, respectively, flowing alone in the channel. The Reynolds number Re_{L}, which is needed for the calculation, is determined from the liquid flow rate.

The experimental data are within the dashed region in Figure 4.

The factors such as material of the heat transfer surface, its roughness, contact angle (see Contact Angle), and fouling layer thickness on the heat transfer surface may exert an additional effect on the heat transfer coefficient α_{tp}.

When suppresion of nucleate boiling is observed (i.e., heat from the heating wall is transmitted by heat conduction through a thin liquid film to the interface, where vaporization occurs), the heat transfer coefficient α_{tp} can be determined by

where α_{L} is calculated from the liquid phase velocity. Alternatively, the value of α_{tp} can be calculated from Equation (3) with α_{mic} = 0.

When the film is thin and when heat transfer through the film is governed by conduction rather than by turbulent convection, α_{tp} can be calculated from:

where λ_{L} is the thermal conductivity of liquid phase and δ_{L} the thickness of liquid film (see Annular Flow). This approach yields good results in estimating heat transfer in boiling of molten metals (high λ_{L}).

In the dispersed flow regime (zone FG in Figure 1) heat from the heating wall is first transmitted to the vapor and then from the vapor to the evaporating droplets. Within the framework of this two-stage process, the thermal resistance is mainly concentrated in transmission of heat to the vapor, and the heat transfer coefficient in pipes and channels is calculated by *Miropolskii's formula*

where Nu_{G} = α_{tp}d/λ_{G},
, ρ_{L} and ρ_{G} are the densities of liquid and vapor phase, respectively, η_{G} and λ_{G} the dynamic viscosity and thermal conductivity of the vapor phase and d the channel diameter.

As was noted above, at vapor quality x = x_{b} (Figure 1) the wall is dried out, which involves drastic deterioration of heat transfer and elevation of temperature of heat release surface. At sufficiently high heat flux densities burnout (transition heat flux) may also occur in other sections of the vapor-generating channel (zone B' C D E in Figure 1). However, the nature of burnout turns out to be different. It will be associated not with the drying of the wall liquid film, but with coagulation of vapor bubbles into a continuous vapor film separating the wall from the main flow (see Burnout (forced convection)).

The overall pressure drop in the channel, where a two-phase mixture flows Δp_{tp}, is the sum of three components (see Pressure Drop, Two-Phase Flow).

Here Δp_{f} is the pressure loss due to friction, Δp_{ac} the component due to the flow acceleration (of liquid and vapor phases) owing to the change of vapor quality, pressure or change of the flow cross section of the channel, Δp_{h} the pressure drop brought about by overcoming the hydrostatic pressure.

Either the homogeneous model or the separated flow model is used most often to describe two-phase flows and calculate pressure losses. In the first model the two-phase flow is treated as a homogeneous medium with averaged parameters (the velocity of the gaseous phase uG and of the liquid uL are equal, 1/ρ_{H} = x/ρ_{G} + 1 – x/ρ_{L}). The most effective description is furnished by the homogeneous model for bubbly and dispersed flow regimes which are characterized by a fairly uniform distribution of the dispersed phase in a carrier medium (the liquid or vapor flow, respectively). This model is also efficient for high pressures when the densities of liquid and vapor phases approach each other.

The separated flow model allowing for the difference in phase velocities, and their force, and energy interaction are most efficient in describing flows with extended interface boundaries, viz., stratified (in a horizontal channel), annular, annular-dispersed, wave, and other flows.

The components Δp_{ac} and Δp_{h} for the homogeneous flow model are calculated by integrating the equations:

and

where θ is the angle of inclination of the channel axis to the horizontal. Δp_{f} is obtained for the homogeneous model by integrating:

where the resistance coefficient .

The presence of bubbles at the wall causes an increase in friction which is taken into account using the factor ψ. Rather cumbersome calculation formulas or the corresponding graphs are available for determining ψ. Generally ψ is a function of , pressure, and geometric characteristics of the channel.

For a discussion of the separated flow model, see the article Pressure Drop, Two-Phase Flow

Boiling of liquid in channels is often accompanied by fluctuations of flow characteristics, which in engineering practice leads untimely to failure of equipment and significantly hampers its running. Fluctuations of parameters are inherent in some regimes, e.g., slug and plug ones, by virtue of their nature; in other cases they are an undesirable side effect which is to be suppressed (see Flow Instabilities). Static and dynamic instabilities of channels with boiling heat transfer agent are distinguished. The static (Ledinegg) instability is related to the fact that, at a constant input of thermal power, the same pressure drop in a vapor-generating channel may correspond to different combinations of flow rate and vapor quality. This may lead to a spontaneous reduction of flow rate of the liquid, growth of vapor quality, and development of off-design, often emergency, thermal conditions in the channel.

Dynamic instability most frequently manifests itself as density waves in which the channel exhibits pulsations of flow velocity with a certain frequency. This is due to the fact that the response of the system to variation of inlet parameters comes with a certain phase shift. The appearance of density waves is also promoted by pulsating heat release.

The pulsations can also be caused by compressed volumes (e.g., reservoirs filled with gas, vapor, or vapor-liquid mixture) as a result of acoustic effects. Stabilization of parameters in vapor-generating channels should receive primary consideration.

The behavior of vapor-generating channels in unsteady regimes, particularly following drastic increases of heat flux, has been inadequately investigated. In this case, the fluid dynamics, the heat transfer coefficient, and the conditions for achieving burnout may differ, sometimes substantially, from those implemented under steady regimes.

#### REFERENCES

Butterworth, D., and Hewitt, G. Eds. (1977) *Two-Phase Flow and Heat Transfer*, Oxford Univ. Press.

Collier, J. G. and Thome, J. (1994) *Forced Convective Boiling and Condensation* (3rd edn.) Oxford University Press, Oxford.

#### References

- Butterworth, D., and Hewitt, G. Eds. (1977)
*Two-Phase Flow and Heat Transfer*, Oxford Univ. Press. - Collier, J. G. and Thome, J. (1994)
*Forced Convective Boiling and Condensation*(3rd edn.) Oxford University Press, Oxford.