Frequently, in industrial processes, liquid mixtures of two or more components have to be evaporated in order to separate them from one another. It is known from experiments that heat transfer coefficients during evaporation of mixtures can be substantially smaller than those of the pure components of the mixture. On the other hand, marked improvements of heat transfer have been noted if one of the components of the mixture is surface-active. Mixtures of organic or inorganic liquids, however, contain surface-active components only in certain cases (soaps, addition of wetting agents), so that a decrease of the heat transfer coefficient is inevitable in comparison with that of the pure components.
Bonilla and Perry (1941) were the first to investigate a large number of binary mixtures of organic liquids and of water with organic liquids. Detailed discussions of the phenomena and experimental results are given by Collier (1972), van Stralen and Cole (1979) and Stephan (1992). As an example, heat transfer coefficients for ethanol-water mixtures are plotted in Figure 1 for a pressure of 1 bar and a heat flux of 10^{5} W/m^{2}. As is recognized, the heat transfer coefficients, α = , of the mixture, where is the heat flux and ΔT the difference between wall and saturation temperature, are clearly smaller than the values α_{id} that would be obtained if one were to interpolate linearly between the heat transfer coefficients of the pure components. One recognizes also a clear decrease of the heat transfer coefficient in the region, in which vapor and liquid compositions are strongly different, as is seen from a comparison with the upper curve in Figure 1.
Figure 1. Heat transfer coefficients for boiling ethanol-water mixtures. is the mole fraction of ethanol in the vapor and is the mole fraction of ethanol in the liquid.
As shown by experiments, the heat transfer decreases with the difference in concentration more strongly at high pressures than at low pressures. This can be explained by the fact that the number of vapor bubbles being formed per surface unit increases with the pressure and there is less surface available for the mass flow rate in the liquid space.
One notices several peculiarities during the boiling of oil-refrigerant-mixtures, which can be encountered frequently in the evaporators of refrigeration plants, because the refrigerant always drags along some lubricating oil from the compressor into the evaporator. According to Figure 2, depending upon the type of oil and the heat flux, small amounts of oil can lead to a slight decrease of the heat transfer coefficient, and amounts up to 3% by mass fraction can even lead to an increase, resulting in a better heat transfer than that of the pure refrigerant. In general, one can conclude that mass fractions of oil over 5% lead to large reductions in heat transfer.
Figure 2. Ratio of the heat transfer coefficients α/α_{0}. α is the heat transfer coefficient of the oil-refrigerant mixture, α_{0} that of the oil-free refrigerant R 114 at 1.285 bar.
One should, therefore, take suitable measures, such as in the use of an oil separator after the compressor, to ensure that the oil content in the evaporator is always below a mass fraction of 5%.
Investigations of the heat transfer in mixtures with more than three components remain unknown, and until now the heat transfer of only a few ternary mixtures has been measured. Basically, the findings for binary mixtures were confirmed.
Essentially two methods have proven reliable for the reproduction of heat transfer data. One starts from empirical correlations for pure substances. Such correlations usually contain dimensionless parameters, which can now be formed with the properties of the mixtures. By use of an additional term, allowance is made for the decrease of heat transfer resulting from the obstruction of bubble growth by diffusion:
In it the heat transfer coefficient α_{0} can be calculated from the equation established for pure substances (see Boiling). One must, however, enter the properties of the mixture.
A considerable amount of the reduction of heat transfer in mixtures, compared to pure substances, is conditioned by the change in the thermal properties, whereas the reduction factor f makes a comparatively small contribution. It lies between 0.8 and close to unity for most hydrocarbon mixtures and the mixtures of hydrocarbons with water. In order to avoid the lengthy computation of the property values of the mixture one prefers simple correlation procedures. Such a procedure starts from the fact that, for the transfer of a definite heat flux to the mixture, a larger wall superheat ΔT = T_{w} - T_{s} is required than for the evaporation of pure substances. The saturation temperature T_{s} in this case is the boiling temperature of the mixture with the mean composition x of the liquid. In order to calculate the superheat, an "ideal" wall superheat ΔT_{id} is defined by
in which the temperature differences ΔT_{i} between the wall and saturation temperatures result from the heat transfer coefficients α_{i} of the pure substances with the heat flux of the mixture according to ΔT_{i} = and can be calculated from pure component-equations. The actual driving temperature difference ΔT is different from the ideal one
with θ = ΔT_{E}/ΔT_{id}. The supplementary term depends mainly on the difference between vapor and liquid composition, and is always positive because of the reduction of the heat transfer in the mixture. Experiments with binary mixtures yielded the simple linear relation
where K_{12} is a positive number, which is approximately independent of the composition. One can interpret K_{12} as a binary interaction parameter that must be determined for a given mixture and pressure. In the pressure range between 1 and 10 bar, the pressure dependence of K_{12} could be reproduced approximately by the empirical equation
with p_{0} = 1 bar and p ≤ 10 bar. The value is different for every mixture, but is independent of pressure. Values for different binary mixtures are given by Stephan (1992). A mean value for for all these mixtures is approximately 1.4.
For a mixture of K components, we have instead of Eq. (4)
Here K_{ik} are interaction parameters that must be ordered in a sequence according to increasing boiling points of the pure substances.
For a ternary mixture, Eq. (6) reduces to
which contains the associated binary mixtures as limiting cases. If one assumes = = 0, then one reduces the ternary to the binary mixture of components 2 and 3, whereas for = = 0, the associated binary mixture consists of components 1 and 3. One recognizes from this that the coefficients K_{iK} with K_{iK} = 0.14 as a mean value, are identical with the values for binary mixtures. Heat transfer coefficients calculated according to this method reproduce quite well the previously known test data for ternary mixtures.
The boiling point rises in an isobaric solution of solids in liquids. The driving temperature difference between wall and saturation temperature thus decreases with the amount of the dissolved solids. The vapor consists of the pure solvent. In the vicinity of the wall, the solvent must overcome a mass transfer resistance in order to get from the liquid to the bubble surface. This impedes bubble growth, and, in comparison with the pure solvent, reduces the heat transfer, exactly as in the case of the binary and multicomponent mixtures of liquids.
Experiments with aqueous solutions of sucrose, sodium chloride, sodium hydroxide, and ammonium nitrate in the pressure range between 1 and 16 bar lead to the empirical correlation
The index L signifies liquid, G gas, and w water. The property values p_{L},c_{p}, η, λ of the solution and the surface tension σ are to be evaluated at the mean temperature T_{m} = (T_{w} + T_{s})/2 and the mass fraction x of the solution.
In a binary mixture consisting of two immiscible liquid phases and the vapor phase, the boiling temperature is clearly determined, according to the Gibbs phase rule, by prescribing the pressure. The boiling temperature of the mixture is less than that of the pure components. If, for example, one mixes water with perchloroethylene, two immiscible liquid phases are formed, whose boiling temperature at a pressure of 1 bar lies at 87.8°C, whereas pure water boils at about 100°C and pure perchloroethylene at about 121°C. If, therefore, one wishes to evaporate a liquid that would decompose at its boiling temperature, one can add an immiscible liquid or its vapor and thus lower the boiling temperature. The heat transfer to such immiscible liquids is determined to a great extent by which of the two liquid phases touches the heating surface. If, for example, a mixture of water/perchloroethylene fills a container with a horizontal heating surface at the bottom, then the lower phase contains mainly the heavier and less volatile perchloroethylene, whereas the upper phase consists primarily of water. At sufficiently large heat fluxes, either nucleate or film boiling occurs, depending upon the heat flux.
In spite of this very complicated phenomenon, the heat transfer during the boiling of immiscible liquids is determined to a great extent by the properties of the liquid at the heating surface. Measurements could be reproduced quite well as long as there was film boiling at the wall with an equation, which starts with the known equations for film boiling and allows for the influence of radiation and the convective heat transport in the case of a subcooled liquid:
a_{f} is the heat transfer coefficient for film boiling, which is calculated from
with Δh = Δh_{LG} + 0.95c_{vG}(T_{w} − T_{s}). The quantity α_{r} is the heat transfer coefficient for radiation, α_{r} = /(T_{w} − T_{s}). The heat transfer coefficient ac is obtained from one of the known equations for heat transfer during turbulent free convection in the vapor film. A subcooling factor θ = (T_{s} − T_{b}) / (T_{w} − T_{s}) plays a role, if the temperature T_{b} in the core of the liquid is below the boiling temperature. The properties are those of the phase at the heated surface. A condition for the use of these equations is that the fluid layers are thicker than 4.5 cm. The above equations are not valid if there is nucleate boiling at the wall. In this case, the heat transfer with boiling immiscible liquids has not yet been sufficiently researched.
As in the case of pure substances, during the boiling of mixtures in forced flow in tubes or channels, there is a decrease in saturation temperature because of the pressure drop along the path of flow. In addition, because the more volatile component is converted into vapor first, the liquid becomes richer in the less volatile component. In many industrial processes where the tube diameter is not very small, and, thus, the pressure drop not very large, the rise in boiling temperature, because of the increase in the less volatile component in the liquid, offsets the decrease in boiling temperature as a consequence of the pressure drop. It is thus possible that the saturation temperature even rises downstream. In a two-fluid heat exchanger, the driving temperature gradient decreases. In general, the heat transfer coefficient also decreases.
Figure 3 shows the course of the vapor temperature T_{G} in the core of an annular flow of an evaporating liquid film which is directed upward in a vertical evaporator tube. The course of the phase interface temperature T_{l} on the vapor side of the liquid film is also plotted in the figure. The mixture evaporated was ethanol/water. The concentration of the higher boiling component in the liquid, in most cases, causes an increase in viscosity, which likewise contributes to a reduction in the heat transfer coefficient.
Figure 3. Temperature T_{G} of the vapor and T_{1} at the phase interface of a boiling ethanol/water mixture with upwards directed annular flow in a vertical tube of 37 mm inner diameter. At inlet: mass flow rate = 0.1 kg/s, quality x* = 0.05, mole fraction of ethanol = 0.041. Heat flux = 2·10^{6} W/m^{2}.
In the case of boiling of pure substance convective boiling is in general governed by two mechanisms (see Boiling). In the extreme case when the heat flux is too low to support nucleate boiling, vapor is generated within the stream from minute nuclei and by evaporation on the vapor liquid interface. We then have "pure convective boiling", in most practical applications, however, heat flux and wall superheat are high enough for the onset of nucleate boiling. Nucleate and pure convective boiling are superimposed according to their relative magnitudes.
The onset of nucleate boiling (see Nucleate Boiling) is given by
where in the critical radius r_{cr} for usual drawn tube materials r_{cr} = 0.3·10^{−6} m is recommended. α_{C} is the convective heat transfer coefficient. It can be obtained from an equation for single-phase heat transfer during turbulent tube flow (see Convective Heat Transfer), for example, from
with the Reynolds Number Re = and the Prandtl Number Pr of the liquid.
Until recently two models of convective boiling were most often used, one by Chen (1963) and one by Shah (1976). Neither model presents a satisfactory solution, as attested by numerous publications. A new model based on an asymptotic addition of the two boiling components was recently presented by Steiner and Taborek (1992) and tested over 13000 data points in vertical convective boiling, mostly of pure substances. It applies to mixtures, as well. One must, however, enter the properties of the mixture. As in the case of pure substances heat fluxes must be below the critical heat flux where film boiling or dry-out of the heated surface occurs. For steam-water no dry-out will occur up to vapor qualities of x* = 0.99. The convective boiling heat transfer coefficient, α_{2Ph}, is composed of a nucleate boiling and a forced convection contribution
Here the nucleate boiling heat transfer coefficient α_{B,o} is the value at normalized conditions , for example = 20000 W/m^{2}, p_{r},o = 0.1 derived from the appropriate equation for pure substances in Nucleate Boiling
The nucleate flow boiling correction factor is given by
with p_{r} ≤ 0.95 and n = 0.8 - 0.1 exp (1.75 p_{r}) for all fluids except cryogenics. For cryogenics we have n = 0.7 - 0.13 exp (1.105 p_{r}). The residual correction f(M) is a function of the molar mass, see Figure 4. An equation representing the values of Figure 4 can be found in the paper of Steiner and Taborek (1992).
The nucleate boiling correction term α_{B,0}F_{nbf} in Equation (13) can only be used if the wall superheat or heat flux is above a certain minimum value q_{0NB} required for the onset of nucleate boiling, Equation (11).
The local convective heat transfer coefficient α_{c} in Eq. (13) can be calculated from one of the standard equations for convective heat transfer, such as Equation (12). F_{2Ph} is two-phase multiplier accounting for the enhancement of heat transfer in the liquid vapor mixture. It is a function of the vapor fraction, x* , and the ratio of liquid to vapor density, ρ_{L}/ρ_{G}, Figure 5. An equation representing the values of Figure 5 is also given in the paper of Steiner and Taborek (1992).
Figure 5. Two-phase flow multiplier F_{2Ph} as a function of vapor fraction x*. Validity restricted if is reached.
Material from Stephan, K. (1992) Heat Transfer in Condensation and Boiling, by permission of Springer Verlag GmbH.
REFERENCES
Bonilla, C. F. and Perry, C. W. (1941) Heat Transmission to Boiling Mixtures. Am. Inst. Chem. Eng. J. 37 685-705.
Chen, J. C. (1966) A Correlation for Boiling Heat Transfer to Saturated Liquids in Convective Flow, Ind. Engng Chem. Process Design Developm. 5 322-329.
Collier, J. G. (1972) Convective Boiling and Condensation, Mac Graw Hill, London.
Shah, M. M. (1976) A New Correlation for Heat Transfer During Boiling Flow Through Pipes, ASHRAE Trans., 82 66-86.
Steiner, D. and Taborek, J. (1992) Flow Boiling Heat Transfer in Vertical Tubes Correlated by an Asymptotic Model, Heat Transfer Eng. 13 43-69.
Stephan, K. (1992) Heat Transfer in Condensation and Boiling, Springer, Berlin.
van Stolen, S. and Cole, R. (1979) Boiling Phenomena, Vol. 1 and 2, Hemisphere, Washington.
Les références
- Bonilla, C. F. and Perry, C. W. (1941) Heat Transmission to Boiling Mixtures. Am. Inst. Chem. Eng. J. 37 685-705.
- Chen, J. C. (1966) A Correlation for Boiling Heat Transfer to Saturated Liquids in Convective Flow, Ind. Engng Chem. Process Design Developm. 5 322-329.
- Collier, J. G. (1972) Convective Boiling and Condensation, Mac Graw Hill, London.
- Shah, M. M. (1976) A New Correlation for Heat Transfer During Boiling Flow Through Pipes, ASHRAE Trans., 82 66-86.
- Steiner, D. and Taborek, J. (1992) Flow Boiling Heat Transfer in Vertical Tubes Correlated by an Asymptotic Model, Heat Transfer Eng. 13 43-69. DOI: 10.1080/01457639208939774
- Stephan, K. (1992) Heat Transfer in Condensation and Boiling, Springer, Berlin.
- van Stolen, S. and Cole, R. (1979) Boiling Phenomena, Vol. 1 and 2, Hemisphere, Washington.