Forster-Zuber equation

The starting point for the Forster-Zuber theory (F4, F5, F6) is the Rayleigh equation (Rl) for a bubble growing in a liquid medium. In this 

It is assumed that the bubble is spherical, that the liquid is incompressible, that viscosity effects may be neglected, and that the usual Thompson equation is applicable during the growth period (time from the appearance of a nucleus of size Ro, Ro = 2c/(pv — p ), to some later time). By the use of these assumptions it is possible to treat the mathematical case of a moving, spherical heat transfer boundary (L2), to combine this with the 

Clausius-Clapeyron equation, and to obtain a theoretical expression for a vapor bubble growing in a superheated liquid. The equation (F5, F6) is a second-order differential equation which is so complex as to be of limited usefulness without serious modification. Fortunately, the equation becomes enormously simpler if the inertia of the liquid can be ignored during bubble growth. Forster and Zuber give a careful discussion of the physical requirements for neglecting inertia of the liquid. These are that either the bubble must be very small or the temperature of the bubble 

Theoretical rate of growth for vapor bubbles. Two arbitrary starting radii are shown (F6). 

The temperature of a bubble growing in a superheated liquid changes with the bubble size. If liquid inertia is negligible, the Forster-Zuber derivation gives the expression 

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Another Nusselt-type equation has been proposed by Forster and Zuber ... 

The correlation given by Forster and Zuber (1955) can be used to estimate pool boiling coefficients, in the absence of experimental data. Their equation can be written in the form ... 

In one respect Eq. (20) is satisfying. The exponents on the Reynolds and Prandtl numbers are roughly the same as those used for ordinary forced-convection heat transfer. The negative sign on the Prandtl-number exponent in Rohsenow s equation has seemed illogical to many scientists. The logical exponents found by Forster and Zuber... 

One objection to a Forster and Zuber assumption has been given by Zwick (Zl). Forster and Zuber state that the principal mechanism for heat transfer to a growing bubble is conduction across the film resistance. Zwick points out that heat can also flow by mass transport and that this convection should be included in the equations. 

In addition to the equations of Rohsenow and of Forster and Zuber, which are the most theoretical equations now available, there exist many equations which are simply empirical correlations. Little pretense is made that these additional equations are defensible on theoretical grounds beyond simple dimensional analysis. A few of these are given below. 

Data from a single laboratory for tubes of different diameters are needed. Tubes are thicker than the bubbles produced, but the reverse is true for wires. The diameter effects may not be the same in the two cases. The equations of Rohsenow and of Forster and Zuber predict that the geometric arrangement is of no consequence. The prediction is not proved at present. 

It is not necessary to assume the liquid film to be completely stagnant. Radial motion can be allowed for, but with some difficulty. It was noted in Sec. IIB2 that Forster and Zuber state that conduction is the chief mode of heat transfer (compared with convection due to radial motion). Eddies or motions of the liquid tangent to the bubble are neglected. The Zwick-Plesset theory likewise excludes eddies. The derivation is lengthy therefore the final typical equations are presented here without proof. 

In the limiting case of S = F = 1, this equation converts into that for saturated boiling, (4.156). The heat transfer coefficients aB for nucleate boiling in free flow were taken by Chen from an equation from Forster and Zuber [4.93]. More recent investigations [4.94], etc. showed however, that this gives somewhat inaccurate results. It therefore seems more sensible to calculate aB using one of the formulae from section 4.2.6. 

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