The heat transfer during a boiling process is governed by the convective heat transfer equation;
Q = h A Δt
Where, Q is the heat flow rate, h is the boiling film coefficient, A is the surface area and Δt is the temperature differential between the heating surface and the saturated liquid; Δt = ts – tsat.
The boiling is a phase change process; the latent heat of fluid is absorbed and it involves changes in density, viscosity, specific heat and thermal conductivity of the fluid. The fluid behaviour is very difficult to describe and as such there is no adequate analytical solution for boiling heat transfer. Because of this aspect, most of the engineering calculations involving boiling are made from empirical relations.
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1. When evaporation takes place at the liquid-vapour interface, the heat transfer is solely due to free convection and the film coefficient follows the relation
Nu = f1(Gr) f2(Pr)
The functions f1 and f2 depend upon the geometry of the heating surface. In terms of heat flux, this relation may be recast as-
Q/A = C k/l (GrPr)m Δt …(12.32)
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The constant C and the exponent m are determined through experiments.
Since Gr = l3 βg Δt/v2 and since the exponent m is usually 1/4 for laminar flow and 1/3 for turbulent flow, the heat transfer in this regime varies with (Δt)5/4 for laminar and (Δt)5/3 for turbulent flow conditions.
Fritz correlated the data of different investigators and formulated the following simplified correlation for water boiling at atmospheric pressure in free convection in a vertical tube heated from outside.
This correlation considers the fact that the bubbles rising in a narrow space become more and more crowded and displace more liquid. Evidently the fluid changes in composition, density and velocity. With forced convection, the fraction of evaporation is small and the formation and motion of bubbles do not significantly disturb the turbulent flow pattern. The situation then corresponds to forced convection of a single-phase liquid.
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With water as the boiling liquid, an approximate correlation is of the form-
Nu = 0.028 (Re)0.8 (Pr)0.4 …(12.34)
2. The nucleate boiling regime is of great engineering importance because of the very high heat fluxes possible with moderate temperature differences.
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The experimental data in the nucleate boiling regime is correlated by modifying the basic expression, Nu = f1(Gr) f2(Pr), where f1 and f2 are the appropriate functions. The Reynolds number is replaced by a modulus significant of the agitation (mixing and turbulent motion) of the fluid particles in nucleate boiling. Such a dimensionless modulus is defined by the relation.
Based on extensive experimental data, Rosenhow has developed the following empirical expression for nucleate pool boiling.
The subscripts f and g refer to fluid (liquid) and vapour (gas) states respectively. The surface-fluid constant Csf depends upon particular combination of the fluid and heating surface involved in the boiling situation and is a function of the surface roughness (number of nucleating sites) and of the angle of contact between the heating surface. For example Csf = 0.013 for water-copper combination and Csf = 0.006 for water-brass combination.
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3. Under the peak heat flux conditions, the vapour coming out of the vapour film is thought to be in the form of pulsating jet or vortex sheet. The relative velocity at the interface produces an instability that causes the jet to break into spheres. Taking this stipulation into account, Zuber has proposed the following relation for the maximum heat flux in nucleate boiling.
It may be observed that the peak heat flux is independent of the heating element.
Another correlation for maximum heat flux, as suggested by Rosenhow and Griffth is of the form-
Equation 12.43 is applicable for water and a variety of organic liquids.
4. The minimum heat flux during film boiling can be calculated from the following expression suggested by Zuber –
Heat transfer during stable film boiling is due to both heat conduction and radiation from the heating surface through the vapour film. Stable film boiling on the surface of horizontal tubes and vertical plates has been studied both analytically and experimental by Bromley and his correlations for the convective and radiative film coefficients are-
D is the outside diameter of the tube, and the vapour properties are taken at the mean film temperature, tf = (ts + tsat)/ 2
Example 1:
An electric wire of 1.25 mm diameter and 250 mm long is laid horizontally and submerged in water at 7 bar. The wire has an applied voltage of 2.2 V and carries a current of 130 amperes. If the surface of the wire is maintained at 200°C, make calculations for the heat flux and boiling heat transfer coefficient.
Solution:
Electrical energy input to wire,
Example 2:
A 0.10 cm diameter and 15 cm long wire has been laid horizontally and submerged in water at atmospheric pressure. The wire has a steady state voltage drop of 14.5 V and a current of 42.5 A. Determine the heat flux and the excess temperature of the wire.
The following equation applies for water boiling on a horizontal submerged surface:
h = 1.54 (Q/A)0.75 = 5.58 (Δt)3 W/m2-K where Q/A is the heat flux rate in W/m2 and At is the temperature difference between surface and saturation.
Solution:
Electrical input to wire,
