Rate of condensation

Not all molecules striking a surface necessarily condense, and Z in Eq. VII-2 gives an upper limit to the rate of condensation and hence to the rate of evaporation. Alternatively, actual measurement of the evaporation rate gives, through Eq. VII-2, an effective vapor pressure Pe that may be less than the actual vapor pressure P. The ratio Pe/P is called the vaporization coefficient a. As a perhaps extreme example, a is only 8.3 X 10" for (111) surfaces of arsenic [11]. 

The derivation that follows is essentially that given by Langmuir [9] in 1918, in which one writes separately the rates of evaporation and of condensation. The surface is assumed to consist of a certain number of sites S of which S are occupied and Sq = S - S arc free. The rate of evaporation is taken to be proportional to 5, or equal tokiSi, and the rate of condensation proportional to the bare surface So and to the gas pressure, or equal to k PSo. At equilibrium. 

If the fraction of sites occupied is 0, and the fraction of bare sites is 0q (so that 00 + 1 = 0 then the rate of condensation on unit area of surface is OikOo where p is the pressure and k is a constant given by the kinetic theory of gases (k = jL/(MRT) ) a, is the condensation coefficient, i.e. the fraction of incident molecules which actually condense on a surface. The evaporation of an adsorbed molecule from the surface is essentially an activated process in which the energy of activation may be equated to the isosteric heat of adsorption 4,. The rate of evaporation from unit area of surface is therefore equal to... 

When extended to the second layer, the Langmuir mechanism requires that the rate of condensation of molecules from the gas phase on to molecules already adsorbed in the first layer, shall be equal to the rate of evaporation from the second layer, i.e. 

Secondary smoke is produced mosdy by the condensation of water in humid or cold air. The presence of hydrogen chloride or hydrogen fluoride in the combustion products increases the extent and rate of condensation. Composition modifications to reduce primary smoke may reduce secondary smoke to some extent, but complete elimination is unlikely. The relatively small amount of smoke produced in gun firings by modem nitrocellulose propellants, although undesirable, is acceptable (102—109). 

Because of their very low boiling points, helium, neon, and hydrogen are noncondensable under the conditions at the top of the nitrogen column, and they concentrate in the nitrogen gas there. Because they cut down on the rate of condensation of nitrogen and thereby reduce the thermal efficiency of the process, they must be withdrawn. The noncondensable stream withdrawn may have a neon, helium, or hydrogen content that varies from 1 to 12%... 

Zinc dust is smaller in particle size and spherical in shape, whereas zinc powder is coarser in size and irregular in shape. The particle size of zinc dust, important in some appHcations, is controUed by adjusting the rate of condensation. Rapid cooling produces fine dust, slower condensation coarse dust. In the case of zinc powder, changes in the atomization parameters can be employed to change particle size to some degree. The particle size distributions for commercial zinc powders range from 44 to 841 p.m (325—20 mesh). The purity of zinc powders is 98—99.6%. 

The Reynolds number of the condensate film (falling film) is 4r/ I, where F is the weight rate of flow (loading rate) of condensate per unit perimeter kg/(s m) [lb/(h ft)]. The thickness of the condensate film for Reynolds number less than 2100 is (SflF/p g). ... 

The efficiencies of the modified machines are 2%-4% higher than those achieved with comparable traditional machines. Higher efficiencies enhance the rate of condensate extraction. In a plant the size of Empress, this could result in 100,000 to 200,000 worth of additional production per year. 

Steam traps must be sized to cope with the maximum rate of condensation (which may be on start-up) but must perform effectively over the whole operational range, minimizing the escape of live steam. 

All of us are familiar with the process of vaporization, in which a liquid is converted to a gas, commonly referred to as a vapor. In an open container, evaporation continues until all the liquid is gone. If the container is closed, the situation is quite different. At first, the movement of molecules is primarily in one direction, from liquid to vapor. Here, however, the vapor molecules cannot escape from the container. Some of them collide with the surface and reenter the liquid. As time passes and the concentration of molecules in the vapor increases, so does the rate of condensation. When the rate of condensation becomes equal to the rate of vaporization, the liquid and vapor are in a state of dynamic equilibrium ... 

Figure 9-3 shows this schematically. If the partial pressure of the vapor is less than the equilibrium value (as in Figure 9-3A), the rate of evaporation exceeds the rate of condensation until the partial pressure of the vapor equals the equilibrium vapor pressure. If we inject an excess of vapor into the bottle (as in Figure 9-3Q, condensation will proceed faster than evaporation until the excess of vapor has condensed. The equilibrium vapor pressure corresponds to that concentration of water vapor at which condensation and evaporation occur at exactly the same rate (as in Figure 9-3B). At equilibrium, microscopic processes continue but in a balance that yields no macroscopic changes.

Kinetic-molecular theory provides an explanation on a molecular level for this equilibrium. Evaporation from the liquid occurs as fast moving molecules on the surface escape from the liquid. In turn, molecules in the gas phase strike the liquid and condense, As the concentration (pressure) of gas molecules builds up in the gas phase, the rate of condensation increases. Eventually, a pressure is reached where the rate of condensation and rate of evaporation just balance, and equilibrium is achieved. The equilibrium pressure is denoted by p and is known as the vapor pressure. The magnitude ofp depends upon the substance, composition of the liquid, and any two of our thermodynamic variables such as temperature and total pressure. The criteria for equilibrium that we will now derive provide the thermodynamic relationships that will help... 

Thus the mass rate of condensation on this small area of surface is ... 

In the previous discussion it has been assumed that the vapour is a pure material, such as steam or organic vapour. If it contains a proportion of non-condensable gas and is cooled below its dew point, a layer of condensate is formed on the surface with a mixture of non-condensable gas and vapour above it. The heat flow from the vapour to the surface then takes place in two ways. Firstly, sensible heat is passed to the surface because of the temperature difference. Secondly, since the concentration of vapour in the main stream is greater than that in the gas film at the condensate surface, vapour molecules diffuse to the surface and condense there, giving up their latent heat. The actual rate of condensation is then determined by the combination of these two effects, and its calculation requires a knowledge of mass transfer by diffusion, as discussed in Chapter 10. 

Rate of evaporation = rate of condensation The dynamic equilibrium between liquid water and its vapor is denoted H20(l) H20(g)... 

FIGURE 8.2 When a liquid and its vapor are in dynamic equilibrium inside a closed container, the rate of condensation is equal to the rate of evaporation. 

When the rate of evaporation equals the rate of condensation, the system is in a state of d)rnamic equilibrium. 

A vapor pressure is the pressure exerted by a gas in equilibrium with its condensed phase. When this equilibrium has been reached, the gas is saturated with that particular vapor. Notice in Table 5A that at 25 °C the atmosphere is saturated with water vapor when the partial pressure of H2 O is 23.756 torr. At this pressure, the molecular density of H2 O in the gas phase is sufficient to make the rate of condensation equal to the rate of evaporation. Any attempt to add more water molecules to the gas phase results in condensation to hold the partial pressure of H2 O fixed at 23.756 torr. 

The easiest of the colligative properties to visualize is the effect of solute molecules on the vapor pressure exerted by a liquid. In a closed system, the solvent and its vapor reach dynamic equilibrium at a partial pressure of solvent equal to the vapor pressure. At this pressure, the rate of condensation of solvent vapor equals the rate of evaporation from the liquid. 

Figure 12-11 is a molecular view of how a solute changes this liquid-vapor equilibrium of the solvent. The presence of a solute means that there are fewer solvent molecules at the surface of the solution. As a result, the rate of solvent evaporation from a solution is slower than the rate of evaporation of pure solvent. At equilibrium, the rate of condensation must be correspondingly slower than the rate of condensation for the pure solvent at equilibrium with its vapor. In other words, the vapor pressure drops when a solute is added to a liquid. A solute decreases the concentration of solvent molecules in the gas phase by reducing the rates of both evaporation and condensation. 

The heavy component must diffuse through the lighter components to reach the condensing surface. The rate of condensation will be governed by the rate of diffusion, as well as the rate of heat transfer. 

Empirical methods approximate methods, in which the resistance to heat transfer is considered to control the rate of condensation, and the mass transfer resistance is neglected. Design methods have been published by Silver (1947), Bell and Ghaly... 

When we developed the model for the stirred tank heater, we ignored the dynamics of the heating coil. Provide a slightly more realistic model which takes into consideration the flow rate of condensing steam. 

This section describes the phase change process for a single component on a molecular level, with both vaporization and condensation occurring simultaneously. Molecules escape from the liquid surface and enter the bulk vapor phase, whereas other molecules leave the bulk vapor phase by becoming attached to the liquid surface. Analytical expressions are developed for the absolute rates of condensation and vaporization in one-component systems. The net rate of phase change, which is defined as the difference between the absolute rates of vaporization and condensation, represents the rate of mass... 

An expression for the absolute rate of condensation can be developed readily if the simple kinetic theory of gases and the ideal gas law are applied (S2) ... 

The assumptions inherent in the derivation of the Hertz-Knudsen equation are (1) the vapor phase does not have a net motion (2) the bulk liquid temperature and corresponding vapor pressure determine the absolute rate of vaporization (3) the bulk vapor phase temperature and pressure determine the absolute rate of condensation (4) the gas-liquid interface is stationary and (5) the vapor phase acts as an ideal gas. The first assumption is rigorously valid only at equilibrium. For nonequilibrium conditions there will be a net motion of the vapor phase due to mass transfer across the vapor-liquid interface. The derivation of the expression for the absolute rate of condensation has been modified by Schrage (S2) to account for net motion in the vapor phase. The modified expression is... 

The BET approach is essentially an extension of the Langmuir approach. Van der Waals forces are regarded as the dominant forces, and the adsorption of all layers is regarded as physical, not chemical. One sets the rates of adsorption and desorption equal to one another, as in the Langmuir case in addition, one requires that the rates of adsorption and desorption be identical for each and every molecular layer. That is, the rate of condensation on the bare surface is equal to the rate of evaporation of molecules in the first layer. The rate of evaporation from the second layer is equal to the rate of condensation on top of the first layer, etc. One then sums over the layers to determine the total amount of adsorbed material. The derivation also assumes that the heat of adsorption of each layer other than the first is equal to the heat of condensation of the bulk adsorbate material (i.e., van der Waals forces of the adsorbent are transmitted only to the first layer). If it is assumed that a very large or effectively infinite number of layers can be adsorbed, the following result is arrived at after a number of relatively elementary mathematical operations... 

A dynamic equilibrium is a situation in which two (or more) opposing processes occur at the same rate so that no net change occurs. This is the kind of equilibrium that is established between two physical states of matter, e.g., between a liquid and its vapor, in which the rate of evaporation is equal to the rate of condensation in a closed container ... 

The electrochemical deposition of doped silica, ORMOSIL and related nanocomposite thin films at the surface of electrodes is opening a whole new field of applications. In the latter case, a potential is applied to an alkoxide solution which alters the pH on the electrode surface and thus enhances the rate of condensation.40 The resulting single-step... 

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