Vaporization, absolute rate

G = Gas specific gravity = mol. wt./29 Pi = Valve inlet pressure, psia AP = Pressure drop across valve, psi Q = Gas flow rate, SCFH Qs = Steam or vapor flow rate, Ib/hr T = Absolute temperature of gas at inlet, °R T5I1 = Degrees of superheat, °F... 

By comparing Eqs. (71) and (72) to the non-phase-change equations in Section II,A,2, it can be seen that the only additional parameters to be evaluated are rv and rcl, the absolute rates of vaporization and condensation at the gas-liquid interface. The methods for evaluating all parameters in these model equations are given in Section III,D,2. 

The absolute rates of vaporization and condensation are evaluated by using the rate expressions discussed in Section III,B. The net rate of phase change at the bubble interface or equivalently the rate of bubble growth, has been widely studied for single bubbles in stationary systems. Bankoff (B2) has reviewed the results of these studies. Ruckenstein (R2) has analyzed bubble growth in flowing systems. 

The absolute rates of vaporization and condensation are evaluated from the rate expressions given in Section III,B. In the past, the rate of mass transfer (which is the net rate of phase change) has not been calculated from an understanding of the physics of the phase-change process at the interface. The rate is generally evaluated by applying some simplifying assumptions to the process, rather than from an expression in terms of the dependent variables of the model equations. 

From the design viewpoint, Eq. (78) could be coupled with Eq. (71) to obtain an approximation of the system performance and if the liquid temperature profile can be estimated, the same procedure can be followed with Eq. (80). However, in general the design engineer needs to use analytical expressions for the absolute rates of vaporization and condensation, so that with a knowledge of the rate terms and the other parameters, Eqs. (71) and (72) could be solved for the temperature and mass flow-rate profiles. 

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... 

Ec is a condensation coefficient. Schrage (S2) has shown that the absolute rate of vaporization is given by... 

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 validity of the second assumption has been examined by Maa (M2), who postulated that the liquid surface temperature and not the bulk liquid temperature determines the absolute rate of vaporization. The rate expression using the surface temperature is... 

Fnel when the exposure time is greater than 10 5 sec, and the gas phase is saturated. From this discussion it can be concluded that the net rate of phase change, that is, the rate of mass transfer, can be calculated from either Eq. (65) or Eq. (66) with an equal degree of accuracy. However, the absolute rates of vaporization and condensation can only be calculated from the rate expressions based on the molecular interchange process. It is shown later in this chapter that the absolute rates must be determined to describe accurately the energy transfer that accompanies phase change. 

Tuning can be troublesome with the vapor inlet scheme if flow across the valve changes from noncritical to critical upon reboiler turndown (67, 68, 362). As boilup falls, so does the absolute pressure downstream of the valve. When the ratio of upstream to downstream pressure exceeds a critical value, critical flow is established through the valve, and the downstream pressure ceases to affect the vapor flow rate. The controller dynamics differ under critical and noncritical flow. A loop tuned for noncritical flow tends to be unstable when flow becomes critical, while a loop tuned for critical flow tends to be sluggish when flow becomes noncritical (67, 68). 

There is an essential difference between the decomposition rates expressed by the quantities J and k. Unlike J, which does not depend on the particle size, k is inversely proportional to the initial dimensions of the particle. For pro = 1 (e.g., for p = 2,000 kg m and rg = 0.5 mm = 5 x 10 m), the rates J and k are numerically equal. The difference between these rates increases proportionately with increasing size and density of the particles. Equation 3.32 permits conversion from relative values of the rate constants k expressed in per second to the absolute rates J in units of kg m s. This opens up an attractive possibility for the interpretation of data obtained by traditional measurement of the a—t kinetic curves in terms of the Langmuir vaporization equations. 

Third, these equations permit the calculation of the absolute rates of a process, a possibility that had been believed unrealizable before their first application in 1981 to the kinetics of solid decomposition [25], The interest in theories of the transition state and of the activated complex was primarily stimulated by the possibility of calculating absolute reaction rates, although the attempts to use them in studies of heterogeneous processes met with only limited success [1, 2]. In contrast, the first comparison of theoretical with experimental values of the A parameters performed within the framework of Langmuir vaporization equations was much more successful [25]. 

As follows from the consideration of the third-law method (Chapter4), its use for determining the reaction enthalpy requires estimation of the equivalent pressure Peqp of the gaseous product under the conditions of free-surface vaporization of the reactant. This, in turn, involves determination of the absolute rate of decomposition J (kg m and, hence, of the effective surface area of the decomposing sample. This problem, as applied to crystals, powders, and melts, is discussed below. 

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