Bubble curve

The parts of the CPC where the incipient phase has higher or lower density than the original phase are sometimes referred to as dew and bubble curve, respectively. 

Two bubbles, one of radius 1 cm and the other of radius 2 cm, stick together. In which direction does the interface between the two bubbles curve ... 

Fig. 3.98. Plot of contact angle 0 vs. bubble radius Rb) curve 1 - 0.32 mol dm 3 NaCl, one bubble curve...

Air unusual type of low-pressure VLE belravior is tlrat of double azeotropy, in wlriclr the dew and bubble curves are S-shaped, tlrus yielding at different compositions both a miniimim-pressure and a inaximum-pressure azeotrope. Assuming that Eq. (12.11) applies, detennine under wliat circumstances double azeotropy is likely to occur. 

In order to better demonstrate whether a system follows Raoult s law, a diagram of the phase equilibrium called T-x-y should be plotted. This plot (Figure 2) shows the equilibrium temperatures at which either a liquid solution will start bubbling (bubble curve) or a vapor mixture starts condensing (dew curve). The two systems with their experimental data and the calculation curve of the ideal solution is shown in Figure 2. In Figure 2, the system of hexane-benzene at the pressure of 101.33 kPa [10] and the system of ethylacetate-benzene [11] show negative deviations from RaoulTs law. 

Experimental data (bubble points) Experimental data (dew points) —Bubble curve (UNIFAC)... 

At high Peclet numbers, for an nth-order surface reaction withn=l/2, 1,2, Eq. (5.1.5) was tested in the entire range of the parameter ks by comparing its root with the results of numerical solution of appropriate integral equations for the surface concentration (derived in the diffusion boundary layer approximation) in the case of a translational Stokes flow past a sphere, a circular cylinder, a drop, or a bubble [166, 171, 364], The comparison results for a second-order surface reaction (n = 2) are shown in Figure 5.1 (for n = 1/2 and n = 1, the accuracy of Eq. (5.1.5) is higher than for n = 2). Curve 1 (solid line) corresponds to a second-order reaction (n = 2). One can see that, the maximum inaccuracy is observed for 0.5 < fcs/Shoo < 5.0 and does not exceed 6% for a solid sphere (curve 2), 8% for a circular cylinder (curve 3), and 12% for a spherical bubble (curve 4). 

The essential features of vapor-liquid equilibrium (VLE) behavior are demonstrated by the simplest case isothermal VLE of a binary system at a temperature below the critical temperatures ofboth pure components. Forthis case ( subcritkaT VLE), each pure component has a well-defined vapor-liquid saturation pressure ff, and VLE Is possible for the foil range of liquid and vapor compositions xt and y,. Figure 1.5-1 ffiustrates several types of behavior shown by such systems. In each case, (he upper solid curve ( bubble curve ) represents states of saturated liquid (he lower solid curve ( dew curve ) represents states of saturated vtqtor. 

Figure 1.5-la is for a system that obeys Raoutt s Law. The significant feature of a Raoult s Law system is rite linearity of the isothermal bubble curve, expressed for a binaty system as... 

Although Raonli s Law is rarely obeyed by real mixtures, it serves as a useful standard against which real VLE behavior can be compared. The dashed lines in Figs. I,5-16-1.5- e are the Rnoult s Law bubble curves preduced by (be vapor pressure FT and FJ". 

Figares 1.5-16 and 1,5-lc illustrate negative deviations from Raoult s Law the actual bubble curves lie below the Raoult s Law bubble curve. In Fig. 1.5-lb the deviations are moderate, but in Fig. 1.5-lc the deviations are so pronounced diet the system exhibits a minimum-pressure (maximum-boiling) homogeneous azeotrope. 

The liquid line in vaporrliquid equilibrium diagrams is also referred to as the liq-uidus, the bubble point curve, or simply the bubble curve. The last two names arise as follows. Consider an equimolar mixture of hexane and triethylamine at-60°C and a pressure of 0.8 bar. Based on Fig. 10.1-3, this mixture is a liquid at these conditions... 

In Figure 9.5 we see that, except near the critical point, the slope of the bubble curve has the same sign as the slope of the dew curve. We now prove that this is usually the case. First, use Table 6.3 to write the Gibbs-Duhem equation for the fugacities in each phase ... 

From the measurables T, P, x , and jcP, five common phase equilibrium problems can be contrived, depending on which quantities are known and which are unknown. For example, the problem introduced in the previous paragraph involves P and x as known and requires us to solve (11.1.1) for T and xP. When phase a is liquid and phase 3 is vapor, this problem is called a bubble-T calculation, for we are to compute a point on the bubble curve of an isobaric Txy diagram. This along with the other four common problems are listed in Table 11.1. 

The systems of Figs. 1.5-Id and 1.5-le show positive deviations ftom Raouit s Law, for which the true bubble curves lie above the Raouit s Law line. In Fig. 1.5-Id, the deviations are modest in Fig. 1.5-Ie they are large, and a maximum-pressure (minimum-boiling) homogeneous azeotrope occurs. 

Detailed reviews of this fairly complex issue exist, see I velt Sengers et ai [1], [17], [18]. Here, 1 will briefly summarize the most useful results. In [ 1 j. Fig. 22, the behavior of the P-x dew-bubble curve is shown on an isotherm just above the solvent critical temperature. It is seen to develop a sharp "birds beak", dew and bubble curve (db) having a common tangent at the solvent s critical point. It can be shown that... 

For pure fluids, it is most common to represent the saturated vapor and saturated liquid transport properties as simple polynomial functions in temperature, although polynomials in density or pressure could also be used. Exponential expansions may be preferable in the case of viscosity (Bmsh 1962 Schwen Puhl 1988). For mixtures, the analogous correlation of transport properties along dew curves or bubble curves can be similarly regressed. In the case of thermal conductivity, it is necessary to add a divergent term to account for the steep curvature due to critical enhancement as the critical point is approached. Thus, a reasonable form for a transport property. 

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