Critical expansion ratio

In the flow of a gas through a nozzle, the pressure falls from its initial value Pi to a value P2 at some point along the nozzle at first the velocity rises more rapidly than the specific volume and therefore the area required for flow decreases. For low values of the pressure ratio P2/P1, however, the velocity changes much less rapidly than the specific volume so that the area for flow must increase again. The effective area for flow presented by the nozzle must therefore pass through a minimum. It is shown that this occurs if the pressure ratio P2/P1 is less than the critical pressure ratio (usually approximately 0.5) and that the velocity at the throat is then equal to the velocity of sound. For expansion... 

The subscript s denotes an isentropic path for ideal nozzle flow. For ideal gas with Pok = constant, substitution of this isentropic expansion law into Eq. (23-98) yields the following critical pressure ratio PJP and critical flow rate Gc ... 

The analysis in [2] indicates that for a foam at hydrostatic equilibrium that is in contact with the foaming solution, Eqs. (4.15) and (4.16) cannot be employed to calculate the average expansion ratio or the critical foam height, which gives the boundary between the ability of a foam either to drain or to suck in liquid. That is so because the maximum volume of the liquid in a foam is a function of the 5-layer and actually does not depend on the whole foam column height. 

The foam column height influences strongly the drainage rate. As it was mentioned, the initial moment of liquid outflow from the foam (drainage initiation) is determined by the critical values of expansion ratio, dispersity and height of the foam. It follows from Eq. (5.60) that if two foams have equal expansion ratio and dispersity, then their initial drainage rates should not depend on foam column height. However, with the decrease in foam column... 

It should be noted that the formula about the modulus of bulk elasticity of a foam refers to deformation at both compression and expansion. At large deformations, however, their effects differ significantly. When the foam is compressed the gas volume can be reduced so that to become comparable to the liquid volume. The expansion of a foam cannot be unlimited depending on its initial expansion ratio, the volume of the foam can increase only until the border pressure reaches a critical value (see Section 6.5.2). The latter is related to foam dispersity and surfactant adsorption, and decreases with the increase in surface area. 

The main factor determining the critical pressure of a foam moving in a porous medium is the increase in the foam expansion ratio. Here again the critical pressure depends... 

Further experiments show a second crilical expansion ratio corresponding to a satu ration ratio of about eight. At higher saturation ratios, dense clouds of fine drops form, the number increasing with the supersaturation. The number of drops produced between the two critical values of the saturation ratio i.s small compared with the number produced above the second limit. 

We may observe, first of all, that no additional expansion will take place at the nozzle outlet if the pressure ratio is above the nozzle s lower critical pressure ratio, (P crin)lPot- The large size of the exhaust chamber will have no effect on the measured efficiency when... 

The tendencies for the orifice expansion factor are the same as those for the Venuui but the values are higher—at a pressure ratio of 0.6 and p equal to 0.7, T is 0.7 while To is 0.86. The critical pressure ratio for air is 0.53 at which point the flow is sonic and these equations are inapplicable. 

The self-similar solution of an unsteady rarefaction wave in a gas-vapour mixture with condensation is investigated. If the onset of condensation occurs at the saturation point, the rarefaction wave is divided into two zones, separated by a uniform region. If condensation is delayed until a fixed critical saturation ratio Xc > 1 is reached, a condensation discontinuity of the expansion type is part of the solution. Numerical simulation, using a simple relaxation model, indicates that time has to proceed over more then two decades of characteristic times of condensation before the self-similar solution can be recognized. Experimental results on heterogeneous nucleation and condensation caused by an unsteady rarefaction wave in a mixture of water vapour, nitrogen gas and chromium-K)xide nuclei are presented. The results are fairly well described by the numerical rdaxation model. No plateau formation could be observed. 

A typical experiment is shown in Fig. 4. Pressure, temperature, vapour mass fraction and saturation ratio are compared with numerical calculation. The characteristic time r, required for numerical evaluation, is obtained by a fit of the experimental vapour mass fraction signal, resulting in r = 5 ms. The chosen values of critical saturation ratio and piston velocity are rather arbitrary. The experimental saturation ratio is calculated from pressure, temperature and vapour mass fraction. When no liquid mass can be detected, the vapour mass fraction is set to the initial value. Experiment and numerical simulation agree fairly well. The first part of the expansion of the gas-vapour mixture is isentropic and accounts for an increase of the saturation ratio. Condensation on the heterogeneous nuclei starts at a value of the measured saturation ratio of about three. After the onset of condensation a rise in temperature is observed due to the release of latent heat. The saturation ratio tends to unity as time increases. The plateau formed in the numerical solution is not observed in the experimental signal. Obviously, the experimental condition is far from self-similarity, and the expansion process is still in its early stage, where relaxation is dominant. The simple numerical model does not describe accurately the... 

Of critical importance, analysis of poly(methyl methacrylate) (PMMA) showed that at a saturation temperature, T, of 40°C, a saturation pressure, P%, of 1,500 psig (at these conditions, carbon dioxide is considered a supercritical fluid), and a saturation time, ts, of 24 h, a 1 mm thick disk absorbed 16.4 wt% carbon dioxide. Additionally, at a foaming temperature, Tf, of 120°C and a foaming time, tf, of 1 min, PMMA had a stable volumetric expansion ratio of 20. Other polymers also absorbed significant quantities of carbon dioxide, such as polystyrene (PS) and poly(vinylidene chloride-co-acrylonitrile) (P(VDC-AN)), which absorbed 8.9 and 2 wt% carbon dioxide, respectively, yet the stable foams that were formed had expansion ratios of less than 2 at the same conditions used to form the PMMA samples. Another polymer poly(vinyl methyl ketone) (PVMK) achieved an expansion ratio of 20. However, the foams were unstable, readily collapsed, and exhibited large voids ( 5 mm diameter), which are inconsistent with microcellular foams. The fact that PVMK readily collapsed after the foaming process made it difficult to determine the concentration of carbon dioxide in the sample. These results led to the eventual incorporation of the MMA monomer into the polymer formulation from the standpoint of carbon dioxide-induced microcellular foamability. 

Equations 10.85 to 10.88 define what has been called a two-term crossover Landau model (CLM). In the classical limit (A/ic 1) the crossover function Y approaches unity and one recovers from eq 10.85 the classical expansion of eq 10.83. In the critical region (A/k 1) the crossover function approaches zero as Y x k/uA) and one recovers from eq 10.85 the power-law expansions specified in Table 10.5 with expressions for the critical amplitudes listed in Table 10.8. The values for the critical-amplitude ratios implied by the crossover Landau model are included in Table 10.3. The nonasymptotic critical behaviour is governed by u and A/cJ or, equivalently by u and by Nq, known... 

The motive nozzle is shaped like a Laval nozzle. This means there is an enlargement of the diameter after the smallest cross section. This is necessary to achieve velocities higher than sonic speed. For steam an expansion pressure ratio of only Pi/Po = is sufficient to just achieve sonic velocity (critical pressure ratio). At higher expansion ratios (supercritical pressure ratios), the exact critical pressure and sonic speed is achieved in the smallest cross section. In these cases in the divergent part of the motive nozzle, a supersonic velocity results from a continuing expansion. Owing to the blocking of the velocity to the sonic speed in the smallest cross section, the mass flow rate of such a supersonic nozzle only depends on the state of the motive media in front of the nozzle and of course on the diameter d.. Here the mass flow rate is proportional to the motive pressurep. ... 

These works have limited use in industrial processes, however. In the first place, the conditions under which they measured the pressure profiles are very different from those in industrial foaming processes. These include differences in die pressure, BA type, BA content, processing temperature, among others. Secondly, they were tmsure that the BA was dissolved, which is critical for the production of high-cell-density and imiform cell structures. Thirdly, they did not relate the nonlinear pressure profiles to the viscosity of either the prrre polymer or the polymer/BA solutiom Finally, the effects of extensional flow on the cell derrsity and on the foam expansion ratio were not addressed. 

It will be seen when the pressure ratio Pi/Pi is less than the critical value (wr — 0.607) the flow rate becomes independent of the downstream pressure P2. The fluid at the orifice is then flowing at the velocity of a small pressure wave and the velocity of the pressure wave relative to the orifice is zero. That is the upstream fluid cannot be influenced by the pressure in the downstream reservoir. Thus, the pressure falls to the critical value at the orifice, and further expansion to the downstream pressure takes place in the reservoir with the generation of a shock wave, as discussed in Section 4.6. 

The mechanism of the inhibitive action of LiOH proposed by Stark et al. [7] is attributed to the formation of lithium silicate that dissolves at the surface of the aggregate without causing swelling [7], In the presence of KOH and NaOH the gel product incorporates Li ions and the amount of Li in this gel increases with its concentration. The threshold level of Na Li is 1 0.67 to 1 1 molar ratio at which expansion due to alkali-silica reaction is reduced to safe levels. Some workers [22] have found that when LiOH is added to mortar much more lithium is taken up by the cement hydration products than Na or K. This would indicate that small amounts of lithium are not very effective. It can therefore be concluded that a critical amount of lithium is needed to overcome the combined concentrations of KOH and NaOH to eliminate the expansive effect and that the product formed with Li is non-expansive. 

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