Expansion ratio, dependencies

Turboexpander thermal effieieney (TTE isentropie) is defined as the ratio of aetual work produeed by the fluid, divided by the work produeed from the isentropie expansion proeess. Depending on gas eomposition, expansion ratio, and liquid formation, TTE varies between 80%-90%. These high effieieneies are the result of improvements to the thermodynamie and meehanieal design of turboexpanders sinee the early 1960s and their use in gas proeessing plant applieations. 

Flame speed was observed to be nearly constant, but increased with the scale of the experiment. Because mixing with air was limited, a volumetric expansion ratio of approximately 3.5 was observed. The maximum pressure observed was found to be scale dependent (Figure 4.5). 

A combination of a Pd-catalyzed nucleophilic substitution by a phenol and a ring expansion was described by Ihara and coworkers [127] using cis- or trans-substituted propynylcyclobutanols 6/l-262a or 6/l-262b. The product ratio depends on the stereochemistry of the cyclobutanols and the acidity of the phenol 6/1-263. Thus, reaction of 6/l-262b with p-methoxyphenol 6/1-263 (X = pOMe) led exclu-... 

Fig. 12.11 shows the structure of a rocket plume generated downstream of a rocket nozzle. The plume consists of a primary flame and a secondary flame.Fil The primary flame is generated by the exhaust combustion gas from the rocket motor without any effect of the ambient atmosphere. The primary flame is composed of oblique shock waves and expansion waves as a result of interaction with the ambient pressure. The structure is dependent on the expansion ratio of the nozzle, as described in Appendix C. Therefore, no diffusional mixing with ambient air occurs in the primary flame. The secondary flame is generated by mixing of the exhaust gas from the nozzle with the ambient air. The dimensions of the secondary flame are dependent not only on the combustion gas expelled from the exhaust nozzle, but also on the expansion ratio of the nozzle. A nitropolymer propellant composed of nc(0-466), ng(0-369), dep(0104), ec(0 029), and pbst(0.032) is used as a reference propellant to determine the effect of plume suppression. The burning rate characteristics of the propellants are shown in Fig. 6-31. Since the nitropolymer propellant is fuel-rich, the exhaust gas forms a combustible gaseous mixture with the ambient air. This gaseous mixture is ignited and afterburning occurs somewhat downstream of the nozzle exit. The major combustion products in the combustion chamber are CO, Hj, CO2, N2, and HjO. The fuel components are CO and H2, the mole fractions of which at the nozzle throat are co(0.47) and iH2(0.24).

An asymmetric Schmidt ring expansion of the 4-substituted cyclohexanones 312 using chiral azido alcohols 313 gave the azepan-2-ones 314 in high yields and good diastereomeric ratios depending on the nature and position of R1 (Scheme 40) <2003JA7914>. 

The dependence of the pressure and ratio upon the area ratio is a function only of the ratio of the specific heats. Since, for a given propellant combination, the specific heat ratio does not vary greatly during the expansion process, the relation between the area and pressure expansion ratios takes a particularly simple form. 

The dependence of expansion ratio of a foam formed through a gauze on gas consumption passes through an extremum [41,42],... 

The volume and shape of Plateau borders depend on the expansion ratio of the foam. In a spherical monodisperse foam with close packing of bubbles all air/liquid interfaces are spherical and the liquid volume which belongs to one cell can be derived from the difference between the volumes of the corresponding polyhedron (for example, a dodecahedron) and the inscribed in it sphere, having in mind the co-ordination number of the foam cell. 

If the condition for polyhedricity R/r 1 is not fulfilled, the radius of curvature and the area of the cross-section become dependent on the co-ordinates along the length of the Plateau border. Analytical dependence of the radius of curvature on the co-ordinates (the border profile) at different foam expansion ratio is not found. 

A comparison of the dependence Af / As vs. (p(Af is film area, (p is volume fraction of dispersed phase) reported by Princen for emulsions, with the data obtained for a dodecahedral model [83] is given in Fig. 1.12. The figure shows that these dependences coincide within the whole range of expansion ratios studied. 

The formulae given so far are correct only under the assumption that the foam is monodisperse. In order to describe the expansion ratio of a real polydisperse foam it is necessary to use the average values of a average linear ab, average surface as and average volume a. The relation between them depends on the type of bubble size distribution function... 

Applying the relation of parameters n, a, r (Eqs. (4.9) and (4.10)) and the dependence of hydrostatic pressure on foam column height (Eq. (1.39)), it is possible to derive the distribution of local foam expansion ratio along the height H. Assuming a border foam (i.e. neglecting the amount of liquid in films) one obtains from Eq. (4.10) and Eqs. (1.40) and (1.42) which account for the p dependence on r, the following relation... 

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. 

Table 4.1 gives a comparison of the values of the real border foam expansion ratio and these calculated from Eqs. (4.10) and (4.25). It can be seen that choosing appropriate K and accounting for the excess part of vertexes, the calculated values of the expansion ratio are close to the real ones even at n = 20. However, for large expansion ratio ranges the correction (5Knr[ in Eq. (4.25)) depends on r differently. 

Fig. 4.1. Dependence ra/r (1) and ri,/rn (2,3) versus foam expansion ratio curve 3 is calculate with the...

The average foam dispersity in the experiments performed varied within the limits of aVL = 6.10"2-3.5.10 1 mm the degree of polydispersity significantly increased in the process of foam coarsening. It can be seen that curves 1 and 2 fit well at expansion ratio n > 300. At low expansion ratio (20 < n < 40) the difference between rjr (n) and rjrn(n) grows to 15% but if the longitudinal curvature is accounted for then this difference is about 7%. This means that the difference in size of the individual bubbles in a polydisperse foam does not influence strongly the course of the ra lrn (n) dependence as compared to the monodisperse model system. 

Another technique for foam expansion ratio determination involves taking samples and can be applied also for foams produced by foam generators. The precision of these direct volume-mass methods depends to a considerable extent on foam uniformity with respect to its expansion ratio and on the sampling conditions [20,21,22], The relative error in the evaluation of n for low expansion ratio foams (n < 100) is about 1-2%. However, for high expansion ratio foams (n > 1000) these methods become inapplicable. 

Here the coefficient k depends on the foam liquid content which means that Eq. (4.49) is valid only for a narrow interval of foam expansion ratios and dispersities. 

In the middle of cells and in faces that are perpendicular to the flowing direction, the borders are branched, which means that the effective number of borders, equivalent to that in a real system, is different from five. The number of independent borders with constant by height radius and length L can be determined by the electro-hydrodynamic analogy between current intensity and liquid flow rate through borders, both being directly proportional to the cross-sectional areas [6,35]. This analogy indicates that the proportionality coefficients (structural coefficients B = 3) in the dependences border hydroconductivity vs. foam expansion ratio and foam electrical conductivity vs. foam expansion ratio, are identical [10]. From the electrical conductivity data about foam expansion ratio it follows... 

This constant wo characterises the initial volumetric flow rate referring to a unit cross-sectional foam area. A rigorous analytical dependence of the constant wq on the structural parameters of a low expansion ratio foam and the properties of the solution (dp IdH, H, r, R, n, a) has not been derived, so that this constant cannot be calculated. 

Analysis of experimental data about drainage of low expansion ratio foams [24,67] in which the flowing process does not start at the moment of foam formation, as well as the fact that the AVL t (t) curve lacks an inflection point (or, respectively, a maximum of rate dVJdx), proves that Eq. (5.46) is one of the simplest and physically well grounded kinetic dependences of foam drainage in gravitational field. 

A typical dependence of drainage onset on foam column height at foam expansion ratio n = 70 is given in Fig. 5.14. [6,22], For high foam columns (H > 16 cm) zb is small and does not practically depend on H. It is mainly determined by the hydrodynamic properties of the system (borders size and viscosity), i.e. of the microsyneresis rate. For small foam column heights t0 strongly depends on H and is determined by the rate of internal foam collapse. These dependences indicate that for a quantitative description of drainage detailed... 

The experimental results found in [2,56] were used to derive the time dependence of the expansion ratio profile along the height for foam layers situated above the level of constant expansion ratio H0. Introducing the data about the upper foam layers microsyneresis in this dependence one can determine the onset of liquid outflow from the foam. 

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