Molar expansion factors

Estimates of Model Parameters. The reactor models for FFB, MAT and riser include important features for translating the MAT and FFB data to steady state riser performance. A series of key parameters specific to a given zeolite and matrix component are needed for a given catalyst. Such key parameters are intrinsic cracking anc( coking activities (kj, A ), activation energies and heats of reaction (Ej, AHj), coke deactivation rate (exponents nj), and axial dispersion in the FFB unit (DA). Other feedstock dependent parameters include the inhibition constants (kHAj), the coking constants (XAj), and the axial molar expansion factor (a). 

By defining as the molar expansion factor. Equation (1.2.14) can be written as ... 

A variable-volume batch reactor is a constant-pressure (piston-like) closed tank. On the other hand, a variable-pressure tank is a constant-volume batch reactor (Fogler, 1999). Thus, in batch reactors, the expansion factor is used only in the case of a constant-pressure tank whereas and not in a constant-volume tank, even if the reaction is realized with a change in the total moles. However, in continuous-flow reactors, the expansion factor should be always considered. In the following section and for the continuous-flow reactors, the volume V can be replaced by the volumetric flow rate Q, and the moles N by the molar flow rate F in all equations. 

In Fiery s theory of the excluded volume (27), the chains in undiluted polymer systems assume their unperturbed dimensions. The expansion factor in solutions is governed by the parameter (J — x)/v, v being the molar volume of solvent and x the segment-solvent interaction (regular solution) parameter. In undiluted polymers, the solvent for any molecule is simply other polymer molecules. If it is assumed that the excluded volume term in the thermodynamic theory of concentrated systems can be applied directly to the determination of coil dimensions, then x is automatically zero but v is very large, reducing the expansion to zero. 

Fig. 4 Plot of static expansion factor (as) as a function of relative temperature 0/T, where a is defined as Rg(T)/Rg((9) symbols are our measured results and the lines are calculated data with three different values of r. If choosing M = 113 (molar mass of monomer NIPAM), we have r 105 for both the PNIPAM samples used [32]...

The effectiveness factor is related to the Thiele modulus for different molar expansion modulus 0 [12]. The molar expansion modulus 0 expresses the intensity of the molar expansion, 0=(m-l),X R. Two simple cases may be considered, whether the cracking reaction is initially controlled by diffusion of reactants in the uncoked catalyst, or not. 

The expansion factor a is a ratio varying from 1 to 2 a is dependent on the chain length (and thus on the molar mass). An empirical approximation is the following ... 

The expansion factor is defined as the molar density of the reagents divided by the molar density of the products in an explosive mixture. The expansion factor is a measure of the increase in volume resulting from combustion. The maximum value of the expansion factor is for adiabatic combustion. 

This nr1 dependence of the second virial coefficient has been already predicted in the Flory excluded volume theory [6] let a be the expansion factor of a chain molecule and Vs the molar volume of solvents. Following Flory, one has... 

The I/O expansion factor will not be much different from the molar expansion fee-tor but it may be measurable. If so, it may provide a simple way of measuring net desorption rates. The kind of device that would make this technique viable is a differential flow rate or differential flow velocity meter. Since temperature does not enter into equation 5.69, this device would compare the flow rates of the inlet and a cooled outlet stream at a convenient common temperature, say the temperature at which inlet flow is metered. 

We may suppose we are considering one mole of feed, or that the input vector a is in partial pressures so that its components are in the mole fractions directly. Due to volume expansion caused by reaction, temperature, and pressure variations, the outlet molar composition vector P, will be the same as the outlet partial pressures but scaled by some expansion factor. We will see what that factor is after we consider fractional conversion and yields. 

Following the derivatimi in Section 11.3.3, the Gibbs energy of elasticity depends on the effective molar chain concattration v, in the cross-linked system before cross-linking as well as on the expansion factor... 

The ratio of [tj]/[tj]0 is called the viscosity expansion factor a. The viscosity-averaged molar mass (A/ ) obtained from the MHSS equation may also be used to get information about the degree of molar mass dispersion from the comparison of the M values of a polymer sample in two different solvents [5,6],... 

Figure 1.18 shows the expansion factor for the end-to-end distance a = R )o(xm)/(R )o(O) (solid lines) and the total coverage00 (broken Unes) plotted against the molar fraction Xm of methanol. Here, (/f )o(O) is the value in pure water. 

The expansion factor (a) is also affected by temperature and type of solvent. It has been shown both theoretically and experimentally that a is a function of the molar mass in good solvents ... 

The volumetric properties of fluids are conveniently represented by PVT equations of state. The most popular are virial, cubic, and extended virial equations. Virial equations are infinite series representations of the compressibiHty factor Z, defined as Z = PV/RT having either molar density, p[ = V ), or pressure, P, as the independent variable of expansion ... 

Deterioration of electrode performance due to corrosion of electrode components is a critical problem. The susceptibility of MHt electrodes to corrosion is essentially determined by two factors surface passivation due to the presence of surface oxides or hydroxides, and the molar volume of hydrogen, VH, in the hydride phase. As pointed out by Willems and Buschow [40], VH is important since it governs alloy expansion and contraction during the charge-discharge cycle. Large volume changes... 

In 1901, H. Kamerlingh Onnes introduced a fundamentally new and improved description of real gas PVT properties in terms of the virial equation of state. [The word virial, deriving from the Latin word viris ( force ) was introduced into physics by R. Clausius, whom we shall meet later.] This equation expresses the compressibility factor Z(Vm, T) in terms of a general power series expansion in inverse molar volume Vm. The starting point for the virial expansion is the ideal limiting behavior (2.12), which can also be expressed as... 

Effect of molar volume change on diffusion factor. In cases where the chemical reaction is accompanied by a volume change, the diffusive flow may be hindered or aided by hydrodynamic flow resulting from volume expansion or contraction, respectively. This again results essentially in an appropriate shift of the abscissa. Inspection of the results of calculations shows that the same values of tj will be obtained, instead of at 1.0 function applicable to no volume change,... 

Table 2. Data of the thermal expansion coefficient a (Kh K ), affinity factor and molar volume of the hquid-hke adsorbate, Vm (cm mol )-...

The combinatorial factor Q is related to the fraction of the three-dimensional space taken by polymeric chains and the other part taken by the solvent and is independent of the compactness (or otherwise) of the chain. The free volume factor Qfv depends on the experimental values of molar volume V (or segmental volume v), of isobaric expansivity a and of isothermal compressibility top. The configurational (or interactional, or potential... 

Virial Equations of State The virial equation in density is an infinite series expansion of the compressibility factor Z in powers of molar density p (or reciprocal molar volume about the real gas state at zero density (zero pressure) ... 

A virial equation is a power-series expansion of the compressibility factor isotherm in molar density from the zero-density ideal gas, z = 1 at p = 0, which is the common point of intersection of aU z isotherms. 

Virial equations of stale are infinite-series representations of lhe gas-phase compressibility factor, with either molar density or pressure Lakco as the independent variable for expansion ... 

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