Density, molar

Approximating the molar density of the waste stream to be that of pure water (i.e., 56kmolm ), then... 

The problem with figure A2.5.6 and figure A2.5.7 is that, because it extends to infinity, volume is not a convenient variable for a graph. A more usefiil variable is the molar density p = 1 / V or the reduced density p. = 1 / Fj. which have finite ranges, and the familiar van der Waals equation can be transfonned into an alternative although relatively unfamiliar fonn by choosing as independent variables the chemical potential p and the density p. 

Extensive tables and equations are given in ref. 1 for viscosity, surface tension, thermal conductivity, molar density, vapor pressure, and second virial coefficient as functions of temperature. 

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 ... 

Equations 175 through 179 allow calculation of thermodynamic properties from volume-expHcit equations of state, ie, equations expHcitiy solvable for volume. If an equation of state is solvable expHcitiy for pressure but not for volume, then alternative formulas must be used, where p is molar density and subscript p/n = 1/E indicates constancy of total volume. Eor equations 180, 181, and 183, T and x are constant for equation 182, Tis constant. 

Density is defined as the mass of a substance contained in a unit volume. In the SI system of units, the ratio of the density of a substance to the density of water at I5°C is known as its relative density, while the older term specific gravity is the ratio relative to water at 60°F. Various units of density, such as kg/m, Ib-mass/fF, and g/cm, are commonly used. In addition, molar densities, or the density divided by the molecular weight, is often specified. This section briefly discusses methods of correlation of density as a function of temperature and presents the most common accurate methods for prediction of vapor, liquid, and solid density. 

The molar volume may be eliminated in favor of the molar density p = V"/ to give... 

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

Pi Average molar density of liquid phase kmoPm (lbmol)/fF... 

Riazi-Whitson They presented a generahzed correlation in terms of viscosity and molar density that was apphcable to both gases and liqmds. The average absolute deviation for gases was only about 8 percent, while for liquids it was 15 percent. Their expression relies on the Chapman-Enskog correlation [Eq. (5-194)] for the low-pressure diffusivity and the Stiel-Thodos correlation for low-pressure viscosity ... 

The mass density (p) of the mixture is the ratio of its total mass (m) to its total Volume (V), whereas the molar density (c) of the mixture is defined as the ratio of the total number of moles (n) to the total volume of the mixture. Thus,... 

For an ideal gas, the total molar concentration Cj is constant at a given total pressure P and temperature T. This approximation holds quite well for real gases and vapours, except at high pressures. For a liquid however, CT may show considerable variations as the concentrations of the components change and, in practice, the total mass concentration (density p of the mixture) is much more nearly constant. Thus for a mixture of ethanol and water for example, the mass density will range from about 790 to 1000 kg/m3 whereas the molar density will range from about 17 to 56 kmol/m3. For this reason the diffusion equations are frequently written in the form of a mass flux JA (mass/area x time) and the concentration gradients in terms of mass concentrations, such as cA. 

For gas-phase reactions, the molar density is more useful than the mass density. Determining the equation of state for a nonideal gas mixture can be a difficult problem in thermod5mamics. For illustrative purposes and for a great many industrial problems, the ideal gas law is sufficient. Here it is given in a form suitable for flow reactors ... 

To integrate this, u is needed. When there is no change in the number of moles upon reaction. Equation (3.2) applies to the total molar density as well as to the mass density. Thus, for constant A, ... 

A gas-phase CSTR with prescribed values for Pout and Tout is particularly simple when ideal gas behavior can be assumed. The molar density in the reactor will be known and independent of composition. 

Then add all these together, noting that the sum of the component concentrations is the molar density ... 

The ideal gas law says that the molar density is determined by pressure and temperature and is thus known and constant in the reactor. Setting the time derivative of molar density to zero gives an expression for Qom at steady state. The result is... 

A simpler method arbitrarily picks values for oq and reacts this material in a batch reactor at constant V and T. When the reaction is complete, P is calculated from the molar density of the equilibrium mixture. As an example, set = 22.2 (P=l atm) and react to completion. The long-time results from integrating the constant-volume batch equations are a = 5.53, 5 = c= 16.63, = 38.79mol/m, and y =0.143. The pressure at equili-... 

This result is perfectly general for a constant-volume reactor. It continues to apply when p, Cp, and H are expressed in mass units, as is normally the case for liquid systems. The current example has a high level of inerts so that the molar density shows little variation. The approximate heat balance... 

In the model equations, A represents the cross sectional area of reactor, a is the mole fraction of combustor fuel gas, C is the molar concentration of component gas, Cp the heat capacity of insulation and F is the molar flow rate of feed. The AH denotes the heat of reaction, L is the reactor length, P is the reactor pressure, R is the gas constant, T represents the temperature of gas, U is the overall heat transfer coefficient, v represents velocity of gas, W is the reactor width, and z denotes the reactor distance from the inlet. The Greek letters, e is the void fraction of catalyst bed, p the molar density of gas, and rj is the stoichiometric coefficient of reaction. The subscript, c, cat, r, b and a represent the combustor, catalyst, reformer, the insulation, and ambient, respectively. The obtained PDE model is solved using Finite Difference Method (FDM). 

Cj i = concentration of Component i in the liquid at the interface (kmol m-3) pL = molar density of the liquid phase (kmol m 3)... 

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