Mixture mass density

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 ... 

In these equations x and y denote independent spatial coordinates T, the temperature Tib, the mass fraction of the species p, the pressure u and v the tangential and the transverse components of the velocity, respectively p, the mass density Wk, the molecular weight of the species W, the mean molecular weight of the mixture R, the universal gas constant A, the thermal conductivity of the mixture Cp, the constant pressure heat capacity of the mixture Cp, the constant pressure heat capacity of the species Wk, the molar rate of production of the k species per unit volume hk, the speciflc enthalpy of the species p the viscosity of the mixture and the diffusion velocity of the A species in the y direction. The free stream tangential and transverse velocities at the edge of the boundaiy layer are given by = ax and Vg = —ay, respectively, where a is the strain rate. The strain rate is a measure of the stretch in the flame due to the imposed flow. The form of the chemical production rates and the diffusion velocities can be found in (7-8). 

Similar considerations concern the irreversible processes of diffusion and reaction in mixtures [5]. A system of M different molecular species is described by the three components of velocity, the mass density, the temperature, and (M — 1) chemical concentrations and is ruled by M + 4 partial differential equations. The M — 1 extra equations govern the mutual diffusions and the possible chemical reactions... 

For ideal gases the mole densities do not change if the reaction produces more or less moles of product than the reactants. However, the mass density of the gas mixture changes, and this causes the volumetric flow rate and velocity of gases through the reactor to change as the conversion increases. 

In these equations pt is the mass density (g. cm.-3) of the fth chemical species, fc is the rate of production of the fth chemical species by chemical reaction (g. cm.-3 sec.-1), and Fi is the external body force per unit mass acting on the ith species. The velocity v is the local mass average velocity (that velocity measured by a Pitot tube), p is the over-all density of the fluid, and U is the local thermodynamic internal energy (per unit mass) of the mixture. The j, are the fluxes of the various chemical species in g. cm.-2 sec.-1 with respect to the local mass average velocity, v. It should be noted that 2j, = 0, 2/c,- = 0, and = p these relations are used in deriving the over-all equation of continuity [Eq. (4)] by adding up the individual equations of continuity given in Eq. (24). 

A more detailed study of fuel cloud dispersion, though one lacking direct exptl verification, was made by Rosenblatt et al (Ref 23). The purpose of their study was to develop and use physically based numerical simulation models to examine the cloud dispersion and cloud detonation with fuel mass densities and particle size distributions as well as the induced air pressures and velocities as the principal parameters of interest. A finite difference 2-D Eulerian code was used. We quote The basic numerical code used for the FAE analysis was DICE, a 2-D implicit Eulerian finite difference technique which treats fluid-particle mixtures. DICE treats par-... 

Generally speaking, a fluid can be a liquid or a gas, where an important difference is in the equation of state that provides a relationship among the pressure, temperature, and mass density. Gases, of course, are compressible in the simplest case an ideal gas law provides the equation of state for a multicomponent mixture as... 

In the treatment which follows, it will be assumed that the mass density of the reaction mixture is constant throughout a series of stirred tanks. Thus, if the volumetric feed rate is v, then in the steady state, the rate of outflow from each tank will also be v. Material balances may then be written on a volume basis, and this considerably simplifies the treatment. In practice, the constancy of the density of the mixture is a reasonable assumption for liquids, and any correction which may need to be applied is likely to be small. 

When a series of stirred-tanks is used as a chemical reactor, and the reactants are fed at a constant rate, eventually the system reaches a steady state such that the concentrations in the individual tanks, although different, do not vary with time. When the general material balance of equation 1.19 is applied, the accumulation term is therefore zero. Considering first of all the most general case in which the mass density of the mixture is not necessarily constant, the material balance on the reactant A is made on the basis of FA moles of A per unit time fed to the first tank. Then a material balance for the rth tank of volume V (Fig. 1.17) is, in the steady state ... 

Stirred tanks are usually employed for reactions in liquids and, in most cases, the mass density of the reaction mixture may be assumed constant. Material balances may then be taken on the basis of the volume rate of flow v which is constant throughout the system of tanks. The material balance on A over tank r may thus be written, in the steady state ... 

A Swiss patent of 1932 to Stettbacher nr covers the conversion of PETN into a plastic mass by means of 10-30% of a fluid nitric ester such as nitroglycerin or nitroglycol. It states that a mixture of 80% PETN and 20% nitroglycerin is a plastic mass, density 1.65, which does not separate into its components and which is suitable for loading shells and detonators. For the latter purpose it is initiated with 0.04 gram of lead azide. 

We consider a mixture of n reacting constituents which are endowed with a microstructure and assume that every place x in the body is simultaneously occupied by a material particle of each constituent which is present at time r. Each constituent has its own bulk mass density (>,. 

Let pf and ps denote the apparent mass densities of the fluid and the solid then the mean velocity v of the mixture is defined by... 

In conclusion to this section, research in the RTD area is always active and the initial concepts of Danckwerts are gradually being completed and extended. The population balance approach provides a theoretical framework for this generalization. However, in spite of the efforts of several authors, simple procedures, easy to use by practitioners, would still be welcome in the field of unsteady state systems (variable volumes and flow rates), multiple inlet/outlet reactors, variable density mixtures, systems in which the mass-flowrate is not conserved, etc... On the other hand, the promising "generalized reaction time distribution" approach could be developed if suitable experimental methods were available for its determination. 

The characteristic nozzle flow assumptions are made, i.e. the flow is laminar, steady, one-dimensional and there are no dissipative or external forces of any kind. The reacting gas is considered to be composed of fx chemical species, each of which is present at a concentration xt (moles per unit mass of mixture). The usual flow variable temperature T, density p, pressure P and velocity u then make a total of jx + 4 variables. The cross-sectional area ratio e is generally specified as a function of the axial distance z and the axial distance z along the flow direction becomes the independent variable. A mass flow rate W per unit reference... 

For liquid droplets, requirement (1) typically means that the spray must be dilute (that is, the ratio of the volume occupied by the condensed phase to the volume occupied by the gas must be small) because collisions tend to be frequent when the volume of particles per unit volume of space becomes too large. Since the mass density of the particles greatly exceeds that of the gas in many sprays and the stoichiometry of most hydrocarbon-oxidizer systems is such that the mass of the fuel is considerably less than that of the gaseous oxidizer in stoichiometric mixtures, the hypothesis of a dilute spray often is valid in hydrocarbon spray combustion. 

Here c(r) is the concentration c at the radial position r (measured from the centrifuge axis), a is the radial distance of the meniscus, M is the molecular mass in daltons, and v is the partial specific volume in ml/gram. For most proteins v varies from 0.69 - 0.75. It is the reciprocal of the density of the particle. Rho (p) is the density (g/ml) of the solvent. A plot of log c(r) against is a straight line of slope M (1 - p) / 2RT. The computer can also accommodate mixtures of proteins of differing molecular masses, interacting mixtures,... 

In practice, in describing a binary mixture of charged particles, another set of dynamic variables is widely used, namely, instead of partial densities nk,a or the set (43), the mass density pk and the charge density qk are utilized. However, it should be mentioned that due to the electroneutrality constraint the charge density qk can be simply connected with the mass-concentration density xk, introduced above. In particular, one has,... 

We proceed by evaluating the mass density of the gas mixture from = MP/RT... 

The average molecular weight of the gas mixture is 0.00619 kg/mol. The mass density of the gas mixture is estimated from the ideal gas law as... 

We require the density of the vapor mixture in order to calculate the low flux mass transfer coefficients. The molar density of the vapor may be estimated using the ideal gas law and, since the system is almost isothermal, may safely be assumed to be nearly constant. The mass density, however, is likely to vary considerably between the bulk and interface, since the molar masses of the three components in the vapor phase cover such a wide range. The mass density should, therefore, be evaluated with the average molar mass... 

The concentration of the various species in a multi-component mixture may be expressed in numerous ways. In this book the equations are formulated in terms of mass fluxes, thus mass concentrations are used. However, the equations could as well be formulated in terms of molar fluxes, and molar concentrations as usually applied in basic textbooks in Chemical Engineering (e.g., [11] [169] [13]). The mass concentration, pc, is the mass of species c per unit of volume of solution. The species c mass density relates to the familiar molar concentration by the simple formula Cc = is the molecular... 

In a set of introductory steps, the mixture composition and the temperature are calculated. A set of scalar transport equations on the form (12.183) is generally solved for the species mass densities and the mixture enthalpy at the next time level n - - 1. However, in reactor simulations, the enthalpy balance is frequently expressed in terms of temperature. The discrete form of the governing equations is thus written as ... 

In general, a mixture mass density can be defined by g ix = Q + Bu and satisfies the mixture continuity equation ... 

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