Density Variable-Volume Systems

Part A Derive a relationship between the composition of the gas mixture leaving the reactor and (V/Fao), for an ideal, isothermal PFR. 

The solution to this problem will be developed using the fractional conversion xa as die conqiosition variable. Be sure that you can solve the problem using the extent of reaction 

a stoichiometric table will be constructed in order to express Ca as a function of xa- Then the design equation will be solved to obtain the required relationship between (V/Fao) and the composition of the stream leaving the reactor. 

Species Molar feed rate (mol/time) Molar flow rate (any position in direction of flow) (mol/time) 

The mole fraction of A. a. at any position in the reactor is given by the moles of A divided by the total number of moles, i.e.. 

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Note that the material balances for fixed beds are valid for die case of constant-density (constant volume) systems. The important term here is the one including the fluid velocity, i.e. the term uJdCIdz. For a variable volume system,... 

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. 

Here A - Ajg is the excess Helmholtz free energy with respect to an ideal gas at the same temperature, volume, and number density of each species. Thus, because of the minus sign, the factor kT, and the factor V in the first equality, si can be regarded as a negative dimensionless excess free energy density for the system. Since both A and Aig are extensive thermodynamic properties of the system, A/V and A JV are functions only of the intensive independent variables. Thus si has been expressed as a function of only the temperature and the number density of each species. (Moreover, we have chosen to use j8 = l/Ztr, rather than T, as the independent temperature variable.) It is this quantity si which has a simple representation in terms of graphs, which will be given below. If si can be calculated (exactly or approximately), this leads to (exact or approximate) results for A and hence for all the thermodynamic properties. 

The total mass balance gives the rate of change of volume with inlet and outlet flow rates for a well-mixed constant-density system. A fed batch is a special case of a variable-volume CSTR operation It has been defined as a bioreactor with inflowing substrate but without outflow. For this system, the equation becomes... 

For convenience, thermodynamic systems are usually assumed closed, isolated from the surroundings. The laws that govern such systems are written in terms of two types of variables intensive (or intrinsic) that do not depend on the mass and extensive that do. By definition, extensive variables are additive, that is, their value for the whole system is the sum of their values for the individual parts. For example, volume, entropy, and total energy of a system are extensive variables, but the specific volume (or its reciprocity - the density), molar volume, or molar free energy of mixing are intensive. It is advisable to use, whenever possible, intensive variables. 

The mixer with two components, can be described by two basic balances, the mass balance and the component balance, in which the density is a function of the composition. Because the density p and the height h are defined as measurable output variables, the system is transformed into a combination of the volume and density balance. The volume balance (compare with Eqn. (4.26)) can be written as ... 

As for classical systems, the isobaric-isothermal ensemble is used to characterize the macroscopic state of a closed, isothermal qirantum system with a variable volume and fixed pressme. The semiclassical (part classical, part quantum) density operator p for a qirantum isobaric-isothermal ensemble is given by... 

In order to specify fhe size of fhe sysfem, af leasf one of fhese variables ought to be extensive (one that is proportional to the size of the system, like n or the total volume V). In the special case of several phases in equilibrium several extensive properties, e.g. n and Vfor two phases, may be required to detennine the relative amounts of the two phases. The rest of the variables can be intensive (independent of the size of the system) like T,p, the molar volume V = V/n,or the density p. For multicomponent systems, additional variables, e.g. several ns, are needed to specify composifion. 

This quantity, which is often referred to as magma density or soHds concentration (mass of crystals per unit system volume), is often an important process variable. A cumulative mass fraction of crystals having a size less than U can also be defined as... 

We have considered volume changes resulting from density changes in liquid and gaseous systems. These volume changes were thermodynamically determined using an equation of state for the fluid that specifies volume or density as a function of composition, pressure, temperature, and any other state variable that may be important. This is the usual case in chemical engineering problems. In Example 2.10, the density depended only on the composition. In Example 2.11, the density depended on composition and pressure, but the pressure was specified. 

VOF or level-set models are used for stratified flows where the phases are separated and one objective is to calculate the location of the interface. In these models, the momentum equations are solved for the separated phases and only at the interface are additional models used. Additional variables, such as the volume fraction of each phase, are used to identify the phases. The simplest model uses a weight average of the viscosity and density in the computational cells that are shared between the phases. Very fine resolution is, however, required for systems when surface tension is important, since an accurate estimation of the curvature of the interface is required to calculate the normal force arising from the surface tension. Usually, VOF models simulate the surface position accurately, but the space resolution is not sufficient to simulate mass transfer in liquids. 

The variable p (r) denotes the nuclear charge density at a point r with coordinates r = xi,X2,x ), and V r) is the Coulomb potential set up at that point by all other charges (the Coulomb constant k = l/(47t o) is dropped in this description). The integration variable in (4.1) is the volume element dr = Ax dr2dx3. The origin of the coordinate system is chosen to coincide with the center of the nuclear charge. A more convenient expression can be obtained by expanding V r) at f = (0,0,0) in a Taylor series, that is,... 

The state (or behaviour) of a system is described by variables or properties which may be classified as (a) extensive properties such as mass, volume, kinetic energy and (b) intensive properties which are independent of system size, e.g., pressure, temperature, concentration. An extensive property can be treated like an intensive property by specifying that it refers to a unit amount of the substance concerned. Thus, mass and volume are extensive properties, but density, which is mass per unit volume, and specific volume, which is volume per unit mass, are intensive properties. In a similar way, specific heat is an intensive property, whereas heat capacity is an extensive property. 

If the system is not of constant density, we must use the more general form of the equation for reaction time (12.3-2) to determine t for a specified conversion, together with a rate law, equation 12.3-3, and an equation of state, equation 2.2-9. Variable density implies that the volume of the reactor or reacting system is not constant. This may be visualized as a vessel equipped with a piston V changes with the position of the piston. Systems of variable density usually involve a gas phase. The density may vary if any one of T, P or n, (total number of moles) changes (so as to alter the position of the piston). 

A material balance for A around the ith tank of volume Vi in the N-tank series (all tanks of equal size), in the case of unsteady-state behavior of a variable-density system,... 

If kinetic data are to be used, it is necessary to transform the variables to conform with those of the partial equilibrium model. The units used in the model equations for and nj are moles formed/kg of solution. Thus the mass of solution in the reacting system from which the kinetic data comes must be known. Frequently, one will know the volume and have to approximate the density. A relation between and t is also needed. For this, the mass of solid originally present must be known. The amount of solid reacing, -ANg, for a time interval At can be obtained from rate curves or calculated from an integrated rate equation. The fraction of the original mass reacting in the time interval gives an approximate value of 5, e.g.,... 

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