Unsteady State Mass Balances

Most real situations are, however, such that conditions change with respect to time. Under these circumstances, a steady-state mass balance is inappropriate and must be replaced by a dynamic or unsteady-state mass balance, expressed as... 

Note that since there are two independent variables of both length and time, the defining equation is written in terms of the partial differentials, 3C/dt and 3C/dZ, whereas at steady state only one independent variable, length, is involved and the ordinary derivative function is used. In reality the above diffusion equation results from a combination of an unsteady-state mass balance, based on a small differential element of solid length dZ, combined with Pick s Law of diffusion. 

The mathematical model then is a set of coupled, nonlinear, one dimensional, unsteady-state mass balances of the form... 

An unsteady-state mass balance for ammonia around stage 1 gives L2X2 + Voyo — L X — Viyi = d(Hxi)/dt... 

Writing an unsteady state mass balance on the tracer... 

An unsteady state mass balance over the typical reactor element lying between x and x + 5x gives rise to the partial differential equation... 

Now consider the same constant volume reactor with no reaction occurring in it but with a time-varying inlet concentration (0 which causes the outlet concentration Ca (0 to vary. Writing an unsteady-state mass balance on species A... 

Unsteady-State Mass Balance Method One widely used technique for determining Kj a in bubbling gas-liquid contactors is the physical absorption of oxygen or COj into water or aqueous solutions, or the desorption of such a gas from a solution into a sparging inert gas such as air or nitrogen. The time-dependent concentration of dissolved gas is followed by using a sensor (e.g., for O2 or CO2) with a sufficiently fast response to changes in concentration. 

Therefore, the final unsteady-state mass balance equation, or the dynamic model derived earlier in (4.56), takes the form... 

An unsteady-state mass balance for solids in a differential element with a small perturbation in a system otherwise in equilibrium yields... 

It should be noted that Eq.(2-113) may be looked upon as an "unsteady state probability balance" on city k in the internal circle J is the total number of cities in the external circle and K in the internal circle. In other words, the equation gives the rate of change of the probability of occupying state k at a certain time. Note that the origin of the equation was an unsteady state "mass balance" on the transition of inhabitants through city k. 

Unsteady-State Mass Balance Method One widely used technique for determining KLa in bubbling gas-liquid contactors is the physical absorption of oxygen or C02... 

In many unsteady-state mass balance problems, itiis convenient to take as the system the fluid in some container. Thus, as the mass of fluid increases or decreases, the volume of the system changes. 

Consider a differential length, dz, of cylindrical conduit as shown in Figure 4.7. Fluid with a uniform velocity u in the axial direction is passing through the differential volume Adz, where A is the cross section of the tube. At a time t = 0, a step input of tracer is introduced uniformly across the cross section at the entrance to the tube at a concentration Cq. We write the general unsteady-state mass balance for the differential volume in the normal fashion ... 

This analysis begins with the unsteady-state mass balance for component i in the well-mixed reactor. At high-mass-transfer Peclet numbers, which are primarily a function of volumetric flow rate q, the rate processes of interest are accumulation, convective mass transfer, and multiple chemical reactions. Generic subscripts are... 

Solution. Four unsteady-state mass balances are written and solved numerically to characterize the composition of the exit stream for each reactor. Initially, all of the parameters are declared. The average residence times are ti = 15 min and T2 = 10 min. The third-order kinetic rate constant for the forward step in the first independent elementary reaction is ki = 0.5 L / mol min. The equilibrium constant, based on molar densities, for the first independent elementary reaction is eq.c = 10 L /mol. The second-order kinetic rate constant for the second independent elementary reaction is kj, = 0.2 L/ mol min. The molar densities of all four components in the inlet stream to the first CSTR, for a stoichiometric feed of reactants A and B, are... 

Unsteady-state mass balances for all four components in the first CSTR are... 

In the unsteady-state mass balances for all four components in the second CSTR, the input terms due to convective mass transfer are based on the unsteady-state solutions in the exit stream of the first CSTR [i.e., C,i(t)] ... 

Solution (a). The generic unsteady-state mass balance with one chemical reaction (i.e., 7 = 1) is written for reactant A in each CSTR ... 

The coefficient of dNi/dt in the summation of (6-4) is defined as the partial molar internal energy of species i, because differentiation with respect to mole numbers of component i is performed at constant T, p and mole numbers of all other species in the mixture. The unsteady-state mass balance for species i in a batch reactor,... 

The unsteady-state mass balance for species i in a batch reactor with one chemical reaction was presented above as... 

The parameters n and X(T) which characterize the rate law are evaluated via the differential method of reaction-rate data analysis based on the unsteady-state mass balance ... 

The unsteady-state mass balance (6-21) is used to replace ViR in (6-33) so that temperature and reactant conversion can be related analytically at any time during... 

These coupled ODEs describe the time dependence of molar density for each component in the reactive mixture. Specifically, for the problem of interest, the unsteady-state mass balances reduce to... 

Stoichiometry and the unsteady-state mass balance with chemical reaction yield the following relation between component partial pressures ... 

Now, it is instructive to re-analyze the unsteady-state macroscopic mass balance on an isolated solid pellet of pure A with no chemical reaction. The rate of output due to interphase mass transfer from the solid particle to the liquid solution is expressed as the product of a liquid-phase mass transfer coefficient c, liquids a Concentration driving force (Ca, — Ca), and the surface area of one spherical pellet, 4nR. The unsteady-state mass balance on the solid yields an ordinary differential equation for the time dependence of the radius of the peUet. For example,... 

---