Mass balances unsteady

Differential and Integral Balances. Two types of material balances, differential and integral, are applied in analyzing chemical processes. The differential mass balance is valid at any instant in time, with each term representing a rate (i.e., mass per unit time). A general differential material balance may be written on any material involved in any transient process, including semibatch and unsteady-state continuous flow processes ... 

Equations (1.1) to (1.3) are diflerent ways of expressing the overall mass balance for a flow system with variable inventory. In steady-state flow, the derivatives vanish, the total mass in the system is constant, and the overall mass balance simply states that input equals output. In batch systems, the flow terms are zero, the time derivative is zero, and the total mass in the system remains constant. We will return to the general form of Equation (1.3) when unsteady reactors are treated in Chapter 14. Until then, the overall mass balance merely serves as a consistency check on more detailed component balances that apply to individual substances. 

An unsteady-state component mass balance, Eq. (20-68), can be written for batch operation by assuming a uniform average retentate concentration c, within the system. Assuming a constant solvent concentration and a 100 percent passage, the solvent balance becomes Eq. (20-69). 

Equation (20-70) is the unsteady-state component mass balance for fed-batch concentration at constant retentate volume. Integration yields the equations for concentration and yield in Table 20-19. 

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... 

In the previous section, we detailed diffusion equations and generalized mass balance equations. We now turn to their practical uses in the pharmaceutical sciences. Mass transport problems can be classified as steady or unsteady. In steady mass transport there is no change of concentration with time [3], characterized mathematically by... 

The TDE moisture module (of the model) is formulated from three equations (1) the water mass balance equation, (2) the water momentum, (3) the Darcy equation, and (4) other equations such as the surface tension of potential energy equation. The resulting differential equation system describes moisture movement in the soil and is written in a one dimensional, vertical, unsteady, isotropic formulation as ... 

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

The mass balance equation for the SBR with slow fill resembles that of unsteady-state CMFR with variable volume. As originally conceived, SBR operation includes a react period after fill. Thus, a slow fill system s represented by a CMFR followed by a PFR, die minimum volume configuration for an activated sludge system capable of achieving the desired overall treatment performance (Irvine and Ketchum, 1989). 

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... 

The complete unsteady-state model for adsorption-desorption and surface reactions of the plug flow laboratory reactor was based on the following equations NH3 mass balance on the catalyst suiface ... 

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. 

A mass balance over the tank in the unsteady state yields ... 

From the mass balance equation (4.49) we can obtain the unsteady-state equations for a dynamic model by using an accumulation term as follows. 

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

In this section we develop a dynamic model from the same basis and assumptions as the steady-state model developed earlier. The model will include the necessarily unsteady-state dynamic terms, giving a set of initial value differential equations that describe the dynamic behavior of the system. Both the heat and coke capacitances are taken into consideration, while the vapor phase capacitances in both the dense and bubble phase are assumed negligible and therefore the corresponding mass-balance equations are assumed to be at pseudosteady state. This last assumption will be relaxed in the next subsection where the chemisorption capacities of gas oil and gasoline on the surface of the catalyst will be accounted for, albeit in a simple manner. In addition, the heat and mass capacities of the bubble phases are assumed to be negligible and thus the bubble phases of both the reactor and regenerator are assumed to be in a pseudosteady state. Based on these assumptions, the dynamics of the system are controlled by the thermal and coke dynamics in the dense phases of the reactor and of the regenerator. 

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

The mass balance in the reactor is derived under the following assumptions (i) unsteady state operation, (ii) convective laminar Newtonian flow in the axial direction z (the Re)molds number is below the transition regime), (iii) diffusion in the z direction is negligible with respect to convection, (iv) symmetry in the y direction (the lamp length is much larger than the reactor width), and (v) constant physical properties. The local mass balance for a species i in the reactor and the corresponding initial and boundary conditions are... 

Considering that the tank operates under unsteady state and well-stirred conditions, the mass balance and the initial condition for a species i yields... 

The reactor was simulated for both steady and transient behaviour. The steady-state model is straightforward and will not be discussed in detail. The unsteady-steady state simulation took advantage of the fact that the rate of reaction is much faster than the thermal response rate. The concentration transient response can thus be modelled as pseudo-steady state in the actual fluidized bed this pseudo-steady state then follows the slowly changing temperature profile. A mass balance on the species, j, for each region (see Figure 2) is written as ... 

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. 

A different numerical strategy to simulate multiphase mixing was introduced by Mann and Mann and Hackett. The idea of the method, called the network-of-zone, is to subdivide the flow domain in a set of small cells assumed to be mixed perfectly. The cells are allowed to exchange momentum and mass with their neighboring cells by convective and diffusive fluxes. Brucato and Rizzuti and Brucato et al. applied this idea to the modeling of solid-liquid mixing. An unsteady mass balance for the particles was derived to estimate the solid distribution in the vessel, namely ... 

The modeling of solid-liquid mixing requires an additional equation to predict the dispersion of the solid phase in the vessel. As mentioned before, the net-work-of-zones approach was used in this work, which is based on unsteady mass balances on the solid phase carried out on a set of cells. In the literature, such mass balances are predominantly made on regular cells (finite volumes) in structured grids. The cells typically consist of quadrangles in 2-D and hexahedra in 3-D. In the present work, due to the use of unstructured... 

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