Steady-State Mass Balance

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. 

Though we frequently use the word equilibrium in our life, equilibrium is not a common state for any real system. The common thermodynamic state is called steady state when a system s parameters are not changing because the output and input fluxes of energy and mass are balanced. For example, the temperature of a human body is remarkably constant due to a delicately balanced steady state, and this temperature, 36.6°C, is quite different from any common temperature in the surroundings. 

Knowing where waste is going is the key to reducing it. When reducing waste from process operations, a steady-state mass balance is not usually comprehensive enough. A balance that takes into account start-up, shutdown, and product changeovers is required. 

Stea.dy-Sta.teFeedforwa.rd, The simplest form of feedforward (FF) control utilizes a steady-state energy or mass balance to determine the appropriate manipulated variable adjustment. This form of feedforward control does not account for the process dynamics of the disturbance or manipulated variables on the controlled variable. Consider the steam heater shown ia Figure 15. If a steady-state feedforward control is designed to compensate for feed rate disturbances, then a steady-state energy balance around the heater yields ... 

For the air—water system, the humidity is easily measured by using a wet-bulb thermometer. Air passing the wet wick surrounding the thermometer bulb causes evaporation of moisture from the wick. The balance between heat transfer to the wick and energy requited by the latent heat of the mass transfer from the wick gives, at steady state,... 

Under steady state, the mass balance on both solids and hquid yield, respectively ... 

This is an old, familiar analysis that applies to any continuous culture with a single growth-limiting nutrient that meets the assumptions of perfect mixing and constant volume. The fundamental mass balance equations are used with the Monod equation, which has no time dependency and should be apphed with caution to transient states where there may be a time lag as [L responds to changing S. At steady state, the rates of change become zero, and [L = D. Substituting ... 

Figure 24-23 is a sketch of continuous culture with recycle. The symbols for flow rates and organism concentrations are F and X, respec tively Assuming perfect mixing and steady state so that the derivatives can be set to zero, mass balances lead to ... 

The equations that have been developed for design using these pseudo constants are based on steady-state mass balances of the biomass and the waste components around both the reactor of the system and the device used to separate and recycle microorganisms. Thus, the equations that can be derived will be dependent upon the characteristics of the reactor and the separator. It is impossible here to... 

Pipe Flow For steady-state flow through a constant diameter duct, the mass flux G is constant and the governing steady-state momentum balance is ... 

The fractional conversions in terms of both the mass balance and heat balance equations were calculated at effluent temperatures of 300, 325, 350, 375, 400, 425, 450, and 475 K, respectively. A Microsoft Excel Spreadsheet (Example6-ll.xls) was used to calculate the fractional conversions at varying temperature. Table 6-7 gives the results of the spreadsheet calculation and Eigure 6-24 shows profiles of the conversions at varying effluent temperature. The figure shows that die steady state values are (X, T) = (0.02,300), (0.5,362), and (0.95,410). The middle point is unstable and die last point is die most desirable because of die high conversion. 

It is also useful to make a steady-state balance with respect to airflow, heat flow, and mass flow of substances into and out of the enclosure. See Chapter 7 for details with respect to calculation of heat and contaminant emission. 

On the other hand, in the steady state the mass balance for the gas in a tube with a constant cross-sectional area is simply... 

Several features of this treatment are of interest. Compare the denominators of Eqs. (3-147) and (3-149) Miller has pointed out that the form of Eq. (3-147) is usually seen in chemical applications of the steady-state approximation, whereas the form of Eq. (3-149) appears in biochemical applications. The difference arises from the manner in which one uses the mass balance expressions, and this depends upon the type of system being studied and the information available. 

Steady-state material balance is used for cell mass. 

For the subsequent tanks, use the steady-state mass balance on substrate ... 

Substituting into the mass balance yields, the cell mass balance is arranged. At steady-state condition ... 

One thus obtains the following differential steady-state mass balance for the surface concentration, Ci, of the promoting species ... 

Consider the water balance of a lake with a constant source flux Q. The outlet is the "threshold" type where the sink is proportional to the mass of water above a threshold value Mi S = k(M — Ml). Calculate the turnover time of water at steady state and the response time relative to changes in Q. 

Equations (4.1) or (4.2) are a set of N simultaneous equations in iV+1 unknowns, the unknowns being the N outlet concentrations aout,bout, , and the one volumetric flow rate Qout- Note that Qom is evaluated at the conditions within the reactor. If the mass density of the fluid is constant, as is approximately true for liquid systems, then Qout=Qm- This allows Equations (4.1) to be solved for the outlet compositions. If Qout is unknown, then the component balances must be supplemented by an equation of state for the system. Perhaps surprisingly, the algebraic equations governing the steady-state performance of a CSTR are usually more difficult to solve than the sets of simultaneous, first-order ODEs encountered in Chapters 2 and 3. We start with an example that is easy but important. 

Closure normally begins by satisfying the overall mass balance i.e., by equating the input and outlet mass flow rates for a steady-state system. For the present case, the outlet flow was measured. The inlet flow was unmeasured so it must be assumed to be equal to the outlet flow. We suppose that A and B are the only reactive components. Then, for a constant-density system, it must be that... 

Operating Modes. The component and mass balances are quite general and apply to any operating mode e.g., batch, semibatch, or steady state. Table 11.2 gives examples for the various modes. 

In the above reactions, I signifies an initiator molecule, Rq the chain-initiating species, M a monomer molecule, R, a radical of chain length n, Pn a polymer molecule of chain length n, and f the initiator efficiency. The usual approximations for long chains and radical quasi-steady state (rate of initiation equals rate of termination) (2-6) are applied. Also applied is the assumption that the initiation step is much faster than initiator decomposition. ,1) With these assumptions, the monomer mass balance for a batch reactor is given by the following differential equation. 

Rate of Formation of Primary Precursors. A steady state radical balance was used to calculate the concentration of the copolymer oligomer radicals in the aqueous phase. This balance equated the radical generation rate with the sum of the rates of radical termination and of radical entry into the particles and precursors. The calculation of the entry rate coefficients was based on the hypothesis that radical entry is governed by mass transfer through a surface film in parallel with bulk diffusion/electrostatic attraction/repulsion of an oligomer with a latex particle but in series with a limiting rate determining step (Richards, J. R. et al. J. AppI. Polv. Sci.. in press). Initiator efficiency was... 

---