Mass balance derivation

Often in plant operations condensate at high pressures are let down to lower pressures. In such situations some low-pressure flash steam is produced, and the low-pressure condensate is either sent to a power plant or is cascaded to a lower pressure level. The following analysis solves the mass and heat balances that describe such a system, and can be used as an approximate calculation procedure. Refer to Figure 2 for a simplified view of the system and the basis for developing the mass and energy balances. We consider the condensate to be at pressure Pj and temperature tj, from whence it is let down to pressure 2. The saturation temperature at pressure Pj is tj. The vapor flow is defined as V Ibs/hr, and the condensate quality is defined as L Ibs/hr. The mass balance derived from Figure 2 is ... 

The Equation of Continuity by Differential Mass Balance Derive the equation of continuity in cylindrical coordinates by making a mass balance over the differential volume Ar(rA6)Az. 

Major elements in melts formed from mantle rocks are by definition compatible, and most of them are well buffered by the residual minerals, so that their concentrations usually vary by factors of less than two in the melts. In contrast, trace elements, particularly those having very low partition coefficients, may vary by as many as three orders of magnitude in the melt, depending on the degree of melting. This is easily seen from the mass-balance-derived equation for the equilibrium concentration of a trace element in the melt, Cl, given by (Shaw, 1970)... 

For a gas-particle separation device, define total efficiency and grade efficiency. Using these definitions and the mass balance, derive an expression relating the size... 

This equation can be integrated between different limits to yield tiie algebraic mass balances derived in Illustration 2.3 by performing integral solute balances over the column. They are... 

We now illustrate the mass balance derivation for i -out using a gradient of concentfa-tion in a single soil layer. When there is a concentration gradient in a single soil layer, the concentration C(nfg) at the lower boundary of the soil layer is C(ds) = C(0)e . So with a concentration gradient. Equation 8.22 becomes... 

Equimolar Counterdiffusion. Just as unidirectional diffusion through stagnant films represents the situation in an ideally simple gas absorption process, equimolar counterdiffusion prevails as another special case in ideal distillation columns. In this case, the total molar flows and are constant, and the mass balance is given by equation 35. As shown eadier, noj/g factors have to be included in the derivation and the height of the packing is... 

Phase II Mass Balance. (/) Determine raw material iaputs. 2) Record water usage. 3) Assess present practice and procedures. (4) Quantify process outputs. (5) Account for emissions to atmosphere, to wastewater, and to off-site disposal. (6) Assemble iaput and output information. (7) Derive a preliminary mass balance. (8) Evaluate and refine the mass balance. 

As an alternative to deriving Eq. (8-2) from a dynamic mass balance, one could simply postulate a first-order differential equation to be valid (empirical modeling). Then it would be necessary to estimate values for T and K so that the postulated model described the reactor s dynamic response. The advantage of the physical model over the empirical model is that the physical model gives insight into how reactor parameters affec t the v ues of T, and which in turn affects the dynamic response of the reac tor. 

Nonlinear versus Linear Models If F, and k are constant, then Eq. (8-1) is an example of a linear differential equation model. In a linear equation, the output and input variables and their derivatives only appear to the first power. If the rate of reac tion were second order, then the resiilting dynamic mass balance woiild be ... 

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... 

This is accomplished by measuring the rate at constant temperature and at various concentrations by varying the feed rate. Calculating 0, multiplying by the measured slope at the calculated 0, and then adding one gives the derivative of the mass balance rate with regard to concentration. 

For example, if 40 percent of stack emissions of the reported substance were derived using monitoring data, 30 percent by mass balance, and 30 percent by emission factors, you would enter the code letter M" for monitoring. 

The following differential equation (or something similar), derived from a mass balance for the room, is solved to find the correlation between flow rates, source rate, contaminant concentrations, cleaning efficiency, and time. 

The requirement for oxygen and carbon source for cell biosynthesis are calculated using nitrogen-limited mass balance equations for growth during exopolysaccharide production 01 res (nitrogen-limited cultures). These balances are derived from experimentally determined values of ... 

Table 10-13 Mass balance calculation for removal of river-derived constituents from the ocean (all units in lO mmol)...

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. 

In the limit of small pressure perturbations, any kinetic equation modeling the response of a catalyst surface can be reduced to first order. Following Yasuda s derivation C, the system can be described by a set of functions which describe the dependence of pressure, coverage amplitude, and phase on T, P, and frequency. After a mass balance, the equations can be separated Into real and Imaginary terms to yield a real response function, RRF, and an Imaginary response function, IRF ... 

Spectroscopy. In the methods discussed so far, the information obtained is essentially limited to the analysis of mass balances. In that re.spect they are blind methods, since they only yield macroscopic averaged information. It is also possible to study the spectrum of a suitable probe molecule adsorbed on a catalyst surface and to derive information on the type and nature of the surface sites from it. A good illustration is that of pyridine adsorbed on a zeolite containing both Lewis (L) and Brbnsted (B) acid sites. Figure 3.53 shows a typical IR ab.sorption spectrum of adsorbed pyridine. The spectrum exhibits four bands that can be assigned to adsorbed pyridine and pyridinium ions. Pyridine adsorbed on a Bronsted site forms a (protonated) pyridium ion whereas adsorption on a Lewis site only leads to the formation of a co-ordination complex. 

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. 

Derivation. For any extravascular drug administration, the mass balance equation can be written as amount absorbed (A) equals amount in body (W) plus amount eliminated (E), or... 

When membrane retention of the solute needs to be considered, one can derive the appropriate permeability equations along the lines described in the preceding section Eqs. (7.1)—(7.3) apply (with P designated as the effective permeability, Pe). However, the mass balance would need to include the membrane compartment, in addition to the donor and acceptor compartments. At time t, the sample distributes (mol amounts) between three compartments ... 

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