Balance yields

Generally, q is small because the outside area is not large in comparison to the amount of heat being transferred, and the energy balance can be simplified. In these conditions it is also convenient to write balances over a differential section of the column. These balances yield the following ... 

If we represent the moles of vapor by moles of liquid in the pot by M, the mole fraction of the more volatile component in this hquid by X, and the mole fraction of the same component in the vapor by y, 2l material balance yields... 

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

In this section we shall use the standard notation employed by biochemical engineers and industrial microbiologists in presenting the material. Thus if we denote by Xv the viable cell (cells/L) or biomass (mg/L) concentration, S the limiting substrate concentration (mmol/L) and P the product concentration (immol/L) in the bioreactor, the dynamic component mass balances yield the following ODEs for each mode of operation ... 

Continuous Stirred Tank Reactor (CSTR). The conversion degree of the azo-dye, the reaction volume (V) and the volumetric flow rate (Q) of the dye-bearing stream are related to each other through the material balance referred to the dye and extended to the reactor volume. Assuming an unstructured model for the biophase, the material balance yields... 

The CSTR model, on the other hand, is based on a stirred vessel with continuous inflow and outflow (see Fig. 1.2). The principal assumption made when deriving the model is that the vessel is stirred vigorously enough to eliminate all concentration gradients inside the reactor (i.e., the assumption of well stirred). The outlet concentrations will then be identical to the reactor concentrations, and a simple mole balance yields the CSTR model equation ... 

Here, Fg plays a similar role to that of the unknown force, in Campbell s models. Fg is resolved into forces Fj and F, and acts at an angle W to F (the flight edge). These forces are in a plane parallel to the screw root surface. The new static force balance yields ... 

Measurements of the photophoretic force on crystalline ammonium sulfate particles were made by Lin and Campillo (1985) using an electrodynamic balance. The measurement procedure is identical to that for any such force, that is, the levitation voltage is measured in the absence of the photophoretic force and then when the force is exerted. A force balance yields an equation for the photophoretic force similar to Eq. (31) ... 

Chromatographic measurements discussed here are based on the evaluation of the effect of the concentration of the complexing agent (hetaeron) in the eluent on the retention factor of the eluite aS treated in Section VI. If the stoichiometry of the hetaeron-solute association process is 1 1. then Eq. (89) of Section VI can be used to calculate the retention factor. A more general expression is available for the scheme presented in Fig. 59. In view of the equilibria involved mass balance yields an expression for the retention factor as follows... 

Mass balance, yields, and conversion were calculated. The yield to conversion functions were determined using linear regression. Tests with mass balances between 95 and 99 wt% were accepted. 

For a particle of arbitrary shape a mass balance yields... 

Figure 4.6 illustrates the force balance on a differential length dz of a circular tube of diameter D. The pressure p(z), which is a continuous, differentiable function of z, may be expanded in a Taylor series. The net force that results from the wall shear is represented as dF = 7iDrwdz. A force balance yields... 

In the nonisothermal case, the reaction rate is a function of both the concentration c and the temperature T and it is not as easy to see what characteristic temperature should be used because with an exothermic reaction the heat may be trapped within the particle. We will not go through the derivation in detail (Chapter 2 of [I] goes into it quite thoroughly), but an energy balance yields... 

In the fully developed H2 starvation region, where H2 is completely depleted, the cathode electrode potential reaches its maximum value as determined by the mixed potential of all electrochemical reactions involved at the cathode, namely, COR and OER. The reaction current therein is determined by 02 crossover (ixthrough membrane to anode, where only ORR takes place in the absence of H2. Charge balance yields... 

Stage 9 A tray-by-tray energy balance is also possible. Those factors considered are the heat of reaction from nitrogen monoxide oxidation, heat of reaction from any shift in the dioxide/tetroxide equilibrium, heat of solution as weak acid flows from the tray above onto the current plate, and the sensible heat to cool both reaction gases and the acid solution passing through each tray. This heat balance yields the required cooling duty on each sieve tray. A simpler derivation of the equation (Ref. Al, p.184) is employed. 

Thus, solvent deasphalting allows for the removal of sulfur compounds as well as nitrogen compounds and metallic constituents by balancing yield with the desired feedstock properties (Table 7-18 Figure 7-22) (Flynn et al., 1961). 

For each kinetic scheme, the relevant mathematical model is given by the mass balances written for the four species involved. In detail, for the scheme (3.55), the mass balance yields... 

The capillary rise method is a classical method for determining surface tension that has important applications. For capillary rise in a narrow capillary (Figure 3.7) a force balance yields ... 

In a polymeric system, it would be reasonable to examine the possibility that the free volume concept described earlier, which explains so well phenomena like molecular diffusion and viscosity, might perhaps also explain nucleation phenomena. The critical radius re of a stable bubble can be obtained from a simple mechanical-force balance, yielding the Laplace equation ... 

If the free energy of adsorption is taken as equal to the net energy of interaction between the layers, then the energy balance yields ... 

An unsteady state mole balance yields — = -Vdot... 

In turn, the outputs of the turbine, driven by the high-pressure steam, are shaft-work and low-pressure steam. A money balance yields the total cost of these products as... 

Theoretical balance yield BA, is given by the ratio between the moles of the target product and the total moles of the primary raw materials (PRM), including all reactants involved in the stoichiometry of the synthesis route. 

Real balance yield BA is the ratio of the target product to the total amount of materials, including secondary raw materials (SRM) as solvents and catalysts, and given by ... 

The ratio of the above indices, called specific balance yield, is a measure of the raw material efficiency ... 

The calculation shows that the stoichiometric yield RY is acceptable, but the theoretical balance yield BAt poor, because catalyst complex lost after reaction. A significant improvement would be the use of solid catalyst. Other alternative is regeneration of A1C13 complex by recycling. The two solutions would lead to the same theoretical yield, but with different costs. Therefore, a deeper investigation should take into account a cost flow analysis too. More details can be found in Christ [2]. 

This is the one-dimensional heat-conduction equation. To treat more than onedimensional heat flow, we need consider only the heat conducted in and out of a unit volume in all three coordinate directions, as shown in Fig. l-3a. The energy balance yields... 

Here it is noted that the differential surface area for convection is the product of the perimeter of the fln and the differential length dx. When we combine the quantities, the energy balance yields... 

Extraction expriments mass balance (yield recovery)... 

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