Equations, balancing calculations from

The yield stress may then be calculated on the basis of a simple force-balance calculation from the following equation ... 

Mass and Energy Balances. The formulation of mass and energy balances follows procedures outlined ia many basic texts (2). The use of solubihties to calculate crystal production rates from a cooling crystallizer was demonstrated by the discussion of equations 1 and 2. Subsequent to determining the yield, the rate at which heat must be removed from such a crystallizer can be calculated from an energy balance ... 

Application of these equations gives the results in Table 13-12. A set of T is calculated from the normahzed by bubble-point calculations. Corresponding values of are obtained from y = K x. Once newA. andT are available, new values of Vn are calculated from energy balances by using data from Maxwell (Data Book on Hydiocaihons, Van Nostrand, Princeton, N.J., 1950). First, an estimate of condenser duty is computed from an energy balance around the condenser. 

Another view is given in Figure 3.1.2 (Berty 1979), to understand the inner workings of recycle reactors. Here the recycle reactor is represented as an ideal, isothermal, plug-flow, tubular reactor with external recycle. This view justifies the frequently used name loop reactor. As is customary for the calculation of performance for tubular reactors, the rate equations are integrated from initial to final conditions within the inner balance limit. This calculation represents an implicit problem since the initial conditions depend on the result because of the recycle stream. Therefore, repeated trial and error calculations are needed for recycle... 

Temperature gradient normal to flow. In exothermic reactions, the heat generation rate is q=(-AHr)r. This must be removed to maintain steady-state. For endothermic reactions this much heat must be added. Here the equations deal with exothermic reactions as examples. A criterion can be derived for the temperature difference needed for heat transfer from the catalyst particles to the reacting, flowing fluid. For this, inside heat balance can be measured (Berty 1974) directly, with Pt resistance thermometers. Since this is expensive and complicated, here again the heat generation rate is calculated from the rate of reaction that is derived from the outside material balance, and multiplied by the heat of reaction. 

The air exchange rate, G , required for temperature control in the occupied zone can be calculated from the room heat balance equation ... 

For once-through natural circulation reboilers, the liquid backup height is calculated from the pressure balance equation. If this height, plus an allowance for froth, reaches the bottom tray level, flooding of the tower will occur. 

Chromic oxide, Cr,03, is used as a green pigment and is often made by the reaction between NajCrjOjfs) and NH(ClfsJ to give Cr >rfs), NaCl(s), Nifg), and HjO(g). Write a balanced equation and calculate how much pigment can be made from 1.0 X 101 kg of sodium dichromate. 

D. Balancing method (Section 17.5). This method forms the basis of all colorimeters of the plunger type, e.g. in the Duboscq colorimeter. The comparison is made in two tubes, and the height of the liquid in one tube is adjusted so that when both tubes are observed vertically the colour intensities in the tubes are equal. The concentration in one of the tubes being known, that in the other may be calculated from the respective lengths of the two columns of liquid and the relation [equation (11)] ... 

The kinetic energy attributable to this velocity will be dissipated when the liquid enters the reservoir. The pressure drop may now be calculated from the energy balance equation and equation 3.19. For turbulent flow of an incompressible fluid ... 

Section 6.11, when we calculated the enthalpy change for an overall physical process as the sum of the enthalpy changes for a series of two individual steps. The same rule applied to chemical reactions is known as Hess s law the overall reaction enthalpy is the sum of the reaction enthalpies of the steps into which the reaction can be divided. Hess s law applies even if the intermediate reactions or the overall reaction cannot actually be carried out. Provided that the equation for each step balances and the individual equations add up to the equation for the reaction of interest, a reaction enthalpy can be calculated from any convenient sequence of reactions (Fig. 6.30). 

The time derivative is zero at steady state, but it is included so that the method of false transients can be used. The computational procedure in Section 4.3.2 applies directly when the energy balance is given by Equation (5.28). The same basic procedure can be used for Equation (5.25). The enthalpy rather than the temperature is marched ahead as the dependent variable, and then Tout is calculated from Hout after each time step. 

The basic scheme for the numerical solution is the same as that used for the 1 -D model, except that in this case the solid temperature field used to solve the DAE system for each monolith channel must be calculated from the three-dimensional solid-phase energy balance equation. The three-dimensional energy balance equation can be solved by a nonlinear finite element solver (such as ABAQUS) for the solid-phase temperature field while a nonlinear finite difference solver for the DAE system calculates the gas-phase temperature and... 

Entropy changes are important in every process, but chemists are particularly interested in the effects of entropy on chemical reactions. If a reaction occurs under standard conditions, its entropy change can be calculated from absolute entropies using the same reasoning used to calculate reaction enthalpies from standard enthalpies of formation. The products of the reaction have molar entropies, and so do the reactants. The total entropy of the products is the sum of the molar entropies of the products multiplied by their stoichiometric coefficients in the balanced chemical equation. The total entropy of the reactants is a similar sum for the reactants. Equation... 

The values of the fluxes that can be calculated from these two equations differ substantially. Therefore, an apparent contradiction exists between the balance requirements with respect to charges and substances. This contradiction is particularly... 

Figure 9. A schematic and ideal model showing how the residence time of the magma in a steady-state reservoir of constant mass M, replenished with an influx O of magma and thoroughly mixed, can be calculated from disequilibrium data, in the simplifying case where crystal fractionation is neglected (Pyle 1992). The mass balance equation describing the evolution through time of the concentration [N2] (number of atoms of the daughter nuclide per unit mass of magma) in the reservoir is ...

The other state variables are the fugacity of dissolved methane in the bulk of the liquid water phase (fb) and the zero, first and second moment of the particle size distribution (p0, Pi, l )- The initial value for the fugacity, fb° is equal to the three phase equilibrium fugacity feq. The initial number of particles, p , or nuclei initially formed was calculated from a mass balance of the amount of gas consumed at the turbidity point. The explanation of the other variables and parameters as well as the initial conditions are described in detail in the reference. The equations are given to illustrate the nature of this parameter estimation problem with five ODEs, one kinetic parameter (K ) and only one measured state variable. 

It must be emphasised that it is unnecessary to correct a heat of reaction to the reaction temperature for use in a reactor heat-balance calculation. To do so is to carry out two heat balances, whereas with a suitable choice of datum only one need be made. For a practical reactor, the heat added (or removed) Qp to maintain the design reactor temperature will be given by (from equation 3.10) ... 

This section is a general discussion of the techniques used for the preparation of flowsheets from manual calculations. The stream flows and compositions are calculated from material balances combined with the design equations that arise from the process and equipment design constraints. 

The number of moles of KCIO, may be calculated from the number of moles of O, by means of the balanced chemical equation, and that value is then converted to mass. 

The Eulerian gas velocity field required in both the mass balance and the above transport equation for nh is found by an approximate method first, the complete field of liquid velocities obtained with FLUENT is adapted downward because the power draw is smaller under gassed conditions next, in a very simple way of one-way coupling, the bubble velocity calculated from the above force balance is just added to this adapted liquid velocity field. This procedure makes a momentum balance for the bubble phase redundant this saves a lot of computational effort. 

The remaining step is to compute the system s bulk composition, if it is not fully known, according to the mass balance equations. The mole numbers Mw, Mi, and are not known when the modeler has constrained the corresponding variable nw, mt, or tig. In these cases, the mole numbers are determined directly from Equations 4.3 1.5. Where gases appear in the basis, the mole numbers Mm of gas components are similarly calculated from Equation 4.6. 

If T is specified, V can be calculated from the material-balance (equation 143-5 or -15) without the need for the energy balance. 

Table 19.4 gives the calculations of E(t), t, and of based on the histogram method. Column (4) lists the values of c j calculated from (A). Column (5) gives At, required for the calculations in subsequent columns, from equations 19.3-12 to -14. Column (6) gives values for the tracer material-balance (see below). Column (7) gives values of E t) from equation 19.3-4 with C(t) = E(t). Columns (8) and (9) give values required for the calculation of Tin equation 19.3-7, and of of in 19.3-8, respectively. 

The mass balance can be used to calculate the pressure rise in a reactor from a reaction by the use of the ideal gas law or the Van der Waals equation [47]. The evaporation of solvents is another cause for a rise in pressure, and, in fact, the evaporation of volatiles is a major factor to be controlled. As an example, the vapor pressure of acetone at three temperatures, as calculated from Weast [47], is given in Table 3.1. The example shows that an increase in temperature of 100°C leads to an increase of pressure by a factor of 23. 

The indirect method uses Eq. (10.9) to estimate F. This procedure requires the value of the covariance matrix, , which can be calculated from the residuals using the balance equations and the measurements. 

It is essential to appreciate that the pressure difference measured by. a manometer automatically eliminates the static head difference. This is shown in Figure 8.1(b). The static head pg(zi — z2) in the pipe is exactly balanced by the extra static head above the right hand limb of the manometer. Consequently, if Ah is calculated from Azm using equation 8.4, no further correction for the static head should be made. 

The number of gas molecules can be measured either directly with a balance (gravimetric method) or calculated from the pressure difference of the gas in a fixed volume upon adsorption (manometric method). The most frequently apphed method to derive the monolayer capacity is a method developed by Brunauer, Emmett, and Teller (BET) [1], Starting from the Langmuir equation (monolayer adsorption) they developed a multilayer adsorption model that allows the calculation of the specific surface area of a sohd. The BET equation is typically expressed in its linear form as... 

If the solute is a liquid (somewhat rare), the weight calculated from Equation (4.11) can, but rather inconveniently, be measured on a balance. However, this weight can also be converted to milliliters by using the density of the liquid. In this way, the volume of the liquid can be measured, rather than its weight, and it can be pipetted into the container. 

It is also possible to prepare a standard solution of calcium carbonate accurately using an analytical balance and a volumetric flask, as was suggested previously for EDTA, and pipetting an aliquot (a portion of a larger volume) of this solution into the reaction flask in preparation for the standardization of the EDTA solution. In this case, the concentration of the calcium carbonate solution is first calculated from the weight-volume preparation data (refer back to Example 4.2) and then the molarity of the EDTA solution is determined using Equation (5.54). 

Balancing the equation using the ion-electron method results in the following (check it) 5 H202 + 6 H 2 Mn04 ->502 + 2 Mn2+ + 8 H20 Calculating the percent hydrogen peroxide utilizes Equation (4.37) from Chapter 4 ... 

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