Differential mole balance

CDP1-Aa Calculate the time to consume 80% of species A in a constant-volume batch reactor for a first- and a second-order reaction. (Includes Solution] CDPl-B Derive the differential mole balance equation for a foam reactor. [2nd Ed. P1-10b]... 

This equation can be combined with the differential mole balance... 

Step I Perform a differential mole balance on a particular species A. Step 2 Substitute for F/ in terms of... 

Perform a differential mole balance on a particular species. 

In complex reaction systems consisting of combinations of parallel and series reactions, the as ailability of software packages (ODE solvers) makes it much easier to solve problems using moles or molar flow rates Fj rather than conversion. For liquid systems, concentration is usually the preferred variable used in the mole balance equations. The resulting coupled differential mole balance equations can be easily solved using an ODE solver. In fact, this section has been developed to take advantage of the vast number of computational techniques now available on personal computers (Polymath). For gas systems, the molar flow rates are usually the preferred variable in the mole balance equation. 

The last statement is the typical form of a liquid-phase component mass balance. When this is divided through by the molecular weight of species i, this becomes a differential mole balance since the concentrations are expressed in molarity units ... 

The mole balances for the various types of reactors we have been studying are shown in Table 8-1, The resulting coupled differential mole balance... 

The fundamental equations of a pit model are the differential mole balances for chemical species dissolving in the pit. The mole balance for species i is... 

The conversion and yield of carbon dioxide and methane are defined in the opposite sense of the yield and conversion used in steam reformers in equations (7.125) and (7.126) and the same differential mass balance equations (7.127) and (7.128) are used. The mole fractions of all components are computed from similar relations as given before. 

The mole balance for A applied to a differential slice of the reactor is as follows. Remember that there are two inlet terms for A, one term expressing A carried forward with the main flow inside the reactor and a second term through the wall 7X4 CAo-... 

Kinetic models utilize a set of algebraic or differential equations based on the mole balances of the main species involved in the process (ozone in water and gas phases, compounds that react with ozone, presence of promoters, inhibitors of free radical reactions, etc). Solution of these equations provides theoretical concentration profiles with time of each species. Theoretical results can be compared with experimental results when these data are available. In some cases, kinetic modeling allows the determination of rate constants by trial and error procedures that find the best values to fit the... 

After substituting Equations 3.1.2 and 3.1.3 into Equation 3.1.1, the oxygen mole balance reduces to Equation 3.1.4 in Table 3.1.1. Because Equation 3.1.4 is an unsteady-state, first-order differential equation, we need an initial condition to calculate the constant of integration. Initially, the tank contains air, which has an oxygen concentration of approximately 21 % by volume. We could also write the mole balance for nitrogen, but in this case it is more convenient to write the total mole balance, which results in Equation 3.1.5. Once we write Equations 3.1.4 to 3.1.6, the nitrogen mole balance is not an independent equation. Equation 3.1.7 states that the molar flow rate is equal to the product of the molar density and the volmnetric flow rate. 

Equation (6.3.4) and taking the limit as Ax approaches zero gives the following differential equation for the mole balance ... 

A differential balance applies to an instant of time and each term is a rate (mass/time or moles/time). An integral balance applies to a time interval and each term is an amount (mass or moles). Balances may be applied to total mass, individual species, or energy. (They may also be applied to momentum, but we will not consider momentum balances in this text.)... 

When using an ordinary differential equation (ODE) solver such as POLYMATH or MATLAB, it is usually easier to leave the mole balances, rate laws, and concentrations as separate equations rather than combining them into a single equation as we did to obtain an analytical solution. Writing She equations separately leaves it to the computer to combine them and produce a solution. The formulations for a packed-bed reactor with pressure drop and a semibatch reactor are given below for two elementary reactions. 

In most instances it is not possible to eliminate the concentration of the active intermediate in the differential forms of the mole balance equations to obtain closed-form solutions, However, an approximate solution may be obtained. Ihe active intermediate molecule has a very short lifetime because of... 

For batch reactors we are interested in deterjnining how long to leave the reactants in the reactor to achieve a certain conversion X To determine this length of time, we transform the mole balance. Equation (2-5), in terms of conversion by differentiating Equation (2-4),... 

We now must determine the ratio F/Fq a function of volume V or the catalyst weight, W to account for pressure drop. We then can combine the concentration, rate law, and design equation. However, whenever accounting for the effects of pressure drop, the differential form of the mole balance (design equation) must be used. 

Liquid Phase. For liquid-phase reactions in which there is no volume change, concentration is the preferred variable. The mole balances are shown in Table 4-5 in terms of concentration for the four reactor types we have been discussing. We see from Table 4-5 that we have only to specify the parameter values for the system (CAo,Uo,etc.) and for the rate law (i.e., ifcyv. .3) to solve the coupled ordiaaiy differential equations for either PFR, PBR, or batch reactors or to solve the coupled algebraic equations for a CSTR. 

Substituting Equation (4-55) in tenns of V and differentiating, tihe mole balance on B becomes... 

To outline the procedure used in the differential method of analysis, we consider a reaction carded out isothermally in a constant-volume batch reactor and the concentration recorded as a function of time. By combining the mole balance with the rate law given by Equation (5-1), we obtain... 

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