Material balance coupled

The differential equations governing a non-isothermal batch reaction describe the material balance coupled to the heat balance (see also Section 2.4.2) ... 

Separate liquid phase and vapor phase component material balances, coupled with... 

Material balance coupling with vapor and liquid flow... 

Based upon this set of elementary reactions, a series of coupled kinetic differential equations may be derived by taking material balances over the various reaction species, as shown below ... 

The reality of the situation is that the maximum discharge rate of gas occurs when the leak first occurs, with the discharge rate decreasing as a function of time as the pressure within the tank decreases. The complete dynamic solution to this problem is difficult, requiring a mass discharge model cross-coupled to a material balance on the contents of the tank. An equation of state (perhaps nonideal) is required to determine the tank pressure given the total mass. 

These rate laws are coupled through the concentrations. When combined with the material-balance equations in the context of a particular reactor, they lead to uncoupled equations for calculating the product distribution. For a constant-density system in a CSTR operated at steady-state, they lead to algebraic equations, and in a BR or a PFR at steady-state, to simultaneous nonlinear ordinary differential equations. We demonstrate here the results for the CSTR case. 

The solution of this set of equations, 18.4-26 (with expression (A) incorporated) to -29, must be coupled with the set of three independent material-balance or continuity equations to determine the concentration profiles of three independent species, and the temperature profile, for either a specified size (V) of reactor or a specified amount of reaction. A nu-merical solution of the coupled differential equations and property relations is required. Equations (A), (B), and (C) in Example 18-6 illustrate forms of the continuity equation. 

For reactions involving heat effects, the total and component material balance equations must be coupled with a reactor energy balance equation. Neglecting work done by the system on the surroundings, the energy balance is expressed by where each term has units of kj/s. For steady-state operation the accumulation... 

The component material balance, when coupled with the heat balance equation and temperature dependence of the kinetic rate coefficient, via the Arrhenius relation, provide the dynamic model for the system. Batch reactor simulation examples are provided by BATCHD, COMPREAC, BATCOM, CASTOR, HYDROL and RELUY. 

Subsequently, the condition of complete separation has to be coupled with the material balances derived for the nodes of the SMB unit and implemented in the Equilibrium Theory Model for Langmuir-type systems. That leads to the set of mathematical conditions given below, which the flow rate ratios have to fulfil in order to achieve complete separation, in particular ... 

The fraction of gas or vapour in the two-phase mixture entering the pressure relief system is an important parameter if the detailed pressure/ time history for the vented runaway reaction is to be predicted. This type of calculation is performed by a number of computer codes (see Annex 4). In order to predict the gas or vapour fraction in the two-phase flow entering the relief system, the "coupling equation", a material balance for the vessel and relief system, needs to be solved iteratively. This is discussed in detail by DIERS[5J and is beyond the scope of this Workbook. 

The coupled equations, Eq. (103) and Eq. (104), will be the general model for all three-dimensional electrodes. The relationship between the charge and material balances can be derived as... 

According to the coupled equation, Eqs. (103) and (104), with the polarization Eq. (41), a charge balance coupled with a material balance for a redox pair in a volume element will be... 

These basic rate models were Incorporated Into a differential mass balance In a tubular, plug-flow reaction. This gives a set of coupled, non-llnear differential equations which, when Integrated, will provide a simulation model. This model corresponds to the Integral reactor data provided by experimentation. A material balance Is written for each of the four components In our system ... 

This energy balance when coupled with material balances. Equations... 

The coupling of diffusion and the ion-exchange process is obtained by the material balance... 

Once formed, the dihydrofuryl radicals may undergo a variety of reactions as known from homogeneous chemistry including disproportionation, addition to double bonds, electron transfer and dimerization. Surprisingly, the latter pathway is followed to about 90 % as indicated by the complete material balance [145]. It is therefore likely that C-C coupling occurs between radicals within the water-2,5-DHF surface layer. This is corroborated by the quadratic dependence of the initial... 

Other materials balance equations may be coupled with Eq. (5) in an way analogous to the monomer balance equation. This is necessary if the availability of some species changes significantly with time, e.g., when the initiator has a half-life comparable to (or shorter than) the polymerization half-life. 

The coupling of the flux and the time dependence of the counter-ion concentration is achieved through the material balance condition and Pick s Second Law (often termed the condition of continuity). For spherical particles and a constant interdifjusion coefficient, D, the foregoing considerations give the following partial differential equation for particle diffusion kinetics ... 

On the other hand, if you cannot avoid solving a set of simultaneous coupled equations because the set you are working with is tightly coupled in its initial structure and cannot be simplified even by successive substitution of one equation into another, you should read Sec. 2.7, in which computer-aided solutions are discussed. As explained in that section, you can use readily available software in computer center libraries or the computer codes in the pocket in the back of this book to solve sets of independent material balances. 

In Sec. 2.3 you learned how to formulate uncoupled material balances that could be solved one equation at a time for the values of the unknown variables. All the equations were quite simple, and you did not have to solve two or more equations (called irreducible equations because of the nature of their coupling) simultaneously. You should have observed in Sec. 2.3 that the information provided with respect to the problems such as flows and concentrations was of a special structure that led to the... 

Many of the problems that you will work with will lead to sets of equations that do have to be solved simultaneously. The purpose of this section is to show you that the formulation of such problems follows the same strategy as used in Sec. 2.3, the strategy outlined in Table 2.4. If only two or three coupled linear material balances are written for a problem, the unknown variables can be solved for by substitution of one equation into another. If the material balances consist of large sets of linear equations, you will find suggestions for solving them in Appendix L, and Fortran computer codes to solve them on the disk in the pocket in the back of this book. A single material balance that is a nonlinear equation of the form... 

The simplicity of equilibrium-based theories results from the fact that the coupled hrst-order partial differential equations (material balances) can be recast as two ordinary differential equations (but these are still coupled). The mathematical technique employed is called the method of characteristics. The results are... 

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