Ordinary mass balance

The sum of k i and ki2 is so high relative to the rate of micromixing that A and B will not coexist. Thus the number of ordinary mass balance differential equations requiring integration can be reduced from five to four by defining the following composite variables ... 

A different approach to decribe the bubbly liquid uses mean fluid properties of the liquid-bubble mixture and starts with the ordinary mass balance... 

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. 

Here, a class of biochemical processes whose mass balances can be described by the following nonlinear ordinary differential equations are considered. 

This same analysis is then performed for the energy balance for the gas, the energy balance for the thermal well, the two mass balances, and the continuity equation. The final coupled system of algebraic and differential equations consists of 5N ordinary differential equations and N + 6(NE + 1) algebraic equations, where... 

Just as an ordinary chemical equation is a shortened version of the complete thermochemical equation which expresses both energy and mass balance, each nuclear equation has a term (written or implied) expressing energy balance. The symbol Q is usually used to designate the net energy released when all reactant and product particles of matter are at zero velocity. Q is the energy equivalent of the mass decrease (discussed above) accompanying the reaction. Q is usually expressed in MeV. 

The governing equations - that is, mainly the component and the total mass balances in the anode channels - are provided here in dimensionless form. The five ordinary differential equations (ODE) with respect to the spatial coordinate describe the development of the five unknowns in one single anode channel, namely the mole fractions, with i = CH4, H2O, H2, CO2, as well as the molar flow density inside the anode channel, y. Here, the Damkohler numbers, Da/, are the dimensionless reaction rate constant of the reforming and the oxidation reaction, respectively, the rj are the corresponding dimensionless reaction rates, and the v, j are the stoichiometric coefficients ... 

Quite often chemical engineering systems are encountered with widely different time constants, which give rise to both long-term and short-term effects. The corresponding ordinary differential equations have widely different eigenvalues. Differential equations of this type are known as stiff systems. Seader and Henley (1988) derived the expressions for maximum and minimum eigenvalues for the differential component mass balance equations related to intermediate plates and reboiler respectively. 

Any standard method of matrix inversion, such as the Gauss-Jordan method (N13), may be used to solve the equations. The coefficients in equations 4.11-4.14 may be used without serious error for most ordinary Portland cement clinkers in which the alite composition is not too different from that assumed here. As a byproduct of the calculation described in this section, and using the full compositions of the phases given in Table 1.2, one may calculate a mass balance table (Table 4.3) showing the distributions of all the oxide components among the phases. 

For the mass balance, the following assumptions and operating conditions are considered (i) steady state, (ii) unidirectional, incompressible, continuous flow of a Newtonian fluid under laminar flow regime, (iii) only ordinary diffusion is significant for a mixture where the main component is water, (iv) azimuthal symmetry, (v) axial diffusion neglected as... 

When only the feed side and permeate side mass balance equations are considered under the isothermal condition, the resulting equations arc a set of first-order ordinary differential equations. Furthermore, a co-current purge stream renders the set of equations an initial value problem and well established procedures such as the... 

The CNMMR model with laminar flow liquid stream in the annular region consists of three ordinary differential equations for the gas in the tube core and two partial differential equations for the liquid in the annular region. These equations are coupled through the diffusion-reaction equations inside the membrane and boundary conditions. The model can be solved by first discretizing the liquid-phase mass balance equations in the radial direction by the orthogonal collocation technique. The resulting equations are then solved by a semi-implicit integration procedure [Harold etal., 1989]. 

For a CSTR the mass balance for the gas phase is the following ordinary differential equation ... 

As the plant to be optimized considers a process operating at steady state, then the variation of the phase concentrations with time is zero. For this reason, the mathematical model that describes the plant is a set of ordinary differential equations, as the phase concentrations depend only on the module axial position. In the tanks, the concentrations are constant. The differential-algebraic nonlinear optimization (DNLP) problem PI to be solved includes the ordinary differential equations that represent the mass balances for the phases in the membrane module. The objective function to be maximized is the amount of metal processed FeC , where Fe is the effluent flow rate whose Cr(VI) concentration after dilution from wastewaters is C . The problem has the following form ... 

Data evaluation Model parameters were obtained by fitting of experimental time dependencies of pressure in the lower cell compartment to theory. Obtaining of theoretical time - pressure courses represents integration of mass balance (partial differential equation, or, assuming pseudo-steady-state, ordinary differential equation). 

These reactions can be mathematically represented in terms of mass balance of each species using a set of ordinary differential equations as ... 

With respect to a total mass balance, in this book the generation and consumption terms are zero whether a chemical reaction occurs in the system or not (we neglect the transfer between mass and energy in ordinary chemical processing) hence... 

The basis of the flow models are ordinary differential mass balances for each component on either side of the membrane. The mechanism for permeation is substituted in the mass balances. When reaction occurs the kinetic expression is also added to the balances. The chemical reaction is assumed to take place in... 

To describe this problem in mathematical terms, either the differential species mass balance (1.39) can be reduced appropriately or alternatively a species mass shell balance over a thin layer, Az, can be put up and combined with Pick s law. The resulting equation for steady diffusion in the thin layer is of course the same in both cases. The simple ordinary differential equation is integrated twice with the appropriate boundary conditions in order to get a relation for the concentration profile that is needed to determine the diffusive flux. 

To develope a model for this mathematical problem we can either simplify the differential species mass balance equation (1.39) appropriately or combine the transient shell species mass balance written for the thin layer Az with Pick s law for binary diffusion. The resulting partial differential equation is called Pick s second law. A simple way to obtain a solution for this differential equation is to adopt the method of combination of variables. It is then necessary to define a new independent variable that enable us to transform the partial differential equation into an ordinary differential equation. 

Recently, Agrawal et al. (2006) had modeled the LDPE reactor as an ideal plug flow reactor and presented all the model equations and parameters for use by researchers. The model equations include ordinary differential equations for overall and component mass balances, energy balance and momentum balance. The reactor model of Agrawal et al. (2006) is adopted, and cost expressions and economic objectives are... 

Based on mass balances, a reaction network can be described in terms of linear equations, with parameters representing the stoichiometric coefficients and variables, the metabolic fluxes. The result is a system of ordinary differential equations,... 

The Matlab Simulink Model was designed to represent the model stmctuie and mass balance equations for SSF and is shown in Fig. 6. Shaded boxes represent the reaction rates, which have been lumped into subsystems. To solve the system of ordinary differential equations (ODEs) and to estimate unknown parameters in the reaction rate equations, the inter ce parameter estimation was used. This program allows the user to decide which parameters to estimate and which type of ODE solver and optimization technique to use. The user imports observed data as it relates to the input, output, or state data of the SimuUnk model. With the imported data as reference, the user can select options for the ODE solver (fixed step/variable step, stiff/non-stiff, tolerance, step size) as well options for the optimization technique (nonlinear least squares/simplex, maximum number of iterations, and tolerance). With the selected solver and optimization method, the unknown independent, dependent, and/or initial state parameters in the model are determined within set ranges. For this study, nonlinear least squares regression was used with Matlab ode45, which is a Rimge-Kutta [3, 4] formula for non-stiff systems. The steps of nonlinear least squares regression are as follows ... 

Several of these simple mass balances with basic rate expressions were solved analytically. In the case of multiple reactions with nonlinear rate expressions (i.e., not first-order reaction rates), the balances must be solved numerically. A high-quality ordinary differential equation (ODE) solver is indispensable for solving these problems. For a complex equation of state and nonconstant-volume case, a differential-algebraic equation (DAE) solver may be convenient. 

A dynamic model of (his process contains two ordinary differential equations, which arise from the total mass balance on each of the tanks. We assume constant density. 

The temperature-dependent physical constants in the mass balance (i.e., the kinetic rate constant and the equilibrium constant) are expressed in terms of nonequilibrium conversion x using the linear relation (3-42). The concept of local equilibrium allows one to rationalize the definition of temperature and calculate an equilibrium constant when the system is influenced strongly by kinetic changes. In this manner, the mass balance is written with nonequilibrium conversion of CO as the only dependent variable, and the problem can be solved by integrating only one ordinary differential equation for x as a function of reactor volume. 

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