Mass balance differential form

Schmid et al. studied in detail the sulfonation reaction of fatty acid methyl esters with sulfur trioxide [37]. They measured the time dependency of the products formed during ester sulfonation. These measurements together with a mass balance confirmed the existence of an intermediate with two S03 groups in the molecule. To decide the way in which the intermediate is formed the measured time dependency of the products was compared with the complex kinetics of different mechanisms. Only the following two-step mechanism allowed a calculation of the measured data with a variation of the velocity constants in the kinetic differential equations. 

In considering the flow in a pipe, the differential form of the general energy balance equation 2.54 are used, and the friction term 8F will be written in terms of the energy dissipated per unit mass of fluid for flow through a length d/ of pipe. In the first instance, isothermal flow of an ideal gas is considered and the flowrate is expressed as a function of upstream and downstream pressures. Non-isothermal and adiabatic flow are discussed later. 

A dynamic model should be consistent with the steady-state model. Thus, Eqs (1) and (4) should be extended to dynamic form. For the better convergence and computational efficiency, some assumption can be introduced the total amounts of mass and enthalpy at each plate are maintained constant. Then, the internal flow can be determined by total mass balance and total energy balance and the number of differential equations is reduced. Therefore, the dynamic model can be established by replacing component material balance in Eq. (1) with the following equation. 

The Navier-Stokes equations have a complex form due to the necessity of treating many of the terms as vector quantities. To understand these equations, however, one need only recognize that they are not mass balances but an elaboration of Newton s second law of motion for a flowing fluid. Recall that Newton s second law states that the vector sum of all the forces acting on an object ( F) will be equal to the product of the object s mass (m) and its acceleration (a), or XF = ma. Now consider the first of the three Navier-Stokes equations listed above, Eq. (10). The object in this case is a differential fluid element, that is, a small cube of fluid with volume dx dy dz and mass p(dx dy dz). The left-hand side of the equation is essentially the product of mass and acceleration for this fluid element (ma), while the right-hand side represents the sum of the forces... 

Definitions for the variables and constants appearing in eqns. 1 and 2 are given in the nomenclature section at the end of this paper. The first of these equations represents a mass balance around the reactor, assuming that it operates in a differential manner. The second equation is a species balance written for the catalyst surface. The rate of elementary reaction j is represented by rj, and v j is the stoichiometric coefficient for component i in reaction j. The relationship of rj to the reactant partial pressures and surface species coverages are given by expressions of the form... 

Generally for modelling chromatograph systems, component mass balances are required for each component in each phase. The differential liquid phase component balances for a chromatographic column with non-porous packing take the partial differential equation form... 

Another key point of differentiation is the fact that nearly all PSA separations are bulk separations and any investigator interested in a high fidelity description of the problem of adsorption must solve a mass balance equation such as Eq. (9.9), the bulk separation equation, together with the uptake rate model and a set of thermal balance equations of similar form. In addition to the more complicated pde and its attendant boundary and initial conditions the investigator must also solve some approximate form of a momentum balance on the fluid flow as a whole. 

An alternative scheme, proposed by Garside et al. (16,17), uses the dynamic desupersaturation data from a batch crystallization experiment. After formulating a solute mass balance, where mass deposition due to nucleation was negligible, expressions are derived to calculate g and kg in Equation 3 explicitly. Estimates of the first and second derivatives of the transient desupersaturation curve at time zero are required. The disadvantages of this scheme are that numerical differentiation of experimental data is quite inaccurate due to measurement noise, the nucleation parameters are not estimated, and the analysis is invalid if nucleation rates are significant. Other drawbacks of both methods are that they are limited to specific model formulations, i.e., growth and nucleation rate forms and crystallizer configurations. 

This is the differential form of the mass balance equation in three dimensions. Since J can be written as pu, where u is the flow velocity of the fluid, the above equation can also be written as... 

Another potential solution technique appropriate for the packed bed reactor model is the method of characteristics. This procedure is suitable for hyperbolic partial differential equations of the form obtained from the energy balance for the gas and catalyst and the mass balances if axial dispersion is neglected and if the radial dimension is first discretized by a technique such as orthogonal collocation. The thermal well energy balance would still require a numerical technique that is not limited to hyperbolic systems since axial conduction in the well is expected to be significant. 

The computer-reconstructed catalyst is represented by a discrete volume phase function in the form of 3D matrix containing information about the phase in each volume element. Another 3D matrix defines the distribution of active catalytic sites. Macroporosity, sizes of supporting articles and the correlation function describing the macropore size distribution are evaluated from the SEM images of porous catalyst (Koci et al., 2006 Kosek et al., 2005). Spatially 3D reaction-diffusion system with low concentrations of reactants and products can be described by mass balances in the form of the following partial differential equations (Koci et al., 2006, 2007a). For gaseous components ... 

We restrict our discussion to those systems of n linear differential equations that evolve from the construction of mass balance models for one or several chemicals in one or several environmental compartments (boxes). Such systems are always of the form ... 

The high activation energies resulting from the chemisorption hypothesis appear to favor its validity. However, it is possible that such a high activation energy could be caused by an equilibrium between the undecomposed coal molecules and the diffusion species resulting from thermal decomposition. That is, for diffusion in the Z dimension only, a simplified mass balance would lead to a differential equation of the form... 

Substitution of the flux equations and the equations of state in the equations for the mass balance leads to set of non-linear differential equations. Dimensional analysis [1] shows that these equations, expressed in terms of pressure and concentration, can be reduced to a simple form when the storage parameters, i.e. soil compressibility a and liquid compressibility f3, are relatively small ... 

The model-based controller-observer scheme requires to solve online the system of differential equations of the observer. The phenol-formaldehyde reaction model is characterized by 15 differential equations, and it is, thus, unsuitable for online computations. To overcome this problem, one of the reduced models developed in Sect. 3.8.1 can be adopted. In order to be consistent with the general form of nonchain reactions (2.27) adopted to develop the controller-observer scheme, the reduced model (3.57) with first-order kinetics has been used to design the observer. The mass balances of the reduced model are given by... 

It should be understood that this rate expression may in fact represent a set of diffusion and mass transfer equations with their associated boundary conditions, rather than a simple explicit expression. In addition one may write a differential heat balance for a column element, which has the same general form as Eq. (17), and a heat balance for heat transfer between particle and fluid. In a nonisothermal system the heat and mass balance equations are therefore coupled through the temperature dependence of the rate of adsorption and the adsorption equilibrium, as expressed in Eq. (18). 

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 ... 

One source of nonlinear compartmental models is processes of enzyme-catalyzed reactions that occur in living cells. In such reactions, the reactant combines with an enzyme to form an enzyme-substrate complex, which can then break down to release the product of the reaction and free enzyme or can release the substrate unchanged as well as free enzyme. Traditional compartmental analysis cannot be applied to model enzymatic reactions, but the law of mass-balance allows us to obtain a set of differential equations describing mechanisms implied in such reactions. An important feature of such reactions is that the enzyme... 

The macroscopic approach, in which it is not taken into account what happens inside the cell in detail, but only an overall view of the system is described. In fact, the system is considered as a black box from the fluid dynamic point of view and then, it is assumed that the cell behaves a mixed tank reactor (the values of the variables only depend on time and not on the position since only one value of every variable describes all positions). This assumption allows simplifying directly all the set of partial differential equations to an easier set of differential equations, one for each model species. For the case of a continuous-operation electrochemical cell, the mass balances take the form shown in (4.5), where [.S, ]... 

Reactive Organic Chemical Mass Balance (Friedlander). In the original formulation of the CMB receptor model (1) it was recognized that the fractional amounts of various chemical species emitted by a source are not necessarily conserved during the transport of the species to the receptor site. This could occur through both physical (differential dispersion or deposition) or chemical (removal due to atmospheric reactions) processes. This possibility was acknowledged by writing the CMB equations in the form... 

Employing a high recirculating flow rate in this small laboratory reactor, the following assumptions can be used (i) there is a differential conversion per pass in the reactor, (ii) the system is perfectly stirred, (iii) there are no mass transport limitations. Also, it can be assumed that (iv) the chemical reaction occurs only at the solid-liquid interface (Minero et al., 1992) and (v) direct photolysis is neglected (Satuf et al., 2007a). As a result, the mass balance for the species i in the system takes the following form (Cassano and Alfano, 2000) ... 

Frequently the integral form of the conservation la v of the property is particularized as total and partial mass balance and also as energy or thermal balance [3.7]. For each particularization, a control volume must be selected in order to have a form capable of permitting the computation of each integral from the relation (3.5). As an initial condition, we have to declare the property, the transport vector and the property generation rate. Figure 3.2 presents the way to obtain the equations of the differential balance of total mass, mass species and energy (heat). The... 

Consider first the tubular reactor. From the material balance (Table 3.5.1), it is clear that in order to solve the mass balance the functional form of the rate expression must be provided because the reactor outlet is the integral result of reaction over the volume of the reactor. However, if only initial reaction rate data were required, then a tubular reactor could be used by noticing that if the differentials are replaced by deltas, then ... 

The differential rate equation is — dc Jdt = UcaCk, and the mass balance equation is c° - Ca = Cr - Cr. Eliminating Ca between these equations and integrating gives the usual second-order integrated equation, which can be written in this form ... 

The distributed model is completed by forming a differential mass balance for the drug solute in a manner completely analogous to that shown previously in deriving Equation 9.14, except for the inclusion of an additional term describing convective flow ... 

A total mass balance necessarily has the form [accumulation = input - output], since mass can neither be generated nor consumed. The accumulation term is always dM < dt. where M t) is the mass of the system contents. Once you have determined M(t) by solving the differential balance equation, you may have to verify that the mathematical solution remains within the bounds of physical reality—that it does not become negative, for example, or that it does not exceed the total capacity of the system. 

It is often convenient to assume a closed chemical system, which imposes one mass balance constraint for each chemical component. In differential form, each equation is... 

This is the conservation of mass relation in differential form, which is also known as the continuity equation or mass balance for steady two-dimensional flow of a fluid with constant density. 

The differential forms of the equations of motion in the velocity boundary layer are obtained by applying Newton s second law of motion to a differential conlrol volume element in the boundary layer. Newton s second law is an expression for momenluin balance and can be slated as the net force acting on the control volume is equal to the mass times the acceleration of the fluid element wilhht the control volume, which is also equal to the net rate of momentum outflow from the control volume. 

For a given waste treatment process configuration, models can be developed by combining the basic kinetic expressions (Equations 1 through 5) with materials balances on substrate and microbial mass for the system. While the mass balances are written in differential form, it is common to use the steady-state form of these expressions—i.e., dX/dt = 0 or dS/dt = 0. This approach appears to be quite satisfactory for design considerations. 

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 ... 

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