Dynamic balance equation

The coefficients of the above equations are the partial differentials of the two dynamic balance equations and are given by... 

The steady-state approach, however, provides no information on the initial transient conditions, whereby the extractor achieves eventual steady state or on its dynamic response to disturbances. For this it is necessary to derive the dynamic balance equations for the system. 

Equation (11) represents the time-discrete dynamic equivalent of the steady-state balance equations (2). The dynamic balance equations (11) present some characteristic properties of the sampled-data input and output relationship, that are not present in the corresponding steady-state equations 1) There are as many equations as the number of outputs 2) Each equation contain only one output 3) Each equation contain, except for special cases, all the inputs variables. 

The degrees of freedom of a process at dynamic state are equal in number or more than those at steady state. This is due to the fact that the dynamic balance equations contain the accumulation terms, whereas for steady-state balances the accumulation is zero. An incorrect estimate of the number of degrees of freedom can have a profound effect on the design of the appropriate controller. Consider the simple liquid holding tank of Example 10.1. The dynamic mass balance yields... 

The dynamic balance equation for the bubble, known as Rayleigh s equation, looks like... 

The electronic density n x) satisfies the Budd-Vaimimenus sum rule [158], which is the quantum mechanics version of the dynamic balance equation... 

In section 11.4 Che steady state material balance equations were cast in dimensionless form, therary itancifying a set of independent dimensionless groups which determine ice steady state behavior of the pellet. The same procedure can be applied to the dynamical equations and we will illustrate it by considering the case t f the reaction A - nB at the limit of bulk diffusion control and high permeability, as described by equations (12.29)-(12.31). 

Solution The initial liquid-phase concentration of oxygen is 0.219mol/m as in Example 11.1. The final oxygen concentration will be 1.05 mol/m. The phase balances. Equations (11.11) and (11.12), govern the dynamic response. The flow and reaction terms are dropped from the liquid phase balance to give... 

Dynamic differential equation balances were written to calculate the molar concentration of each species in the reactor. These equations consist of inflow, outflow, accumulation, and reaction terms for a CSTR. If there are no outflow terms, the equations reduce to semibatch... 

A dynamic differential equation energy balance was written taking into account enthalpy accumulation, inflow, outflow, heats of reaction, and removal through the cooling jacket. This balance can be used to calculate the reactor temperature in a nonisothermal operation. 

Liquid is fed continuously to a stirred tank, which is heated by internal steam coils (Fig. 1.21). The tank operates at constant volume conditions. The system is therefore modelled by means of a dynamic heat balance equation, combined with an expression for the rate of heat transfer from the coils to the tank liquid. 

Liquid flows continuously into an initially empty tank, containing a full-depth heating coil. As the tank fills, an increasing proportion of the coil is covered by liquid. Once the tank is full, the liquid starts to overflow, but heating is maintained. A total mass balance is required to model the changing liquid volume and this is combined with a dynamic heat balance equation. 

Force and velocity are however both vector quantities and in applying the momentum balance equation, the balance should strictly sum all the effects in three dimensional space. This however is outside the scope of this text and the reader is referred to more standard works in fluid dynamics. 

For complex reaction systems, the heats of reaction of all individual reactions have to be estimated and the dynamic heat balance equations must include the heats of all the reactions. 

The dynamic total mass balance equation is represented by... 

The component mass 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. 

Adding the above two component balance equations gives the dynamic equation for the complete stage as... 

An alternative approach to the solution of the system dynamic equations, is by the natural cause and effect mass transfer process as formulated, within the individual phase balance equations. This follows the general approach, favoured by Franks (1967), since the extractor is now no longer constrained to operate at equilibrium conditions, but achieves this eventual state as a natural consequence of the relative effects of solute accumulation, solute flow in, solute flow out and mass transfer dynamics. 

The coupling of the component and energy balance equations in the modelling of non-isothermal tubular reactors can often lead to numerical difficulties, especially in solutions of steady-state behaviour. In these cases, a dynamic digital simulation approach can often be advantageous as a method of determining the steady-state variations in concentration and temperature, with respect to reactor length. The full form of the dynamic model equations are used in this approach, and these are solved up to the final steady-state condition, at which condition... 

Dynamic Difference Equation for the Component Balance Dispersion Model... 

Model relationships include the mass balance equations for the gas and liquid phases and equations representing the measurement dynamics. 

Most chemical reactions do not progress completely from reactants to products. Instead, the net reaction stops in the forward direction when equilibrium is established. Analysis of the contents of the reaction vessel would show a constant concentration of monomers and polymer once equilibrium is reached. This situation is actually a dynamic equilibrium, where the monomers are forming polymers at the same rate as the polymers depolymerize to monomer. Therefore, at equilibrium, the net concentrations of any one species remains constant. The amount of monomer converted into polymer will be defined by the equilibrium constant, K. This constant is the ratio of the concentration of the products to the reactants, with each concentration raised to the stoichiometric coefficients in the balanced equation. For Eq. 3.5 ... 

The dynamic total material balance equation is given by... 

The dynamic component balance equations for each of the two phases in turn. 

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