Rate equation assuming steady-state

This method can also be used to calculate the catalyst retention factor. The above equations assume steady-state operation, constant unit inventory, and constant addition and loss rate. 

Closure normally begins by satisfying the overall mass balance i.e., by equating the input and outlet mass flow rates for a steady-state system. For the present case, the outlet flow was measured. The inlet flow was unmeasured so it must be assumed to be equal to the outlet flow. We suppose that A and B are the only reactive components. Then, for a constant-density system, it must be that... 

It is important to note however that Equation (10) assumes steady state in the Th distribution so that production truly is balanced by decay and export. It is easy to imagine a scenario after a phytoplankton bloom, when the export of POC (and " Th) has decreased or even ceased, such that the water column " Th profile would still show a deficit with respect to caused by prior high flux events. This relief deficit will disappear as " Th grows into equilibrium with on a time scale set by the " Th half-life. The magnitude by which the Th flux is over- or under-estimated depends on whether deficits are increasing or decreasing and at what rate. 

Continuity and rate equations can also be written in a cylindrical coordinate system for the two-dimensional annular chromatography. Assuming steady state and neglecting velocity and concentration variations in the radial direction, the above-mentioned equations may then be written as... 

Ion-molecule association reactions and the collisional deactivation of excited ions have been the subjects of recent reviews.244-246 Several systematic studies have been performed in which the relative deactivating efficiencies of various Mf species have been determined. By applying the usual kinetic formulations for the generalized reaction scheme of equation (11.31), and assuming steady-state conditions for (AB+), an expression for the low-pressure third-order rate coefficient can be derived ... 

Estimates of dosing rate and average steady-state concentrations, which may be calculated using clearance, are independent of any specific pharmacokinetic model. In contrast, the determination of maximum and minimum steady-state concentrations requires further assumptions about the pharmacokinetic model. The accumulation factor (equation [7]) assumes that the drug follows a one-compartment body model (Figure 3-2 B), and the peak concentration prediction assumes that the absorption rate is much faster than the elimination rate. For the calculation of estimated maximum and minimum concentrations in a clinical situation, these assumptions are usually reasonable. 

Under direct Lnm excitation and assuming that the population of the ground state (Ng) of the emitting 4f ion decays rapidly to populate the first excited state (Ne), the rate equations reduce to those of a two-level system, and further assuming steady state, one gets... 

The procedure, in analyzing kinetic data by numerical integration, is to postulate a reasonable kinetic scheme, write the differential rate equations, assume estimates for the rate constants, and then to carry out the integration for comparison of the calculated concentration-time curves with the experimental results. The parameters (rate constants) are adjusted to achieve an acceptable fit to the data. Carpen-tejAs. pp. 76-81 si Q s some numerical calculations. Farrow and Edelson and Porter and Skinner used numerical integration to test the validity of the steady-state approximation in complex reactions. 

The kinetic rate expression for hydrolysis is derived by assuming steady-state for all reaction Intermediates. Assiimlng further that the rate of hydrolysis Is first-order In ether, the following equations are obtained ... 

Starting from total reflux conditions, the distillate rate is incremented from zero to D, thereby lowering the reflux rate to Lq - D. Since negligible fluid dynamic lags are assumed, all the liquid rates are instantly lowered to Lj - D. The vapor rates are maintained at Vj. These flow rates and the steady-state total reflux mole fractions are used to calculate the mole fraction derivatives by Equations 17.29 through 17.31. The molar holdups in these equations are calculated from the assumed constant volume holdups multiplied by calculated molar densities. 

Assuming steady state but no rate determining step, Langmuir adsorption and single site reactions leads to the following equation for the rate of disappearance of Aj (viz Appendix A) ... 

For a heterogeneous perfectly mixed system working continuously and after steady state conditions have been reached, the rate equation assuming constant density... 

At the start of the polymerization, the rate of formation of radicals greatly exceeds the rate at which they are lost by termination. However, [M"] increases rapidly and so the rate of loss of radicals by termination increases. A value of [M ] is quickly attained at which the latter rate exactly equals the rate of radical formation. The net rate of change in [M ] is then zero and the reaction is said to be under steady-state conditions. In practice, most free-radical polymerizations operate under steady-state conditions for all but the first few seconds. It is, therefore, quite satisfactory to assume steady-state conditions and to set d[M ]/d/ = 0 in Equation (1.3). This yields the following equation for the steady-state concentration of chain radicals ... 

The rate equation, assuming a steady-state population of the precusor complex... 

In this equation is the purely transport controlled limiting current, D is the diffusion coefficient assumed to be identical for all species, and k is the first-order rate constant for the chemical reaction step. A plot of the observed limiting currents versus 5 determined from the first one-electrode process is shown in Fig. 10. A theoretical line can be fitted to the experimental points and a rate constant of k = 600 100 s-i is in good agreement with the value reported in the literature, k = 740 200 s [67]. For the reduction of ortho-bromonitrobenzene, a rate constant ofk = 200 50 s was determined, which is also in close agreement with the literature value of k = 250 s [71]. The implication is that ultrasound facilitates the measurement of fast rate constants under steady state conditions at... 

After inserting the expressions for the rates of the termination reaction [(20-42)], of the propagation reaction [(20-55)], and of the transfer reaction [equation (20-113)], and assuming steady-state conditions (u pp = =... 

In view of the comnents above it is not surprising that solution polymerization dominates most discussions of polymerization kinetics, e.g. in ref. 8 the extensive discussion includes four sentences on solid-state polymerization, which is dismissed as only of academic interest. The application of diacetylenes as time-tenqperature indicators seems likely to change this. The basic ideas of conventional discussions are, however, relevant even in the solid state. The polymerization is vie% as three distinct processes, (a) initiation, (b) propagation and (c) termination of the polymer chain. The details of these processes depend on the type of polymerization reaction involved. The polymerization kinetics are then deduced by solving the rate equations for the three processes. In order to do this it is usual to assume steady-state conditions, when the population of propagating polymer chains remains constant as a result of an exact balance between the rates of initiation and termination (8). 

The balance equations [APP 64] enable us to estabhsh the entropy production rate, from which we deduce the form of the phenomenological relations by way of linearized TIP. Note that with regard to the chemical reactions, there is no need to linearize the stem, because the hterature generally gives us nonlinear chemical kinetic laws with known coefficients (the specific reaction rates). In addition, the balance laws give us the system of equations to be solved in order to determine the fields of variables characterizing the evolution of the plasma in space and time, by means of the knowledge of the physical coefficients. Here, we shall assume steady-state one-dimensional evolution. 

MJ and [AJ are the concentrations of monomer and alkyl at the surface. If we assume steady-state surface concentrations of both monomer and alkyl and equate the rates of adsorption and desorption of each we get... 

Table III lists the material properties of the components. Fibers and matrix were considered as isotropic, and in a first step the Norton-equation for steady state creep was assumed. Because of missing parameters for compression creep, tension creep data for the fibers were adapted from elsewhere The data for the matrix was estimated as follows. Based on the experimental results for the 0° and 90 fiber orientation the overall creep rate for the matrix was chosen higher and the stress exponent lower than for the fibers.

The (CEF) model (see Chapter 1) provides a simple means for obtaining useful results for steady-state viscometric flow of polymeric fluids (Tanner, 1985). In this approach the extra stress in the equation of motion is replaced by explicit relationships in terms of rate of strain components. For example, assuming a zero second normal stress difference for veiy slow flow regimes such relationships arc written as (Mitsoulis et at., 1985)... 

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