Quasi-steady process

Fig. 8. The ratio of the drag force to the weight of an a-pinene droplet with initial diameter 29.8 /tm evaporating in nitrogen at 293 K. The solid line is the prediction based on Stokes law for the drag force on a sphere, assuming a quasi-steady process.

The issue of activity coefficient measurement for binary droplets was addressed by Allen et al. (1990). For low-vapor-pressure species, their diffusional fluxes in the gas phase are independent because of their low gas phase concentrations, and for a quasi-steady process each flux may be written... 

For a quasi-steady process, the mass flux is obtained from Eq. (97), for Jj = pidaldt, where pi is the droplet density. For Pa = 0 one obtains... 

For the subcritical pressure range of interest, gas-phase heat and mass diffusion rates are of the order of 10 -1 cm /sec while the liquid-phase heat transfer rate is of the order of 10" cm /sec, and the liquid surface area regression rate is approximately 10" -10" ctn /sec. Inasmuch as the gas-phase transfer rates are much faster than all of the liquid-phase transfer rates, gas-phase heat and mass transfer can be represented as quasi-steady processes. The validity of this quasi-steady approximation has been substantiated by the numerical study of Hubbard et al. (10). Furthermore, Law and Sirignano (6) have demonstrated that effects caused by the hquid surface regression during the droplet heating period are negligible relative to the liquid-phase heat conduction rate. 

Fuchs theory is restricted to a quasi-steady process, where the time dependence is neglected. The quasi-steady state approach of electrostatic retardation in adsorption will be described in Section 7.3. 

A maximum occurs in the characteristic displacement curves only when the surface tension relaxation is not as fast as compared to the capillary filling. More simplistic scaling estimates in the capillary rise phenomenon can readily be obtained by noting that within certain limits, the capillary rise represents a quasi-steady process, in which the amount of surfactant adsorbed to the solid/liquid interface per unit time is equal to that transported to the liquid/vapor interface by diffusion, which implies... 

The system under consideration is a pinned sessile droplet of a liquid (water in these simulations) oti a solid substrate open to ambient air. The problem is taken axisymmetric with a cylindrical system of coordinates r and z (Fig. 1). We focus attention on relatively small droplets, neglecting gravity force. Thus the droplet is a spherical cap. Diffusimi model of evaporation is taken to describe the quasi-steady process of droplet evaporation. The characteristic time scales of heat, ty gat 10s, and momentum, Is, transfer processes inside the droplet are smaller than the droplet evaporatimi time (lO s) at least by one order of magnitude. For this reason all those processes are taken as steady state processes. Convection in air caused by evaporation is neglected, because the experiments [21] did not reveal any difference in evaporation regimes with and without forced convection in the ambient air. 

The respiratory quotient (RQ) is often used to estimate metabolic stoichiometry. Using quasi-steady-state and by definition of RQ, develop a system of two linear equations with two unknowns by solving a matrix under the following conditions the coefficient of the matrix with yeast growth (y = 4.14), ammonia (yN = 0) and glucose (ys = 4.0), where the evolution of C02 and biosynthesis are very small (o- = 0.095). Calculate the stoichiometric coefficient for RQ =1.0 for the above biological processes ... 

Our treatment of chain reactions has been confined to relatively simple situations where the number of participating species and their possible reactions have been sharply bounded. Most free-radical reactions of industrial importance involve many more species. The set of possible reactions is unbounded in polymerizations, and it is perhaps bounded but very large in processes such as naptha cracking and combustion. Perhaps the elementary reactions can be postulated, but the rate constants are generally unknown. The quasi-steady hypothesis provides a functional form for the rate equations that can be used to fit experimental data. 

U-shaped curve, we have mixtures that can be ignited for a sufficiently high spark energy. From Equation (4.25) and the dependence of the kinetics on both temperatures and reactant concentrations, it is possible to see why the experimental curve may have this shape. The lowest spark energy occurs near the stoichiometric mixture of XCUi =9.5%. In principle, it should be possible to use Equation (4.25) and data from Table 4.1 to compute these ignitability limits, but the complexities of temperature gradients and induced flows due to buoyancy tend to make such analysis only qualitative. From the theory described, it is possible to illustrate the process as a quasi-steady state (dT/dt = 0). From Equation (4.21) the energy release term represented as... 

It is useful to point out here that we frequently encounter partial steady-states. An important example is the case where the diffusion process is much faster than a surface process, and thus a quasi-steady-state is reached for the diffusion concentration profile at each changing concentration of the surface. This distinction between different timescales of the processes can lead to a significant simplification of complex problems, see end of Section 4.3 or Chapter 4 in this volume. 

All of the previous ideas are developed further in Chapter 8, where the analysis of dynamic and quasi-steady-state processes is considered. Chapter 9 is devoted to the general problem of joint parameter estimation-data reconciliation, an important issue in assessing plant performance. In addition, some techniques for estimating the covariance matrix from the measurements are discussed in Chapter 10. New trends in this field are summarized in Chapter 11, and the last chapter is devoted to illustrations of the application of the previously presented techniques to various practical cases. 

All the previous ideas are developed further in Chapter 8, where the analysis of dynamic and quasi-steady-state processes is considered. 

The system is some physical object, and its behavior can normally be described by equations. The system can be dynamic (discrete or continuous) or static. Here, we will refer to a process under steady-state behavior. Later in this book we will extend our attention to considering dynamic or quasi-steady-state situations. 

In this chapter, the data reconciliation problem for dynamic/quasi-steady-state evolving processes is considered. The problem of measurement bias is extended to consider dynamic situations. Finally in this chapter, an alternative approach for nonlinear dynamic data reconciliation using nonlinear programming techniques will be discussed. 

In the previous chapters the data reconciliation problem was analyzed for systems that could be assumed to be operating at steady state. Consequently, only one set of data was available. In some practical situations, the occurrence of various disturbances generates a dynamic or quasi-steady-state response of the process, thus nullifying this steady-state assumption. In this chapter, the notions previously developed are extended to cover these cases. 

Under dynamic or quasi-steady-state conditions, a continuously monitored process will reveal changes in the operating conditions. When the process is sampled regularly, at discrete periods of time, then along with the spatial redundancy previously defined, we will have temporal redundancy. If the estimation methods presented in the previous chapters were used, the estimates of the desired process variables calculated for two different times, t and t2, are obtained independently, that is, no previous information is used in the generation of estimates for other times. In other words, temporal redundancy is ignored and past information is discarded. 

The second problem to be tackled is data reconciliation for applications in which the dominant time constant of the dynamic response of the system is much smaller than the period in which disturbances enter the system. Under this assumption the system displays quasi-steady-state behavior. Thus, we are concerned with a process that is essentially at steady state, except for slow drifts or occasional sudden transitions between steady states. In such cases, the estimates should be consistent, that is, they should satisfy the mass and energy balances. 

In this chapter different aspects of data processing and reconciliation in a dynamic environment were briefly discussed. Application of the least square formulation in a recursive way was shown to lead to the classical Kalman filter formulation. A simpler situation, assuming quasi-steady-state behavior of the process, allows application of these ideas to practical problems, without the need of a complete dynamic model of the process. 

T. A. Rapoport, R. Heinrich, and S. M. Rapoport, The regulatory principles of glycolysis in erythrocytes in vivo and in vitro, a minimal comprehensive model describing steady states, quasi steady states and time dependent processes. Biochem J. (1976). 

Diffusion. The transport process may consist of two parts, diffusion and convection. When the liquid is stagnant and resting relative to the particle the transport is done by diffusion only. A steady state is quickly established in the solution around the particle (4 ). (Strictly it is a quasi-steady state since the particle is growing ( 5)). At the particle surface the concentration gradient becomes equal to (c-cs)/r, which leads to the growth rate... 

When one gas diffuses into another, as A into B, even without the quasi-steady-flow component imposed by the burning, the mass transport of a species, say A, is made up of two components—the normal diffusion component and the component related to the bulk movement established by the diffusion process. This mass transport flow has a velocity Aa and the mass of A transported per unit area is pAAa. The bulk velocity established by the diffusive flow is given by Eq. (6.58). The fraction of that flow is Eq. (6.58) multiplied by the mass fraction of A, pA/p. Thus,... 

An interesting approach to the spray problem has been suggested by Chiu and Liu [29], who consider a quasi-steady vaporization and diffusion process with infinite reaction kinetics. They show the importance of a group combustion number (G), which is derived from extensive mathematical analyses and takes the form... 

Equation 8.4 predicts that aerobic respiration should release dissolved inorganic nitrogen and phosphorus into seawater in the same ratio that is present in plankton, i.e., 16 1. As shown in Figure 8.3, a plot of nitrate versus phosphate for seawater taken from all depths through all the ocean basins has a slope close to 16 1. Why do both plankton and seawater have an N-to-P ratio of 16 1 Does the ratio in seawater determine the ratio in the plankton or vice versa Current thinking is that the N-to-P ratio of seawater reflects a quasi steady state that has been established and stabilized by the collective impacts of several biological processes controlled by marine plankton. 

The conversion system is of batch type, over-fired, updraft air, and has a maximum capacity of 300 kWt. The methodology, described in Paper II, is based on the assumption of a steady-state conversion process, which is not the case for a batch reactor. Consequently, main assumption one above needs to be reconsidered and modified despite the fact that a batch conversion system is studied, which implies unsteady conditions, the process is assumed to be quasi-steady that is, the rate of change in the process variables in the range of interest is assumed to be slow compared with the response rate of the measurement system. 

Lazman M., Algebraic geometry methods in analysis of quasi steady state and dynamic models of cataljdic reactions. Proceedings of the 4th International Conference on Unsteady-State Processes in Catalysis USPC-4, Montreal, Quebec, Canada, October 26-29, 2003, Dr. H. Sapoundjiev (Ed.), Natural Resources Canada, 92-93 (2003a). 

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