Quasi-steady predictions

This procedure constitutes an application of the steady-state approximation [also called the quasi-steady-state approximation, the Bodenstein approximation, or the stationary-state hypothesis]. It is a powerful method for the simplification of complicated rate equations, but because it is an approximation, it is not always valid. Sometimes the inapplicability of the steady-state approximation is easily detected for example, Eq. (3-143) predicts simple first-order behavior, and significant deviation from this behavior is evidence that the approximation cannot be applied. In more complex systems the validity of the steady-state approximation may be difficult to assess. Because it is an approximation in wide use, much critical attention has been directed to the steady-state hypothesis. 

Thus, it should be noted that the flame propagation in combustible vortex rings is not steady, but "quasi-steady" in the strict sense of the word. This may explain why prediction 9, based on the momentum flux conservation can better describe the flame speed for large values of Vg than prediction 4, which adopts the Bernoulli s equation on the axis of rotation. 

A paper by Ozturk, Palsson, and Dressman (OPD), reporting a refinement of the MMSH model, did create some controversy. OPD developed a film model with reaction in spherical coordinates and applied quasi-steady-state assumptions to the boundary conditions at the solid surface [11], They theorized that the flux of all species at the solid surface must be zero, except for HA, or the other species (A-, H+, OH ) would penetrate the solid surface. A debate by correspondence in the Letters to the Editor columns of Pharmaceutical Research ensued [12,13], The reader is invited to evaluate which author s arguments are more convincing. What is difficult to evaluate is whether the OPD model produces dissolution results which are different from those which would be predicted using the MMSH model cast in comparable spherical geometry. Simply, these authors never graphically demonstrate how their model predictions compare to the MMSH model. Algebraically, the solutions to both models appear comparable. 

Recently, Razumovskid441 studied the shape of drops, and satellite droplets formed by forced capillary breakup of a liquid jet. On the basis of an instability analysis, Teng et al.[442] derived a simple equation for the prediction of droplet size from the breakup of cylindrical liquid jets at low-velocities. The equation correlates droplet size to a modified Ohnesorge number, and is applicable to both liquid-in-liquid, and liquid-in-gas jets of Newtonian or non-Newtonian fluids. Yamane et al.[439] measured Sauter mean diameter, and air-entrainment characteristics of non-evaporating unsteady dense sprays by means of an image analysis technique which uses an instantaneous shadow picture of the spray and amount of injected fuel. Influences of injection pressure and ambient gas density on the Sauter mean diameter and air entrainment were investigated parametrically. An empirical equation for the Sauter mean diameter was proposed based on a dimensionless analysis of the experimental results. It was indicated that the Sauter mean diameter decreases with an increase in injection pressure and a decrease in ambient gas density. It was also shown that the air-entrainment characteristics can be predicted from the quasi-steady jet theory. 

Similar to Yamamoto et al. [60], Tennikova and co-workers [55], used the so-called quasi-steady state approach to predict SMC chromatography. The basic equation used in their modeling was the dependence of the zone migration on the composition of the mobile phase and on the gradient function, which in its differential form is given by ... 

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. 

Once precipitation begins, a quasi-steady state will eventually be attained in which the soil pe and pH are poised by the redox and precipitation equilibria operating. In the transition to the steady state, protons will be provided by dissociation of acids in the soil solution—e.g. H2CO3 derived from C02-and by reactions with the soil exchange complex. The course of reduction and the eventual steady state will depend on these reactions and it is therefore necessary to allow for them in predicting what the steady state conditions will be. 

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.

These findings validate the approach of Krier et al. (53) when they adopted the collapsed A/PA—GDF model to predict low frequency instability behavior of composite solid propellants at normal rocket pressure. Since m = 1.5 X 10 5 sec. at these pressures, their quasi-steady treatment of the O/F flame reaction time is valid for low freqeuncies above about 5000 c.p.s., the dynamic lag of the O/F flame must be taken into account—the dynamic lag of the A/PA flame need be considered only when frequencies approach 100 k.c.p.s. 

When calculating the characteristic coefficient, ku, kn, k2[, k22, of Box 23.3 you will find that the response velocity of the sediment reservoir is much smaller than that of the open water column. In order to predict the decrease of both concentrations, Cssc and C°p, you can assume a quasi-steady state between the two concentrations in which the system is controlled by the decrease of the slowly reacting sediment reservoir. 

The basic parameters which determine the kinetics of internal oxidation processes are 1) alloy composition (in terms of the mole fraction = (1 NA)), 2) the number and type of compounds or solid solutions (structure, phase field width) which exist in the ternary A-B-0 system, 3) the Gibbs energies of formation and the component chemical potentials of the phases involved, and last but not least, 4) the individual mobilities of the components in both the metal alloy and the product determine the (quasi-steady state) reaction path and thus the kinetics. A complete set of the parameters necessary for the quantitative treatment of internal oxidation kinetics is normally not at hand. Nevertheless, a predictive phenomenological theory will be outlined. 

The paper outlines a general procedure for predicting the dynamic response, including resonance, of hyperbolic cooling towers to turbulent winds. Pressure spectra on the tower surface were measured in a boundary layer wind tunnel. Application to full-scale tower is examined. It is concluded that while the quasi-steady response increases with the wind velocity squared, the resonant response increases faster than wind velocity cubed. 10 refs, cited. 

Results from a quasi steady-state model (QSSM) valid for long crystals and a constant melt level (if some form of automatic replenishment of melt to the crucible exists) verified the correlation (equation 39) for the dependence of the radius on the growth rate (144) and predicted changes in the radius, the shape of the melt-crystal interface (which is a measure of radial temperature gradients in the crystal), and the axial temperature field with important control parameters like the heater temperature and the level of melt in the crucible. Processing strategies for holding the radius and solid-... 

The dynamic stability of the quasi steady-state process suggests that active control of the CZ system has to account only for random disturbances to the system about its set points and for the batchwise transient caused by the decreasing melt volume. Derby and Brown (150) implemented a simple proportional-integral (PI) controller that coupled the crystal radius to a set point temperature for the heater in an effort to control the dynamic CZ model with idealized radiation. Figure 20 shows the shapes of the crystal and melt predicted without control, with purely integral control, and with... 

Numerical simulations that combine the details of the thermal-capillary models described previously with the calculation of convection in the melt should be able to predict heat transfer in the CZ system. Sackinger et al. (175) have added the calculation of steady-state, axisymmetric convection in the melt to the thermal-capillary model for quasi steady-state growth of a long cylindrical crystal. The calculations include melt motion driven by buoyancy, surface tension, and crucible and crystal rotation. Figure 24 shows sample calculations for growth of a 3-in. (7.6-cm)-diameter silicon crystal as a function of the depth of the melt in the crucible. 

A computer model has been developed to provide numerical simulations of fluidized bed coal gasification reactors and to yield detailed descriptions, in space and time, of the coupled chemistry, particle dynamics and gas flows within the reactor vessels. Time histories and spatial distributions of the important process variables are explicitly described by the model. With this simulation one is able to predict the formation and rise of gas bubbles, the transient and quasi-steady temperature and gas composition, and the conversion of carbon throughout the reactor. 

Group Combustion of Droplets in Fuel Clouds. I. Quasi-steady Predictions... 

Quasi-steady theory predicts isolated particle flame radii which are greater than those experimentally observed. While a discussion of causes of this descrepancy is beyond the scope of this chapter, the reader should keep in mind that absolute errors are associated with equations such as Equation 32. Nevertheless the functional dependency on ambient oxidizer concentration is of principal interest here. 

In real cells, multiple transmembrane pumps and channels maintain and regulate the transmembrane potential. Furthermore, those processes are at best only in a quasi-steady state, not truly at equilibrium. Thus, electrophoresis of an ionic solute across a membrane may be a passive equilibrative diffusion process in itself, but is effectively an active and concentra-tive process when the cell is considered as a whole. Other factors that influence transport across membranes include pH gradients, differences in binding, and coupled reactions that convert the transported substrate into another chemical form. In each case, transport is governed by the concentration of free and permeable substrate available in each compartment. The effect of pH on transport will depend on whether the permeant species is the protonated form (e.g., acids) or the unprotonated form (e.g., bases), on the pfQ of the compound, and on the pH in each compartment. The effects can be predicted with reference to the Henderson-Hasselbach equation (Equation 14.2), which states that the ratio of acid and base forms changes by a factor of 10 for each unit change in either pH or pfCt ... 

J. Relaxed steady-state or sliding regime (T rj. When the input varies rapidly relative to the characteristic response time, the state oscillates with a very small amplitude. The quasi-steady-stale approximation can be applied to the state using the time-averaged value of the control. The performance of the system can be predicted using the performance in comparable steady-state operation. 

E. Hesstvedt, O. Hov and I.S.A. Isaksen, Quasi-Steady-State Approximations in Air Pollution Modeling Comparison of Two Numerical Schemes for Oxidant Prediction, Int. J. Chem. Kinet. 10 (1978) 971-994. 

---