Physical properties, variable

The treatment of viscosity variations included the possibility of variable density. Equations (8.12) and (8.52) assumed constant density, constant a, and constant otj-. We state here the appropriate generalizations of these equations to account for variable physical properties. 

For completeness, axial dilfusion and variable cross-section terms were included in Equations (8.74) and (8.75). They are usually dropped. Also, the variations in and X are usually small enough that they can be brought outside the derivatives. The primary utility of these equations, compared with Equations (8.12) and (8.52), is for gas-phase reactions with a signihcant pressure drop. 

---

Thus, in the three-layer model, with the intermediate layer having variable physical properties (and perhaps also chemical), subscripts f, i, m and c denote quantities corresponding to the filler, mesophase, matrix and composite respectively. It is easy to establish for the representative volume element (RVE) of a particulate composite, consisting of a cluster of three concentric spheres, that the following relations hold ... 

The sum of squares as defined by Equation 7.8 is the general form for the objective function in nonlinear regression. Measurements are made. Models are postulated. Optimization techniques are used to adjust the model parameters so that the sum-of-squares is minimized. There is no requirement that the model represent a simple reactor such as a CSTR or isothermal PER. If necessary, the model could represent a nonisothermal PFR with variable physical properties. It could be one of the distributed parameter models in Chapters 8 or 9. The model... 

Solution A rigorous treatment of a reversible reaction with variable physical properties is fairly complicated. The present example involves just two ODEs one for composition and one for enthalpy. Pressure is a dependent variable. If the rate constants are accurate, the solution will give the actual reaction trajectory (temperature, pressure, and composition as a function of time). If ko and Tact are wrong, the long-time solution will still approach equilibrium. The solution is then an application of the method of false transients. 

Phase Balances for Components. Material balances can be written for each phase. For the general case of unsteady operation and variable physical properties, the liquid-phase balance is... 

In general, hazard identification criterion represents the deviation of one or more measured variables from specified values. This is the basis upon which a significant percentage of risk analyses are done. For a chemical process, a number of measurable variables, physical properties, and states or positions of various parts of the overall equipment, e.g., pumps, valves, and motors, can be specified for every time or phase of the process. Certain deviations from the "standard" recipe or settings can then be defined in advance as hazardous, and thus can be used for initiation of an alarm at the early stage of a runaway or upset condition. 

The solids forwarding angle q> is related to rate by Eq. 5.7, and the force and torque balances provide the relationship between and the pressure gradient and q> using Eq. 5.23. Here, the equation is written in differential form so that integration can be performed numerically in the z direction using variable physical property data. 

TREATMENT OE VARIABLE PHYSICAL PROPERTIES BY DIMENSIONAL ANALYSIS... 

Treatment of Variable Physical Properties by Dimensional Analysis... 

If the scale-up is performed in the pi-space with variable physical properties, the requirement idem also concerns the form of the dimensionless formulated material function. This aspect can make the choice of the model material system considerably more difficult. This requirement is fulfilled, a priori, only if the interesting range Au in the standard representation w = (u) lies close to the standardization point, see the explanation concerning Fig. 9 b in the text. 

With variable physical properties the relevance list widens by both key or transformation parameters a and b as well as by the reference point. In the dependence p(T) this includes the quantities... 

Treatment of Variable Physical Properties by Dimensional Analysis The pi-set of eight remodeled pi-numbers now reads ... 

---