Order-of-magnitude scaling

In order to perform order-of-magnitude scaling for this problem, we need a dimensionless distance, dimensionless time, and dimensionless volumetric flow rate, which we define as... 

The dimensionless equations using order-of-magnitude scaling become... 

However, often the real problem is not with the numerical algorithm but with the engineer developing the equation set. If one is interested in the slower dynamic parts of the problem, a quasi-steady-state assumption should be made for the fast parts of the problem. On the other hand, if one is interested in the fast parts of the problem, the value of the slower parts essentially remains constant over these very short time periods. Therefore, stiff systems of equations should not arise in most properly formulated simulations that use order-of-magnitude scaling in model formation. 

If the chlorination of benzene example is to be solved using an IMSL integration routine, then the describing equations should first be order-of-magnitude scaled. To do this we want to create dimensionless variables of order one. For this problem we can choose a characteristic concentration of Nbo and a characteristic time associated with the primary... 

We will develop an appropriate mathematical model for the pilot scale reactor studied by Clough and Ramirez (1976). The mathematical model is derived by simplification of the general conservation equations. The reduction of the general equations is made possible by order-of-magnitude scaling agreements based upon reasonable values for system dependent and independent variables. An important approximation which is implied in the model development is that the flow through the packed bed reactor can be described by both axial and radial dispersion mechanisms. 

Figure 10.2 Scanning electron microscopy (SEM) images of electrospun nano- and micro-fibrous poly(E-caprolactone) scaffolds demonstrating fiber dimensions of different orders of magnitude (scale bars 5 pm for subset and 50 pm for full images).

The Landolt reaction (iodate + reductant) is prototypical of an autocatalytic clock reaction. During the induction period, the absence of the feedback species (Irere iodide ion, assumed to have virtually zero initial concentration and fomred from the reactant iodate only via very slow initiation steps) causes the reaction mixture to become kinetically frozen . There is reaction, but the intemiediate species evolve on concentration scales many orders of magnitude less than those of the reactant. The induction period depends on the initial concentrations of the major reactants in a maimer predicted by integrating the overall rate cubic autocatalytic rate law, given in section A3.14.1.1. 

Figure Bl.11.1. Resonance frequencies for different nuclei in a field of 14.1 T. Widths indicate the quoted range of shifts for each nucleus, and heights mdicate relative sensitivities at the natural isotopic abundance, on a log scale covering approximately six orders of magnitude. Nuclei resonatmg below 140 MHz are not shown.

At somewhat longer times the modulus undergoes a gradual transition to a value lower by 3 or 4 orders of magnitude than its glassy value. Observed on this time scale, the material has a leathery consistency. 

The MTO process employs a turbulent fluid-bed reactor system and typical conversions exceed 99.9%. The coked catalyst is continuously withdrawn from the reactor and burned in a regenerator. Coke yield and catalyst circulation are an order of magnitude lower than in fluid catalytic cracking (FCC). The MTO process was first scaled up in a 0.64 m /d (4 bbl/d) pilot plant and a successfiil 15.9 m /d (100 bbl/d) demonstration plant was operated in Germany with U.S. and German government support. 

Scale-up is the process of developing a plant design from experimental data obtained from a unit many orders of magnitude smaller. This activity is considered successful if the commercial plant produces the product at plaimed rates, for plaimed costs, and of desired quaUty. This step from pilot plant to full-scale operation is perhaps the most precarious of all the phases of developing a new process because the highest expenses ate committed at the stages when the greatest risks occur. 

When two reactants in a catalytic process have such different solubiUty properties that they can hardly both be present in a single Hquid phase, the reaction is confined to a Hquid—Hquid interface and is usually slow. However, the rate can be increased by orders of magnitude by appHcation of a phase-transfer catalyst (40,41), and these are used on a large scale in industrial processing (see Catalysts, phase-TRANSFEr). Phase-transfer catalysts function by faciHtating mass transport of reactants between the Hquid phases. Often most of the reaction takes place close to the interface. 

Ionic Equilibria.. The ion product constant of D2O (see Table 3) is an order of magnitude less than the value for H2O (24,31,32). The relationship pD = pH + 0.41 (molar scale 0.45 molal scale) for pD ia the range 2—9 as measured by a glass electrode standardized ia H2O has been established (33). For many phenomena strongly dependent on hydrogen ion activity, as is the case ia many biological contexts, the difference between pH and pD may have a large effect on the iaterpretation of experiments. 

Contrac tors bids offer the most rehable information on cost. Order-of-magnitude costs, however, may be required for preliminary studies. One way of estimating them is to obtain cost information from similar facihties and scale it to the proposed installation. Costs of steel storage tanks and vessels have been found to vaiy approximately as the 0.6 to 0.7 power of their weight [see Happel, Chemical Process Economics, Wiley, 1958, p. 267 also Williams, Chem. Eng., 54(12), 124 (1947)]. AU estimates cased on the costs of existing eqiiipment must be corrected for changes in the price index from the date when the equipment was built. Considerable uncertainty is involved in adjusting data more than a few years old. 

Three criteria for scale-up are that the laboratory and industrial units have the same mass-transfer coefficients /cg and E/cl and the same ratio of the specific interfacial surface and liquid holdup Tables 23-9 and 23-10 give order-of-magnitude values of some parameters that may be expected in common types of liquid/gas contactors. 

Wind speed has velocity components in all directions so that there are vertical motions as well as horizontal ones. These random motions of widely different scales and periods are essentially responsible for the movement and diffusion of pollutants about the mean downwind path. These motions can be considered atmospheric turbulence. If the scale of a turbulent motion (i.e., the size of an eddy) is larger than the size of the pollutant plume in its vicinity, the eddy will move that portion of the plume. If an eddy is smaller than the plume, its effect will be to difhise or spread out the plume. This diffusion caused by eddy motion is widely variable in the atmosphere, blit even when the effect of this diffusion is least, it is in the vicinity of three orders of magnitude greater than diffusion by molecular action alone. 

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