Nondimensionalized values

There is a further parameter required, which may not be immediately apparent from the governing equation itself. The actual value of the radius where the gap is located must enter the problem. The fact that r itself appears in the differential equation is a clue that such a parameter could be needed. In this problem we already know the analytic solution and that it contains a lnr term. Thus the solution depends on the values of r in a nonlinear way. Here we define the extra parameter as the nondimensional value of the inner radius, that is, the nondimensional rod radius or the inner radius in terms on numbers of gap thicknesses,... 

Then, by analogy with equation (2.167), we get nondimensional value of the partition coefficient... 

Conclusive parameters of the problem, as it has been stated, are four nondimensional values the Mach number of incident shock wave Mj, the thickness of the specimen d, the density of porous medium p porosity ttQ. 

Takahashi et al obtained the impulsive pressure coefficient ai by reanalyzing the results of comprehensive sliding tests, being a nondimensional value representing the impulsive pressure component, which should be regarded as an additional effect to the slowly varying pressure component. The effect of the dynamic (impulsive) pressure indicated by 0 2 in Goda s formula does not under all conditions accurately estimate the effective pressure (equivalent static pressure) due to impulsive pressure, and therefore, ai was introduced. 

Reynolds Number. The Reynolds number, Ke, is named after Osborne Reynolds, who studied the flow of fluids, and in particular the transition from laminar to turbulent flow conditions. This transition was found to depend on flow velocity, viscosity, density, tube diameter, and tube length. Using a nondimensional group, defined as p NDJp, the transition from laminar to turbulent flow for any internal flow takes place at a value of approximately 2100. Hence, the dimensionless Reynolds number is commonly used to describe whether a flow is laminar or turbulent. Thus... 

The hardening index /I is a nondimensional scalar which has the same value... 

To determine the deterioration in component performance and efficiency, the values must be corrected to a reference plane. These corrected measurements will be referenced to different reference planes depending upon the point, which is being investigated. Corrected values can further be adjusted to a transposed design value to properly evaluate the deterioration of any given component. Transposed data points are very dependent on the characteristics of the components performance curves. To determine the characteristics of these curves, raw data points must be corrected and then plotted against representative nondimensional parameters. It is for this reason that we must evaluate the turbine train while its characteristics have not been altered due to component deterioration. If component data were available from the manufacturer, the task would be greatly reduced. 

The nondimensionalized side-on peak overpressures and their respective positive-phase durations can be transformed into real values for side-on peak overpressures and positive-phase durations by calculating ... 

These substitutions replace the eight dimensional parameters in the original equations by the four nondimensional parameters above. The parameter Pe is the Peclet number (37) and shows the relative importance of convection compared to diffusion. The advantage of this formulation becomes obvious when typical parameter values are substituted into the equations. 

Another commonly used approximation for which there is an analytical solution is to assume that the root acts as a zero sink for uptake. Here the solute concentration at the root surface is taken to be zero and uptake is therefore completely controlled by the diffusive flux to the root (21,40,41). The implicit assumption is that root uptake is very rapid in comparison to resupply by transport and hence the root very rapidly depletes the solute concentration at the root surface to zero and maintains it there. The validity of this assumption depends on the value of X and it is inapplicable unless X is greater than or about 10 (38). For such large X, there is a nondimensional critical time (/,.) after which it is reasonable to assume a zero sink (38,42). Approximate values of /, are... 

Figure 3.9 compares the separated-to-intermittent flow regime boundaries for both 8-in.- and 4-in.-pipe experiments. The conditions are presented in terms of nondimensional superficial velocities J = Jk[pk/(pL - pc)gZ)]1/2 (or the density-modified Froude numbers). For 4-in.-pipe experiments, data have been obtained only for 3 MPa, while for the 8-in.-pipe experiments, data have been obtained for pressures of 3.0, 5.0, and 7.3 MPa. Note the value of J at the flow regime transition increased with pressure for the 8-in.-pipe experiments. 

Such spatial variations in, e.g., mixing rate, bubble size, drop size, or crystal size usually are the direct or indirect result of spatial variations in the turbulence parameters across the flow domain. Stirred vessels are notorious indeed, due to the wide spread in turbulence intensity as a result of the action of the revolving impeller. Scale-up is still an important issue in the field of mixing, for at least two good reasons first, usually it is not just a single nondimensional number that should be kept constant, and, secondly, average values for specific parameters such as the specific power input do not reflect the wide spread in turbulent conditions within the vessel and the nonlinear interactions between flow and process. Colenbrander (2000) reported experimental data on the steady drop size distributions of liquid-liquid dispersions in stirred vessels of different sizes and on the response of the drop size distribution to a sudden change in stirred speed. 

Fig. 12. Snapshot from a two-phase DNS of colliding particles in an originally fully developed turbulent flow of liquid in a periodic 3-D box with spectral forcing of the turbulence. The particles (in blue) have been plotted at their position and are intersected by the plane of view. The arrows denote the instantaneous flow field, the colors relate to the logarithmic value of the nondimensional rate of energy dissipation.

Where dissolution or precipitation is sufficiently rapid, the species concentration quickly approaches the equilibrium value as water migrates along the aquifer the system is said to be reaction controlled. Alternatively, given rapid enough flow, water passes along the aquifer too quickly for the species concentration to be affected significantly by chemical reaction. The system in this case is transport controlled. The relative importance of reaction and transport is described formally by the nondimensional Damkohler number, written Da. 

The greatest value can be found at the emission level at z=2 m with C/C0=3, 45. The nondimensional concentration decreases to the ground and to the height with regard to the emission level. Looking at equation (3.17) this is expressed by theory, too. It must be mentioned that the horizontal planes are covered with an equal distant grid of 15 m— width, while in the vertical planes the z-distance step of 1 m is enlarged by the factor 15 in the plots of fig. 8 and fig. 9. 

From equation (3.13) we can deduct a rough approximation of the location where maximum ground-level concentration occurs. It is argued that the turbulent diffusion acts more and more on the emitted substances, when the distance from the point source increases therefore the downwind distance dependency of the diffusion coefficients is done afterwards. If we drop this dependency, equation (3.13) leads to xmax=34,4 m for AK=I (curve a) and xmax=87,7 m for AK=V (curve b), what is demonstrated in fig n The interpolated ranges of measured values are lined in. Curve a overestimates the nondimensional concentration maximum, but its location seems to be correct. In the case of curve b the situation is inverted. Curve c is calculated with the data of AK=II. The decay of the nondimensional concentration is predicted well behind the maximum. Curve d is produced with F—12,1, f=0,069, G=0,04 and g=l,088. The ascent of concentration is acceptable, but that is all, because there is no explanation of plausibility how to alter the diffusivity parameters. Therefore it must be our aim to find a suitable correction in connection with the meteorological input data. 

The nondimensionalized frequencies are related to linear and angular frequencies by equation 3.36. The conversion factor from linear frequencies in cm to undimension-alized frequencies is chik = 1.4387864 cm (where c is the speed of hght in vacuum). Acoustic branches for the various phases of interest may be derived from acoustic velocities through the guidelines outlined by Kieffer (1980). Vibrational modes at higher frequency may be derived by infrared (IR) and Raman spectra. Note incidentally that the tabulated values of the dispersed sine function in Kieffer (1979c) are 3 times the real ones (i.e., the listed values must be divided by 3 to obtain the appropriate value for each acoustic branch see also Kieffer, 1985). 

The effect of heating on the azimuthal and streamwise components of the vorticity field is shown in Fig. 11.1 the effect on the radial component is comparable to that on the azimuthal component, and is therefore not separately shown. The vorticity distributions at the same nondimensional time t = 35 are plotted side by side for the unheated and heated case for each component. The positive and negative values of vorticity are shown by solid and dotted lines, respectively. 

Gliksman s approach The result of the conversion of equations into nondimensional ones is a set of dimensionless parameters (Froude number, velocity, particle size, diameter ratios, etc.) that should be matched in both small and large systems. It is not necessary for the values of the parameters to be equal in each system. Instead, the dimensionless number ratios have to remain the same. To achieve this, the particle size and/or the particle density of the solids have to be changed appropriately in the small unit. It usually results in a smaller particle size in the small unit compared to the large one. 

The fu st term is a modified Archimedes number, while the second one is the Froude number based on particle size. Alternatively, the first term can be substituted by the Reynolds number. To attain complete similar behavior between a hot bed and a model at ambient conditions, the value of each nondimensional parameter must be the same for the two beds. When all the independent nondimensional parameters are set, the dependent parameters of the bed are fixed. The dependent parameters include the fluid and particle velocities throughout the bed, pressure distribution, voidage distribution of the bed, and the bubble size and distribution (Glicksman, 1984). In the region of low Reynolds number, where viscous forces dominate over inertial forces, the ratio of gas-to-solid density does not need to be matched, except for beds operating near the slugging regime. 

Transfer velocity across gaseous boundary layer typically between 0.1 and 1 cm s"1 (up to 5 cm s 1, see Fig. 20.2). Km is the nondimensional liquid/gas distribution coefficient (for air-water interface inverse nondimensional Henry s law coefficient, i.e., Jfr w) with typical values between 10-3 and 103. DA is the molecular gaseous diffusivity, typical size 0.1 cm2s . 

Figure 19.18 shows how for different values of y sorption proceeds as a function of the nondimensional time t defined by Eq. 19-79. As y increases from the infinite bath case, we see that the time required to reach equilibrium decreases. This can be understood by recognizing that while the chemical is diffusing into the sphere, the concentration in the surrounding fluid drops. Hence, the total mass exchange needed to attain equilibrium between the fluid and the sphere is smaller than in the infinite bath case in which the external concentration remains constant. 

Consider two aldehydes at neutral pH, formaldehyde and acetaldehyde. The hydration/ dehydration (pseudo-) first-order rate constants and the nondimensional Henry s law constants are summarized below. Since in the following discussion you are interested in orders of magnitude only, you assume that aqueous molecular diffusiv-ities of all involved species are the same as the value for C02, (DIW = 2 x 10 5 cm2s 1) and that the corresponding values in air are the same as the value for water vapor (Dwateri = 0.26 cm2s 1). This allows us (as a rough estimate) to calculate v,w and v,a directly from Eqs. 20-15 and 20-16, respectively. 

Step input. All the conclusions drawn for the pulse input can directly be transferred to the step input. The concentrations, if expressed for the nondimensional coordinates , and 9, are not affected by sorption the shape of the concentration curve along x for a fixed time is independent of sorption. Yet, when the front passes by a fixed location x, the time needed for the concentration to increase from, say, 5% to 95% of the maximum concentration, grows as (/w)-1. This can be directly deduced from Table 25.2, where the duration of the passage of the front is quantified by the nondimensional time interval A0 - 095 -05. This value does not depend on sorption, but after transformation back into real time it does (see Eq. 25-42) ... 

The reference and the scale have the same units as x. A nonzero reference indicates that the difference between x and xref is important to the specific problem at hand rather than the absolute value of x itself. xref is determined from the problem and is typically some value of x at the boundary or some initial time. The scale is some combination of other dimensional quantities that are relevant to the physical problem, such as boundary and initial conditions and physical constants. If chosen properly, the scale provides a measure of the range of values that the variable x — xref will take for the particular physical problem. The ideal choice of the reference and scale quantities for a given problem will result in order-unity values for the nondimensional quantities. 

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