Spatial meanings

In many cases, some boundary conditions are not well known or not known at all. Temperature boundary conditions can be obtained from thermal building-dynamics programs that allow the capture of spatial mean temperatures during a time period as long as a whole year. Some of these programs yield surface temperature values (e.g., TRNSYS), which can be used as temperature boundary conditions at the time of CFD study. 

The graphical integration method is based on graphical presentation of the average flow profile. For a circular duct, the cross-section is virtually divided into several concentric ring elements. The spatial mean velocity of such an element is determined as an arithmetical mean of local velocities along the circumference of the corresponding radius. For a circular cross-section the flow rate can be expressed as... 

How important is spatially resolved volatilisation in comparison with spatial means ... 

Atmospheric concentrations in the model were compared to observations made over various ocean surfaces [Iwata et al (1993)] and on a floating ice island in the Canadian Arctic [Patton et al (1989)]. For the comparison with Iwata et al (1993) spatial means of the relevant ocean regions were derived and compared with the reported data. For the comparison with Patton et al (1989) the corresponding model grid box was identified. 

Because the velocity u contains the random component u, the concentration c is a stochastic function since, by virtue of Eq. (2.2), c is a function of u. The mean value of c, as expressed in Eq. (2.5), is an ensemble mean formed by averaging c over the entire ensemble of identical experiments. Temporal and spatial mean values, by contrast, are obtained by averaging v ues from a single member of the ensemble over a period or area, respectively. The ensemble mean, which we have denoted by the angle brackets ( ), is the easiest to deal with mathematically. Unfortunately, ensemble means are not measurable quantities, although under the conditions of the ergodic theorem they can be related to observable temporal or spatial averages. In Eq. (2.7) the mean concentration (c) represents a true ensemble mean, whereas if we decompose c as... 

It has been shown recently [25] that concentrations of NOj, tend to reduce with increase in the amplitude of discrete-frequency oscillations. The mechanisms remain uncertain, but may be associated with the imposition of a near-sine wave on a skewed Gaussian distribution with consequent reduction in the residence time at the adiabatic flame temperature. Profiles of NO, concentrations in the exit plane of the burner are shown in Fig. 19.6 as a function of the amplitude of oscillations with active control used to regulate the amplitude of pressure oscillations. At an overall equivalence ratio of 0.7, the reduction in the antinodal RMS pressure fluctuation by 12 dB, from around 4 kPa to 1 kPa by the oscillation of fuel in the pilot stream, led to an increase of around 5% in the spatial mean value of NO, compared with a difference of the order of 20% with control by the oscillation of the pressure field in the experiments of [25]. The smaller net increase in NO, emissions in the present flow may be attributed to an increase in NOj due to the reduction in pressure fluctuations that is partly offset by a decrease in NOj, due to the oscillation of fuel on either side of stoichiometry at the centre of the duct. 

Dispersion is the enhanced mixing of material through spatial variations in velocity. When it is of interest (when we are not keeping track of the three-dimensional mixing), dispersion is typically one or two orders of magnitude greater than turbulent diffusion. The process of dispersion is associated with a spatial mean velocity. The means used in association with diffusion, turbulent diffusion, and dispersion are identified in Table 6.2. 

The means by which diffusion (and possibly turbulent diffusion) is combined with a spatial mean velocity to result in dispersion is illustrated in Figure 6.10. A velocity profile over space with mixing due to diffusion (and possibly turbulent diffusion) is... 

Table 6.2 Temporal or spatial means and scales used in association with various mixing processes...

A similar spatial mean velocity (bulk mean velocity) is used for the plug flow reactor model. Thus, plug flow with dispersion is a natural match, where the mixing that truly occurs in any reactor or environmental flow is modeled as dispersion. This is the model that will be applied to utilize dispersion as a mixing model. 

Any flow with a nonuniform velocity profile will, when spatial mean velocity and concentration are taken, result in dispersion of the chemical. For laminar flow, the well-described velocity profile means that we can describe dispersion analytically for some flows. Beginning with the diffusion equation in cylindrical coordinates (laminar flow typically occurs in small tubes) ... 

Thus the mere presence of metaphorical language does not by itself tell us whether the space-time metaphor is a psychologically real conceptual mapping. For example, the temporal and spatial meanings could be represented as alternate meaning senses or even as separate homophonic lexical entries. The apparent systematicity would then be illusory, the result of post hoc regularization. 

Let us assume that all physical quantities in our star can be well characterised by their spatial mean values (indicated by bars over the variables), i.e., that our star is not structured by nuclear shell sources. For such more complicated objects, the following considerations hold nevertheless for their cores. We approximate the potential energy of our star as... 

Fig. 8. Final spatial mean gas density as a function of inlet gas temperature.

Experimental data from an approximately adiabatic container [1] are compared with the theory in Fig. 8, The parameter for comparison is the final spatial mean gas density which is obtained from the equation of state [8] and a mean temperature found by a spatial integration oiT[x,6), (8). Because these data do not involve heat transfer with an amMent they are employed to determine a reasonable value for Ascanbeseen ranges between 1 and 3 Btu/hr-ft - F. It is thought that this range is on the low side and that somewhat increased values... 

Jackendoff (1996) describes how cultures differ in the ways that speakers orient themselves in geographic space. These orientations define a viewpoint that must be re-negotiated when speakers confront new systems of geographic and spatial meaning. See also Levelt, 1996 O Keefe, 1996. 

These curves for a reactor in which the solid is perfectly mixed would also be applicable to reactors in which gas conversion is not small. In this case is calculated using the spatial mean gas concentration in the bed calculated as described at the end of Section 7.5.5. 

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