Uniform coordinate system

L describes lightness and extends from 0 (black) to 100 (white). The a coordinate represents the redness-greenness of the sample. The b coordinate represents the yellowness-blueness. The coordinates a and b have a range of approximately [—100,100]. Notice the cube root in the above equation. The cube root was introduced in order to obtain a homogeneous, isotropic color solid (Glasser et al. 1958). Visual observations correspond closely to color differences calculated in this uniform coordinate system. A transformation of the sRGB color cube to the L a b color space is shown in Figure 5.5. 

Measurement of Whiteness. The Ciba-Geigy Plastic White Scale is effective in the visual assessment of white effects (79), but the availabihty of this scale is limited. Most evaluations are carried out (ca 1993) by instmmental measurements, utilising the GIF chromaticity coordinates or the Hunter Uniform Color System (see Color). Spectrophotometers and colorimeters designed to measure fluorescent samples must have reversed optics, ie, the sample is illuminated by a polychromatic source and the reflected light passes through the analy2er to the detector. 

Consider the specific example of a spherical electrode having the radius a. We shall assume that diffusion to the spherical surface occurs uniformly from all sides (spherical symmetry). Under these conditions it will be convenient to use a spherical coordinate system having its origin in the center of the sphere. Because of this synunetry, then, aU parameters have distributions that are independent of the angle in space and can be described in terms of the single coordinate r (i.e., the distance from the center of the sphere). In this coordinate system. Pick s second diffusion law becomes... 

The PFTR was in fact assumed to be in a steady state in which no parameters vary with time (but they obviously vary with position), whereas the batch reactor is assumed to be spatially uniform and vary only with time. In the argument we switched to a moving coordinate system in which we traveled down the reactor with the fluid velocity , and in that case we follow the change in reactant molecules undergoing reaction as they move down the tube. This is identical to the situation in a batch reactor ... 

Diffusive transport with convection occurring simultaneously can be solved more easily if we orient our coordinate system properly. First, we must orient one axis in the direction of the flow. In this case, we will choose the v-coordinate so that u is nonzero and v and w are zero. Second, we must assume a uniform velocity profile, u = U = constant with y and z. Then, equation (2.33) becomes... 

We noted in the preceding section that the polarizability of an ellipsoid is anisotropic the dipole moment induced by an applied uniform field is not, in general, parallel to that field. This anisotropy originates in the shape anisotropy of the ellipsoid. However, ellipsoids are not the only particles with an anisotropic polarizability in fact, all the expressions above for cross sections are valid regardless of the origin of the anisotropy provided that there exists a coordinate system in which the polarizability tensor is diagonal. 

Let us first treat the simplest case in which diffusion is occuring along the x direction of a Cartesian coordinate system, with conditions being uniform in the y and z directions. Such a situation, termed planar diffusion, is depicted in Fig. 10. Consider the small region of volume Adx and the concentration changes that occur within this region in the time interval between t and t + df. We can write... 

The problem is extended by Crank to the case where the slab is initially at a uniform temperature below the fusion point, for which a mass transfer analog involves site mobility HI). Note that the center line of the slab is no longer a fixed point in the X coordinate system. A second spatial coordinate is introduced for the unmelted region... 

Example 5.5 Continuous Heating of a Thin Sheet Consider a thin polymer sheet infinite in the x direction, moving at constant velocity Vq in the negative x direction (Fig. E5.5). The sheet exchanges heat with the surroundings, which is at T = T0, by convection. At x = 0, there is a plane source of heat of intensity q per unit cross-sectional area. Thus the heat source is moving relative to the sheet. It is more convenient, however, to have the coordinate system located at the source. Our objective is to calculate the axial temperature profile T(x) and the intensity of the heat source to achieve a given maximum temperature. We assume that the sheet is thin, that temperature at any x is uniform, and that the thermophysical properties are constant. 

In the present analysis it will be assumed that both walls of the channel are heated to the same uniform temperature and that the flow is therefore symmetrical about the channel center line. The coordinate system shown in Fig. 8.16 will therefore be used in the analysis and, because of the assumed symmetry, the solution will only be obtained for y values between 0 and W/2, IV being, as indicated in Fig. 8.16, the full width of the channel. 

Similar equations apply to cylindrical and spherical coordinate systems. Finite difference, finite volume, or finite element methods are generally necessary to solve (5-15). Useful introductions to these numerical techniques are given in the General References and Sec. 3. Simple forms of (5-15) (constant k, uniform S) can be solved analytically. See Arpaci, Conduction Heat Transfer, Addison-Wesley, 1966, p. 180, and Carslaw and Jaeger, Conduction of Heat in Solids, Oxford University Press, 1959. For problems involving heat flow between two surfaces, each isothermal, with all other surfaces being adiabatic, the shape factor approach is useful (Mills, Heat Transfer, 2d ed., Prentice-Hall, 1999, p. 164). 

Consider a cylindrical soft particle, that is, an infinitely long cylindrical hard particle of core radius a covered with an ion-penetrable layer of polyelectrolytes of thickness d in a symmetrical electrolyte solution of valence z and bulk concentration (number density) n. The polymer-coated particle has thus an inner radius a and an outer radius b = a + d. The origin of the cylindrical coordinate system (r, z, cp) is held fixed on the cylinder axis. We consider the case where dissociated groups of valence Z are distributed with a uniform density N in the polyelectrolyte layer so that the density of the fixed charges in the surface layer is given by pgx = ZeN. We assume that the potential i/ (r) satisfies the following cylindrical Poisson-Boltz-mann equations ... 

A coordinate system that is stationary with respect to the wave will be adopted, and it will be assumed that the flow is in the - -x direction and that properties are uniform in planes normal to the x axis. Upstream conditions (at X = — oo) will be identified by the subscript 0, while downstream conditions (at x = -h oo) are denoted by the subscript oo. The flow is illustrated schematically in Figure 2.1. Equations (l-26)-(l-29) govern the system. 

In particular, the one-step chemical process v F -h v O products will be investigated, where and are the stoichiometric coefficients for fuel F and oxidizer 0 appearing as reactants. It will be convenient to adopt a coordinate system in which the combustion wave is at rest, the combustible mixture approaches from x = — oo and equilibrium reaction products move away toward x = -h oo conditions become uniform as x oo in Figure 11.6. All the conservation equations derived in Section 11.4 will be needed here, and all the simplifications in Section 11.4 are assumed to be valid. Since the initial relative velocity of the droplets and the gas is zero and the velocity gradients may not be too large, all droplets will be assumed to travel at the same velocity as the gas (t = u). Estimates of the droplet acceleration using equation (71) indicate that this additional approximation is valid in the present problem if the droplets are not too large. Other assumptions will be stated in the course of the illustrative analysis. 

Note that the dimensionality of the flow also depends on the choice of coordinate system and its orientation. The pipe flow discussed, for example, is one-dimensional in cylindrical coordinates, but two-dimensional in Cartesian coordinates illustrating the importance of choosing the most appropriate coordinate system. Also note that even in this simple flow, the velocity cannot be uniform across the cross section of the pipe because of the no-slip condition. However, at a well-rounded entrance to the pipe, the velocity profile may be approximated as being nearly uniform across the pipe, since the velocity is nearly constant at all radii except very close to the pipe wall. 

Cartesian coordinates are a convenient alternative representation for a spatial distribution function. Being uniform over the local space, the data structure obtained is easy to represent (access), to normalize, and to visualize. Use of a Cartesian representation becomes a necessity for complex or very flexible molecules. The principal drawbacks of this coordinate system are the size of the data structure it generates (typically about 1,000,000 elements), its inherent inefficiency (since the grid size is determined by the shortest dimension of the smallest feature one hopes to capture), and the fact that its sampling pattern is usually not commensurate with the structures one wants to represent (which can cause artificial surface features or textures when visualized). Obtaining sufficiently well-averaged results in more distant volume elements can be a problem if the examination of more subtle secondary features is desired. See Figures 7, 8 and 9 for examples of SDFs that have utilized Cartesian coordinates. 

The motion of a rod in laminar flow given by Eq. (1.3) is not uniform. However, on the averse the rod rotates around the z axis with a constant cingular velocity g /2. This can be seen by evaluating the moment on the rod or from decomposing the laminar flow into two parts as we shall do in Appendix II. Of course, we consider that the center of rod is at the origin of the coordinate system. The rod is composed oi2n+ 1 spheres separated from each other with a bond of lei th h. The spheres are assumed to have a radius a. 

Let us suppose that the uniform dc field Fo is directed along the Z axis of the laboratory coordinate system and that a small probing field Fi, having been applied to the assembly of dipoles in the distant past (t = — oo) so that... 

The steady drag is the component of the hydrod3mamic force acting on the particle surface in the continuous phase flow direction. One might, for example, imagine a uniform velocity in the z-direction as sketched in Fig 5.1, and describe the external flow using the Cartesian coordinate system, then the steady drag force is defined by [14, 30] ... 

The following assumptions were made in formulating this model 1) there is no solute adsorption to the stationary phase, 2) the porous particles which form the stationary phase are of uniform size and contain pores of identical size, 3) there are no interactions between solute molecules, 4) the mobile phase is treated as a continuous phase, 5) the intrapore diffusivity, the dispersion coefficient and the equilibrium partition coefficient are independent of concentration. The mobile phase concentration. Cm, is defined as the mass (or moles) per interstitial volume and is a function of the axial coordinate z and the angular coordinate 0. The stationary phase concentration, Cs, is defined as the mass per pore volume and depends on z, 6 and the radial coordinate, r, of a spherical coordinate system whose origin is at the center of one of the particles. 

Strains and stresses were computed for the joined specimen cooled uniformly to room temperature from an assumed stress-free elevated temperature using numerical models described in detail previously [19, 20]. The coordinate system and an example of the finite element mesh utilized are shown in Figure 3. Elastie-plastic response was permitted in both the Ni and Al203-Ni composite materials a von Mises yield condition and isotropic hardening were assumed. 

Problem 9-17. Heat Transfer From an Ellipsoid of Revolution at Pe S> 1. In a classic paper, Payne and Pell. J. Fluid Meek 7, 529(1960)] presented a general solution scheme for axisymmetric creeping-flow problems. Among the specific examples that they considered was the uniform, axisymmetric flow past prolate and oblate ellipsoids of revolution (spheroids). This solution was obtained with prolate and oblate ellipsoidal coordinate systems, respectively. 

---