Distance-time plane

As in the gravity sedimentation problem, characteristic solutions also exist for nondiffusive sedimentation in an ultracentrifuge. We recall the definition of a characteristic as a line in the distance-time plane, here the r-t plane, on which an ordinary differential equation may be written. Such an equation must be expressed as a relation connecting total differentials in which partial derivatives do not appear. Since we wish to obtain relations involving total differentials, we write... 

Results of the lattice constants and density measurements are given in Table III their variations as a function of composition are shown in Figures 8 and 9. At the present time, two crystal structures only slightly different from one another have been proposed (4, 9). They differ only in the origin of z, the distances between planes of titanium and of sulfur, and in the distribution of titanium among the available sites (Figure 10). In the present case, density measurements do not permit making a decision they simply show that lattice defects are related to the variable number of titanium atoms per unit cell, and that the number of sulfur atoms per unit cell is constant and equal to 4. 

We have been concerned hitherto with so much solvent that its amount for practical computations of D can be reckoned as infinite. Such a supposition limits the applicability of the equations given, for it will be much more usual to work with small amounts of diffusion media. The solutions now to be given will concern themselves wdth problems such as the following. A solute diffuses from a solution bounded between the planes x =0 x = h into a solvent bounded between the planes x and x = 1, The concentration-distance-time curves may be measured readily enough and have now to be interpreted so that the diffusion constants, 2), may be evaluated. The new solutions of Fick s law will apply to numerous cases of the interdiffusion of metals, and salts, so long as D does not depend on the concentration, and whenever the amount of metal or salt is limited. 

Figure 5.44 The progress of regular and Mach waves in the normalized distance and time planes, a is half the distance between the line generators.

Suppose we define the rate of radial growth of the crystalline disks as r. Then disks originating from all nuclei within a distance rt of an arbitrary point, say, point X in Fig. 4.6a, will reach that point in an elapsed time t. If the average concentration of nuclei in the plane is N (per unit area), then the average number of fronts F which converge on x in tliis time interval is... 

Each axial mode has its own characteristic pahem of nodal planes and the frequency separation Av between modes is given by Equation (9.4). If the radiation in the cavity can be modulated at a frequency of cjld then the modes of the cavity are locked both in amplitude and phase since t, the time for the radiation to make one round-trip of the cavity (a distance 2d), is given by... 

So, we consider the shallow shell with the distances on the mid-surface coinciding with those on the plane. At the same time the curvatures are not equal to zero, in general. The shells like these are called the weakly curved plates. 

The properties required of a material in order for it to support a stable shock wave were listed and discussed. Rarefaction, or release waves were defined and their behavior was described. The useful tool of plotting shocks, rarefactions, and boundaries in the time-distance plane (the x-t diagram) was introduced. The Lagrangian coordinate system was defined and contrasted to the more familiar Eulerian coordinate system. The Lagrangian system was then used to derive conservation equations for continuous flow in one dimension. 

If we say that, with a given applied field, in unit time this excess consists of n electrons, the current density will be Ne, since we are dealing with unit area. In Fig. 16 let us suppose that the excess flow of electrons is in the downward direction we can then, to show the character of the flow, make the following construction. Parallel to the plane AB, consider a plane CD, also of unit area and let the distance between CD and AB be chosen such that the total number of conduction electrons in the volume between CD and AB at any moment is n. 

In accordance with Ohm s law, if we were to double the intensity X of the electric field, the current would be doubled that is to say, the plane CD would have to be placed at twice the distance from AB. If the number of conduction electrons per unit volume is p, and the distance between the planes CD and AB is denoted by v, we have n = pv, since we are discussing the unit area. Hence the net resultant charge transported in unit time across AB, that is, the current density, is given by... 

If we were to forget that the flow of current is due to a random motion which was already present before the field was applied—if we were to disregard the random motion entirely and assume that each and every electron, in the uniform field X, moves with the same steady velocity, the distance traveled by each electron in unit time would be the distance v used in the construction of Fig. 16 this is the value which would lead to a current density j under these assumptions, since all electrons initially within a distance v of the plane AB on one side would cross AB in unit time, and no others would cross. Further, in a field of unit intensity, the uniform velocity ascribed to every electron would be the u of (34) this quantity is known as the mobility of the charged particle. (If the mobility is given in centimeters per second, the value will depend on whether electrostatic units or volts per centimeter are used for expressing the field strength.)... 

The quantity of solute B crossing a plane of area A in unit time defines the flux. It is symbolized by J, and is a vector with units of molecules per second. Fick s first law of diffusion states that the flux is directly proportional to the distance gradient of the concentration. The flux is negative because the flow occurs in a direction so as to offset the gradient ... 

Considering the particular problem of the unidirectional flow of heat through a body with plane parallel faces a distance l apart, the heat flow is normal to these faces and the temperature of the body is initially constant throughout. The temperature scale will be so chosen that this uniform initial temperature is zero. At time, t = 0, one face (at x — 0) will be brought into contact with a source at a constant temperature 9 and the other face (at x = () will be assumed to be perfectly insulated thermally. 

Temperatures at off-centre locations within the solid body can then be obtained from a further series of charts given by Heisler (Figures 9.17-9.19) which link the desired temperature to the centre-temperature as a function of Biot number, with location within the particle as parameter (that is the distance x from the centre plane in the slab or radius in the cylinder or sphere). Additional charts are given by Heisler for the quantity of heat transferred from the particle in a given time in terms of the initial heat content of the particle. 

Monochromatic Waves (1.14) A monochromatic e.m. wave Vcj r,t) can be decomposed into the product of a time-independent, complex-valued term Ucj r) and a purely time-dependent complex factor expjojt with unity magnitude. The time-independent term is a solution of the Helmholtz equation. Sets of base functions which are solutions of the Helmholtz equation are plane waves (constant wave vector k and spherical waves whose amplitude varies with the inverse of the distance of their centers. 

The Eulerian approach requires a measurement of the temperature or the progress variable at many sample points at a given normal distance from the ignition plane, at a given time elapsed since ignition. The progress variable introduced here can be for instance a normalized temperature or concentration that varies from... 

---