Plane sources

The mathematical translation of the plane-source problem is as follows. Initially, there is a finite amount of mass M but very high concentration at a = 0, i.e., the density or concentration at a = 0 is defined to be infinite (which is unrealistic but merely an abstraction for the case in which initially the mass is concentrated in a very small region around a = 0). The initial condition is not consistent with that required for Boltzmann transformation. Hence, other methods must be used to solve the case of plane-source diffusion. Because this is the classical random walk problem, the solution can be found by statistical treatment as the following Gaussian distribution ... 

If the plane source is on the surface of a semi-infinite medium, the problem is said to be a thin-film problem. The diffusion distance stays the same, but the same mass is distributed in half of the volume. Hence, the concentration must be twice that of Equation 3-45a ... 

The extended source can be viewed as a summation (or integral) of point plane sources. The mass density at each plane e ( 8,8) is Cod. At position x, which is distance x - away from this plane, according to Equation 3-45a, the concentration due to this plane source is... 

Therefore, the concentration at x due to the extended instantaneous source can be found by summing all the plane sources ... 

AS. 1.1 Plane source for one-dimensional infinite medium (-oo[Pg.570]

A3.1.2. Plane source for one-dimensional semi-infinite medium (0 < x> co)... 

Initial condition Plane source at a = 0 with total mass M. 

Point source in ID Line source in 2D Plane source in 3D... 

Obtain the instantaneous plane-source solution in Table 5.1 by representing the plane source as an array of instantaneous point sources in a plane and integrating the contributions of all the point sources. 

Figure 9.11 shows a typical diffusion penetration curve for tracer self-diffusion into a dislocated single crystal from an instantaneous plane source at the surface [17]. In the region near the surface, diffusion through the crystal directly from the surface source is dominant. However, at depths beyond the range at... 

A comparison of guarded hot plate, transient plane source and modified hot wire methods has been made54 using polyurethane foam, and the strengths and weaknesses of the techniques discussed,... 

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. 

The value of depends on the intensity of the heat source. Heat generated at the plane source is conducted in both the x and — x directions. The fluxes c/t and cy2. and in these respective directions are obtained from Eqs. E5.5-8 and E5.5-9 ... 

Thus the maximum excess temperature is proportional to the intensity of the source, and it drops with increasing speed Vo, and increases in the thermal conductivity and the heat transfer coefficient. From Eqs. E5.5-8 and E5.5-9 we conclude that the temperature drops quickly in the positive x direction as a result of the convection (Vo < 0) of the solid into the plane source, and slowly in the direction of motion. Again, in this chapter we encounter... 

Galya DP (1987) A horizontal plane source model for ground-water transport. Ground Water 25... 

Beeause the eoneentration of metal ions at the interface is far in excess of that in the bulk of the solution, diffusion into the solution begins. Since the source of the diffusing ions is an ion layer parallel to the plane electrode, it is known as a plane source and since the diffusing ions are produced in an instantaneous pulse, a fuller deseription of the souree is eontained in the term instantaneous plane source. 

As the ions from the instantaneous plane source diffuse into the solution, their concentration at various distances will change with time. The problem is to calculate the distance and time variation of this concentration. 

This is the solution to the instantaneous-plane-source problem. When n/n,otai is plotted against X for various times, one obtains curves (Fig. 4.31) that show how the ions injected into the x = 0 plane at f = 0 (e.g., ions produced at the electrode in an impulse of metal dissolution) are distributed in space at various times. At any particular time t, a semi-bell-shaped distribution curve is obtained that shows that the ions are mainly clustered near the x = 0 plane, but there is a spread. With increasing time, the spread of ions increases. This is the result of diffusion, and after an infinitely long time, there are equal numbers of ions at any distance. 

Fig. 4.34. When diffusion occurs from an instantaneous plane source (set up, e.g., by a pulse of electrode dissolution), then 68% of the ions produced in the pulse lie between x = 0 and x = x g, after the time t.

Fig. 4.35. If it were possible for diffusion to occur in the +xand -xdirections from an instantaneous plane source at x= 0, then one-third of the diffusing species would lie between x = 0 and x= +x g and a similar number would lie between x=Oandx=-xj g.

When the current, or flux, is a single impulse, an instantaneous-plane source for diffusion is set up and the concentration variation is given by... 

Ions are pumped into a system electrochemically. At r = 0, a short burst of dissolution of an electrode is caused, giving rise to totai ions, which then begin to diffuse away from the source. Seek in the text the appropriate equation by which one may know the number of ions at a distance x and time t. This is a plane-source problem. Thus, Cu ions could be dissolved from a Cu plate filling the end of a tube of solution. The question is how many ions would have diffused... 

Consider the case of the moving rectangular plane source of length 21 and width 2b, as illustrated in Fig. 15-1. The steady state is attained at time t - the center of the source at that time is chosen as the origin of the coordinate system. At some given time previous, the center of the source was at the point 0, 0), 1/ being the velocity of translation. To find the temperature at time t = a for a given point (x, y, z) in the area bounded by x = and y = b, we must calculate the contributions of heat from every element of area dz dy within that area from -t = 0 to t = Equation 15-3 becomes... 

The case just considered is a very simple one in which the concentration has been forced to vary linearly from Cj to c. Another case of interest is when there is an instantaneous plane source at a particular plane in a liquid. A simple way of arriving at the equations applicable to this case is to note that a general solution of (11.41) for Pick s second law is... 

Figure 11.9 shows plots of cjn against x, for three different. i alues of Dt. This plot shows how the solute raolecules spread out from the instantaneous plane source located at x == 0. 

Plots of c// o against distance from an instantaneous plane source at X = 0. At this plane there are initially molecules of solute, at infinite concentration (volume = 0), The numbers on the curves are values of Dt... 

The mathematical expression for a is derived as follows (Fig. 8.4). A plane source of area emitting Sq particles/(m s), isotropically, is located a distance d away from a detector with an aperture equal to A. Applying the definition given by Eq. 8.2 for the two differential areas dA and dAj and integrating, one obtains ... 

Figure 8.4 Definition of the solid angle for a plane source and a plane detector parallel to the source.

Equation 8.4 is valid for any shape of source and detector. In practice, one deals with plane sources and detectors having regular shapes, examples of which are given in the following sections. 

Crete wall and use cylindrical attenuation, a good approximation can be made using the plane-source method (case 41). DJD (actual) = 5 X lOVlO = 5 X 10. JVom Fig. 10-3, using E = 2 Mev for ordinary concrete, = 3.5 ft. If this had been done by cylindrical attenuation (case 31), guess x 3.3 ft, or 40 in. 

TPS (Transient Plane Source) technique has been shown to be effective method to measure the thermal conductivity, diffusivity of rare earth oxide powder such as gadolinium oxide, samarium oxide, and yttrium oxide. The details of the measutrement are described in Ref 66. The experimental results of effective thermal conductivity as well as thermal diffusivity of the above described three rare earth oxides are tabulated in Table 5-7. 

The time-dependent temperature distribution in a transient experiment is governed by Eq. 4, and usually the related parameter, thermal diffusivity. is obtained. However, under certain circumstanees the solution to the heat equation contains the thermal conductivity as well as the thermal diffusivity, and by choosing a suitable method the diffusivity can be eliminated from the answer. The more important methods are the line and plane source heater methods and arc described below. These arc not Standard methods, but they can be used where speed is more imp .>rtant than absolute accuracy, to give a conductivity value more quickly than the Standard methods. They can also be used to compare a range of materials. 

This solution is derived from Equation 10.14 assuming an instantaneous plane source (layer thickness D is subsequently constant because of the restriction in concentration. 

Transient-state or unsteady-state methods make nse of either a line source of heat or plane sources of heat. In both cases, the usual procedure is to apply a steady heat flux to the specimen, which mnst be initially in thermal eqnUibrinm, and to measnre the tanperatnre rise at some point in the specimen, resnlting from this applied flux [83]. The Fitch method is one of the most common transient methods for measuring the thermal conductivity of poor conductors. This method was developed in 1935 and was described in the National Bureau of Standards Research Report No. 561. Experimental apparatus is commercially available. 

This is just twice the value, given previously using the assnnption that the thermal shield is an infinite slab source instead of an infinite plane source with a cosine distribution. The calculations are considered sufficiently conservative that the factor of 2 is neglected. Thus after using approximations for B and (b) as shown previously, the y flux at P is, as given in Eq. (4a),... 

We first consider a plane source of neutrons at x = 0. Evidently, the angular distribution of neutrons will be independent of y and and have at every point axial symmetry with respect to the x axis. The number of neutrons per unit volume for which the velocity component in the x direction Vx lies between fiv and (fi + dfji)v will denoted by... 

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