The Radial Differential System

Solutions of the Radial Differential System for the Discrete Spectrum... 

Applying the first law to a radially differential system within the pipe walls leads to the same governing equation as for the solid cylinder, with different boundary conditions, The solution given by Eq. (2.94), inserted into the first condition, gives... 

The surface tension is very insensitive to changes in supersaturation. While the planar surface tension is, of course, temperature-dependent, the ratio of the small system surface tension to the planar limit is nearly independent of temperature. This simplicity in the functional dependence of the radially dependent surface tension has been anticipated by a number of workers in terms of the parameter defined by Equation 18. Tolman obtained the following differential equation... 

Another potential solution technique appropriate for the packed bed reactor model is the method of characteristics. This procedure is suitable for hyperbolic partial differential equations of the form obtained from the energy balance for the gas and catalyst and the mass balances if axial dispersion is neglected and if the radial dimension is first discretized by a technique such as orthogonal collocation. The thermal well energy balance would still require a numerical technique that is not limited to hyperbolic systems since axial conduction in the well is expected to be significant. 

The first step in the solution procedure is discretization in the radial dimension, which involves writing the three-dimensional differential equations as an enlarged set of two-dimensional equations at the radial collocation points with the assumed profile identically satisfying the radial boundary conditions. An examination of experimental measurements (Valstar et al., 1975) and typical radial profiles in packed beds (Finlayson, 1971) indicates that radial temperature profiles can be represented adequately by a quadratic function of radial position. The quadratic representation is preferable to one of higher order since only one interior collocation point is then necessary,6 thus not increasing the dimensionality of the system. The assumed radial temperature profile for either the gas or solid is of the form... 

Since the resulting system after radial collocation is still too complex for direct mathematical solution, the next step in the solution process is discretization of the two-dimensional system by orthogonal collocation in the axial direction. Although elimination of the spatial derivatives by axial collocation increases the number of equations,8 they become ordinary differential equations and are easily solved using traditional techniques. Since the position and number of points are the only factors affecting the solution obtained by collocation, any set of linearly independent polynomials may be used as trial functions. The Lagrangian polynomials of degree N based on the collocation points... 

Note that this equation still retains the radial coordinate r. Therefore, unlike wedge case, there is not a unique ordinary differential that applies at any radius. Rather, there is an ordinary differential for every r position. Such local similarity behavior certainly represents a simplification compared to the original partial-differential-equation system. Nevertheless, the differential equation is more complex than that for the wedge case. 

An important issue in the boundary-layer problem, and in differential-algebraic equations generally, is the specification of consistent initial conditions. We think first of the physical problem (not in Von Mises form), since the inlet profiles of u, v, and T, as well as pressure p, must be specified. However, all the initial conditions are not independent, as they would be for a system of standard-form ordinary differential equations. So assuming that the axial velocity u and temperature T profiles are specified, the radial velocity must be required to satisfy certain constraints. 

It is possible and beneficial to reduce the system to index-one by replacing A with a new dependent variable , where A = 3/31 [13], The initial condition for is arbitrary, since itself never appears in the equations—a suitable choice is 4> = 0. Anywhere A appears, it is simply replaced with 3/31, which is conveniently done in the DAE software interface. The index reduction can be seen from the following procedure The continuity at the inlet boundary (an algebraic constraint) can be differentiated once with respect to t to yield an equation for dV/dt. Then dV/dt is replaced by substitution of the radial-momentum equation. This substitution introduces A = 34>/31, which makes the continuity equation (at the inlet boundary) an independent differential equation for 4>. Thus the modified system is index-one. This set of substitutions is not actually done in practice—it simply must be possible to do them to achieve the index reduction. 

The complex phase shift can be obtained from exact numerical solution of the radial Schrodinger equation.2 The following quantities can immediately be given in terms of 8r The differential elastic cross section in the center-of-mass system... 

The passage of electrons or other particles with charge q and mass m through an electrostatic lens system is governed by their motion under the action of the electric field. In the case considered here, cylindrical symmetry around the optical axis (z-axis) and paraxial rays will be assumed. Of the cylindrical coordinates only the transverse radial coordinate p and the distance coordinate z are of relevance, and the electrostatic potential of the lens is given by q>(p, z). As shown in Section 10.3.1, in the paraxial approximation the potential q>(p, z) is fully determined by the potential symmetry axis. Hence, the equations of motion and the fundamental differential equation of an electrostatic lens depend only on this potential. The fundamental lens equation is given by (see equ. (10.38))... 

This allows us to represent partial differential equations as found in the balance equations using the collocation method. Equation (11.47) is a solution to a partial differential equation represented by a system of linear algebraic equations, formed by the interpolation coefficients, oij, and the operated radial functions. The interpolation coefficients are solved for using matrix inversion techniques to approximately satisfy the partial differential equation... 

Discretization of the partial differential equation system in axial (z) and radial (r) direction by means of the orthogonal collocation method (7) leads to the following system of ordinary differential equations. 

The location of the collocation points is shown in Figure 4. The parameters of the differential equation system were estimated by applying a modified version of the method described by Van den Bosch and Hellincks (8). For the interpolation it was assumed that the radial profiles could be described by the polynomial functions given in Eqs. 18) and 19), respectively. 

Differential diagnoses of peripheral neuropathy were entertained. Laboratory tests revealed that serum parameters for electrolytes and proteins were all within the normal range. Urine porphyrinogen and porphobilinogen levels were normal. Tests were negative for serum rheumatoid factor and antinuclear antibodies, the latter used in detection of connective tissue diseases such as systemic lupus erythematosus and polyarteritis nodosa that could present with features of peripheral neuropathy. Nerve conduction studies of the radial, ulnar, and median nerves revealed delayed conduction. Biopsies of the ulnar and radial nerves showed loss of nerve fibers and sudanophilic (indicating lipid) deposits in the Schwann cells of the neurons. Similarly, the yellowish plaques of the pharynx showed abundant macrophages filled with sudanophilic material. These deposits were not membrane-bound. 

For the bulk polymerization of styrene using thermal initiation, the kinetic model of Hui and Hamielec (13) was used. The flow model (Harkness (1)) takes radial variations in temperature and concentration into account and the velocity profile was calculated at every axial point based on the radial viscosity at that point. The system equations were solved using the method of lines with a Gear routine for solving the resulting set of ordinary differential equations. 

The Producing Gas-Oil Ratio Equation. This equation, which is also known as the instantaneous gas-oil ratio equation, is of fundamental importance in reservoir analysis. To derive this equation consider a radial flow system shown in Figure 97. Consider a da thickness within the system at radius x. If the pressure gradient at this radius is dP/dx, Darcy s Law in differential form for the simultaneous flow of oU and gas in the reservoir may be written... 

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