Pulse of solutions

A closed-form solution of Eq. 6.24 has been derived by Lapidus and Amundson [3], Levenspiel and Smith [17], Carberry and Bretton [18], Reilley et al. [19], and Wicke [20]. All these authors used an "open-open" boundary condition, i.e., conditions assuming that the column stretches to infinity in both directions (z —00, dC/dz = 0 z oo, dC/dz = 0), and that a Dirac 6 z) pulse of solute is injected at z = 0. With these boundary conditions, the solution of Eq. 6.24 is given by... 

In most cases, chromatography is performed with a simple initial condition, C(f = 0,z) = q t = 0,z) = 0. TTie column is empty of solute and the stationary and mobile phases are under equilibrium. There are some cases, however, in which pulses of solute are injected on top of a concentration plateau (see Chapter 3, Section 3.5.4). The behavior of positive concentration pulses injected xmder such conditions is similar to that of the same pulses injected in a column empty of solute and they exhibit similar profiles. Even imder nonlinear conditions (high plateau concentration), a pulse that is sufficiently small can exhibit a quasi-linear behavior and give a Gaussian elution profile. Its retention time is linearly related to the slope of the isotherm at the plateau concentration. Measuring this slope is the purpose of the pulse method of measurement of isotherm data. Large pulses may also be injected and they will give overloaded elution profiles similar to those obtained with a column empty of solute. 

Both buffer and sample must be loaded with care to prevent introducing air bubbles into the cells, which can lead to erratic baselines. Samples are loaded into the syringes slowly to prevent cavitation and then slowly injected into the cells. Both sample and reference cells are overloaded so that sufficient solution accumulates at the top to permit withdrawal of approximately 100 p-1 of solution without introducing air into the capillaries or cells. Rapid pulses of solution in and out of the cells created by oscillation of the syringe barrel back and forth quickly between the forefinger and thumb should dislodge and expell air bubbles trapped in the cell and loading capillaries. 

We shall use this solution to describe response to a pulse of solute, (c) Find the constant K by noting that must be conserved at any time, t > 0 to get ... 

Consider first diffusion in one dimension, initiated by injection of a pulse of solute (no moles) at zero time into a tube of unit cross-section, as in Figure 3.2. Let p dx be the probability that at time t a particular molecule has undergone displacement to a distance between x and (x -I- dx) from its initial position. This probability is the ratio of the number of molecules at distances between x and x -H dx, which is cAa dx, to the number woIVa originally introduced at zero time, i.e., (c/no)dx whence p = c/no- Referring to Equation (3.12), we then have ... 

Consider a rectangular separator vessel of dimensions (/ ly, Iz) in the (x,y,z)-coordinate directions, respectively (Figure 3.2.1). Let the separator contain a single-phase system, say a solvent. At time f = 0, we subject the system to a pulse of solute species 1 and solute species 2, the total number of moles of each species being mi and m2, respectively. The pulse is located at the vessel side characterized by X = 0, lx, y = 0, ly z = 0. Assume the following. 

As a first example, we consider the diffusion away from a sharp pulse of solute. This example is the third truly important problem for diffusion. It complements the cases of a thin film and the semi-infinite slab to form the basis of perhaps 95 percent of all the diffusion problems which are encountered. The initially sharp concentration gradient relaxes by diffusion in the z direction into the smooth curves shown in Fig. 2.4-1. We want to calculate the shape of these curves. This calculation illustrates the development of a differential equation and its solution using Laplace transforms. 

The specific example concerns the fate of a sharp pulse of solute injected into a long, thin tube filled with solvent flowing in laminar flow (Fig. 4.4-1). As the solute pulse moves through the tube, it is dispersed. We want to calculate the concentration profile resulting from this dispersion. 

Fig. 4.4-1. Taylor dispersion. In this case, solvent is passing in steady laminar flow through a long, thin tube. A pulse of solute is injected near the tube s entrance. This pulse is dispersed by the solvent flow, as shown.

We now turn to more complex and more expensive methods, which can also be easier to run or which give more accurate results. The first of these is Taylor dispersion, illustrated schematically in Fig. 5.6-3 (Ouano, 1972). This method, which is valuable for both gases and liquids, employs a long tube filled with solvent that slowly moves in laminar flow. A sharp pulse of solute is injected near one end of the tube. When this pulse comes out the other end, its shape is measured with a differential refractometer. Except for the refractometer, which can be purchased off the shelf, the apparatus is inexpensive and moderately easy to build. This apparatus can be used routinely by those with little training. It can be operated relatively easily at high temperature and pressure. It has the potential to give results accurate to better than one percent. 

The analysis of elution chromatography is a close parallel to that for adsorption. The best theoretical development, due to Golay, assumes the process takes place not in a packed bed but in a thin cylindrical tube with adsorbent-coated walls. Solvent moves steadily in laminar flow through the tube. At time zero a pulse of solutes is injected at one end of the tube. Each solute elutes with a concentration profile given by Eq. 4.4-31... 

Glaser and Lichtenstein (G3) measured the liquid residence-time distribution for cocurrent downward flow of gas and liquid in columns of -in., 2-in., and 1-ft diameter packed with porous or nonporous -pg-in. or -in. cylindrical packings. The fluid media were an aqueous calcium chloride solution and air in one series of experiments and kerosene and hydrogen in another. Pulses of radioactive tracer (carbon-12, phosphorous-32, or rubi-dium-86) were injected outside the column, and the effluent concentration measured by Geiger counter. Axial dispersion was characterized by variability (defined as the standard deviation of residence time divided by the average residence time), and corrections for end effects were included in the analysis. The experiments indicate no effect of bed diameter upon variability. For a packed bed of porous particles, variability was found to consist of three components (1) Variability due to bulk flow through the bed... 

Siemes and Weiss (SI4) investigated axial mixing of the liquid phase in a two-phase bubble-column with no net liquid flow. Column diameter was 42 mm and the height of the liquid layer 1400 mm at zero gas flow. Water and air were the fluid media. The experiments were carried out by the injection of a pulse of electrolyte solution at one position in the bed and measurement of the concentration as a function of time at another position. The mixing phenomenon was treated mathematically as a diffusion process. Diffusion coefficients increased markedly with increasing gas velocity, from about 2 cm2/sec at a superficial gas velocity of 1 cm/sec to from 30 to 70 cm2/sec at a velocity of 7 cm/sec. The diffusion coefficient also varied with bubble size, and thus, because of coalescence, with distance from the gas distributor. 

Reactions such as these are of interest in themselves. Beyond that, one can use the pulse radiolysis experiment as a preparative technique for other species. Thus, the reactions of numerous aliphatic, carbon-centered radicals have been evaluated.22 If one employs a reasonably high concentration of solute, say 0.1-1 M CH3OH, the formation of CH2OH is complete within the electron pulse. Following that, reactions such as the following can be studied ... 

In the salt injection method(3l) a pulse of salt solution is injected into the line and the time is measured for it to travel between two electrode pairs situated a known distance apart, downstream from the injection point. 

In a recent study of the transport of coarse solids in a horizontal pipeline of 38 mrrt diameter, pressure drop, as a function not only of mixture velocity (determined by an electromagnetic flowmeter) but also of in-line concentration of solids and liquid velocity. The solids concentration was determined using a y-ray absorption technique, which depends on the difference in the attenuation of y-rays by solid and liquid. The liquid velocity was determined by a sail injection method,1"1 in which a pulse of salt solution was injected into the flowing mixture, and the time taken for the pulse to travel between two electrode pairs a fixed distance apart was measured, It was then possible, using equation 5.17, to calculate the relative velocity of the liquid to the solids. This relative velocity was found to increase with particle size and to be of the same order as the terminal falling velocity of the particles in the liquid. 

Participation of adsorbed intermediates can also be shown by the prolonged decay of the potential 011 interruption of the current (Conway and Vijh, 1967a) or by measurement of the time-dependence of the formation of products by carrying out the reaction with pulses of potential of controlled duration (Fleischmann et al., 1966). Thus the formation of ethane in the Kolbe reaction of acetate ions in acid solutions is initially proportional to the square of time as would be predicted for the rate of the step (27) (Fleischmann et al., 1965). 

Solution This solution illustrates a possible definition of the delta function as the limit of an ordinary function. Disturb the reactor with a rectangular tracer pulse of duration At and height A/t so that A units of tracer are injected. The input signal is Cm = 0, t < 0 = A/Af, 0 < t < At ... 

In addition to sample rotation, a particular solid state NMR experiment is further characterized by the pulse sequence used. As in solution NMR, a multitude of such sequences exist for solids many exploit through-space dipolar couplings for either signal enhancement, spectral assignment, interauclear distance determination or full correlation of the spectra of different nuclei. The most commonly applied solid state NMR experiments are concerned with the measurement of spectra in which intensities relate to the numbers of spins in different environments and the resonance frequencies are dominated by isotropic chemical shifts, much like NMR spectra of solutions. Even so, there is considerable room for useful elaboration the observed signal may be obtained by direct excitation, cross polarization from other nuclei or other means, and irradiation may be applied during observation or in echo periods prior to... 

---