Laplace Instruments

There are a variety of feedback controller tuning methods. Probably 80 percent of all loops are tuned experimentally by an instrument mechanic, and 75 percent of the time the mechanic can guess approximately what the settings will be by drawing on experience with similar loops. We will discuss a few of the time-domain methods below. In subsequent chapters we will present other techniques for tinding controller constants in the Laplace and frequency domains. 

Spiegel MR (1965) Theory and problems of Laplace transforms. McGraw-Hill, New York Nicholson RS, Olmstead ML (1972) Numerical solutions of integral equations. In Matson JS, Mark HB, MacDonald HC (eds) Electrochemistry calculations, simulations and instrumentation, vol 2. Marcel Dekker, New York, p 119... 

When Lavoisier invited the younger Laplace to collaborate with him in a quantitative study of heat, the invitation was a great compliment to the younger scientist. The results of their joint research were presented to the Royal Academy of Sciences in 1783. The collaboration was symbolic of the union of chemistry with physics and a landmark in the quantification of what Lavoisier regarded as an imponderable substance, that is, a substance without weight. Both scientists were committed to the search for precise measurements, at a time when instrument makers were producing apparatus of unprecedented accuracy. 

The measurements that Lavoisier and Laplace obtained with their instrument,... 

From the experimental standpoint, the use of a.c. techniques offers many advantages. Sensitivity is much higher than in d.c. measurements, since phase-sensitive detection can be used and very small probe signals can be employed ( 5mV). The technique is therefore a truly equilibrium one, unlike cyclic voltammetry. An alternative approach to the commonly used sinusoidal signal superimposed on the selected d.c. potential is to use a potential step and to employ Laplace transform methods. Instrumentally, this is rather more demanding and the advantages are not clear [51]. Fourier transform methods have also been considered and their use will have advantages in terms of the time-scale for an experiment, especially at very low frequencies. 

The surface tension value in the maximum bubble pressure method is calculated via the Laplace equation. For the instrument under discussion, the capillary radius is small and the bubble shape is thus assumed to be spherical. Thus the deviation of the bubble shape from a spherical one can be neglected and needs no correction. Hence, the following equation results. 

Porosimeter An instrument for the determination of pore size distribution by measuring the pressure needed to force liquid into a porous medium and applying the Young-Laplace equation. If the surface tension and contact angle appropriate to the injected liquid are known, pore dimensions can be calculated. A common liquid for this purpose is mercury hence, the term mercury porosimetry. 

The functioning of the instrument, as described in detail also in the book chapter mentioned above [176], is different from most of the other instruments. Due to the large internal gas volume (about 35 cm ) an easy procedure for determining the effective adsorption time in the moment of maximum pressure was derived (see below). The surface tension y can be calculated from the measured maximum capillary pressure P and the known capillary radius r jp using the Laplace equation in the simplified form for spherical drop/bubble shapes... 

A versatile interfacial and film rheometer has been developed in our laboratory (7—10). In this technique, a curved, spherical cap-shaped fluid interface or liquid film is formed at a capillary tip and the interfacial tension (IFT) of the single interface or the film tension of the film can be determined by measuring the capillary pressure of the interface or film (Fig. 1). The IFT or film tension is related to the capillary pressure and the radius of the interface or film curvature by the Young-Laplace equation. The IFT and film tension can be measured not only in equilibrium, but also in dynamic conditions as well. The automated apparatus makes it possible to change the interfacial or film area in virtually any mode (expansion or contraction) at various rates (Fig. 2). This instrument is now made available through our laboratory. 

A.A. Pilla, Laplace plane analysis of electrode reactions, in Electrochemistry. Calculations, Simulations, and Instrumentations, ed. J.S. Mattson, H.B. Mark, C. MacDonald, Jr., (Marcel Dekker, New York, 1972)... 

Drop and bubble shape tensiometry is a modem and very effective tool for measuring dynamic and static interfacial tensions. An automatic instrument with an accurate computer controlled dosing system is discussed in detail. Due to an active control loop experiments under various conditions can be performed constant drop/bubble volume, surface area, or height, trapezoidal, ramp type, step type and sinusoidal area changes. The theoretical basis of the method, the fitting procedure to the Gauss-Laplace equation and the key procedures for calibration of the instrument are analysed and described. 

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