Laplace filter

Surfactants aid dewatering of filter cakes after the cakes have formed and have very Httle observed effect on the rate of cake formation. Equations describing the effect of a surfactant show that dewatering is enhanced by lowering the capillary pressure of water in the cake rather than by a kinetic effect. The amount of residual water in a filter cake is related to the capillary forces hoi ding the Hquids in the cake. Laplace s equation relates the capillary pressure (P ) to surface tension (cj), contact angle of air and Hquid on the soHd (9) which is a measure of wettabiHty, and capillary radius (r ), or a similar measure appHcable to filter cakes. 

For the design of the actively compensated RF pulse, experimental and numerical determination of the response function h(t) of the circuit is necessary. We should also keep in mind that modification to the circuit, such as probe timing, insertion or removal of RF filters, and so on, can alter h(t). In practice, it is convenient to measure the response y t) to a step excitation u(t) instead of that to the impulse excitation. By performing Laplace transformation to... 

A transfer function, defined as the Laplace transfer of the impulse response of a linear system, can be obtained from the model. This can be very useful, because with a transfer function the influence of extra-column effects (detector, amplifier, filter) on the peak shape can be easily calculated. The transfer function is ... 

The fact that the permeability kp is taken to be a constant scalar indicates that the filter cake is homogeneous, isotropic, and stationary in properties with time. Since the filtrate is incompressible, Eq. (157) leads immediately to Laplace s equation ... 

This very simple and established method has become a standard technique used by suppliers to measure the largest active pores (as well as cracks or pinholes) in a membrane. The principle is to measure the pressure needed to force air through a liquid-filled membrane. The bottom of the filter is in contact with air and, as the air pressure is gradually increased, air bubbles penetrate through the membrane at a certain pressure. The pressure and pore radius are related by the Laplace equation... 

Vanderveen et al. studied D2O in Nafion membranes using MQ-filtered deuteron experiments as a function of the hydration level. They evaluated the T2 relaxation times and interpreted the results in terms of a hydration model with two water domains. Ohkubo and co-workers studied Nafion, as well as sulfonated poly(ethersulfone) membranes. They used diffusion-weighted H inversion-recovery measurements followed by Laplace-trans-form analysis of the distribution of longitudinal relaxation times and were able to distinguish water molecules in larger and smaller channels. 

The filtering behavior of the passive RC low-pass filter can be understood easily by using the potential divider approach and Laplace equivalents in the s domain for the circuit components. Thus, using the potential divider approach and rearranging, we have... 

The output of the simple moving average filter is the average of the M -F 1 most recent values of x(n). Intuitively, this corresponds to a smoothed version of the input, but its operation is more appropriately described by calculating the frequency response of the filter. First, however, the z-domain representation of the filter is introduced in analogy to the s- (or Laplace-) domain representation of analog filters. The z transform of a causal discrete-time signal x(n) is defined by... 

A structured model of the process (typically a Laplace tramsfer function) is used directly in a design method such as pole-placement or internal model control (IMG) to yield expressions for the controfier parameters that are functions of the process model parameters and some user-specified parameter related to the desired performance, e.g. a desired closed-loop time constant. These approaches to PID design carry restrictions on the allowable model structure, although it has been shown that a wide range of types of processes can be accommodated if the PID controller is augmented with a first order filter in series. An example of this design approach may be found in Rivera et al. (1986). 

Another characterization method that can be included with those based on the Laplace equation is mereury intrusion porosimetry. The method (also proposed by Washburn [101]) was developed by Ritter and Drake [52] and was applied for the first time to the characterization of membrane filters by Honold and Skau [102]. It has been shown to be a reliable method for the characterization of pore size distributions, pore strueture, and speeific surface areas. Here, a Hg-air interface appears inside each pore. Thus, Eq. (4) is also followed. However, in this case, Hg does not wet practically any kind of sample (the corresponding contact angles ranging from 112° to 150°) [52, 101]. 

Separation and direct analysis in one of the phases. If the aqueous phase can be separated and analyzed it is possible to measure the proportion of BLI associated with the droplets (surface and core) by difference. Unfortunately, fine particles are hard to filter or separate by sedimentation. More aggressive separation procedures (e.g., solvent extraction) often destroy the equihbrium under investigation (especially surface and Laplace pressure effects). Alternatively, a volatile BLI can be determined in the headspace allowing its activity coefficient in the emulsion to be determined non-destructively (Ghosh et al, 2006). 

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