Independent portions

Equations (22-86) and (22-89) are the turbulent- and laminar-flow flux equations for the pressure-independent portion of the ultrafiltra-tion operating curve. They assume complete retention of solute. Appropriate values of diffusivity and kinematic viscosity are rarely known, so an a priori solution of the equations isn t usually possible. Interpolation, extrapolation, even precuction of an operating cui ve may be done from limited data. For turbulent flow over an unfouled membrane of a solution containing no particulates, the exponent on Q is usually 0.8. Fouhng reduces the exponent and particulates can increase the exponent to a value as high as 2. These equations also apply to some cases of reverse osmosis and microfiltration. In the former, the constancy of may not be assumed, and in the latter, D is usually enhanced very significantly by the action of materials not in true solution. 

These compounds contain two or more independent portions that are not bonded to each other by any valence forces but nevertheless must remain linked. Catenanes are made up of two or more rings held together as links in a chain, while in rotaxanes a... 

In addition, deletion of units allows a network to divide into multiple independent portions, without interconnections, and thus describe more accurately datasets in which one part consists of sample patterns that are very unlike the remainder. Once the performance of the network is not noticeably improved by the addition of a unit, the expansion and contraction may be brought to a halt. The network can then be used as an interpretive tool in exactly the same fashion as a SOM. 

Algebraic equations (AEs) describe relations among variables of the system which are independent of time or space changes. They represent variables not related by material energy or momentum balances. They may characterize physicochemical or other type of relationships between physically independent portions of the system, as the vapor and liquid spaces of a reaction vessel. 

The poly-L-proline II configuration propagates outward from these nuclei along single gelatin chains. This process is responsible for the more rapid, concentration-independent portion of the mutarotation phenomenon. 

In this equation, gives the solvent-independent portion of the pre-exponential factor. Separation of the solvent contributions to the exponential and pre-exponential terms in the rate constant is much more difficult for adiabatic reactions. However, if the outer sphere contribution to A G° is small, then the contribution from Xl in the pre-exponential factor dominates. Under these circumstances a plot of the logarithm of the electron transfer rate constant In ket should be linear in In xl-... 

An alternative explanation for the small pH-independent portion of the seawater a s is that some side reactions are occurring which affect the fitted parameters. 

An important feature of most simulation languages is that they permit control of the pseudorandom numbers through random number streams or seeds, which permit the user to access different, and thus apparently independent, portions of the pseudorandom number sequence. Such control allows the user to induce dependence where desired or obtain independence where necessary. 

FIGURE 4.13 Flow-volume curves for increasing effort level during expiration, including maximal effort Note eflbit-independent portion of curves. 

The available data on the fractional coverage of platinum anodes by adsorbed formic acid are summarized in Figure g 39,176-178 coverage-potential curves had similar shapes as determined by several investigations however, the potential-independent portions vary considerably. 

From equation 11, the ionic conductivity (a) can be calculated from the frequency-independent portion of the loss factor data, that is, where a log-log plot of loss factor versus frequency has a slope of -1, using the relation... 

Typically, redundancy is an entirely independent portion of the system that can be switched on in the event of a failure of the identical, originally used counterpart. The use of redundancy increases the reliability of the system. An example of a simple circuit that has a redundant resistor/capacitor portion is shown in Fig. 4.5. 

In this formulation, the denotes the usual energy-independent portion of the scattering cross section and the a[p the resonance structure. We will assume that the scattering cross section for the nonfuel materials has no resonances and can be well represented by an energy-independent function. Let us define, then, the cross section... 

The entropic structural parameters ri and T2 in Eqs. 22b-22d are the temperature independent portions of Xy that arise from chains of blend components 1... 

An additional dependence on the copolymer compositions x and y beyond that predicted by simple random copolymer FH theory appears in Eq. 24b through the front conversion factor C/S1S2 as well as through factors of 51/52 and 52/51. The main novel feature of Eq. 24b, however, lies in the presence of the temperature-independent portion of xsans. This term represents the influence of monomer structural asymmetry on the nonrandom chain packing and coincides with the leading contribution of the LCT for binary blends in the incompressible, high molecular weight, athermal, fully flexible chain limit. 

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