Analogy Solutions

While the ion pair combination 1 is found to be a successful example to afford artificial ion channels, it may not be difficult to find analogous solutions. Illustrated in Figure 7 is a small expanded scheme for constructing artificial supramolecular ion channels from synthetic amphiphilic pairs of hydrophilic and hydrophobic counterions (note that the combination 2a-2e corresponds to 1). All of these compounds gave stable single-channel currents when incorporated into a bilayer lipid membrane. 

Most solutions of the diffusion equation (7-12) are taken from analogous solutions of the heat conductance equation that has been known for many years ... 

ANALOGY SOLUTIONS FOR HEAT TRANSFER IN TURBULENT FLOW... 

In outlining the main steps in obtaining an analogy solution, attention will here be given to two-dimensional flow. The total shear stress and total heat transfer rate are made up of the molecular and turbulent contributions, i.e. ... 

As discussed in the previous chapter, most early efforts at trying to theoretically predict heat transfer rates in turbulent flow concentrated on trying to relate the wall heat transfer rate to the wall shear stress [1],[2],[3],[41. The reason for this is that a considerable body of experimental and semi-theoretical knowledge concerning the shear stress in various flow situations is available and that the mechanism of heat transfer in turbulent flow is obviously similar to the mechanism of momentum transfer. In the present section an attempt will be made to outline some of the simpler such analogy solutions for boundary layer flows, attention mainly being restricted to flow over a flat plate. 

The simplest analogy solution is that due to Reynolds. This analogy has many limitations but does have some practical usefulness and serves as the basis for more refined analogy solutions. 

The analogy solutions discussed in the previous section use the value of the wall shear stress to predict the wall heat trans er rate. In the case of flow over a flat plate, this wall shear stress is given by a relatively simple expression. However, ir, general, the wall shear stress will depend on the pressure gradient and its variation has to >e computed for each individual case. One approximate way of determining the shear stress distribution is based on the use of the momentum integral equation that was discussed in Chapter 2 [1],[2],[3],[5]. As shown in Chapter 2 (see Eq. 2.172), this equation has the form ... 

These two equations effectively constitute the turbulence model. They are solved simultaneously with Eq. (6.86) to give the variation of rjpu with x. The selected analogy solution equation is then used to give the local heat transfer rate variation. 

ANALOGY SOLUTIONS FOR FULLY DEVELOPED PIPE FLOW... 

The ideas used in the previous chapter to derive analogy solutions for boundary layer flows can easily be extended to obtain such analogy solutions for turbulent 304... 

Eqs. (7.122) to (7.125) can be simultaneously solved to give the variationsheat transfer rate, the Reynolds analogy, for example, giving ... 

Some simple methods of determining heat transfer rates to turbulent flows in a duct have been considered in this chapter. Fully developed flow in a pipe was first considered. Analogy solutions for this situation were discussed. In such solutions, the heat transfer rate is predicted from a knowledge of the wall shear stress. In fully developed pipe flow, the wall shear stress is conventionally expressed in terms of the friction factor and methods of finding the friction factor were discussed. The Reynolds analogy was first discussed. This solution really only applies to fluids with a Prandtl number of 1. A three-layer analogy solution which applies for all Prandtl numbers was then discussed. 

The so-called Taylor-Prandtl analogy was applied to boundary layer flow in Chapter 6. Use this analogy solution to derive an expression for the Nusselt number in fully developed turbulent pipe flow. 

Consider the flow of water through a 65-mm diameter smooth pipe at a mean velocity of 4 m/s. The walls of the pipe are kept at a uniform temperature of 40°C and at a certain section of the pipe the mean water temperature is 30°C. Find the heat transfer coefficient for this situation using both the Reynolds analogy and the three-layer analogy solution. 

Analogy Solutions for Heat Transfer in Turbulent Flow 244... 

Analogy Solutions for Fully Developed Pipe Flow 304... 

The silicate species discussed in the preceding section can react with aluminate anions, Al(OH)4 to produce aluminosilicate anions. Si NMR spectra of solid silicates and aluminosilicates indicate that the replacement of Si by A1 in the second coordination sphere of a give Si causes a low-field shift of about 5 ppm. Since each Si atom can have up to four metal atoms in its second coordination spere, fifteen possible Qn(mAl) structural units can be envisioned. The estimated chemical shift ranges for these units are given in Table 3. It is apparent from this table that the 29si spectrum of an aluminosilicate solution in which A1 and Si atoms were statistically distributed would be much more complex than that of an analogous solution containing only silicate species. 

Schreiber utilized intra-site olefin metathesis as a key step in synthesizing C2-symmetric fhnctionalized dimeric molecules for the close association of proteins. Both (2a) and (4a) were effective at 5 mol% giving very high conversions to the dimerized product (equation 66). The E. Z ratios of the linked bead dimerized molecules were similar to what was found in analogous solution reactions. 

Concentration is variable with time, Pick s second law Most interactions involving mass transfer between the packaging and food behave under non-steady state conditions and are referred to as migration. A number of solutions exist by integration of the diffusion equation 8.7 that are dependent on the so-called initial and boundary conditions of special applications. Many solutions are taken from analogous solutions of the heat conductance equation that has been known for many years ... 

Chloro(/7-cyclopentadienyl)thorium(lV) is a white solid, less oxygen-sensitive than the uranium analog but considerably more moisture-sensitive. Exposure to air causes the appearance of an ocher color. The compound is less soluble in all organic solvents than the uranium analog solutions are extremely air-sensitive. The nmr spectrum in benzene-de exhibits a sharp singlet at t3.81. The infrared spectrum (Nujol mull) shows 7t-cyclopentadienyl bands at 1016 (m), 812 (sh), and 788 (s) cm i. 

This is identical in form to Eq (6.5). with the constant equal to —and we can write down the analogous solutions... 

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