Concentration modulus

The value of the concentration modulus depends on the convective velocity and the mass-transfer coefficient of the concentration boundary layer (D/ i) that means that on the membrane structure and the hydrodynamic conditions. If the retention coefficient is equal to 1, then c /ch = exp(Pe). The larger convective velocity (or smaller diffusion coefficient) causes higher concentration polarization on the membrane interface. 

In ultrafiltration where dissolved species are separated, a solution is flowing parallel to the membrane surface under pressure as shown in the schematic in Figure 12.3. For the solute which is rejected by the membrane, there is a concentration gradient created across the stagnant boundary layer next to the surface of the membrane. The ratio of concentration of solute at the membrane surface (cm) to that of concentration in the bulk solution (ci,) is called concentration modulus indicative of concentration polarization. 

In Figure 5.24 the predicted direct stress distributions for a glass-filled epoxy resin under unconstrained conditions for both pha.ses are shown. The material parameters used in this calculation are elasticity modulus and Poisson s ratio of (3.01 GPa, 0.35) for the epoxy matrix and (76.0 GPa, 0.21) for glass spheres, respectively. According to this result the position of maximum stress concentration is almost directly above the pole of the spherical particle. Therefore for a... 

With appropriate caUbration the complex characteristic impedance at each resonance frequency can be calculated and related to the complex shear modulus, G, of the solution. Extrapolations to 2ero concentration yield the intrinsic storage and loss moduH [G ] and [G"], respectively, which are molecular properties. In the viscosity range of 0.5-50 mPa-s, the instmment provides valuable experimental data on dilute solutions of random coil (291), branched (292), and rod-like (293) polymers. The upper limit for shearing frequency for the MLR is 800 H2. High frequency (20 to 500 K H2) viscoelastic properties can be measured with another instmment, the high frequency torsional rod apparatus (HFTRA) (294). 

Another difference between hot and cold elastomeric SBR latices is that hot types are carried to < 90% conversion and not normally shortstopped. The cold latices are usually shortstopped at ca 60—80% conversion. Again the desired physical properties of the contained copolymer are responsible for these differences. Cold latices are used in applications where the modulus, eg, in foam, or retention of physical properties at high filler loadings, eg, in fabric backing, are required. The cold latices are generally suppHed at a higher soHds concentration than the hot series because of these uses. 

The characteristics of soluble sihcates relevant to various uses include the pH behavior of solutions, the rate of water loss from films, and dried film strength. The pH values of sihcate solutions are a function of composition and concentration. These solutions are alkaline, being composed of a salt of a strong base and a weak acid. The solutions exhibit up to twice the buffering action of other alkaline chemicals, eg, phosphate. An approximately linear empirical relationship exists between the modulus of sodium sihcate and the maximum solution pH for ratios of 2.0 to 4.0. 

Example 5 Application of Effectiveness For a second-order reaction in a plug flow reactor the Thiele modulus is ( ) = SVQ, and inlet concentration is C50 = 1.0. The equation will he integrated for 80 percent conversion with Simpsons rule. Values of T) are... 

A guide to tire stabilities of inter-metallic compounds can be obtained from the semi-empirical model of Miedema et al. (loc. cit.), in which the heat of interaction between two elements is determined by a contribution arising from the difference in work functions, A0, of tire elements, which leads to an exothermic contribution, and tire difference in the electron concentration at tire periphery of the atoms, A w, which leads to an endothermic contribution. The latter term is referred to in metal physics as the concentration of electrons at the periphery of the Wigner-Seitz cell which contains the nucleus and elecUonic structure of each metal atom within the atomic volume in the metallic state. This term is also closely related to tire bulk modulus of each element. The work function difference is very similar to the electronegativity difference. The equation which is used in tire Miedema treatment to... 

A high modulus gradient at the interface is also be avoided in materials Joined as a result of the interdiffusion of materials to form a fractal surface [32]. The effect is to produce an interfacial composite region. This strengthens the interface and leads to a more gradual change in modulus and avoids the sharp concentrations of stress which would occur at a smooth interface. 

Class and Chu [34] have studied the tackification of natural rubber and SBR over a wide range of resin concentrations for several tackifiers. From their graphical data it can be estimated that 1 1 tackification (by weight) with a poly(/-butyl styrene) resin, MW 850 and Tg = 59°C, gives a PSA with Tg about — 13°C, and storage modulus, G about 8.8 x 10 Pa, well within the PSA window. 

In the earlier art, there was some consideration that partial incompatibility of the tackifier resin with the rubber was responsible for the appearance of tack, but this no longer is seriously held in light of continuing studies by many investigators. Aubrey [38] has addressed this in his review of the mechanism of tackification and the viscoelastic nature of pressure sensitive adhesives. Chu [39] uses the extent of modulus depression with added tackifier as a measure of compatibility. Thus in a plot of modulus vs. tackifier concentration, the resin that produces the deepest minimum is the most compatible. On this basis, Chu rates the following resins in order of compatibility for natural rubber rosin ester > C-5 resin > a-pinene resin > p-pinene resin > aromatic resin. 

Langley, N.R. and Polmanteer, K.E., Relation of elastic modulus to crosslink and entanglement concentrations in rubber networks. J. Polym. Sci. Polym. Phys. Ed., 12(6), 1023-1034 (1974). 

No one steel exceeds the tensile modulus of mild steel. Therefore, in applications in which rigidity is a limiting factor for design (e.g., for storage tanks and distillation columns), high-strength steels have no advantage over mild steel. Stress concentrations in mild steel structures are relieved by plastic flow and are not as critical in other, less-ductile steels. 

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