Modulus enhancement factor

Tahir and Birley24 have considered the tangent modulus at intervals on the stress relaxation curve and compared it to the modulus of the initial loading curve to derive what they termed the Modulus Enhancement Factor, hence adding to the information which can be gained from a stress relaxation test. 

Figure 2.2 Modulus enhancement factor for reinforced thermoplastics as a function of fiber volume fraction, orientation angle, and aspect ratio. (Adapted from Ref. [18].)...

Polymer sample Test direction Maximum modulus (GPa) Maximum strength (MPa) Modulus enhancement factor Strength enhancement factor... 

Figure 5.20 Modulus enhancement factor, G (blend)/G (niatrix), as a function of volume fiaction rubber Nonreactive (O) PS-Ox/EP (A) PS/EP-MA Reactive ( ) PS-Ox/EP-MA. T = 180°C lj— 0.1 rad/s...

Figure 1.2 Modulus of the field enhancement factor versus the aspect ratio a = b and wavelengths X for SPM tips of different materials (a) gold, (b) platinum, (c) silver, (d) p-doped silicon, (e) tungsten. Reprinted with permission from J. Jersch, Applied Physics A, 66, 29 (1998). Copyright 1998, Springer-Verlag.

As the modulus is greater than unity under such conditions, it is conveniently called enhancement factor (34) because it measures the maximum possible increase in retention upon ion-pair formation in a given chromatographic system. The behavior of the enhancement factor as a function... 

These parameters, such as the coefficient of diffusion, D, mass-transfer coefficient in the gas and liquid phase or film, kg and k], Thiele modulus, Hatta number, Ha, and enhancement factor, E, are all dependent on the pressure. 

The mass-transfer coefficient with a reactive solvent can be represented by multiplying the purely physical mass-transfer coefficient by an enhancement factor E that depends on a parameter called the Hatta number (analogous to the Thiele modulus in porous catalyst particles). 

Solving the diffusion-reaction equation in the liquid, the enhancement factor can be related to the Hatta number Ha, which is similar to the Thiele modulus defined for heterogeneous gas-solid catalysts. Thus, the Hatta number and its relation to the controlling regime are... 

Introductory paragraphs similar to the above can be found in hundreds of nanocomposite papers. With the exception of reinforced elastomers, nanocomposites have not lived up to expectations. Although claims of modulus enhancement by factors of 10 exist, these claims are offset by measurements that show little or no improvement... The lackluster performance of nanocomposites has been attributed to a number of factors including poor dispersion, poor interfacial load transfer, process-related deficiencies, poor alignment, poor load transfer to the interior of filler bundles, and the fractal nature of filler clusters [5]. 

The connection between chemical reaction engineering and transport phenomena also stems from multiphase reactions. For solid catalyzed gas or liquid reactions, mass transfer in the bulk or on the surface may become a problem. For gas-liquid reactions, the transport of the species to the reaction zone has to be considered. Similar problems arise for liquid-liquid reactions. Thns, we intend to give a brief introduction to these problems and, in the process, introdnce dimensionless qnantities such as the Thiele modulus, Damkohler number, Hatta modulus, effectiveness factor, and enhancement factor, and nse them in designing reactors. 

Define, evaluate, and use the concepts of enhancement factor and Hatta modulus. 

If the rate is slow, Hatta modulus is small. If the rate is fast, the Hatta modulus is large. For very large Hatta modulus, the enhancement factor is equal to Hatta modulus, since tanh oo = i. Finally, for instantaneous reactions, the enhancement factor has a limiting value given by Equation 6.80. A qualitative plot of enhancement factor versus Hatta modulus is given in Figure 6.12. The merits of the figure and how it can be used for equipment selection will be discussed in Chapter 11. 

Figure 6.12 Enhancement factor as a function of the Hatta modulus.

Considering that the emitted intensity is proportional to the squared electric field modulus in the far-field region (see Eq. (1.222)), we can also write the enhancement factor G along a direction 9, cp as [74] ... 

As might be expected from a consideration of the factors discussed in Section 4.2, the imidisation process will stiffen the polymer chain and hence enhance Tg and thus softening points. Hence Vicat softening points (by Procedure B) may be as high as 175°C. The modulus of elasticity is also about 50% greater than that of PMMa at 4300 MPa, whilst with carbon fibre reinforcement this rises to 25 000 MPa. The polymer is clear (90% transparent) and colourless. 

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