Maxwell predictions

James Clerk Maxwell predicted the existence of electromagnetic waves in 1864 and developed the classical sine (or cosine) wave description of the perpendicular electric and magnetic components of these waves. The existence of these waves was demonstrated by Heinrich Hertz 3 years later. 

The physics of Newton and Maxwell predicted that a heated body would emit an infinite amount of very high energy radiation. 

The agreement between the predictions and the experiments is good except for Pe < 20. The prediction of Jeffrey [30] for Dd does not agree with the experimental results. Note that the improvement over the Maxwell prediction, that is, the Jeffrey results for Kt/(pcp)/=edilute concentration, that is, is valid only for e - 1. 

Figure 5.18 Robeson chart for the CO2/CH4 gases and the MMCM membranes obtained from ABS including ACl and AC2 active carbon as filler (2-10% of ACl 20—40% of AC2) showing the best fitting Maxwell predictions for the pure active carbons.

An interesting problem area that occupied Maxwell s attention from 1855 to 1859 was the nature of the rings of Saturn. He showed that certain ideas of the time, such as that Saturn had solid or rigid rings, were not possible. Instead, the stability of the rings require that they consist of concentric circles of small objects, the orbital speed of each circle being dependent on its distance from the planet. In 1895, the differential rotation of the rings Maxwell predicted almost 40 years earlier was confirmed by observation. 

Ions generated in the ion source region of the instrument may have initial velocities isotropically distributed in tliree dimensions (for gaseous samples, this initial velocity is the predicted Maxwell-Boltzmaim distribution at the sample temperature). The time the ions spend in the source will now depend on the direction of their initial velocity. At one extreme, the ions may have a velocity Vq in the direction of the extraction grid. The time spent in the source will be... 

The Maxwell model thus predicts a compliance which increases indefinitely with time. On rectangular coordinates this would be a straight line of slope I/77, and on log-log coordinates a straight line of unit slope, since the exponent of t is 1 in Eq. (3.69). 

As we did in the case of relaxation, we now compare the behavior predicted by the Voigt model—and, for that matter, the Maxwell model—with the behavior of actual polymer samples in a creep experiment. Figure 3.12 shows plots of such experiments for two polymers. The graph is on log-log coordinates and should therefore be compared with Fig. 3.11b. The polymers are polystyrene of molecular weight 6.0 X 10 at a reduced temperature of 100°C and cis-poly-isoprene of molecular weight 6.2 X 10 at a reduced temperature of -30°C. 

About 1902, J. W. Gibbs (1839-1903) introduced statistical mechanics with which he demonstrated how average values of the properties of a system could be predicted from an analysis of the most probable values of these properties found from a large number of identical systems (called an ensemble). Again, in the statistical mechanical interpretation of thermodynamics, the key parameter is identified with a temperature, which can be directly linked to the thermodynamic temperature, with the temperature of Maxwell s distribution, and with the perfect gas law. 

This is the governing equation of the Maxwell Model. It is interesting to consider the response that this model predicts under three common time-dependent modes of deformation. 

Example 2.13 A plastic which can have its creep behaviour described by a Maxwell model is to be subjected to the stress history shown in Fig. 2.43(a). If the spring and dashpot constants for this model are 20 GN/m and 1000 GNs/m respectively then predict the strains in the material after 150 seconds, 250 seconds, 350 seconds and 450 seconds. 

The predicted strain variation is shown in Fig. 2.43(b). The constant strain rates predicted in this diagram are a result of the Maxwell model used in this example to illustrate the use of the superposition principle. Of course superposition is not restricted to this simple model. It can be applied to any type of model or directly to the creep curves. The method also lends itself to a graphical solution as follows. If a stress is applied at zero time, then the creep curve will be the time dependent strain response predicted by equation (2.54). When a second stress, 0 2 is added then the new creep curve will be obtained by adding the creep due to 02 to the anticipated creep if stress a had remained... 

This will in fact be constant for all values of M3 because the Maxwell Model cannot predict changes in strain if there is no stress. The overall variation in strain is shown in Fig. 2.46. 

It is apparent therefore that the Superposition Principle is a convenient method of analysing complex stress systems. However, it should not be forgotten that the principle is based on the assumption of linear viscoelasticity which is quite inapplicable at the higher stress levels and the accuracy of the predictions will reflect the accuracy with which the equation for modulus (equation (2.33)) fits the experimental creep data for the material. In Examples (2.13) and (2.14) a simple equation for modulus was selected in order to illustrate the method of solution. More accurate predictions could have been made if the modulus equation for the combined Maxwell/Kelvin model or the Standard Linear Solid had been used. 

The viscoelastic behaviour of a certain plastic is to be represented by spring and dashpot elements having constants of 2 GN/m and 90 GNs/m respectively. If a stress of 12 MN/m is applied for 100 seconds and then completely removed, compare the values of strain predicted by the Maxwell and Kelvin-Voigt models after (a) 50 seconds (b) 150 seconds. 

Maxwell and Kelvin-Voigt models are to be set up to simulate the creep behaviour of a plastic. The elastic and viscous constants for the Kelvin-Voigt models are 2 GN/m and 100 GNs/m respectively and the viscous constant for the Maxwell model is 200 GNs/m. Estimate a suitable value for the elastic constant for the Maxwell model if both models are to predict the same creep strain after 50 seconds. 

The creep curve for polypropylene at 4.2 MN/m (Fig. 2.5) is to be represented for times up to 2 X 10 s by a 4-element model consisting of a Maxwell unit and a Kelvin-Voigt unit in series. Determine the constants for each of the elements and use the model to predict the strain in this material after a stress of 5.6 MN/m has been applied for 3 x 10 seconds. 

J7 In a tensile test on a plastic, the material is subjected to a constant strain rate of 10 s. If this material may have its behaviour modelled by a Maxwell element with the elastic component f = 20 GN/m and the viscous element t) = 1000 GNs/m, then derive an expression for the stress in the material at any instant. Plot the stress-strain curve which would be predicted by this equation for strains up to 0.1% and calculate the initial tangent modulus and 0.1% secant modulus from this graph. 

Where, the diffusivity D for the transfer of one gas in another is not known and experimental determination is not practicable, it is necessary to use one of the many predictive procedures. A commonly used method due to Gilliland 6 is based on the Stefan-Maxwell hard sphere model and this takes the form ... 

In order to design a zeoHte membrane-based process a good model description of the multicomponent mass transport properties is required. Moreover, this will reduce the amount of practical work required in the development of zeolite membranes and MRs. Concerning intracrystaUine mass transport, a decent continuum approach is available within a Maxwell-Stefan framework for mass transport [98-100]. The well-defined geometry of zeoHtes, however, gives rise to microscopic effects, like specific adsorption sites and nonisotropic diffusion, which become manifested at the macroscale. It remains challenging to incorporate these microscopic effects into a generalized model and to obtain an accurate multicomponent prediction of a real membrane. 

The predictive power of the approach becomes obvious by noting that the dependence of the coupling on T and / /, is completely governed by the requirement of thermodynamic consistency [11] Maxwell s relation,... 

We have developed the idea that we can describe linear viscoelastic materials by a sum of Maxwell models. These models are the most appropriate for describing the response of a body to an applied strain. The same ideas apply to a sum of Kelvin models, which are more appropriately applied to stress controlled experiments. A combination of these models enables us to predict the results of different experiments. If we were able to predict the form of the model from the chemical constituents of the system we could predict all the viscoelastic responses in shear. We know that when a strain is applied to a viscoelastic material the molecules and particles that form the system gradual diffuse to relax the applied strain. For example, consider a solution of polymer... 

In principle this integral could be applied directly to the Maxwell model to predict the decay of stress at any point in time. We can simplify this further with an additional assumption that is experimentally verified, i.e. that the function in the integral is continuous. The first value for the mean theorem for integrals states that if a function f(x) is continuous between the limits a and b there exists a value f(q) such that... 

Incorrect conclusion 1 above is sometimes said to derive from the reciprocity principle, which states that light waves in any optical system all could be reversed in direction without altering any paths or intensities and remain consistent with physical reality (because Maxwell s equations are invariant under time reversal). Applying this principle here, one notes that an evanescent wave set up by a supercritical ray undergoing total internal reflection can excite a dipole with a power that decays exponentially with z. Then (by the reciprocity principle) an excited dipole should lead to a supercritical emitted beam intensity that also decays exponentially with z. Although this prediction would be true if the fluorophore were a fixed-amplitude dipole in both cases, it cannot be modeled as such in the latter case. 

The sum over weighted relaxation times is heavily dominated by the longest time (the reptation time) r gp=L /7T Dp. Because of this the frequency-dependent dissipative modulus, G"(cd) is expected to show a sharp maximum The higher modes do modify the prediction from that of a single-mode Maxwell model, but only to the extent of reducing the form of G"(a>) to the right of the maximum from ccr to In fact, experiments on monodisperse linear polymers... 

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