Modulus—loss factor limitations

The significance of this relation to the acoustic designer is that there are limitations on the combination of modulus and loss factor that can be achieved. It is not possible to select the modulus and loss factor of a material independently. As an example, if it is desired to have a material whose modulus is independent of frequency, then the loss factor must be identically zero. Conversely, if a non-zero loss factor is required, there will.be frequency dependence in the modulus. 

The test results of a material damping test are most useful when placed on a reduced temperature nomograph, which plots the limited number of test results to a graph from which one can obtain the damping properties (modulus and loss factor) at any given combination of temperature and frequency. The WLF equation (51 is used to obtain a nomograph for the results of each test. 

Catalyst Effectiveness. Even at steady-state, isothermal conditions, consideration must be given to the possible loss in catalyst activity resulting from gradients. The loss is usually calculated based on the effectiveness factor, which is the diffusion-limited reaction rate within catalyst pores divided by the reaction rate at catalyst surface conditions (50). The effectiveness factor E, in turn, is related to the Thiele modulus,

first-order rate constant, a the internal surface area, and the effective diffusivity. It is desirable for E to be as close as possible to its maximum value of unity. Various formulas have been developed for E, which are particularly usehil for analyzing reactors that are potentially subject to thermal instabilities, such as hot spots and temperature mnaways (1,48,51). 

This approach is limited, however, to data obtained within the temperature range where the chosen type of function is applicable. A wider range is possible using empirical shift factors as done in this work. Figure 9 depicts the algorithm employed to transform resonant (varying frequency) loss modulus vs temperature curves into calculated constant... 

Loss of stability from increased second-order effects (i.e. changes in the effects of actions or in the forces in members resulting from deformation of the components, members or of the structure as a whole) due to creep may be dealt with by using ultimate limit state loads and the long-term structural stiffness, e.g. with a creep factor applied to the modulus of elasticity. 

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