Temperature nomograph, reduced

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. 

Figure 2 is a reduced temperature nomograph which demonstrates the procedure for reading the nomograph as follows Select a combination of temperature and frequency, for example, 1000 Hz and 75 C. Find the point for 1000 Hz on the right-hand frequency axis. Proceed horizontally to the temperature line for 75 C. At this intersection, draw a vertical line. Then, read the modulus and loss factor values from the appropriate data curve, at the point of intersection with the vertical line. In this example, modulus G(1000 Hz, 75 C) = 8 x 10° U/wr and loss factor (1000 Hz, 75 C) = 1.96. 

The temperature scale at the top of the reduced temperature nomograph shows increasing temperature from right to left. Labeling of the temperature lines is done in a uniform temperature increment. The temperature increment is identified at the top of the nomogram. The nomograph presentation and data reduction procedure have been described in earlier publications (6.71. 

Using a computerized data reduction scheme that incorporates a generalized WLF equation, dynamic mechanical data for two different polymers were correlated on master curves. The data then were related to the vibration damping behavior of each material over a broad range of frequencies and temperatures. The master curves are represented on a novel reduced temperature nomograph which presents the storage modulus and loss tangent plots simultaneously as functions of frequency and temperature. ... 

Figure 2. Viscoelastic master curves represented on reduced temperature nomograph. Key solid symbols, modulus values and open symbols, loss tangent values. Insert at upper left shows the shift factor function, aT, used for data reduction.

Figure 3. Reduced temperature nomograph for a fluorosilicone polymer. Data taken by both resonant beam and DMA instruments. Key upper curve, modulus and lower curve, loss tangent.

Figure 4. Reduced temperature nomograph showing that rheovibron and DMA data for PMMA superpose into single master curves when T — 175°C DMA points are center on each plot. Key A, modulus curve and loss tangent.

Figure 5. Reduced temperature nomograph for PMMA data of Figure 4 plotted with T0 = 0°C. The data do not superpose from left to right data are 110 Hz., 35 Hz, DMA, 11 Hz and 3.5 Hz. Key A, modulus and loss tangent.

The data obtained for the PMMA sample were reduced and are displayed in the nomograph form of Figure 3. The form of the WLF equation used is the "universal" WLF equation with T replaced by Tg and defined as the temperature at the peak value of loss modulus for the 0.01 Hz curve. The constants and C2 were assigned the values of 17.4 and 51.6 respectively. 

The flat appearance of the E" curve is due to the compressed nature of this particular nomograph scale. Both functions appear to fit equally well and therefore satisfy the criteria of curve shape and shift factor consistency for using the reduced variable time-temperature superposition. Additionally, the criterion of reasonable values for a-j- is satisfied by virtue of using the "universal" WLF equation. 

The master curve is then represented on a reduced frequency nomograph which allows direct reading of modulus (E or E") or loss tangent as functions of either frequency or temperature. 

The procedure by which the nomograph is generated is not limited to the WLF equation. Since it is based on the reduced variable concept, any superposition equation that results in the calculation of a temperature shift factor may be used to calculate the needed data to create the master curve and subsequent nomograph. The software can easily be modified to calculate and display a master curve on some other superposition equation. 

Many compounds decompose when heated to their boiling points so they cannot be distilled at atmospheric pressure. In this situation it may be possible to avoid thermal decomposition by carrying out the distillation at reduced pressure. The reduction in the boiling point will depend on the eduction in pressure and it can be estimated from a pressure-temperature nomograph (Fig. 11.11). 

The surface tension has been correlated with other physical parameters such as liquid compressibility, viscosity, molar fractions, and the refractive index. Rao et al. [8] developed a linear relationship between the surface tension at normal boiling point (log a ) and the reduced boiling point temperature (T ). Hadden [9] presented a nomograph for hydrocarbons that enables rapid calculation of a. For cryogenic liquids, Sprows and Prausnitz [10] introduced the equation... 

As illustrated in Figure 10.14, the master curve allows the extrapolation of data over broad temperature and time ranges. Similar master curves can be constructed with frequency as the variable, instead of time. More elegant still is the reduced frequency nomograph, which permits both reduced frequency and temperature simultaneously see Figure 10.15 (42). 

In vibration and acoustic analysis the viscoelastic spectrum are often represented as a master curve known as the reduced frequency nomograph. Figure 14.3 is an example of a nomograph constructed with simulated data. In addition to the thermomechanical information that is usually presented, both absolute temperature and frequency axes are included in the nomograph. The information presented by the inclusion of the temperature and frequency axes is already contained in the master curve of the viscoelastic data. However, it is convoluted within the master curve. In order to obtain frequency data at other tanperatures, a new master curve must be created from the raw data or transformed from the original master curve. 

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