Linear coefficients

Equations of first order with linear coefficients, 

We have two degrees of freedom to eliminate two constants, so we eliminate e and y by selecting 

An inconsistency arises if pa = ab, which is obvious. It is clear that the final result is now of the homogeneous type 

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This table lists values of /3, the cubical coefficient of thermal expansion, taken from Essentials of Quantitative Analysis, by Benedetti-Pichler, and from various other sources. The value of /3 represents the relative increases in volume for a change in temperature of 1°C at temperatures in the vicinity of 25°C, and is equal to 3 a, where a is the linear coefficient of thermal expansion. Data are given for the types of glass from which volumetic apparatus is most commonly made, and also for some other materials which have been or may be used in the fabrication of apparatus employed in analytical work. 

Automated soldering operations can subject the mol ding to considerable heating, and adequate heat deflection characteristics ate an important property of the plastics that ate used. Flame retardants (qv) also ate often incorporated as additives. When service is to be in a humid environment, it is important that plastics having low moisture absorbance be used. Mol ding precision and dimensional stabiUty, which requites low linear coefficients of thermal expansion and high modulus values, ate key parameters in high density fine-pitch interconnect devices. 

The coefficient of cubical expansion may he taken as three times the linear coefficient. In the following table, t is the temperature or range of temperature, and C, the coefficient of expansion. 

E = Young s modulus a = linear coefficient of thermal expansion. 

Linear coefficients of expansion per degree increase in temperature ... 

Plastics can also be combined with other materials such as aluminum, steel, and wood to provide specific properties. Examples include PVC/wood window frames and plastic/ aluminum-foil packaging material. All combinations require that certain aspects of compatibility such as processing temperature and linear coefficient of thermal expansion or contraction exist. 

Figure 3.10. The relationship between the three coefficients A, B, and C for the curves shown in Figure 3.9 the quadratic and the linear coefficients are tightly linked. The intercept suffers from higher variability because it carries the extrapolation penalty discussed in Section 2.2.5 and Figure 2.8.

If a volume expansion is required, then mccisurements in three simultaneous dimensions are needed, a result experimentally difficult to achieve, to say the least. Even a slab of a single crystal does not completely solve the problem since thermal expansion in three dimensions is needed for the volume thermal expansion coefficient. The crystal has three (3) crystallographic axes and may have three (3) linear coefficients of expansion. Only if the crystal is cubic does one have the case where all three values of ol are equal. 

Note that the linear coefficient of expansion, at is obtedned finm the slope of the straight fine. The glass softening point is also easily observed as is the glass transitional temperature (which is the point where the amorphous assy phase begins its transition to a crystalline phase. These glass-points can also be used to cross-check values obtained by the DTA method. 

It is possible for a glaiss to have more than one linear coefficient of expansion. One such case is shown in the following diagram, given as 7.5.6. on the next page. 

I il a soda-lime-silica glass, containing magnesia and boric oxide 1 than 1 per cent), made by the General Electric Co. It is often ibed as GEC X.8. or simply as X.8. The linear coefficient of Jial expansion between 20 and 350°C is 9-65 0-10 x 10-. This I il available as tubing and rod in a wide range of sizes. 

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