Calculation of thermal expansion

The correlation results maybe used for calculation of thermal expansion coefficient of liquid and volumetric flow from thermal expansion. Examples are shown in Table 7-2. 

Thermomechanical analysis allows the calculation of thermal expansivity from the same data set as used to calculate the Tg. Since many materials are used in contact with a dissimilar material in the final product, knowing the rate and amount of thermal expansion helps design around mismatches that can cause failure in the final product. These data are only available when the Tg is collected by thermal expansion, not by the flexure or penetration method. This is in many ways the simplest or most essential form of TMA measurement. A sample is prepared with parallel top and bottom surfaces and is allowed to expand under minimal load (normally 5mN or less, ideally OmN) as it is slowly heated and/or cooled. The CTE is calculated by ... 

Bending beam theory calculation of elastic modulus, 361-362 calculation of glass temperature, 362 calculation of thermal expansion coefficient, 362 layer stress determination, 361 Benzophenone-3,3, 4,4 -tetracarboxydi-anhydride-oxydianiline-m-phenylenediamine (BTDA-ODA-MPDA) polyimide, properties, 115-116 Bilayer beam analysis schematic representation of apparatus, 346,348/ thermal stress, 346 Binary mixtures of polyamic acids curing, 116-124 exchange reactions, 115 Bis(benzocyclobutenes) heat evolved during polymerization vs. 

In Part VII, Greg Rutledge discusses the modeling and simulation of polymer crystals. He uses this as an excellent opportunity to introduce principles and techniques of solid-state physics useful in the study of polymers. The mathematical description of polymer helices and the calculation of X-ray diffraction patterns from crystals are explained. Both optimization (energy minimization, lattice dynamics) and sampling (MC, MD) methods for the simulation of polymer crystals are then discussed. Applications are presented from the calculation of thermal expansion, elastic coefficients, and even activation energies and rate constants for defect migration by TST methods. 

The dcy/dP data and the theoretically derived TOEC were used for calculations of thermal expansion coefficients, temperature dependence of the volume compressibility and the pressure variation of lattice constants, and the agreement with experiment was quite good. For more detail about these calculations see Ramji Rao and Ramanand (1980, 1984) and references given in tables 15 and 16. 

It is quite common that a liquid-carrying line can be isolated between two isolation valves. The isolated liquid is under thermal expansion due to heat gain from high ambient temperature. Small thermal expansion can be adequate to increase the pressme beyond the design pressure limit of the pipe. The exact calculation of thermal expansion and contingency is complex and is often not required. The volume released is small and a nominal 20 x 25 NB relief valve is installed to protect the system. However, the thermal expansion can be established mathematically as follows ... 

No tables of the coefficients of thermal expansion of gases are given in this edition. The coefficient at constant pressure, l/t)(3 0/3T)p for an ideal gas is merely the reciprocal of the absolute temperature. For a real gas or liquid, both it and the coefficient at constant volume, 1/p (3p/3T),, should be calculated either from the equation of state or from tabulated PVT data. 

Example 2.6 The bobbin shown in Fig. 2.16 has been manufactured by sliding the acetal ring on to the steel inner and then placing the end-plate in position. At 20°C there are no stresses in the acetal and the distance between the metal end-plates is equal to the length of the acetal ring. If the whole assembly is heated to 1(X)°C, calculate the axial stress in the acetal. It may be assumed that there is no friction between the acetal and the steel. The coefficients of thermal expansion for the acetal and the steel are 80 x 10 °C and 11 X 10 °C respectively. The modulus of the acetal at 100°C is 1.5 GN/m. ... 

Example 2.7 A nylon ring with a nominal inside diameter of 30 mm, an outer diameter of SO mm and a width of S mm is to be made an interference fit on a metal shaft of 30 mm diameter as shown in Fig. 2.17. The design condition is that the initial separation force is to be 1 kN. Calculate (a) the interference on radius needed between the ring and the shaft and (b) the temperature to which the nylon must be heated to facilitate easy assembly. What will be the maximum stress in the nylon when it is in position on the shaft The coefficient of friction between nylon and steel is 0.2S. The short-term modulus of the nylon is 1 GN/m, its Poisson s ratio is 0.4 and its coefficient of thermal expansion is 100 X 10- °C- . 

When a pipe fitting is tightened up to a 12 mm diameter polypropylene pipe at 20°C the diameter of the pipe is reduced by 0.05 mm. Calculate the stress in the wall of the pipe after 1 year and if the inside diameter of the pipe is 9 mm, comment on whether or not you would expect the pipe to leak after this time. State the minimum temperature at which the fitting could be used. Use the tensile creep curves and take the coefficient of thermal expansion of the polypropylene to be 9.0 X 10- °C . 

Hygroscopic (moisture) effects arise for polymer materials such as some epoxies that absorb moisture chemically after curing and therefore expand. These effects are directly analogous to thermal effects and are characterized by coefficients of moisture expansion and p2 in principal material coordinates in direct analogy to a.( and 02 for coefficients of thermal expansion. All calculations for thermal effects with the a can be replaced by or supplemented with analogous terms for moisture expansion. 

The formula for this calculation is very simple and very accurate. It requires three factors (1) the difference in temperature of the machine housing between the feet and shaft bearings, (2) the distance between the shaft centerline and the feet, and (3) the coefficient of thermal expansion of the machine housing material. The thermal... 

Both of these are equations are approximate and are useful only for giving estimates of the value of the of the polyblend or copolymer. To calculate values of more accurately requires additional information such as the coefficients of thermal expansion of both components in both their liquid and glassy states. Given the uncertainty in the numerical value of T, which as we have seen depends on the method by which has been determined, there is little point in developing such arithmetical refinements. 

The glass transition of the polymer Is Tg. while that of the plasticizer is Te the volume fraction of plasticizer is Fi(b), and its weight fraction js Wg. Typical values of TA are betvaen -50 and - 100°0. To calculate more accurate values of Tg additional information must be available, such as the Tg value of a known mixture or the coefficients of thermal expansion (aA and a ) of the pure components in both their liquid and glassy states (51,95). For each Component i... 

Kujawa and Winnik [209] reported recently that other volumetric properties of dilute PNIPAM solutions can be derived easily from pressure perturbation calorimetry (PPC), a technique that measures the heat absorbed or released by a solution owing to a sudden pressure change at constant temperature. This heat can be used to calculate the coefficient of thermal expansion of the solute and its temperature dependence. These data can be exploited to obtain the changes in the volume of the solvation layer around a polymer chain before and after a phase transition [210], as discussed in more detail in the case of PVCL in Sect. 3.2.2. 

Although we are not dealing with very dilute solutions, we may use the density term in equation 14.5 to estimate a correction to these values. However, we need to bear in mind that this estimate can be considered at several approximation levels. The most simple is to assume that In p is constant in the experimental temperature range, and thus only the reaction entropy is affected. Using the value for the pure solvent, p = 1.6227 kg dm-3 (at 20°C) [48], we obtain ArS 23 = -56.2 J K-1 mol-1. A more accurate calculation would include the variation of In p with the temperature, which, in the absence of experimental data, can be estimated from the coefficient of thermal expansion of the solvent, a ... 

CNT can markedly reinforce polystyrene rod and epoxy thin film by forming CNT/polystyrene (PS) and CNT/epoxy composites (Wong et al., 2003). Molecular mechanics simulations and elasticity calculations clearly showed that, in the absence of chemical bonding between CNT and the matrix, the non-covalent bond interactions including electrostatic and van der Waals forces result in CNT-polymer interfacial shear stress (at OK) of about 138 and 186MPa, respectively, for CNT/ epoxy and CNT/PS, which are about an order of magnitude higher than microfiber-reinforced composites, the reason should attribute to intimate contact between the two solid phases at the molecular scale. Local non-uniformity of CNTs and mismatch of the coefficients of thermal expansions between CNT and polymer matrix may also promote the stress transfer between CNTs and polymer matrix. 

An extension of the procedure for calculating the deton velocities to include those expls which.yield solid carbon as a reaction product has been accomplished by the same investigators (See Ref 32) on the assumption that the volumes of solid and gas are additive, that the gas obeys eq 23 and that the solid has zero coefficients of thermal expansion and basic compression. The composition of the reaction products was assumed to be that of chemical equilibrium at the temp and pressure immediately behind the deton wave, and a numerical procedure, involving successive approximations, was developed for the determination of the composition from a consideration of the simultaneous equilibria involved. This method of calculation was briefly discussed in Ref 39, pp 86-7... 

The thermomechanical behaviour of undrawn semicrystalline polymers above Tg is shown in Fig. 14. The values of the coefficients of thermal expansion calculated from the heat effects agree well with dilatometric results. For PE, the influence of degree of crystallinity on the value of thermal effects and thermal expansion coefficients was also studied 64). 

EXAMPLE 8-6 A sample of reservoir oil was placed in a laboratory cell at 5000psig and 76°F. The volume was 54.74 cc. Temperature was increased to 220°F and pressure was held constant by increasing cell volume to 59.55 cc. Calculate the coefficient of isobaric thermal expansion and calculate the thermal expansion. 

According to Ferry12 the macroscopic coefficient of thermal expansion, al5 above Tg reflects the appearance of the free-volume. In this case the value a( must be equal to af = cq — ag s Aa. Indeed, the experimental values for Aa are in good agreement with the theoretically calculated values for ctf. 

Calculated from these data, the volume coefficient of thermal expansion for crystalline TATB is 30.4 x lO / K... 

Manson and Chin 151) reported that the addition of filler to an epoxy binder reduces the epoxy s permeability coefficient (P), as well as the solubility of water in the resin (S) and that the reduction is stronger than expected from theory 1 2). Diffusion coefficients calculated from P and S for the unfilled resin were found to be somewhat higher than those for filled resin. The difference seems to be due to the formation of ordered layers, up to 4 pm thick, around every filler particle. The layers form because of residual stresses caused by the difference between the binder and filler coefficients of thermal expansion. The effective activation energy for water to penetrate into these materials, calculated in the 0-100 °C temperature range, is 54.3 kJ/mol151). 

Thus Equation (10.33) is solved as the new equilibrium equation. To calculate the thermal expansion behavior of the model, the thermal expansion coefficient is necessary as the calculating parameter. If the temperature of the model is even, the initial temperature and the final temperature are used as just a calculating parameter. If a temperature distribution exists in the model, the temperature distribution data is dispensable for the stress calculation. The temperature is firstly calculated by a CFD and the calculated data is used as the boundary condition in the stress calculation. If the thermal expansion coefficient is temperature dependent, the temperature dependence must be considered in the calculation. Here the temperature data at the nodes is transferred from STAR-CD to the ABAQUS. 

EXAMPLE 15.2. The temperature of the inside wall of a tube is 200 °C and the outside wall temperature is 40 °C. Calculate the stresses at the outside of the wall if the tube is made from a glass having a coefficient of thermal expansion of a = 8 x 10 6/°C, an elastic modulus of 60 GPa, and a Poisson s ratio of 0.3. 

Although the linear coefficient of thermal expansion varies with temperature, it can be considered constant within typical design and processing conditions. It is especially high for polyolefins, where it ranges from 1.5 x 10 4K-1 to 2 x 10 4K 1 however, fibers and other fillers significantly reduce thermal expansion. A rule of mixtures is sufficient to calculate the thermal expansion coefficient of polymers that are filled with powdery or small particles as well as with short fibers. In this case, the rule of mixtures is written as... 

For the calculation of the thermal shock-induced stresses, we consider the plate shown in Fig. 15.1 with Young s modulus E, Poisson s ratio v, and coefficient of thermal expansion (CTE) a, initially held at temperature /j. If the top and bottom surfaces of the plate come into sudden contact with a medium of lower temperature T they will cool and try to contract. However, the inner part of the plate initially remains at a higher temperature, which hinders the contraction of the outer surfaces, giving rise to tensile surface stresses balanced by a distribution of compressive stresses at the interior. By contrast, if the surfaces come into contact with a medium of higher temperature Tm, they will try to expand. As the interior will be at a lower temperature, it will constrain the expansion of the surfaces, thus giving rise to compressive surface stresses balanced by a distribution of tensile stresses at the interior. 

---