Thermal coefficients relations between

The experiments result in an explicit measure of the change in the shock-wave compressibility which occurs at 2.5 GPa. For the small compressions involved (2% at 2.5 GPa), the shock-wave compression is adiabatic to a very close approximation. Thus, the isothermal compressibility Akj- can be computed from the thermodynamic relation between adiabatic and isothermal compressibilities. Furthermore, from the pressure and temperature of the transition, the coefficient dO/dP can be computed. The evaluation of both Akj-and dO/dP allow the change in thermal expansion and specific heat to be computed from Eq. (5.8) and (5.9), and a complete description of the properties of the transition is then obtained. 

Various other relations between the thermal coefficients may easily be obtained if required. Thus ... 

A rather general, approximate relation between Young s modulus E and the coefficient a of linear thermal expansion was proposed by Barker [58] ... 

In a celebrated paper, Einstein (1917) analyzed the nature of atomic transitions in a radiation field and pointed out that, in order to satisfy the conditions of thermal equilibrium, one has to have not only a spontaneous transition probability per unit time A2i from an excited state 2 to a lower state 1 and an absorption probability BUJV from 1 to 2 , but also a stimulated emission probability B2iJv from state 2 to 1 . The latter can be more usefully thought of as negative absorption, which becomes dominant in masers and lasers.1 Relations between the coefficients are found by considering detailed balancing in thermal equilibrium... 

The connection between surface tension and the coefficient of thermal expansion has already been referred to in Chapter I. We have next to consider the relations between surface tension and vapour pressure, which are of considerable importance in a number of physical processes. 

The relation between the transfer coefficient and thermal fluctuations in fluidized-bed heat transfer. 

This introduces a new unknown (and free at this stage) coefficient K. In the case of a system of molecules in a thermal bath (definitely not the one we consider), there is a relation between D and K such that, at thermal equilibrium, the equilibrium density in the potential (h is given by Boltzmann s law. This requires that K = mD/hgT, where kg is Boltzmann s constant and T the absolute temperature. In Eq. (12) the factor p in front of in is to ensure that, if... 

For a cubic site, relations between the cumulants and the coefficients of the OPP model have been derived by Kontio and Stevens (1982), and applied to the Al(4) atom in the alloy VA110 4.2 The coordination of Al(4) is illustrated in Fig. 2.4(a), while the potential along [111], derived from the thermal parameter refinement, is shown in Fig. 2.4(b). It is clear from these figures that higher than third-order terms contribute to the potential, because the deviation from the harmonic curve is not exactly antisymmetric with respect to the equilibrium configuration. The potential appears steeper at the higher temperature, which is opposite to what is expected on the basis of the thermal expansion of the solid. 

Other physical properties. Anisotropy of thermal and electrical conductivity, coefficient of thermal expansion, elasticity, and dielectric constant may also provide information on internal structure. These properties, however, have so far been little used in structure determination, because they are less easily measured than those already considered consequently not very much experimental evidence is available for the purpose of generalizing on the relations between such properties and structural features. For further information on these subjects, see Wooster (1938), Nye (1957). 

Calculations of the relations between the input and output amounts and compositions and the number of extraction stages are based on material balances and equilibrium relations. Knowledge of efficiencies and capacities of the equipment then is applied to find its actual size and configuration. Since extraction processes usually are performed under adiabatic and isothermal conditions, in this respect the design problem is simpler than for thermal separations where enthalpy balances also are involved. On the other hand, the design is complicated by the fact that extraction is feasible only of nonideal liquid mixtures. Consequently, the activity coefficient behaviors of two liquid phases must be taken into account or direct equilibrium data must be available. 

If we are dealing with mutual diffusion of gases which are close in molecular weight (e.g., carbon monoxide and air), it may be shown that the temperature of the flame pellet will prove to be equal to the theoretical combustion temperature of the mixture. This equality depends on the existence in the kinetic theory of gases of a simple relation between the diffusion coefficient (on which the supply of reagents and heat release rate depend) and the thermal conductivity (on which the heat evacuation depends). 

Taking account of the conductive and convective heat transfer is a fundamental necessity since this heat transfer is inseparably linked to the processes which supply fuel and oxygen to the flame. Whatever the relation between the existing and supplied amounts of air and gas, when they are supplied separately the flame is always established in such a state as to have the fuel and oxygen supplied to the surface in stoichiometric relation to one another. For equivalence of the diffusion coefficient and the thermal diffu-sivity (particularly in turbulent motion, which provides such equality) the concentration of the combustion products and the temperature in the flame correspond precisely to combustion of a stoichiometric mixture (for equal losses to radiation) such is the conclusion of our calculations. 

Additions of alkali and alkaline earth elements are used in glasses to lower the viscosity at temperatures low enough for the glass to be economically formed into useful shapes. However, these additions also raise the coefficient of thermal expansion. Figure 15.10 shows the relation between the coefficient of thermal expansion and the temperature at which the viscosity is 107 Pa s, which is considered a temperature for forming. 

The relation between the coefficient of thermal expansion and the temperature at which... 

Mickley, H. S., Fairbanks, D. F. and Hawthorn, R. D. (1961). The Relation Between the Transfer Coefficient and Thermal Fluctuations in Fluidized Bed Heat Transfer. Chem. Eng. Prog. Symp. Ser., 57(32), 51. 

Long ago, Langmuir suggested that the rate of deposition of particles on a surface is proportional to the density of particles in the vicinity of the surface and to the available area on the surface [1], However, the calculation of the available area is still an open problem. In a first approximation, one can assume that the available area is the total area of the surface minus the area already occupied by the adsorbed particles [1]. A better approximation can be obtained if the adsorbed particles, assumed to have the shape of a disk, are in thermal equilibrium on the surface, either because of surface diffusion and/or of adsorption/desorption kinetics. In this case, one can use one of the empirical equations available for the compressibility of a 2D gas of hard disks, calculate the chemical potential in excess to that of an ideal gas [2] and then use the Widom relation between the area available to one particle and its excess chemical potential on the surface (the particle insertion method) [3], The method is accurate at low densities of adsorbed particles, where the equations of state are accurate, but, in general, poor at high concentrations. The equations of state for hard disks are based on the virial expansion and only the first few coefficients of this... 

V1 3 varies in the range I. .. 4/3. From Eq. (15 ) one immediately obtains a relation between the reduced volume V and the thermal expansion coefficient a ... 

Another well-known example is the coupling between mass flow and heat flow. As a result, an induced effect known as thermal diffusion (Soret effect) may occur because of the temperature gradient. This indicates that a mass flow of component A may occur without the concentration gradient of component A. Dufour effect is an induced heat flow caused by the concentration gradient. These effects represent examples of couplings between two vectorial flows. The cross-phenomenological coefficients relate the Dufour and Soret effects. In order to describe the coupling effects, the thermal diffusion ratio is introduced besides the transport coefficients of thermal conductivity and dififusivity. 

S]). The direct piezoelectric effect is the production of electric displacement by the application of a mechanical stress the converse piezoelectric effect results in the production of a strain when an electric field is applied to a piezoelectric crystal. The relation between stress and strain, expressed by Equation 2.7, is indicated by the term Elasticity. Numbers in square brackets show the ranks of the crystal property tensors the piezoelectric coefficients are 3rd-rank tensors, and the elastic stiffnesses are 4th-rank tensors. Numbers in parentheses identify Ist-rank tensors (vectors, such as electric field and electric displacement), and 2nd-rai tensors (stress and strain). Note that one could expand this representation to include thermal variables (see [5]) and magnetic variables. 

First, consider the case when ka is important but fc7 is not. Nothing can be done if the product adp is constant and the Reynolds number is high, because then the friction factor is insensitive to Reynolds number, and G must be nearly constant. If the activity is constant and the Reynolds number is high, dp varies about as the cube of G. With the same assumptions about the effective thermal conductivity and the heat transfer coefficient that were used for the three-dimensional reactor, with the additional assumption that in the prototype reactor hKr2/ke — 2, the relation between U and r2 obtained by eliminating dp is... 

The model presented for a thermal explosion predicts that for a reaction mixture of fixed composition and fixed initial temperature, there will be a critical pressure above which explosion will occur and below which a normal stationary reaction will take place. The relation between the critical pressure and temperature is given by a modified Arrhenius equation with a negative temperature coefficient [Eq. (XIV.3.8)] which is... 

These are the required relations between the thermal coefficients in the two sets of variables T, V, i and T, p,... 

In the last section we derived the relations between the thermal coefficients Cy, i v Ih one hand and on the... 

Just as we were able to estimate an effective diffusion coefficient for the porous pellet, so we can obtain an effective thermal conductivity, such that the heat flux across a unit area within the pellet is times the gradient of temperature normal to that area. Wc shall not go into the relation between kt and the other physical properties of the porous material—this is well covered by Petersen in his fifth chapter—but it is worth pointing out that whereas the diffusion of matter is largely through the pores, that of heat is rather through the solid material. Values of kt for porous materials are of the order of cal/sec cm°C. 

Material Parameters. The key means whereby material specificity enters continuum theories is via phenomenological material parameters. For example, in describing the elastic properties of solids, linear elastic models of material response posit a linear relation between stress and strain. The coefficient of proportionality is the elastic modulus tensor. Similarly, in the context of dissipative processes such as mass and thermal transport, there are coefficients that relate fluxes to their associated driving forces. From the standpoint of the sets of units to be used to describe the various material parameters that characterize solids, our aim is to make use of one of two sets of units, either the traditional MKS units or those in which the e V is the unit of energy and the angstrom is the imit of length. 

We notice, using (A3.1.20) and (A3.1.26), that this method leads to a simple relation between the coefficients of shear viscosity and thermal conductivity, given by... 

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