Temperature equations for

To calculate the closure temperature of any point in a mineral without carrying out the full forward numerical calculations, Dodson (1986) analyzed the problems for different effective shapes systematically and modified his closure temperature equation for the whole minerals slightly to apply to individual points. His formulation is by adding a correction term to Equation 5-75b. This correction term will be referred to as gi in this book and another correction term by Ganguly and Tirone (1999, 2001) will be referred to as g2- The formulation of Dodson (1986) for the calculation of closure temperature at every point of a profile is... 

Crouch, R. F, and A. Cameron Viscosity-temperature equations for lubricants. J. Inst. Petrol. 47, 307 (1961). 

Schematic representation of these three possible boundary conditions at the wall is shown in Fig. 2.5 for the enthalpy/temperature equation. For systems with conjugate heat transfer, continuity of the temperature and the normal component of fluxes are specified at the walls. For systems with reactions occurring on solid surfaces, generally, accumulation of species at the solid surface is neglected and the diffusive flux at the wall is equated to the surface reaction rate.

The averaging of the temperature equation for multiphase reactive systems is not straight forward and numerous forms of the averaged equation can be found in the literature. 

The granular temperature equation for the particle phase is written ... 

The pellet temperature equation for the multiphase gas-solid mixture can be expressed as ... 

The corresponding temperature equation for the interstitial gas is given by (11.6). To define the alternative model versions the effective heat of reaction term St = Pcat — in the basic model heat balance is... 

Thus, expressing the rate constant in Arrhenius form in both the concentration and temperature equations for each stage, the following recursion formulas for the Mh stage can be developed ... 

The temperature equation for the equilibrium constant is derived from thermodynamics using the Gibbs energy of formation, G°, the enthalpy of formation H°, and the temperature dependence as derived from ... 

The outcome of this procedure is, in addition to the previously described kinetic theory of granular flow (KTGF) transport equations with the given flux and source closures, characteristic transport equations for species mass and thermal temperature. It is noted that the use of second order velocity moments or higher moments usually requires some kind of manipulation in order to obtain equations in the desired form. The derivation of the thermal temperature equation for reactive systems is certainly not trivial. The application of this theory to reactive systems is extensively discussed in the following two sections. 

The molecular temperature equation for the continuous phase is defined by ... 

There are five governing equations for the standard non-thermal line contact EHL problem, those of pressure, film thickness, force balance, density and viscosity. In addition there are three equations linked with the thermal model, these being the energy equation and a surface temperature equation for each surface. 

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