Calculation of Temperature

Consider the one degree-of-freedom model H(q, p) = + U(q) on the unbounded domain, with U bounded below and growing sufficiently rapidly as q oo. The associated canonical density is 

We will denote the average with respect to the canonical probability density by Av.  

Thus the temperature can be related to the canonical average of the kinetic energy. If we consider a Nd = Nc degree of freedom system with kinetic energy K = 2nd potential energy U qi,q2, . then we may decompose the unnormalized canonical distribution as 

Then we again And, by the integration argument given above, that 

Thus the average kinetic energy of any individual particle is directly related to the 

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Zonal models are often used in analytical calculation of temperature, concentration, or humidity conditions in ventilated spaces. The space is divided in two or several zones, which typically have different target levels as described in Section 2.1 These typical zones can also be divided into additional subzones. 

Mundt presents a two-zone model for the calculation of temperature gradient within a stratification strategy. 

There are also formulas for calculation of temperature and concentration distribution along and across an air jet. These are based on the similarity profile of the jet. ... 

Table 2.3 Corrected equations for the calculation of temperature-dependent equilibrium constants based on publications of Reimschuessel and co-workers [21]...

Finally some other developments are briefly mentioned for the sake of completeness. These include the work of Sposito and Babcock 184> which bears close resemblance to the partitioned potential methods. Their idea is to solve the quantum mechanical energy spectrum of the complex using only the empirical potential function as the potential operator in the Hamiltonian. This spectrum then leads to calculations of temperature-dependent energies of formation. 

The equations describing the concentration and temperature within the catalyst particles and the reactor are usually non-linear coupled ordinary differential equations and have to be solved numerically. However, it is unusual for experimental data to be of sufficient precision and extent to justify the application of such sophisticated reactor models. Uncertainties in the knowledge of effective thermal conductivities and heat transfer between gas and solid make the calculation of temperature distribution in the catalyst bed susceptible to inaccuracies, particularly in view of the pronounced effect of temperature on reaction rate. A useful approach to the preliminary design of a non-isothermal fixed bed catalytic reactor is to assume that all the resistance to heat transfer is in a thin layer of gas near the tube wall. This is a fair approximation because radial temperature profiles in packed beds are parabolic with most of the resistance to heat transfer near the tube wall. With this assumption, a one-dimensional model, which becomes quite accurate for small diameter tubes, is satisfactory for the preliminary design of reactors. Provided the ratio of the catlayst particle radius to tube length is small, dispersion of mass in the longitudinal direction may also be neglected. Finally, if heat transfer between solid cmd gas phases is accounted for implicitly by the catalyst effectiveness factor, the mass and heat conservation equations for the reactor reduce to [eqn. (62)]... 

K9. Kjaer, J., Measurement and Calculation of Temperature and Conversion in Fixed-Bed Catalytic Converters. Gjellerups Forlag, Copenhagen, 1958. 

Detonation (and Explosion), Temperature Developed On. It may be defined as the maximum temperatures developed on detonation and explosion and must not be confused with Detonation (and Explosion) Temperature described in previous item A. Calculation of Temperature of Detonation (or Explosion). The oldest and simplest method is based on the assumption that expln is an adiabatic process taking place at constant volume and that the heat evolved (Qv), is used exclusively for heating the products of expln. Another assumption is that temp can be calcd by. dividing the heat of expln by specific heats of the products of expln ... 

For purposes of this calculation, latent heats at constant volume and at constant pressure are assumed equal, heat capacities at constant pressure and at constant volume are assumed equal for solids and liquids [See also Calculation of Temperature of Detonation (and Explosion) 1 and Experimental Determination of Temperature of Detonation [and Explosion) , under Detonation (and Explosion) Temperature Developed On in Vol 4 of Encycl, pp D589 L to D601-R]... 

Further advancements in the theory of fixed bed reactor design have been made(56,57) but it is unusual for experimental data to be of sufficient precision and extent to justify the application of sophisticated methods of calculation. Uncertainties in the knowledge of effective thermal conductivities and heat transfer between gas and solid make the calculation of temperature distribution in the bed susceptible to inaccuracies, particularly in view of the pronounced effect of temperature on the reaction rate. 

Section 4.2 deals with the most useful hydrate equilibria—calculations of temperatures and pressures at which hydrates form from gas and free water. In this section, two historical methods, namely, the gas gravity method (Section 4.2.1) and the Kvs, value method (Section 4.2.2), for calculating the pressure-temperature equilibrium of three phases (liquid water-hydrate-vapor, Lw-H-V)1 are discussed. With the gas gravity method in Section 4.2.1.1, a method is given for limits to expansion, as for flow through a valve. In Section 4.2.2 a distribution coefficient (KVSi) method is provided to determine whether a component prefers residing in the hydrate or the vapor phase. These methods provide initial estimates for the calculation and provide a qualitative understanding of the equilibria. A statistical... 

We defined the parameters of our model and made numerical calculations of temperature dependent of two gaps. It is a qualitative agreement with experiments. We proposed a two channel scenario of superconductivity first a conventional channel (intraband gi) whichis is connected with BCS mechanism in different zone and a unconventional channel (interband gi) which describes the tunneling of a Cooper pair between two bands. The tunneling of Cooper pair also stabilizes the order parameters of superconductivity [9-12] and increases the critical temperature of superconductivity. 

M.F. Horstemeyer et al Micromechanical finite element calculations of temperature and void configuration effects on void growth and coalescence. Int J. Plasticity 16, 979-1015 (2000)... 

In most industrial processes coexisting phases are vapor and liquid, although liquid/liquid, vapor/solid, and liquid/solid systems are also encountered. In this chapter we present a general qualitative discussion of vapor/liquid phase behavior (Sec. 12.3) and describe the calculation of temperatures, pressures, and phase compositions for systems in vapor/liquid equilibrium (VLE) at low to moderate pressures (Sec. 12.4).t Comprehensive expositions are given of dew-point, bubble-point, and P, T-flash calculations. 

Due to the constructive peculiarities in the most of widely used high pressure and temperature apparatuses including both lens and belt type ones temperature field in reaction cell has axial symmetry. Thus, one has possibility to analyze temperature conditions in reaction cells of different design by means of finite element method calculation scheme [4], Beneath we discuss the results and application of this kind of calculations of temperature fields for one possible type of lens-shaped high pressure apparatus [1],... 

The calculation of temperatures and equilibrium compositions of gas mixtures involves simultaneous solution of linear (material balance) and nonlinear (equilibrium) algebraic equations. Therefore, it is necessary to resort to various approximate procedures classified by Carter and Altman (Cl) as (1) trial and error methods (2) iterative methods (3) graphical methods and use of published tables and (4) punched-card or machine methods. Numerical solutions involve a four-step sequence described by Penner (P4). 

For the worst case, calculation of temperature rise within the sample due to mechanical energy dissipation is about 1"C. With heat loss from the sample, this value should be lower. 

Conservative calculation of temperature rise within the sample due to mechanical energy dissipation is less than 1"C. 

Figure 9. Numerical calculation of temperature distribution in hot, high pressure,...

The most commonly encountered coexisting phases in industrial practice are vapor and liquid, although liquid/liquid, vaporlsolid, and liquid/solid systems are also found. In this chapter we first discuss the nature of equilibrium, and then consider two rules that give the lumiber of independent variables required to detemiine equilibrium states. There follows in Sec. 10.3 a qualitative discussion of vapor/liquid phase behavior. In Sec. 10.4 we introduce tlie two simplest fomiulations that allow calculation of temperatures, pressures, and phase compositions for systems in vaporlliquid equilibrium. The first, known as Raoult s law, is valid only for systems at low to moderate pressures and in general only for systems comprised of chemically similar species. The second, known as Henry s law, is valid for any species present at low concentration, but as presented here is also limited to systems at low to moderate pressures. A modification of Raoult s law that removes the restriction to chemically similar species is treated in Sec. 10.5. Finally in Sec. 10.6 calculations based on equilibrium ratios or K-values are considered. The treatment of vapor/liquid equilibrium is developed further in Chaps. 12 and 14. 

Relationship (45) for the time of homogenization can be adjusted to allow calculation of temperature effect by introducing D = DoCxp ( /iiT), where is the diffusion activation energy having the approximate value of 142 k J mole (34 kcal mole ). After expressing the constants, one obtains for the above case... 

Original equation for dynamic calculation of temperature on plate j ... 

Henriques developed a two-step method to calculate burn injury. The first step is a calculation of temperature distribution within the skin, while the second determines burn injury based on the time-temperature history. This general approach remains in use although modern burn modeling techniques generally involve sophisticated computer models of temperature distribution that were unavailable to Henriques and Moritz. 

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