Local thermal equilibrium

Whitaker, S, Local Thermal Equilibrium An Application to Packed Bed Catalytic Reactor Design, Chemical Engineering Science 41, 2029, 1986. 

Whitaker, S, Improved Constraints for the Principle of Local Thermal Equilibrium, Industrial and Engineering Chemistry Research 29, 983, 1991. 

Similar convection-diffusion equations to the Navier-Stokes equation can be formulated for enthalpy or species concentration. In all of these formulations there is always a superposition of diffusive and convective transport of a field quantity, supplemented by source terms describing creation or destruction of the transported quantity. There are two fundamental assumptions on which the Navier-Stokes and other convection-diffusion equations are based. The first and most fundamental is the continuum hypothesis it is assumed that the fluid can be described by a scalar or vector field, such as density or velocity. In fact, the field quantities have to be regarded as local averages over a large number of particles contained in a volume element embracing the point of interest. The second hypothesis relates to the local statistical distribution of the particles in phase space the standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell-Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference co-moving with the fluid. Especially the second assumption may break dovm when gas flow at high temperature or low pressure in micro channels is considered, as will be discussed below. 

Assuming local thermal equilibrium, i.e. the equality of the averaged fluid and solid temperature, a transport equation for the average temperature results which still contains and integral over the fluctuating component. In order to close the equation, a relationship between the fluctuating component and the spatial derivatives of the average temperature of the form... 

Secondary Ion Yields. The most successful calculations of secondary in yields are based on the local thermal equilibrium (LTE) model of Andersen and Hinthorne (1973), which assumes that a plasma in thermodynamic equilibrium is generated locally in the solid by ion bombardment. Assuming equilibrium, the law of mass action can be applied to find the ratio of ions, neutrals and electrons, and the Saha-Eggert equation is derived ... 

We have discussed the transition moment (the quantum mechanical control of the strength of a transition or the rate of transition) and the selection rules but there is a further factor to consider. The transition between two levels up or down requires either the lower or the upper level to be populated. If there are no atoms or molecules present in the two states then the transition cannot occur. The population of energy levels within atoms or molecules is controlled by the Boltzmann Law when in local thermal equilibrium ... 

Local thermal equilibrium (LTE) is an assumption that allows for the molecules to be in equilibrium with at least a limited region of space and remains an assumption when using the Boltzmann law for the relative populations of energy levels. The LTE assumption notwithstanding, observation of a series of transitions in the spectrum and measurement of their relative intensities allows the local temperature to be determined. We shall see an example of this in Section 4.4 where the Balmer temperature of a star is derived from the populations of different levels in the Balmer series. 

Atoms and molecules have available to them a number of energy levels associated with the allowed values of the quantum numbers for the energy levels of the atom. As atoms are heated, some will gain sufficient energy either from the absorption of photons or by collisions to populate the levels above the ground state. The partitioning of energy between the levels depends on temperature and the atom is then said to be in local thermal equilibrium with the populations of the excited states and so the local temperature can be measured with this atomic thermometer. 

For N2 molecules in the air at room temperature cr(d is of the order of the speed of sound, 370 ms-1, a is 0.43 nm2 and Z = 5 x 1028 cm-3 s 1. This is a very large number, which means that collisions between molecules occur very frequently and the energy can be averaged between them, ensuring the concept of local thermal equilibrium. Each molecule collides ZAK/NA times per second, which is about 5 x 109 s x once every 0.2 ns. However, in the diffuse ISM where the molecule density is of order 102 cm3 the collision frequency is 5 x 10-8 s-1 or a collision every 1.5 years. 

Local thermal equilibrium (LTE) The assumption that all molecules within a particular environment have acquired their fair share of the local energy. 

Alternate mass-core hard potential channel In the two billiard gas models just discussed there is no local thermal equilibrium. Even though the internal temperature can be clearly defined at any position(Alonso et al, 2005), the above property may be considered unsatisfactory(Dhars, 1999). In order to overcome this problem, we have recently introduced a similar model which however exhibits local thermal equilibrium, normal diffusion, and zero Lyapunov exponent(Li et al, 2004). 

Almost all of the models assume local thermal equilibrium between the various phases. The exceptions are the models of Beming et al., ° who use a heat-transfer coefficient to relate the gas temperature to the solid temperature. While this approach may be slightly more accurate, assuming a valid heat-transfer coefficient is known, it is not necessarily needed. Because of the intimate contact between the gas, liquid, and solid phases within the small pores of the various fuel-cell sandwich layers, assuming that all of the phases have the same temperature as each other at each point in the fuel cell is valid. Doing this eliminates the phase dependences in the above equations and allows for a single thermal energy equation to be written. 

The assumption of local thermal equilibrium also means that an overall effective thermal conductivity is needed, because there is only a single energy equation. One way to calculate this thermal conductivity is to use Bruggeman factors. 

Most of the models use a simplified analogue of eq 68 where eqs 69, 70, and 72 have been substituted into it and local thermal equilibrium is assumed and the equation is summed over phases. The resultant equation is then further simplified for fuel cells... 

If an atomic transition is optically pumped by a beam of laser radiation having the appropriate frequency, the population in the upper state can be considerably enhanced along the path of the beam. This causes an intensification of the spontaneous emission from this state, which contains information about the conditions within the pumped region, since the exponential decay time for the intensified emission depends upon both the electron number density and the electron temperature. The latter can be obtained from the intensity ratio of the fluorescence excited from two different lower levels, if local thermal equilibrium is assumed. This method has been dis-... 

In this summary, the local thermal equilibrium model has been used to derive the energy equation. This model is much simpler than the two-phase model however, the local thermal equilibrium model is most likely not adequate to describe the transport of energy when the temperature of the fluid and solid are undergoing extremely rapid changes. Although such extremely rapid temperature changes are not expected, in most RTM, IP, and AP processes the correctness of the local thermal equilibrium assumption can be verified by following the procedure discussed by Whitaker [28]. 

Such defects, if present, will be assumed to be in local thermal equilibrium at very small concentrations. 

The mass fraction Gi of the mass velocity M differs from the mass fraction of the local mixture when diffusion is taking place. Ki is the ordinary reaction rate of chemical kinetics, expressed as mole/unit volume/unit time, which is measured in a static experiment as a function of temperature and composition if the flame is in local thermal equilibrium. If not, this equation serves only as a measure of the nonequilibrium local reaction rate. 

In the case of local thermal equilibrium B(a, x) is equal to the Planck function. 

A major complication in the analysis of convection and segregation in melt crystal growth is the need for simultaneous calculation of the melt-crystal interface shape with the temperature, velocity, and pressure fields. For low growth rates, for which the assumption of local thermal equilibrium is valid, the shape of the solidification interface dDbI is given by the shape of the liquidus curve Tm(c) for the binary phase diagram ... 

One of the simplified heat transfer models of two-phase flows is the pseudocontinuum one-phase flow model, in which it is assumed that (1) local thermal equilibrium between the two phases exists (2) particles are evenly distributed (3) flow is uniform and (4) heat conduction is dominant in the cross-stream direction. Therefore, the heat balance leads to a single-phase energy equation which is based on effective gas-solid properties and averaged temperatures and velocities. For an axisymmetric flow heated by a cylindrical heating surface at rw, the heat balance equation can be written as... 

The applicability of the preceding pseudocontinuum approach to convective heat transfer of gas-solid systems without heat sources depends not only on the validity of the phase continuum approximation but also on the appropriateness of the local thermal equilibrium assumption. The local thermal equilibrium may be assumed only if the particle-heating... 

Many attempts have been made to quantify SIMS data by using theoretical models of the ionization process. One of the early ones was the local thermal equilibrium model of Andersen and Hinthome [36-38] mentioned in the Introduction. The hypothesis for this model states that the majority of sputtered ions, atoms, molecules, and electrons are in thermal equilibrium with each other and that these equilibrium concentrations can be calculated by using the proper Saha equations. Andersen and Hinthome developed a computer model, C ARISMA, to quantify SIMS data, using these assumptions and the Saha-Eggert ionization equation [39-41]. They reported results within 10% error for most elements with the use of oxygen bombardment on mineralogical samples. Some elements such as zirconium, niobium, and molybdenum, however, were underestimated by factors of 2 to 6. With two internal standards, CARISMA calculated a plasma temperature and electron density to be used in the ionization equation. For similar matrices, temperature and pressure could be entered and the ion intensities quantified without standards. Subsequent research has shown that the temperature and electron densities derived by this method were not realistic and the establishment of a true thermal equilibrium is unlikely under SIMS ion bombardment. With too many failures in other matrices, the method has fallen into disuse. 

There is a local thermal equilibrium among the solid phase, liquid phase and gaseous phase. 

In a real radiation source this perfect equilibrium cannot exist and there are losses of energy as a result of the emission and absorption of radiation, which also have to be considered. However, as long as both only slightly affect the energy balance, the system is in so-called local thermal equilibrium and ... 

At complete ionization of the hydrogen (e.g. when added to a plasma with another gas as the main constituent) ne = p/(2 x k x Te) has a maximum at a wavelength of X — (7.2 x 107)/Te or at a fixed wavelength, the maximum intensity is found at a temperature Te = (5.76 x 107)/2. Thus, the electron temperature can be determined from the wavelength dependence of the continuum intensity. As Te is the electron temperature, absolute measurements of the background continuum emission in a plasma, e.g. for the case of hydrogen, allow determination of the electron temperature in a plasma, irrespective of whether it is in local thermal equilibrium or not. Similar methods also make use of the recombination continuum and of the ratio of the Bremsstrahlung and the recombination continuum. 

The Saha equation is only valid for a plasma which is in local thermal equilibrium, where the temperature in the equation is then the ionization temperature. When this condition is not fullfilled, the equilibrium between the different states of ionization is given by the so-called Corona equation [16],... 

This method can again only be applied for a plasma in local thermal equilibrium, the temperature of which is known. The partition functions Zaj and Zy for the atom and ion species, respectively, are again a function of the temperature and the coefficients of these functions have been calculated for many elements [5], Furthermore, the accuracy of the gA values and of the temperatures is important for the accuracy of the determination of the degree of ionization. One often uses the line pairs Mg II 279.6 nm/Mg I 278.0 nm and Mg II 279.6 nm/Mg I 285.2 nm for determinations of the degree of ionization of an element in a plasma. 

When the plasma is not in local thermal equilibrium (LTE), the electron number densities cannot be determined on the basis of the Saha equation. Irrespective of the plasma being in local thermal equilibrium or not, the electron number density can be derived directly from the Stark broadening of the Hg line or of a suitable argon line. This contribution to broadening is a result of the electrical field of the quasi-static ions on one side and the mobile electrons on the other side. As described in Ref. [17] it can be written as ... 

The plasma temperature in the carbon arc is of the order of 6000 K and it can be assumed to be in local thermal equilibrium. According to the temperatures obtained, it could be expected that in arc emission spectrometry mainly the atom lines will be the most sensitive lines, when considering their norm temperatures. Arcs are usually used for survey trace analysis but also for the analysis of pure substances when the highest power of detection is required. However, they may be hampered by poor precision (RSDs of 30% and higher). 

The ICP is almost in local thermal equilibrium. Indeed, the excitation temperatures (from atomic line intensity ratios) are about 6000 K [382] and the rotation temperatures (from the rotation lines in the OH bands) are 4000-6000 K (see Refs. [383, 384]). From the broadening of the Hg-line, an electron number density of 1016 cm 3 is obtained, whereas from the intensity ratio of an ion and an atom line of the same element the electron number density found is 1014 cm-3. It has also been reported that measured line intensity ratios of ion to atom lines are higher by a factor of 100 than those calculated for a temperature of 6000 K and the electron number density found is 1016 cm 3. This indicates the existence of over-ionization. This can be understood from the excitation processes taking place. They include the following. 

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