Measurement Of Average Kinetic Energy

Quantum Mechanics

Albert Thomas FromholdJr., in Encyclopedia of Physical Science and Technology (3rd Edition), 2003

IX.Due east Metals

The boilerplate kinetic energy for the highest populated energy ring can be estimated by invoking a "particle in a box" model of a metal holding its conduction electrons inside the boundary walls, with no consideration given to the bodily periodic potential energy of the lattice. This simple arroyo, known as the free-electron model, oftentimes yields surprisingly accurate quantitative values for a number of physical properties of metals associated with the conduction electrons. In such cases, a full solution of the Schrödinger equation for the actual periodic potential may not be required.

The kinetic energy of the highest filled country in a given energy band at 0   Kelvin (K) is designated the Fermi free energy. A computation of how the boilerplate free energy changes with increases in the thermodynamic temperature of the organisation yields the specific heat of the conduction electrons. The accurate predictions obtained by quantum mechanics for the specific heat of metals at depression temperatures represents a remarkable success for the theory, that is to be sharply contrasted with the total failure on the part of the classical approach to provide an adequate quantitative estimate of this concrete property of metals.

Quantum mechanics gives great insight into the scattering of conduction electrons by imperfections in a metal. The quantum nature of the scattering of conduction electrons places the restriction on the process that scattering can have place only to vacant quantum states of the system. This ways that at 0   K, an elastic handful event tin can occur merely for a conduction electron having an energy equal to the Fermi energy, because that is the only energy at which both filled and empty states simultaneously be. The situation is not quite and so restrictive at higher temperatures, where there is a statistical probability that nearby states are occupied or unoccupied over a range of energy at least k _B T in width in the neighborhood of the Fermi free energy $\mathcal{E}$ _F [Boltzmann abiding one thousand _B   =   i.38   ×   10⁻²³ J/Yard; T  =   absolute (Kelvin) temperature]. Nonetheless, electron scattering and, hence, the electrical resistivity are still severely restricted in metals by the requirements of the Pauli exclusion principle.

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Fluid Backdrop

John R. Fanchi , in Integrated Reservoir Asset Management, 2010

2.two.i Temperature

Temperature is a measure of the average kinetic energy of a organization. The about commonly used temperature scales are the Fahrenheit and Celsius scales. The relationship between these scales is

(2.2.1) $T_C=\frac{5}{\mathrm{nine}}\left(T_F-32\right)$

where T_C and T_F are temperatures in degrees Celsius and degrees Fahrenheit, respectively.

Applications of the equations of state described in the following require the apply of absolute temperature scales. Absolute temperature may be expressed in terms of degrees Kelvin or degrees Rankine. The absolute temperature scale in degrees Kelvin is related to the Celsius scale by

(2.two.2) $T_{\mathrm{Yard}}=T_C+273$

where T_K is temperature in degrees Kelvin. The absolute temperature scale in degrees Rankine is related to the Fahrenheit scale past

(2.2.3) $T_R=T_F+460$

where T_R is temperature in degrees Rankine.

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Parameters to Depict Air Masses and Vertical Motions

Dario Camuffo , in Microclimate for Cultural Heritage (2nd Edition), 2014

3.3 Potential Temperature

Nosotros have seen that the temperature is representative of the average kinetic energy of the molecules constituting an air parcel but this value changes when the parcel is moved, irresolute its height above the Earth's surface. It is useful to find a parameter that allows for a comparison between the boilerplate kinetic energy of molecules belonging to two distinct air masses, contained of their height, i.e. referring them to a standard level. The potential temperature Θ is the temperature a parcel of air would have if brought dry adiabatically from its initial state to the standard pressure of one thousand   hPa. Note that the round number 1000 is used instead of 1013, in order to make calculations easier. At sea level, the potential temperature is very shut to the actual air temperature and information technology is exactly Θ   = T when p  =   k   hPa. The potential temperature of an air package is independent of the meridian, similar to the temperature of mixed incompressible liquids.

Θ is computed with the help of the Poisson equation utilizing the initial values of T and p, i.e.

(3.18) $\mathrm{\Theta }=T{\left(\frac{1000}{p}\right)}^k$

where Θ is directly proportional to T merely inversely related to p ^k .

A region of the atmosphere in which Θ is abiding along the vertical is named dry adiabatic. A representation in terms of Θ immediately shows where the atmosphere is adiabatic, i.e. dΘ/dz  =   0, with vertical isolines; where it is superadiabatic, dΘ/dz  >   0, with isolines tilted towards left; and where it is subadiabatic, dΘ/dz  <   0, with isolines tilted towards correct (Fig. 3.1). Further details will be given in Chapter vii.

Figure 3.1. Vertical contour of the air temperature T (blue line) and potential temperature Θ (cherry line) in the Venice hinterland, on 25 June 1979 at 10 am. The offset 150   thou show a profile superadiabatic, i.e. with a gradient greater (in the accented value) than the adiabatic one (Θ contour tilted towards left); from 150 to 280   g is the substantially adiabatic region (Θ profile vertical); and above the air is stable (Θ profile tilted towards right).

In meteorology, conditions prediction and pollution transport, the basic equations are more easily solved if appropriate coordinates are used. For this reason, the geopotential, the pressure, the entropy and the potential temperature are used as a vertical coordinate instead of the geometric pinnacle (for a discussion run into Kasahara, 1974). Potential temperature or isentropic coordinates are peculiarly convenient for description of adiabatic motions. As in the definition of Θ, the atmosphere is characterized by adiabatic displacements in the whole region where Θ   =   const, besides δQ  =   0, and trajectories are controlled by isentropic motions. Therefore, all the surfaces with the aforementioned Θ are isentropic and an air package having a given value of Θ will remain on the same surface characterized by a given potential temperature Θ or within the same layer divisional past the ii isentropes Θ and Θ   +   dΘ, unless external work is supplied to it. Therefore, the isentropic representation of the atmosphere is a useful tool to forecast where a pollutant can or cannot be transported and whether it will exist able to cantankerous mount chains or non (Fig. 3.2). In fact, except near the soil where heat is exchanged and in the absenteeism of condensation, an air package tends to maintain the same entropy and potential temperature.

FIGURE 3.ii. Vertical cross-department of potential temperature over a terrain with a mount concatenation. The air layer betwixt Θ _northward and Θ _{n +one} volition remain separated into two distinct parts (i.eastward. yellow and cyan) past the mountain chain.

It is likewise possible to notice a relationship between potential temperature and entropy. Dividing by T both sides of Eqn (three.12) and integrating, ane obtains the entropy $\mathsf{S}$ of the air:

(3.19) $\mathsf{S}=\int \frac{d\mathit{Q}}{T}=\int c_p\frac{dT}{T}-\int {\Re }_a\frac{dp}{p}=c_p\ln T-{\Re }_a\ln p$

This equation holds for unsaturated air, assuming that the mixture of air and vapour behaves as an ideal gas (unsaturated vapour). For saturated vapour, the equation is formally identical, except for an additive constant (Goody, 1995). The relationship is easily obtained substituting in the previous equation the definition of Θ in Eqn (3.eighteen) in logarithmic form, i.eastward.

(iii.20) $\ln \mathrm{\Theta }=\ln T+\frac{{\Re }_a}{c_p}\left(\text{ln}1000-\text{ln}p\right)$

so that the entropy $\mathsf{S}$ is immediately found, i.e.

(3.21) $\mathsf{Due\; south}=c_p\ln \mathrm{\Theta }$

where the condiment arbitrary reference constant has been omitted.

The Poisson equation tin be compared with the supposition that the vertical temperature gradient is constant and given past Eqn (3.17), i.e. dT/dz  = γ, so that

(iii.22) $T=T_{\text{o}}+\mathit{\gamma }z$

Dividing the hydrostatic equation (Eqn (3.16)) by the equation of state for perfect gases (Eqn (ane.ii)), and using the previous equation for T,

(3.23) $\frac{dp}{p}=-\frac{gdz}{{\Re }_a\left(T_o+\gamma z\right)}=-\frac{g}{{\Re }_a\gamma }\frac{d\left(T_o+\gamma z\right)}{T_o+\gamma z}$

or

(3.24) $\text{d}\ln p=-\frac{g}{\gamma {\Re }_a}\text{d}\ln T\text{.}$

Integration gives

(three.25a) $\frac{p_{\mathrm{ii}}}{p_1}={\left(\frac{T_2}{T_{\mathrm{one}}}\right)}^{-g/\gamma {\Re }_a}$

(iii.25b) $\frac{T_2}{T_1}={\left(\frac{p_2}{p_1}\right)}^{-\gamma {\Re }_a/g}$

where $\text{γ}{\Re }_a/g=\mathrm{thousand}=0.286$ .

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Ocean Energy

James VanZwieten , Yufei Tang , in Comprehensive Renewable Energy (Second Edition), 2022

8.03.ii.3.1.ane Southeast US measurements: Southeast Florida

To identify the correlation between potential mooring locations, system operating depth, and available ability near 26°N the average Kinetic Energy Flux (KEF) profiles for four longitudes in the Florida Straits are presented ( Fig. 3) (VanZwieten et al., 2015). ADCPs were moored in an due east-west configuration at the same latitude (26°4.3'North), approximately perpendicular to the shoreline. Locations were selected to approximately mensurate from the western edge of the Gulf Stream current to its core. At the shallowest and nearly Due west location (79°55'Due west) ADCP information were available from 455 days, Center-W location (79°52.5'Due west) ADCP data were available from 382 days, Center-East location (79°l'W) ADCP information were available from 1082 days and Eastern location (79°45'W longitude) ADCP data were available from 294 days. These results all prove that available power decreases quickly with depth, with approximately forty% less power available at a depth of 100   m than at 50   thou (VanZwieten et al., 2015).

Fig. 3. Temporally averaged KEF (height iv figures); flow speed statistic including minimum speed (blue), speed exceeded 75% of the time (magenta), mean speed (greenish), speed exceeded 15% of the time (ruby), and maximum speed (blackness) (bottom iv figures); at the iv evaluated ADCP locations offshore from Florida (VanZwieten et al., 2015).

Fig. 3-top presents the average KEF for these longitudes, showing an increase with distance from the shore until the 2d deepest location (for a depth of 50   m). At this depth the KEF is 1.58, 2.09, 2.twenty, and 2.14   kW/m², moving abroad from shore. This shows that moving farther from the shore across the 2d shallowest location does not significantly change the boilerplate KEF. Moving eastward from shore the standard deviation of KEF for a depth of l   m are 1.28, 1.34, 1.29, and 0.97   kW/thousand^two. This shows that while mean bachelor power in this region does not significantly increase with distance from the shore, the chapters factor of installed devices and power to generate base load ability does increase with distance from the shore (Machado et al., 2016).

Period speeds are important for designing mooring systems and estimating peak loads (Fig. 3-bottom). Minimum currents at this latitude approached 0.0   m/due south at the shallowest 3 locations, where no power would be produced and mooring lines could become slack (a dangerous status, depending on mooring/anchor blueprint). However, at the furthest offshore location, current speeds were not measured below 0.8   g/s for a depth of 50   grand. Maximum electric current speeds are important for estimating maximum loading condition and these measurements bear witness that the middle two locations contained the maximum flow speeds in this expanse, which are around three.0   m/s. However, these current speeds occurred over a ii-24-hour interval window, June 22–23, 2013, where measurements are not bachelor from the shallowest and deepest locations. Besides these 2 days, maximum measured electric current speeds at all locations were approximately 2.5   k/s at a depth of l   yard (VanZwieten et al., 2015).

To evaluate HYCOM predictions, data from three of these locations were used, with dates that marshal with the HYCOM data in Section eight.03.2.2. At the shallowest and most inshore location, (79°55'W longitude) ADCP information were available from 455 days, while data at the center location (79°51'W and 79°50.5'W longitude) ADCP information were available from 742 days and at the easternmost location (79°45'W longitude) ADCP data were bachelor from 294 days. These data show that near the surface, HYCOM estimated mean current speeds were less accurate at filigree points farther from the current׳due south core (Fig. four). From west to eastward, HYCOM calculated mean current is under-predicted by 28%, 16%, and viii% at 50   m below the ocean surface. While mean current predictions converge toward measured values near the cadre of the Gulf Stream, HYCOM significantly under-predicts the variability of the current speed at all locations (VanZwieten et al., 2015). Additionally, maximum current speeds at a depth of fifty m are under-predicted by HYCOM past 37%, 26% and 19%, moving from west to east.

Fig. 4. HYCOM vs ADCP minimum speed (blueish), speed exceeded 75% of the time (magenta), hateful speed (greenish), speed exceeded 15% of the time (reddish), and maximum speed (black) at three ADCP locations off Southeast Florida. The larger symbols (asterisk, diamond and big dot) betoken HYCOM data and the smaller indicate ADCP data (ADCP information look similar lines in left two plots) (VanZwieten et al., 2015).

Extreme weather tin can alter the water velocity profiles of ocean currents, especially near the sea surface. ADCP measurements fabricated during two hurricanes were described in particular in (Baxley et al., 2019) and are summarized here. In 2012, Hurricane Sandy began in the Caribbean and reached maximum wind speeds of 185   km/h while over Cuba, before moving n along the eastern seaboard and ultimately to New York. Although the storm׳s closest point of approach (CPA) to the Florida declension was over 500   km to the east on October 26, 2012, meaning changes in the FC flow speed and direction within the upper 150   m of the h2o column were observed past an ADCP moored near the Gulf Stream core in 340   chiliad of water and slightly closer to shore in 260   m of water.

Subsequent ADCP measurements during Hurricane Irma in 2017 also showed major changes in h2o velocity within 150   m of the bounding main surface (Baxley et al., 2019). Hurricane Irma reached category 5 intensity on the Saffir—Simpson Hurricane Wind Scale. Irma passed between Cuba and the Bahamas, making its CPA to a single ADCP system that was deployed 33   km off southeast Florida in 500   m of water nearly the average FC core at 1800 GMT on September 10, 2017. This hurricane made seven landfalls, including as a category 4 hurricane in the Florida Keys and a category three when striking southwestern Florida (Cangialosi et al., 2018). Information technology then moved northward forth the due west Florida coastline, dissipating on September 11, 2017. When passing through the Florida Keys and then up Florida׳s west coast the counterclockwise rotation of this storm forced water around the southern tip of Florida resulting in extremely strong electric current speeds (Fig. 5 Left). And then, approximately two weeks later, extremely deadening current speeds were recorded (Fig. 5 Right), which were also probable related to the tempest.

Fig. 5. ADCP measured current speed and direction recorded during the passing of Hurricane Irma (Left) and approximately two weeks later (Right), highlighting the impacts of this storm on the Gulf Stream current (Hasankhani et al., 2021). Note: Data inside the top 50   m of the water cavalcade are extrapolated from deeper data.

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Flow in pipes and in conduits of non-circular cross-sections

R.P. Chhabra , J.F. Richardson , in Non-Newtonian Period in the Process Industries, 1999

3.2.3 Average kinetic energy of fluid

In order to obtain the kinetic energy correction factor, α, for insertion in the mechanical energy residue, it is necessary to evaluate the average kinetic energy per unit mass in terms of the average velocity of flow. The calculation procedure is exactly similar to that used for Newtonian fluids, (e.one thousand. see [ Coulson and Richardson, 1999]).

Average kinetic free energy/unit mass

(3.15a) $\mathrm{=}\frac{{\displaystyle \int }\frac{\mathrm{ane}}{\mathrm{2}}{\mathit{V}}_{\mathit{z}}^{\mathrm{2}}\text{d}\dot{\mathit{m}}}{{\displaystyle \int }\text{d}\dot{\mathit{m}}}\mathrm{=}\frac{{{\displaystyle \int }}_{\mathrm{0}}^{\mathit{R}}\frac{\mathrm{ane}}{\mathrm{ii}}{\mathit{5}}_{\mathrm{Z}}^{\mathrm{2}}\mathrm{ii}\pi \mathit{r}{\mathit{V}}_{\mathrm{z}}\rho \text{d}\mathit{r}}{{{\displaystyle \int }}_{\mathrm{0}}^{\mathit{R}}\mathrm{2}\pi \mathit{r}{\mathit{Five}}_{\mathit{z}}\rho \text{d}\mathit{r}}$

(three.15b) $\mathrm{=}\frac{{\mathit{V}}^{\mathrm{two}}}{\mathrm{ii}\alpha }$

where α is a kinetic free energy correction factor to have account of the non-uniform velocity over the cross-section. For ability-law fluids, substitution for 5_z from equation (3.7a) into equation (three.15a) and integration gives

(3.16) $\alpha \mathrm{=}\frac{\left(\mathrm{two}\mathit{n}\mathrm{+}\mathrm{one}\right)\left(\mathrm{5}\mathit{northward}\mathrm{+}\mathrm{3}\right)}{\mathrm{iii}{\left(\mathrm{3}\mathit{n}\mathrm{+}\mathrm{1}\right)}^{\mathrm{2}}}$

The corresponding expression for a Bingham plastic is cumbersome. However, Metzner [1956] gives a simple expression for α which is accurate to within ii.5%:

(3.17) $\alpha \mathrm{=}\frac{\mathrm{1}}{\mathrm{2}\mathrm{-}\varphi }$

Again, both equations (3.16) and (three.17) reduce to α = 1/2 for Newtonian fluid behaviour. Notation that as the degree of shear-thinning increases, i.e. the value of north decreases, the kinetic energy correction gene approaches unity at n = 0 as would be expected, as all the fluid is flowing at the aforementioned velocity (Figure 3.3). For shear-thickening fluids, on the other hand, information technology attains a limiting value of 0.37 for the infinite degree of shear-thickening behaviour (n = ∞).

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Flow in Pipes and in Conduits of Non-circular Cantankerous-sections

R.P. Chhabra , J.F. Richardson , in Non-Newtonian Catamenia and Applied Rheology, 2008

iii.2.3 Average Kinetic Energy of Fluid

In guild to obtain the kinetic energy correction factor, α, for insertion in the mechanical energy balance, it is necessary to evaluate the average kinetic free energy per unit mass in terms of the average velocity of menstruation. The calculation procedure is exactly similar to that used for Newtonian fluids (e.g., run across Coulson and Richardson, 1999).

Boilerplate kinetic energy/unit of measurement mass:

(iii.42) $=\frac{\int \frac{\mathrm{one}}{\mathrm{ii}}{V}_z^2\mathrm{d}\dot{m}}{\int \mathrm{d}\dot{m}}=\frac{{\int }_0^R\frac{1}{2}{\mathrm{Five}}_z^2\mathrm{two}\pi r{\mathrm{Five}}_z\rho \mathrm{d}r}{{\int }_0^R\mathrm{ii}\pi r{V}_z\rho \mathrm{d}r}$

(three.43) $=\frac{V^2}{\mathrm{ii}\alpha }$

where α is a kinetic energy correction factor to take business relationship of the non-uniform velocity over the cross-section. For power-law fluids, substitution for Five_z from equation (three.10) into equation (three.42) and integration gives:

(3.44) $\alpha =\frac{(2n+1)(\mathrm{v}n+3)}{3(3n+\mathrm{i}{)}^{\mathrm{two}}}$

The corresponding expression for a Bingham plastic is cumbersome. Nonetheless, Metzner (1956) gives a unproblematic expression for α which is accurate to within 2.five%:

(3.45) $\alpha =\frac{1}{2-\phi }$

Again, both equations (3.44) and (3.45) reduce to α=i/2 for Newtonian fluid behaviour. Note that equally the caste of shear-thinning increases, i.e., the value of n decreases, the kinetic free energy correction cistron approaches unity at n=0 every bit would be expected, as all the fluid is flowing at the same velocity (Effigy three.3). For shear-thickening fluids, on the other hand, it attains a limiting value of 0.37 for the infinite degree of shear-thickening behaviour (n=∞).

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Dense Matter Physics

Y.C.Leung George , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

II.H Loftier Temperatures

At high temperatures, of the order of millions of degrees Kelvin, the thermal energy of the particles in the organisation becomes comparable to the boilerplate kinetic free energy of the electrons, in which case the zero-temperature equation of state must be corrected for thermal effects. We use the term "finite temperature" to denote a situation where temperature shows a perceptible outcome on the energy state of the arrangement. The proper treatment of finite temperature volition commencement be discussed by ignoring the electrostatic interaction, which will be discussed later.

Without electrostatic interaction, the nuclei behave like a classical ideal gas, and the pressure P _A they contribute to the system is given by the classical ideal gas law:

(31) ${\text{P}}_{\text{A}}=\rho /\text{A}{\text{m}}_{\text{p}}{\text{k}}_{\text{B}}\text{T}$

The electrons, on the other hand, must exist handled past the Fermi gas method. At finite temperatures electrons no longer occupy all the depression-free energy states; instead, some electrons are thermally excited to high-energy states. Through quantum statistics it is found that the probability for occupation of a quantum state of momentum p is given by the function,

(32) $\text{f}\left(\text{p}\right)={\left\{\mathrm{one}+\text{exp}\left[\left(\epsilon \left(\text{p}\right)-\mu \right)/{\text{k}}_{\text{B}}\text{T}\right]\right\}}^{-\mathrm{one}}$

where ɛ(p)   = p ²/2thousand _east and μ is the chemical potential, which in the present noninteracting instance is given by the Fermi energy ɛ_F:

(33) $\mu ={\epsilon }_{\text{F}}={\text{p}}_{\text{F}}^2/2{\text{chiliad}}_{\text{eastward}}$

It accounts for the free energy needed to introduce an boosted particle into the system. The Fermi–Dirac distribution f(p) is the distribution of occupied quantum states in an identical fermion system in thermal equilibrium. Since the distribution that nosotros shall employ depends on only the magnitude of the momentum and not its direction, it shall henceforth be written every bit f(p). The finite temperature parameters of a noninteracting electron system are to exist evaluated from:

(34a) ${\text{n}}_{\text{e}}=\frac{2}{{\text{h}}^{\mathrm{three}}}\int \mathrm{iv}\pi {\text{p}}^{\mathrm{ii}}d\text{p}\text{f}(\text{p})$

(34b) ${\varepsilon }_{\text{east}}=\frac{2}{{\text{h}}^{\mathrm{three}}}\int 4\pi {\text{p}}^{2}d\text{p}\text{f}(\text{p})\varepsilon (\text{p})$

(34c) ${\text{P}}_{\text{e}}=\frac{\mathrm{ii}}{{\text{h}}^3}\int 4\pi {\text{p}}^{2}d\text{p}\text{f}(\text{p})\frac{p\upsilon }{3}$

where the range of the p integration extends from zip to infinity. Note that, for relativistic electrons, ɛ(p) in Eq. (34b) should be replaced by e _k of Eq. (13) and v in Eq. (34c) should be inverse from v  = p/g _eastward to that given in Eq. (15). The integrals in these equations cannot exist evaluated analytically, and tabulated results of these integrals are available.

It is likewise useful to evaluate an entropy density (entropy per unit volume) for the system. The entropy density due to the electrons may be expressed in terms of quantities already evaluated as

(35) ${\text{due south}}_{\text{e}}=\left({\text{P}}_{\text{e}}+{\epsilon }_{\text{eastward}}-{\text{n}}_{\text{east}}\mu \right)/\text{T}$

The entropy density due to the nuclei is

(36) ${\text{s}}_{\text{A}}=\left({\text{P}}_{\text{A}}+{\epsilon }_{\text{A}}\right)/\text{T}=\left(5/3\right){\text{P}}_{\text{A}}/\text{T}$

where ɛ_A is the energy density of the nuclei. The entropy density has the same dimensions every bit Boltznman'southward constant k _B.

The equation of state relates the force per unit area of the system to its density and temperature. Since the pressure level is determined by two independent parameters, it is hard to display the result. Special cases are obtained by holding either the temperature abiding or the entropy of the arrangement constant as the density is varied. An equation of state that describes a thermodynamic process in which the temperature remains constant is called an isotherm, and an equation of land that describes an adiabatic procedure for which the entropy of the organization remains constant is chosen an adiabat. In the example where the total number of particles in the system remains unchanged as its volume is varied, constant entropy means the entropy per particle remains a constant, which is obtained by dividing the entropy density given by either Eq. (35) or Eq. (36) by the particle number density. In Fig. 3, typical isotherms and adiabats for a system of (noninteracting) neutrons at high densities are shown. Such a system of neutrons is called neutron matter, which is an important form of dumbo matter and will exist discussed in detail later on. Neutrons are fermions, and the method of Fermi gas employed for the study of an electron organization may be applied in the same fashion to the neutron system. In Fig. iii it is axiomatic that the adiabats have much steeper rises than the isotherms, and this feature holds for all matter systems in general.

Figure 3. Isotherms (solid lines) and adiabats (dashed lines) for a dense system of neutrons computed by the Fermi gas method with consummate neglect of nuclear interaction. Isotherms shown are computed for temperatures equivalent to m _B T  =   0, 5, and 10   MeV, and the adiabats for entropy per neutron equal to South/k_B  =   2 and 4.

The inclusion of electrostatic interaction to a finite-temperature system can be dealt with by extending the Thomas–Fermi–Dirac method described before. At a finite temperature the electron distribution in the system's quantum states must be modified from that of degenerate electrons to that of partially degenerate electrons every bit adamant by the Fermi–Dirac distribution of Eq. (32). Results of this type have been obtained, just they are too elaborate to be summarized here. Interested readers are referred to references listed in the bibliography, where reference to the original articles on this topic tin can exist constitute.

This section concentrates mainly on simple matter systems with densities lying in the first density domain, merely the concepts discussed are applicable to the study of dense matter at all densities. At higher densities, the electrostatic interaction is replaced past nuclear interaction as the dominant form of interaction. The discussion of nuclear interaction will be deferred to the side by side section.

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Heat transfer in nuclear thermal hydraulics

P.L. Kirillov , H. Ninokata , in Thermal-Hydraulics of Water Cooled Nuclear Reactors, 2017

seven.one.1.2 Estrus conduction mechanisms

Heat transfer medium is always a set of discrete particles, and heat propagation is a reflection of the mechanisms of interaction of the particles. Ratio of the average kinetic free energy Due east_grand of a particle to the bounden energy U betwixt particles is the major criterion to make up one's mind the state of the substance, i.e., gas, liquid, and solid (run into Fig. seven.two).

Fig. 7.2. Heat conduction mechanisms.

The energy transfer in gases is carried out by particle collisions and translational move. From the molecular kinetics theory of molecular dynamics, the thermal electrical conductivity is known to be written for the monatomic gas at normal temperatures and pressures equally:

$\lambda =\frac{1}{3}\rho c_{5}l\overline{w}\text{,}$

where ρ is the density; c_v is the specific heat at constant volume; fifty the mean gratuitous path of the molecules; $\overline{w}=\sqrt{3RT/M}$ the average velocity of the molecules with R the universal gas constant and M the molecular weight. The mean free path of molecules is in changed-proportion to the force per unit area (50  ∝   ane/p), and the density in proportion to the force per unit area (ρ  ∝ p). Hence $\rho \cdot l\sim \mathrm{const}\text{.}$ and the gas thermal conductivity is weakly dependent on pressure. Thermal electrical conductivity of gas is in the range 0.005–0.4   W/(grand   K). Information technology is noted that the highest thermal electrical conductivity of pure gases is attained by hydrogen and helium, which are 0.xv and 0.125   W/(m   M), respectively. Thermal electrical conductivity of air is of the order of ~  0.03   West/(g   Yard). The thermal conductivity of gases increases with increasing temperature.

In liquids, the energy is transferred in the procedure of elastic collisions of oscillating particles. Exceptions are liquid metals in which the thermal electrical conductivity close to that of solid metals in which oestrus is transferred not only by vibrations from one particle to another, but likewise with the free electrons (Kokorev and Farafonov, 1990). The thermal conductivity of the water is of the gild of 0.6   W/(thousand   Chiliad) at 20°C, and liquid sodium 75   W/(k   K) at 300°C.

In solids, the energy transfer mechanism is associated with vibrations of the atoms that plant the solid. The vibrations of atoms are contained of each other and can be transmitted at the speed of sound from one atom to another. The solid body can be regarded as a vessel containing a cloud of electrons and fictitious particles, i.eastward., phonons. The highest thermal conductivity of metal is for silvery ~   430   W/(m   M) and copper ~   400   W/(thou   K) (Fig. 7.3). Impurities reduce the thermal conductivity of pure metals.

Fig. 7.3. Typical values of thermal conductivity from thermal insulators to superconductors.

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DEFINITIONS AND SOURCES OF ENERGY

André Gardel , in Free energy: Economy and Prospective, 1981

d3 The instance of fossil fuels

Everybody knows that a fuel may be burned, and that this operation liberates rut, which is energy. The question is then to determine this primary energy, under possible hypotheses of product, with the assistance of perfect installations but of reasonable telescopic.

Then let us note that the fuel contains chemical energy and that hither information technology is merely considered in the context of an oxidation. The free energy freed may be liberated directly at room temperature in the grade of electricity, in a fuel cell (see Chapter 5, § e1), just this procedure is still costly and its widespread use would lead to installations of unreasonable size: we would exit the realm of current economic reality.

This energy is liberated past combustion in the grade of heat It is this grade simply that is considered in this affiliate.

This heat is apparently well known; however, information technology is only if the combustion weather are exactly defined (for example: combustion at atmospheric pressure, fuel and oxidant taken at room temperature and the combustion products later brought back to information technology).

The free energy liberated will heighten the temperature of the products of combustion and bring them to a temperature which will depend on the nature and temperature of the oxidant (air or oxygen) and on the presence or absence of other unnecessary substances (nitrogen, carbon dioxide, etc.) which would also exist heated up. This temperature volition also depend on the insulation of the furnace and on the presence or absence of unburnt residue. Now, the uses to be made of the oestrus volition depend on the temperature at which that heat is available. Thus, to decide the bachelor master free energy, nosotros must also fix these conditions.

Nosotros shall only consider combustion with air, admitted without excess and at room temperature, complete combustion (no unburnt residue) and in a completely insulated enclosure. Thus nosotros may obtain a flame temperature.

We thus postulate that the fuel equates to this heat of combustion delivered at this flame temperature.

Is this estrus the master energy to exist considered?

Firstly, we should observe that this heat does not exist as such: it is included in the internal free energy of the combustion gas, which has been raised to the flame temperature. It must be recalled that it is impossible to speak of an internal energy as being heat or work; it has been acquired by a estrus input and/or a work input and it volition allow heat and/or piece of work to exist liberated, but in the form of internal energy it is neither i nor the other.

This fact appears immediately if internal energy is examined on the particle scale . Information technology is essentially only the kinetic energy of the atoms or molecules forming the gas, temperature being a measure of the average kinetic energy of these particles and pressure being the force exerted past the particle collisions with the walls of the chamber. ¹ From the chemic energy liberated by the combustion (rearrangement of atoms in molecules), the hateful kinetic free energy of the particles has increased (the temperature has risen) and the full kinetic energy of the particles has increased (the internal free energy has increased). From this greater kinetic energy, heat may be taken past transmitting role of this internal energy to the particles of another body (by standoff of the gas particles with those of this body ² ); besides mechanical energy may be taken out by making the gas work, i.e. by displacing a wall subjected to pressure (when the particles hit the moving wall, office of their kinetic energy is transmitted to the wall).

We are thus led to ask what main energy we take available according to whether we plan to use the internal free energy to give off heat or do work. A priori,it is not obvious that these come up to the same thing, or that the rut or work are equal to the oestrus of combustion of the chemical reaction.

First, permit us indicate the temperature of the flame: this, for the combustion conditions already given, may be adamant exactly when the fuel is precisely known, as is the case for chemically pure substances (carbon, hydrogen, marsh gas, etc.); on the other hand, the temperature may vary if we are dealing with fuels the composition of which is not abiding (coal, petroleum, natural gas). The temperatures given in degrees Celsius are in fact the increase in temperature over the initial ambient conditions. Fixing the ambience temperature at 15° or 288 K, the absolute temperatures are ³ (to the nearest 10 One thousand):

pure fuel	flame temperature
	relative (°)	absolute (Thou)
carbon C	1970	2260
hydrogen Hii	2050	2340
methane CH4	2020	2310
ethane C2Hhalf dozen	1970	2260
propane C3Hviii	1980	2270
butane C4Hx	1980	2270
acetylene C2H2	2040	2330

For real fuels, the flame temperatures are piddling different. In effect, the fuels contain a mixture of pure fuels (for example sulphur in coal, diverse hydrocarbons in petroleum or natural gas) or inert bodies, such as silica or calcium carbonate in coal, nitrogen or carbon dioxide in natural gas. It is rare for the inert bodies to exceed 10% of the weight of the fuel; the combustion gases accept a mass which is a multiple of that of the fuel due to the contribution of the air (of which four-fifths is inert nitrogen). Consequently, the inert portion of the fuel only represents half-dozen to 8‰ of the mass of the gas (for x% of inert thing); heating it upwardly only decreases the flame temperature past some fifteen degrees. Similarly, a water content in coal just requires 7‰ of the heat of combustion for a loftier level of 10% water.

However, it may happen that the fuel limerick is less favourable (a lower proportion of energy-producing components). For instance, the natural gas from Groningen, containing 68% marsh gas and 24% of inert products (by mass) only reaches a relative flame temperature of 1915° or 2200 Yard (with air at 15°), some 110 K lower than that of methane. The gas from Lacq, containing 97.4% methane by volume, had a relative flame temperature of 1940 ° or 2230 K (air at xv°), whilst a heavy oil, with 85.seven% of carbon past weight, 11.seven% of hydrogen and 2.6% of sulphur had 1990° or 2280 1000.

We conclude that a flame temperature of 2200 K corresponds well for coal too as natural gas and that it appears adequate for petroleum products.

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STRUCTURE-ACOUSTIC INTERACTION, HIGH FREQUENCIES

A. Sestieri , in Encyclopedia of Vibration, 2001

Judge of the Response Parameters

The balance equations provide the values of the energies stored in each modal subsystem. From them the parameters of practical involvement can be determined.

For a vibrating structure, the time-average energy doubles the kinetic average free energy. In fact, information technology is:

$E=T+U=2\cdot \frac{\mathrm{ane}}{\mathrm{ii}}M<{\upsilon }^2>$

where M is the total mass of the system and $<{\upsilon }^2>$ the fourth dimension- and space-averaged mean-square velocity of the vibrating organization. Then, ane can write:

$<{\upsilon }^2>=\frac{E}{\mathrm{One\; thousand}}$

and so that the mean-foursquare velocity of the vibrating structure is determined.

For an acoustic cavity, the quantity of interest is the mean-foursquare pressure, averaged in time and space. The energy stored in the acoustic cavity E tin be linked to the free energy density: $D=<p^{\mathrm{ii}}>/\rho c^{\mathrm{two}}$ , where ρ is air density and c the speed of sound in air. Therefore one has:

$\mathrm{East}=\mathit{DV}=\frac{<p^2>}{\rho c^2}\Rightarrow <p^2>=\frac{E}{V}\rho c^2$

Note that the solution of any SEA problem is a unique field value for the whole modal subsystem, so that whatsoever local variable distribution along the system is lost.

In Figure 2 the measured third-octave band average audio transmitted from an engine into an automotive compartment is compared with the sound pressure level determined by Ocean, together with ii curves that would represent the values ±twoσ (σ standard difference) of the Body of water judge.

Figure 2. Sound pressure level (SPL) in an automotive compartment. Continuous line, Sea results; circles, measured results.

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Measurement Of Average Kinetic Energy,

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