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In existographies, Erwin Schrödinger (1887-1961) (IQ:190|#26) [RGM:424|1,500+] (Becker 160:44) (GPE:7) [CR:224] was an Austrian physicist, noted for his 1925 Schrodinger equation, a Lagrangian-based state equation for the wave movement of an electron about an atom, for his 1935 book Science and the Human Temperament, wherein he stated his opinion that the rise of cultures is governed by the second law of thermodynamics, for his ultra-famous 1943 "What is Life?, wherein he attempted, in laymanized terms, to explain "life" in terms of physics and chemistry, a lecture-turned-book, wherein he postulated that “life feeds on negative entropy”, and followup retraction "Note to Chapter 6", wherein he had to admit to error, commenting that he he should have discussed life in terms of "free energy" in stead of entropy, in regards to the theory of life; also noted for his 1946 Statistical Mechanics book. [1] Schrodinger won the 1933 Nobel Prize for his development of the Schrodinger equation.

Influences
Schrodinger was strong influenced by Arthur Schopenhauer.

Do Electrons Think?
In 1949, Schrodinger, in his “Do Electrons Think?” BBC talk, stated the following: [9]

“Twice within two and a half thousand years there was a notable departure from the belief in the strict necessity of physical events. The first time this happened only about 150 years after Leucippus. We gather this from the didactic poem of Lucretius, who was the posthumous mouth-piece of Epicurus, who lived in the fourth century B.C. Their suggestion had no consequences, it was all but forgotten. The second time the strictly causal linkage in the chain of physical happenings was thrown into doubt was only 30 years ago by Franz Exner in Vienna. About ten years later (that is, about twenty years ago, from today [1949]) the disbelief in strict causation became part and parcel of you might call the 'new creed' now adopted by most physicists and called 'quantum mechanics'.

Both times, the alleged break-down of strict causality in the domain of physics was hailed for removing the obstacle in our understanding the spontaneity of the movements of the animals and of man—in understanding free will, as one usually calls it. Let us see whether this claim is justified.

The hypothesis, as reported by Lucretius, was very simple indeed. He just states that the atoms do swerve in a very small but entirely undetermined and unforeseeable way from the courses you would expect them to take from supposed strict physical laws. No theory of the swerves is offered. This amounts to saying that the strict laws are only figments. The actual path of a particle is to a small extent arbitrary in the neighborhood of the fictitiously prescribed path. It is not illogical to surmise that the several little arbitrarinesses of the single atoms collaborate to bring about the apparent arbitrariness in the behavior of the animals and of man. What Lucretius forgets is that he has explained nothing, he has solved no problem. He has only referred the problem back to the ultimate particles, where it has become much more difficult to grapple with. The simplest, spontaneous bodily movement, say lifting of my arm, would require the planned collaboration of billions of single atoms in their undetermined swerves, if they should bring about the integrated action.

In modern physics, the denial of strict causation is of entirely different nature, in two respects. First, there is no question of only small departures from a fictitious exact law of motion. The behavior of primal particles, as electrons for instance, or of small atomic systems composed of only a few of them, is now supposed to be undetermined and unforeseeable within a wide margin of uncertainty. It is thought that in times we have to allow a particle the choice between several entirely different courses to take. Let this for the moment be figurative speech, meaning only that nothing in the observed situation determines the course the particle actually takes. But on the other hand the situation is supposed to determine with rigorous precision the statistics of the various possible “choices.” Given the same situation over and over again, the particle will, for instance, in exactly two thirds of the cases follow one course, in one third of cases the other one; and similarly when there are more than two courses to follow.

Again, the same as 2,000 years ago, it has been suggested that this breach of strict causation leaves room for the display of the spontaneous movements in the animals and in man. Is this claim justified? I think not.”

Here, of note, we recall Gilbert Lewis' 1925 The Anatomy of Science, wherein he discusses the "
behavior of a group of electrons and the behavior of a university faculty":

Human thermodynamics
In 1935, in the context of human thermodynamics, Schrodinger, in his Science and the Human Temperament, stated the following in regards to the second law and rise of human cultures, as cited by Leslie White (1959): [3]

“We are convinced that [the second law of thermodynamics] governs all physical and chemical processes, even if they result in the most intricate and tangled phenomena, such as organic life, the genesis of a complicate world of organisms from primitive beginnings, [and] the rise and growth of human cultures.”

Life feeds on negative entropy
In popular or colloquial use, Schrodinger's cryptic "life feeds on negative entropy" postulate, might be considered as one of the most oft-quoted passages culled from the publications of thermodynamics. In extrapolation to human life, this would imply that people, in some way, feed on negative entropy (or a negative value of entropy S).
In linguistic form, Schrödinger’s postulate is similar to Austrian physicist Ludwig Boltzmann’s 1886 postulate that “the general struggle for existence of animate beings is … a struggle for entropy”. [2]

Negative entropy and order
In terms of entropy and order, in somewhat riddled form, Schrödinger reasoned that living organisms feed on negative entropy. To begin, he states that he will “try to sketch the bearing of the entropy principle (the second law of thermodynamics) on the large-scale behavior of a living organism”. Second, he equates thermodynamic equilibrium, or what he calls “maximum entropy”, as a state in which chemical potentials are equalized, wherein systems become dead, in which no observable changes occur. To avoid decay to this hypothetical death state, Schrödinger reasons that it is not energy that living beings feed on that keeps them at bay from decay but “negative entropy”. In rephrasing this statement, he says “the essential thing in metabolism is that the organism succeeds in freeing itself from all the entropy it cannot help producing while alive.” In making these ball-park statements, Schrödinger calls on the statistical concept of order and disorder, connections that were revealed, as he says, by the investigations of Boltzmann and Gibbs in statistical physics. On this basis, he situates the following definition:

$\text{entropy} = k \log D \,$

where k is the Boltzmann constant and D is a “quantitative measure of the atomistic disorder of the body in question”. Here, to note, he fails to mention that this expression is generally valid only for ideal gases. In any event, Schrödinger reasons that this statistical expression applies to living organisms. Moreover, to make his verbal argument mathematical, he states that “if D is a measure of disorder, its reciprocal, 1/D, can be regarded as a direct measure of order.” In addition, “since the logarithm of 1/D is just the minus of the logarithm of D, we can write can write Boltzmann’s equation thus:

$-\text{entropy} = k \log \frac{1}{D} \,$

Hence, as Schrödinger states, “the awkward expression negative entropy can be replaced by a better one: entropy, taken with the negative sign, is itself a measure of order.” Thus, he concludes “the device by which an organism maintains itself stationary at a fairly high level of orderliness”, a state he equates with a low level of entropy, consists in “sucking orderliness from its environment”.

Difficulties on theory
See main: Note to Chapter 6
The basic difficulty in Schrödinger’s negative entropy theory is that he equates sustenance (metabolism) with measures of entropy. In the correct sense, sustenance is a function of substrate interactions, as studied in the field of surface chemistry. In an appended note to his thermodynamics-life chapter, however, Schrödinger states that:

“The remarks on negative entropy have met with doubt and opposition from physicist colleagues. Let me say first, that if I had been law catering for them alone I should have let the discussion turn on free energy instead. It is the more familiar notion in this context. But this highly technical term seemed linguistically too near to energy for making the average reader alive to the contrast between the two things.”

In other words, in terms of entropy as defined by Clausius, as the “amount of work the molecules of the system do on each other" (that cannot be compensated), this outline by Schrödinger makes little sense. If he had let the discussion turn on free energy, he would have been confronted with the various postulates of energy interactions between people, such as was done by Goethe in his 1809 Elective Affinities (see: Goethe's human chemistry); where before 1882, free energy was called "chemical affinity" or elective affinity.

In 1987 commentary on Schrödinger’s What is Life?, American chemical engineer Linus Pauling noted that Schrödinger was discussing a change in the entropy of the "system", he never defined the system. Pauling wrote, "Sometimes he seems to consider that the system is a living organism with no interaction whatever with the environment; sometimes it is a living organism in thermal equilibrium with the environment; and sometimes it is the living organism plus the environment, that is, the universe as a whole." Pauling wrote that Schrödinger failed to recognize the most important question: "How biological specificity is achieved; that is, how the amino-acid residues are ordered into the well-defined sequence characteristic of the specific organism." [4] Similarly, Austrian-born English molecular biologist Max Perutz argued that we live on free energy and that there was no necessity to postulate negative entropy. [5]

God | Atheism
Schrodinger was a Schopenhauer-rooted silent atheist, who similar to Einstein employed culturally loaded "oh my good lord", similar to the way many now say "OMG" stylized talk, in his speech, and sometimes in his scientific publications; the following is one example:

“When you feel your own equal in the body of a beautiful woman, just as ready to forget the world for you as you for her – oh my good lord – who can describe what happiness then. You can live it, now and again – you cannot speak of it.”
— Erwin Schrodinger (c.1940), Publications [10]

In 1944, Schrodinger, in his What is Life?, made a blurry comment about the “lord’s quantum mechanics”, when discussing chromosomes, clocks, and organisms. The following are some of his other religious-tinged comments:

“Whence came I, whither go I? Science cannot tell us a word about why music delights us, of why and how an old song can move us to tears. Science is reticent too when it is a question of the great Unity – the 'one' of Parmenides – of which we all somehow form part, to which we belong. The most popular name for it in our time is god – with a capital ‘G’. Whence come I and whither go I? That is the great unfathomable question, the same for every one of us. Science has no answer to it.”

In 1954, in his Nature and the Greeks, he states:

“I am very astonished that the scientific picture of the real world around me is deficient. It gives a lot of factual information, puts all our experience in a magnificently consistent order, but it is ghastly silent about all and sundry that is really near to our heart, that really matters to us. It cannot tell us a word about red and blue, bitter and sweet, physical pain and physical delight; it knows nothing of beautiful and ugly, good or bad, god and eternity. Science sometimes pretends to answer questions in these domains, but the answers are very often so silly that we are not inclined to take them seriously. In particular, and most importantly, this is the reason why the scientific worldview contains of itself no ethical values, no esthetical values, not a word about our own ultimate scope or destination, and no god, if you please. Whence came I and whither go I?”

Other comments in his collected quotes fall along these lines. [8] One example of which is: [7]

“I shall quite briefly mention here the notorious atheism of science. The theists reproach it for this again and again. Unjustly. A personal god can not be encountered in a world picture that becomes accessible only at the price that everything personal is excluded from it. We know that whenever god is experienced, it is an experience exactly as real as a direct sense impression, as real as one’s own personality. As such he must be missing from the space-time picture. ‘I do not meet with god in space and time’, so says the honest scientific thinker, and for that reason he is reproached by those in whose catechism it is nevertheless stated: ‘god is spirit’.”

This “we do not find god in spacetime”, supposedly, is reference to Schopenhauer who said on one of his works: "The world extended in space and time is but our representation." (Ѻ)(Ѻ) In other places, Schrodinger explained that he was a silent atheist, because he did not know that atheism was a thing to be proud of:

“I never realized that to be nonbelieving, to be an atheist, was a thing to be proud of. It went without saying as it were.”
— Erwin Schrodinger (c.1940), Publications [10]

“Our creed is indeed a queer creed. You others, Christians (and similar people), consider our ethics much inferior, indeed abominable. There is that little difference. We adhere to ours in practice, you don't.”
— Erwin Schrodinger (c.1940), Publications [10]

Here, we can think of Daniel Scargill, who famously said: “there is a desirable glory in being and being reputed an atheist”.

References
1. (a) Schrödinger, Erwin. (1944). What is Life? (ch. 6 “Order, Disorder, and Entropy", pgs. 67-75). Cambridge: Cambridge University Press.
(c) Schrödinger’s scripture "What is Life?" was based on a course of public lectures delivered under the auspices of the Dublin Institute for Advanced Studies at Trinity College, Dublin, in February 1943 and published in 1944.
2. Thims, Libb. (2007). Human Chemistry (Volume One) (preview), pg. 87. Morrisville, NC: LuLu.
3. (a) Schrodinger, Erwin. (1935). Science and the Human Temperament (thermodynamics, 4+ pgs; quote, 47). George Allen & Unwin, Ltd.
(b) White, Leslie A. (1959). “Energy and Tools”, in: The Evolution of Culture: the Development of Civilization and the Rise and Fall of Rome (pg. 39), McGraw-Hill; in: Readings for a History of Anthropological Theory (editors: Paul A. Erickson and P.A.E. Liam D. Murphy) (§23, pgs. 293-310; quote, pg. 297). University of Toronto Press, 2010.
4. Pauling, Linus. (1987). "Schrödinger's contribution to chemistry and biology", pp. 225–233 in Schrödinger: Centenary Celebration of a Polymath, edited by C. W. Kilmister. Cambridge University Press, Cambridge.
5. Perutz, Max. (1987). “Erwin Schrödinger's What Is Life? and molecular biology”, pp. 234–251 in Schrödinger: Centenary Celebration of a Polymath, edited by C. W. Kilmister. Cambridge University Press, Cambridge.
6. (a) Ubbelohde, Alfred René. (1954). Man and Energy: Illustrated (pg. 184). Hutchinson's Scientific & Technical Publications.
(b) Schrödinger, Erwin. (1944). What is Life? (pg. 85). Cambridge University Press.


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The effect of turbulence on mass and heat transfer rates of small inertial particles

# The effect of turbulence on mass and heat transfer rates of small inertial particles

Nils Erland L. Haugen, Jonas Krüger, Dhrubaditya Mitra and Terese Løvås
###### Abstract

The effect of turbulence on the mass and heat transfer between small heavy inertial particles (HIP) and an embedding fluid is studied. Two effects are identified. The first effect is due to the relative velocity between the fluid and the particles, and a model for the relative velocity is presented. The second effect is due to the clustering of particles, where the mass transfer rate is inhibited due to the rapid depletion of the consumed species inside the dense particle clusters. This last effect is relevant for large Damköhler numbers and it may totally control the mass transfer rate for Damköhler numbers larger than unity. A model that describes how this effect should be incorporated into existing particle simulation tools is presented.

Key words: Reacting multiphase flow, Particle/fluid flow, Combustion, Turbulent reacting flows, Turbulence simulations

## 1 Introduction

Both in nature and in industrial applications, one regularly finds small inertial particles embedded in turbulent flows. By small inertial particles, we mean particles that are smaller than the smallest scales of the turbulence and have significantly higher material density than the fluid. For such particles, there will be momentum exchange between the particles and the turbulent fluid, and, depending on the conditions, there may also be heat and mass transfer. This is particularly so for chemically reacting particles, but there are also a large number of other applications where heat and mass transfer between particles and fluid are important. Here, the main focus will be on reacting particles that consume one or more of the species in the embedding gas through surface reactions. Relevant examples are; chemical reactions on the surface of a catalytic particle, fuel oxidation on the surface of a oxygen carrying particle in a Chemical Looping Combustion (CLC) reactor, condensation of water vapor on cloud droplets and combustion or gasification of char.

The presence of turbulence in a fluid will enhance the transport properties of the flow. This means that the mean-field viscosity, diffusivity and conductivity may be drastically increased from their laminar values. This effect has been studied for many years, and a large number of different models exist in the literature, such as the k- model (Jones & Launder (1972)) and different versions of the Reynolds Stress Models (e.g. Pope (2003)). Turbulence may also modify gas phase combustion, and even though this is somewhat more complicated, a significant number of models have been developed during the last decades. Some examples are the Eddy Dissipation Model (Magnussen & Hjertager (1976)), the Eddy Dissipation Concept (Ertesvåg & Magnussen (2000)) and variations of Probability Density Function (e.g. Dopazo (1994)) models.

With the above knowledge in mind, it is interesting to realize that, except for the recent work of Kruger et al. (2016), there is currently no model describing the effect of turbulence on the heat and mass transfer of small inertial particles. When a reacting particle is embedded in a turbulent flow, the turbulence can potentially influence the mass transfer, and hence the surface reaction rates in two ways. The first way is through particle clustering, where particles form dense clusters due to turbulence, and where the gas phase reactants within the cluster are quickly consumed while there are no particles that can consume the reactants in the particle voids outside the clusters. The main effect of the clustering is to decrease the overall mass transfer rate. The second way turbulence influence the mass transfer rate is by increasing the mean velocity difference between the particle and the gas. This effect will increase the mass transfer rate.

The same two effects are also active for the heat transfer. The similarity between heat and mass transfer can be seen by considering the expressions for the transfer coefficients of mass

 κ=ShDdp (1.0)

and heat

 κth=NuDthdp, (1.0)

where is the particle radius, Sh and are the Sherwood and Nusselt numbers and and are the mass and thermal diffusivities. For single spherical particles in flows with low and medium particle Reynolds numbers, the Sherwood and Nusselt numbers can be approximated by the empirical expressions of Ranz & Marshall (1952)

 ShRM = 2+0.69Re1/2pSc1/3 (1.0) NuRM = 2+0.69Re1/2p1/3Pr.

A well known example where reacting particles are consumed in a turbulent fluid is the case of pulverized coal combustion, where turbulence influences the process in several ways that are understood to varying degrees. The combustion of coal can be divided into four separate processes; 1) drying, 2) devolatilization, 3) combustion of volatiles and 4) burnout of the remaining char. Processes 1 and 2 involve the evaporation of fluids and thermal cracking of hydrocarbons, while process 3 involves homogeneous reactions. In process 4, gas phase species diffuse to the particle surface and react with the solid carbon. This happens via adsorption of e.g. an oxygen radical to a carbon site on the particle surface and a subsequent desorption of carbon monoxide into the gas phase. This makes process 4 dominated by heterogeneous chemical reactions. Many published studies utilize RANS based simulation tools that describe simulations of pulverized coal conversion in the form of combustion or gasification with an Eulerian-Eulerian approach (Gao et al. (2004) and Zhang et al. (2005)) or a Lagrangian-Eulerian approach (Silaen & Wang (2010); Vascellari et al. (2014, 2015); Klimanek et al. (2015); Chen et al. (2012, 2000)). However, none of these papers take the effect of turbulence on the heterogeneous char reactions into account. To the knowledge of the authors, the only studies where account is made for this effect are the papers of Luo et al. (2012); Brosh & Chakraborty (2014); Brosh et al. (2015) and Hara et al. (2015) where the Direct Numerical Simulations (DNS) approach is used. In a DNS, all turbulence scales are explicitly resolved on the computational grid, such that the effect of turbulence is implicitly accounted for. However, the DNS approach is extremely costly and can therefore only be used for small simulation domains. For simulations of large scale applications, the RANS or LES based simulation tools will therefore be the only applicable tools for the foreseeable future.

In the current paper, the same framework as was developed by Kruger et al. (2016) has been used and extended. The aim of the paper is to identify the effect of turbulence on the mass and heat transfer of solid particles, and to develop models that describe this effect for all Damköhler numbers.

## 2 Mathematical model and implementation

In the current work, the so called point-particle direct numerical simulation (PP-DNS) approach is used. Here, the turbulent fluid itself is solved with the direct numerical simulation (DNS) methodology, where all turbulent scales are resolved and no modelling is needed. The particles are however not resolved, but rather treated as point particles where the fluid-particle momentum, mass and heat interactions are modelled. The point particle approach is a simplification that relies heavily on the quality of the models. The alternative approach, which is to resolve the particles and their boundary layer, is extremely CPU intensive and can currently not be done for more than a few hundred particles, even on the largest computers (Deen & Kuipers (2014)).

A number of simplifications are made in this paper. This has been done in order to make the simulations less CPU intensive, and, even more importantly, to isolate the dominating physical mechanisms. The particles are considered to be ever lasting, i.e. they are not consumed. The reaction on the particle surface is converting reactant A to product B;

 A→B (2.0)

isothermally, i.e.; there is no production or consumption of heat, such that only the mass transfer effect is considered. As explained above, the effect on the heat transfer rate will be similar to the effect on the mass transfer rate. As reactant A is converted product B, the thermodynamical and transport properties are not changed.

### 2.1 Fluid equations

The equations determining the motion of the carrier fluid is give by the continuity equation

 ∂ρ∂t+∇⋅(ρ\boldmathu)=0, (2.0)

and the Navier–Stokes equation

 ρD\boldmathuDt=−∇P+∇⋅(2μ\boldmathS)+ρf+\boldmathF. (2.0)

Here, , , and are the density, velocity and dynamic and kinematic viscosities of the carrier fluid, respectively. The pressure and the density are related by the isothermal sound speed , i.e.,

 P=c2sρ, (2.0)

while the trace-less rate of strain tensor is given by

 \boldmathS=12(∇\boldmathu+(∇\boldmathu)T)−13∇⋅\boldmathu. (2.0)

Kinetic energy is injected into the simulation box through the forcing function , which is solenoidal and non-helical and injects energy and momentum perpendicular to a random wave vector whose direction changes every time-step (Haugen et al. 2012; Kruger et al. 2016). Similar kinds of forcing has also previously been used for particle laden flows by other groups (Bec et al. 2007). The energy injection rate is maintained at a level such that the maximum Mach number is always below 0.5. The domain is cubic with periodic boundaries in all directions. The momentum exchange term, , is chosen to conserve momentum between the fluid and the solid particles, i.e.,

 \boldmathF=−1Vcell∑kmk% \boldmathak (2.0)

when is the volume of the grid cell of interest and and are the mass and acceleration (due to fluid drag) of the k’th particle within the grid cell.

The equation of motion of the reactant has the well-known advection-reaction-diffusion form:

 ∂X∂t+∇⋅(X\boldmathu)=D¯Mc∇⋅(∇X)+~R\/, (2.0)

where , and are the mole fraction, the mean molar mass and the diffusivity of the reactant, respectively. The last term in Eq. (2.0), , is the sink term due to the gas-solid reactions on the surface of the solid particles.

### 2.2 Particle equations

The particles that are embedded in the flow are treated as point particles, which means that they are assumed to be significantly smaller than the viscous scale of the fluid and the diffusive scale of the reactant. The motion of the k’th particle is described by the equations for position

 dXkdt=Vk (2.0)

and velocity

 dVkdt=\boldmathak (2.0)

when the particle acceleration due to fluid drag is given by . Note that gravity is neglected in this work. The particle response time is given by (Schiller & Naumann (1933))

 τ=τSt1+fc (2.0)

when is the Stokes time, is a Reynolds number correction term to the classical Stokes time, is the density ratio, is the material density of the particles,

 Rep=|\boldmathu(Xk)−Vk|dpν=ureldpν (2.0)

is the particle Reynolds number and is the particle diameter.

### 2.3 Surface reactions

Let us now model the reactive term. We assume that the reactions are limited to the surface of the particles and that the reactions are diffusion controlled, i.e. that all reactant that reaches the particle surface is consumed immediatelyIt is possible to relax the assumption of diffusion controlled reactions by also accounting for chemical kinetics at the particle surface, see Kruger et al. (2016).. The reactive term can then be written as

 ~R=1Vcell∑kAkpκXK∞ (2.0)

where is the external surface area of the particle, the mass transfer coefficient is given by

 κ=DShdp (2.0)

and Sh is the Sherwood number.

To couple the reactive particle with the continuum equations we use the following prescription; for the -th particle, which is at position , we set

 Xk∞=X(Xk), (2.0)

i.e.; the far field reactant mole fraction is set equal to the reactant mole fraction of the fluid cell where the particle is. In the current work, the particle Sherwood number is determined by the expression of Ranz & Marshall (1952) (see Eq. (1) in the introduction), which is in contrast to the work of Kruger et al. (2016) where the Sherwood number was set to a constant value of 2, which corresponds to the Sherwood number in a quiescent flow. The particle Reynolds number is given by Eq. (2.0) and the Schmidt number, , is the ratio of the fluid viscosity and the mass diffusivity.

### 2.4 The reactant consumption rate

It is useful to define a reactant consumption rate as

 α=−¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(~RX∞)=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯npApκ, (2.0)

when represents the volume average of flow property and is the particle number density. If everything is assumed to be homogeneously distributed over the volume, the reactant consumption rate is given by

 αhom=npApκ=npApShDdp (2.0)

for a given particle size and number density.

In many RANS based simulation tools, where the local fluid velocity is not resolved, it is common to neglect the relative velocity difference between the turbulent eddies and the particles. This implies that . Since the effect of particle clustering is also neglected in such models, the modelled reactant consumption rate becomes;

 αSh,Da=limSh→2,Da→0α=npAp2Ddp. (2.0)

In the following, will be used for normalization.

It is useful to define the Damköhler number, which is the ratio of the typical turbulent and chemical time scales, as

 Da=τLτc (2.0)

where is the integral time scale of the turbulence, is the turbulent forcing scale, is the root-mean-square turbulent velocity and the chemical time scale is

 τc=1/αSh,Da. (2.0)

Particles in a turbulent flow field will tend to form clusters with higher particle number density than the average (Squires & Eaton 1991; Eaton & Fessler 1994; Toschi & Bodenschatz 2009; Wood et al. 2005). If the chemical time scale is short compared to the life-time of the clusters, the reactant concentration within the clusters will be much lower than outside the clusters. On the other hand, if the particle number density is low, the particle clusters will not have enough time to consume a significant fraction of the reactant during the life-time of the cluster, and hence, the reactant concentration will be roughly the same inside as it is outside the clusters. By assuming that the life-time of the clusters is of the order of the turbulent time scale, it is clear that the reactant concentration of particle flows with low Damköhler number will behave as if the particles were homogeneously distributed over the volume, i.e.; for small Damköhler numbers there is no effect of particle clustering on the reactant consumption.

From Eqs. (2.0) - (2.0) it can be deduced that for the homogeneous case, and then also for all cases with low Damköhler numbers, the reactant consumption rate will scale linearly with the Damköhler number for a given turbulent flow field, such that

 αhom=DaτLSh2. (2.0)

When relaxing the restriction to small Damköhler numbers, the effect of particle clustering eventually comes into play. Kruger et al. (2016) have shown that the reactant consumption rate is given by

 α=αcαhomαc+αhom (2.0)

when is a cluster dependent decay rate. (Note that since Kruger et al. assumed the Sherwood number to be 2, their equals our .) From this expression, the following normalized reactant consumption rate is found

 ~αSh=αShαSh,Da=αcτLαcτL+DaSh/2Sh2. (2.0)

when Sh is given by Eq. (1) and the corresponding relative velocity between the particle and the fluid is determined by a model (which will be obtained in the next subsection). For diffusion controlled reactions, the modified reaction decay rate, as given by Eq. (2.0), is a measure of the relative modification to the mass transfer rate due to the effect of turbulence. This means that a modified Sherwood number can now be defined that accounts for the effect of turbulence;

 Shmod=2~α. (2.0)

In the limit of small Damköhler numbers, this expression reduces to , as expected.

By employing the modified Sherwood number given by Eq. (2.0), one can now use the common expression for the reactant consumption rate, as given by Eq. (2.0), to find the real reactant consumption rate. In most cases, however, one needs the particle conversion rate for individual particles, which is closely connected to the reactant decay rate. For diffusion controlled mass transfer, the particle conversion rate is given by , where is the molar concentration of the gas phase and the mass transfer coefficient is now found by using the modified Sherwood number (as given by Eq. (2.0)) in; Eq. (1)

 κ=DShmoddp. (2.0)

In many applications, the mass transfer rate is not purely diffusion controlled. This can be accounted for by including the effect of reaction kinetics at the particle surface. The corresponding particle conversion rate can then be expressed as (Kruger et al. 2016)

 ˙nreac=−λκλ+κX∞Cg, (2.0)

where is the surface specific molar conversion rate. Since the reaction kinetics is only dependent on the conditions at the particle surface, the surface specific molar conversion rate is not affected by the turbulence. This is, as we have already seen, not the case for the mass transfer coefficient, which is now given by Eq. (2.0). In this way, all the common machinery for calculating particle reaction rates can still be used since the effects of the turbulence are incorporated into the modified Sherwood number.

## 3 Results

In all of the following, statistically stationary homogeneous and isotropic turbulence is considered. The Reynolds number is varied by changing the domain size while maintaining constant viscosity and turbulent intensity. The Damköhler number is varied by changing the number density of particles, while keeping everything else the same. All relevant simulations are listed in table 1.

### 3.1 The mean relative particle velocity

In order to predict a representative value of the particle Sherwood number from Eq. (1), the particle Reynolds number is required. From Eq. (2.0) it is clear that this also requires the relative particle velocity , which will be found in this subsection.

Given a particle with a response time that equals the Stokes time;

 τp=Sd2p18ν, (3.0)

such that , where is the Kolmogorov time scale and is the integral time scale. With respect to the particle-turbulence interactions, the turbulent power spectrum may be divided into three distinct regimes, based on the relation between the particle response time and the turbulent eddy turnover time . The first regime is defined as the section of the turbulent power spectrum where the turbulent eddies have turnover times that are much larger than the response time of the particles, i.e. where . All the turbulent eddies in this regime will see the particles as passive tracers, which follow the fluid perfectly. I.e., there will be no relative velocity between the particles and the eddies. The third regime is defined as the part of the power spectrum where the turbulent eddies have much shorter time scales than the particles, i.e. where . The eddies in regime three will see the particles as heavy bullets that move in straight lines, without being affected by the motion of the eddies. Hence, the velocity of these eddies will contribute to the relative particle-fluid velocity. The second regime is now defined as the relatively thin band in-between regimes one and three, where . These are the eddies that are responsible for particle clustering, since they are able to accelerate the particles to a level where they are thrown out of the eddy due to their inertia. In the following, we will refer to a typical eddy in regime two as a resonant eddy, and we define the scale of this eddy as . The resonant eddies are identified by their time scale, , which is of the order of the particle response time, . For convenience, we set the two time scales equal, such that

 τℓ=τp. (3.0)

Based on the definitions above, it is clear that the largest turbulent eddies that yield a relative velocity between the fluid and the particles, are the resonant eddies. By assuming Kolmogorov scaling, the velocity of the resonant eddies is known to be , which can be combined with the above expression for the time scales to yield

 kℓ=kLSt−3/2 (3.0)

when the particle Stokes number is defined as

 St=τpτL (3.0)

and and are the wave-numbers of the resonant eddies and the integral scale, respectively. In obtaining Eq. (3.0), it has also been used that the turnover time of the resonant eddies is , while that of the integral scale eddies is .

Since all scales smaller than will induce a relative velocity between the particles and the fluid, it is reasonable to assume that the relative velocity between the fluid and the particles will be a certain fraction of the integrated turbulent velocity of all scales smaller than , such that

 urel=β~uℓ (3.0)

when is defined as

 12~u2ℓ=∫kηkℓE(k)dk (3.0)

and is the wave-number of the Kolmogorov scale (), where is the dissipation rate of turbulent kinetic energy. Integration of Eq. (3.0) yields

 ~uℓ=urms  ⎷k−2/3ℓ−k−2/3ηk−2/3L−k−2/3η (3.0)

for when it has been used that the total turbulent kinetic energy is given by

 12u2rms=∫kηk1E(k)dk, (3.0)

where is the wavenumber of the largest scale in the simulation. Combining Eqs. (3.0) and (3.0) with Eq. (3.0) finally yields

 urel=βurms  ⎷Stk−2/3L−k−2/3ηk−2/3L−k−2/3η. (3.0)

The unknown constant in this equation, , can be determined numerically from Eq. (3.0), i.e. . Here, is found directly from DNS simulations, while is calculated from Eq. (3.0). It is seen from figure 1 that is close to 0.41 for most Stokes and Reynolds numbers. The main exception is for low Reynolds and Stokes numbers, where is significantly larger. This can be understood by inspecting the left panel of figure 2, where it is seen that for and , we are already far into the dissipative subrange, where our model is not expected to be correct since it relies on a Kolmogorov scaling.

It is surprising to see that Eq. (3.0) reproduces the relative particle velocity for such low Stokes numbers, even for the smaller Reynolds numbers. This may be explained by reconsidering Eq. (3.0), where we assumed that the resonant eddies correspond to the eddies that have exactly the same turnover time as the response time of the particles. This is just an order of magnitude estimate, and a more correct expression would probably be

 τℓ=γτp, (3.0)

where is of the order of unity. More work should, however, be devoted to understanding the coupling between the particles and the turbulent eddies. In particular, a more exact definition of the resonant eddies is needed. We nevertheless believe that is a universal property of the HIP approximation and the Navier-Stokes equations that will have a constant value for all and St as long as the resonant eddies are within the inertial range.

In the right panel of figure 2, the average relative particle velocity, as found from the DNS simulations (symbols), is compared with the predicted values from Eq. (3.0) (solid lines). It is seen that the fit is rather good for most Reynolds and Stokes numbers. This supports the use of Eq. (3.0) for predicting the relative particle velocity.

### 3.2 The cluster size

The typical size of the clusters is assumed to be the size of the resonant eddies. From Eq. (3.0) this yields a cluster size of

 l=LSt3/2. (3.0)

It can be seen from figure 3 that the particle number density distribution does indeed show more small scale variation for the smaller Stokes numbers. This has been quantified in figure 4 where the power spectrum of the particle number density is shown. Here we see that the spectrum peaks at large scales for while the peak is located at much smaller scales for smaller Stokes numbers. The peak in the spectrum does not, however, follow Eq. (3.0) as accurately as expected. The reason for this is most likely that power spectra are not the right diagnostics to study the size of particle clusters, but it may also be partly because of: 1) poor statistics due to too few particles (the smaller clusters are not filled with particles), 2) the constant in the definition of the resonant eddies not being unity (see e.g. Eq. (3.0)), or 3) finite Reynolds number effects.

The power spectrum can be integrated to yield a measure of the strength in the particle number density fluctuations, given by the root-mean-square (rms) particle number density;

 nrms=∫Pdk. (3.0)

It is found that the rms particle number density is decreasing with Stokes number. More specifically, is 1.6, 1.5, 1.2 and 0.8 for Stokes numbers of 1, 0.3, 0.1 and 0.03, respectively. This means that the high density regimes have higher particle number densities for larger Stokes numbers.

### 3.3 Reactant consumption rate

The normalized reactant consumption rate is shown in figure 5. The symbols correspond to the results from the DNS simulations, while the solid lines are given by Eq. (2.0). Here, the Stokes number is found by using the model for the relative velocity, as given by Eq. (3.0), in the expression for the Sherwood number (Eq. (1)). The value of the cluster decay rate, , is the only free parameter and it is chosen by a best fit approach. The values of are found in table 1.

The value of for small Damköhler numbers equals the Sherwood number divided by two, while the Damköhler number for which starts to decrease is determined by the cluster decay rate . Overall, the model seems to follow the results from the DNS simulations rather well.

From figure 5 it can be seen that for large Stokes numbers, the curves for the normalized decay rates of a given Stokes number overlap for different Reynolds numbers if the Reynolds number is high enough. This is because the resonant eddies are at scales larger than the dissipative subrange. So increasing the Reynolds number, which may be considered a shift of the dissipative subrange to smaller scales, is not affecting the resonant eddies, and hence also the clustering is unaffected. If, however, the Reynolds or the Stokes number is small, such that the resonant eddies are in the dissipative subrange, a change in Reynolds number will have an effect on the normalized decay rate ().

Figure 6 shows that by decreasing the Stokes number, the normalized reactant decay rate stays unchanged up to larger Damköhler numbers. This means that the effect of particle clustering is weaker for smaller Stokes numbers. This is expected since the limit of very small clusters corresponds to individual particles, where is independent of Da. From the simulations with small Damköhler numbers and , which are shown in the right panel of figure 6, it can be observed that the normalized decay rate is monotonically decreasing with Stokes number. The reason for this is that for these simulations the particle size is kept constant as the Stokes number is changed, such that the Sherwood number is decreased with decreasing Stokes number. This is, however, not the case for the simulations with , where it is found that the normalized decay rate for small Damköhler numbers is lower for than for . The reason for this is that a smaller particle radius was used for the simulations with . The effect of reducing the particle radius is that the particle Reynolds number, and hence also the Sherwood number, is decreased.

### 3.4 The cluster decay rate

If the chemical time scale is much shorter than the lifetime of the particle clusters, the interior of the clusters will quickly be void of reactants. This means that the reactant consumption rate is controlled by the flux of reactant to the surface of the clusters, where the reactant will be consumed at the exterior of the cluster. Based on this, it is clear that for large Da (small ), the clusters behave as large solid particles, or super-particles. Following Kruger et al. (2016), the reactant decay rate is then given by the so called cluster decay rate;

 αc=ncκcAc (3.0)

when is the number density of clusters (or super-particles), is the reactant diffusion rate to the super-particles, is the surface area of the clusters, is the turbulent diffusivity that carries the reactant from the surrounding fluid to the surface of the clusters and and are constants that depend on the dimensionality of the clusters. It is clear that turbulent eddies larger than , as given by Eq. (3.0), can not participate in the turbulent transport of reactants to the clusters, while eddies slightly smaller than will participate. A first approximation of the turbulent diffusivity to the surface of the clusters is therefore given by

 Dt=uℓl=uLLSt2. (3.0)

By combining the above, taking into account Eq. (3.0), it can be found that

 αcτLStSh=A1A2, (3.0)

where the right hand side should be constant for resonant eddies well inside the inertial range.

From figure 7, it can be seen that the right hand side of the above equation is constant only for Stokes numbers smaller than . Since the value of the right hand side starts to increase already for , this may once again indicate that from Eq. (3.0) is different from unity. The discrepancy may also be due to the fact that when it comes to the shape of the particle clusters, a large scale strain may stretch the particle clusters. For , there are no vortices that are larger than the clusters, and hence the dimensionality of the clusters becomes different. This will inevitably yield different values of . The value of the geometric coefficients can be fitted by

 A1A2=0.08+St/3, (3.0)

but this is just an empirical fit and more work is required in order to understand the fundamentals behind the shape and size of the particle clusters.

## 4 Conclusion

In this work, the effect of turbulence on the mass (and heat) transfer between inertial particles and the embedding fluid is studied. The turbulence is shown to have two effects on the mass transfer. The first effect is active for all Damköhler numbers, and here the turbulence increases the mass transfer rate due to the relative velocity between the particles in the fluid. A corresponding model for the relative velocity between the fluid and the particles is given by Eq. (3.0) which uses basic variables of the flow. With this, adding effects of relative velocity into RANS based simulations is possible.

The second effect with which turbulence influences the mass transfer rate is through the clustering of particles. It is shown that the size of the particle clusters increases with the particle Stokes number, and that the clustering decreases the overall mass transfer rate between the particles and the fluid. This is a confirmation of the findings of Kruger et al. (2016). In addition, a model is developed that takes this effect into account and incorporates it into a modified Sherwood number. This model is shown to give reasonable results for Stokes numbers (based on the turbulent integral scale) less than , while an empirical fit is employed to account for Stokes numbers up to unity. More work is still required in order to fully understand the size and dimensionality of the the particle clusters. As of now, a unique way of characterizing particle clusters does not exist, and very little work has actually been put into the study of large-scale clustering of particles.

## Acknowledgements

The research leading to these results has received funding from the Polish-Norwegian Research Program operated by the National Centre for Research and Development under the Norwegian Financial Mechanism 2009-2014 in the frame of Project Contract No Pol-Nor/232738/101/2014. This work was supported by the grant ”Bottlenecks for particle growth in turbulent aerosolsâ from the Knut and Alice Wallenberg Foundation, Dnr. KAW 2014.0048 and by grant from Swedish Research Council (Dnr. 638-2013-9243). NELH and DM also acknowledge the Research Council of Norway under the FRINATEK grant 231444.

• Jones & Launder  (1972) W. P. Jones and B. E. Launder, Int. J. Heat Mass Transfer 15, 301 (1972).
• Pope  (2003) S. B. Pope, Turbulent flows. Cambridge University Press (2003).
• Schiller & Naumann  (1933) L. Schiller and A. Naumann, Ver. Deut. Ing 77, 318 (1933).
• Crowe  (2012) C. T. Crowe, Multiphase Flows Droplets. CRC Press (2012).
• Magnussen & Hjertager (1976) B. F. Magnussen and B. H. Hjertager, 16’th Symposium (International) on Combustion, pp. 719-729. Pittsburgh: The Combustion Institute (1976).
• Ertesvåg & Magnussen  (2000) I. S. Ertesvåg and B. F. Magnussen , Combust. Sci. Technol. 159, 213 (2000).
• Dopazo  (1994) C. Dopazo, (ed. P. A. Libby and F. A. Williams), pp. 375-474. London: Academic Press. (1994).
• Ranz & Marshall  (1952) W. E. Ranz and W. R. Marshall, Chem Engr. Prog. 48, pps. 141 and 173 (1952).
• Kruger et al.  (2016) J. Kruger, N. E. L. Haugen, T. Lovas and D. Mitra Proc. Comb. Symp., dx.doi.org/10.1016/j.proci.2016.06.187
• Silaen & Wang  (2010) A. Silaen and T. Wang, Int. J. Heat Mass Transfer 53, 2074-2091 (2010).
• Vascellari et al.  (2014) M. Vascellari, S. Schulze, P. Nikrityuk, D. Safronov and C. Hasse, Flow turbulence and combustion 92, 319-345 (2014).
• Vascellari et al.  (2015) M. Vascellari, D. G. Roberts, S. S. Hla, D. J. Harris and C. Hasse, Fuel 152, 58-73 (2015).
• Klimanek et al.  (2015) A. Klimanek, W. Adamczyk, A. Katelbach-Wozniak, G. Wecel and A. Szlek, Fuel 152, 131-137 (2015).
• Chen et al.  (2012) L. Chen, S. Z. Yong and A. F. Ghoniem, Progress in energy and combustion science 38, 156-214 (2012).
• Chen et al.  (2000) C. Chen, M. Horio and T. Kojima, Chemical Engineering Science 55, 3875-3883 (2000).
• Gao et al.  (2004) J. Gao, C. Xu. S. Lin and G. Yang, AIChE Journal 45, 1095-1113 (2004).
• Zhang et al.  (2005) Y. Zhang, X.-L. Wei, L.-X. Zhou and H.-Z. Sheng, Fuel 84, 1798-1804 (2005).
• Luo et al.  (2012) K. Luo, H. Wang, J. Fan and F. Yi, Energy and Fuels 26, 6128-6136 (2012).
• Brosh & Chakraborty  (2014) T. Brosh and N. Chakraborty, Energy and Fuels 28, 6077-0688 (2014).
• Brosh et al.  (2015) T. Brosh, D. Patel, D. Wacks and N. Chakraborty, Fuel 145, 50-62 (2015).
• Hara et al.  (2015) T. Hara, M. Muto, T. Kitano, R. Kurose and S. Komori, Combustion and Flame 162, 4391-4407 (2015).
• Deen & Kuipers  (2014) N. G. Deen and J. A. M. Kuipers, Chemical Engineering Science 116, 645-656 (2014).
• Johansen et al.  (2007) A. Johansen, J. S. Oishi, M.M.Mac Low, H. Klahr, T. Henning, and A. Youdin, Nature 448, 1022 (2007).
• Haugen & Kragset  (2010) N. E. L. Haugen and S. Kragset, J. Fluid Mech. 661, 239 (2010).
• Haugen et al.  (2012) N. E. L. Haugen, N. Kleeoring, I. Rogachevskii and A. Brandenburg, Phys. Fluids 24, 075106 (2012).
• (The Pencil-Code) The Pencil-Code, http://pencil-code.nordita.org/
• Haugen & Brandenburg  (2006) N. E. L. Haugen and A. Brandenburg, Phys. Fluids 18, 075106 (2006).
• Calzavarini et al.  (2008) E. Calzavarini, M. Kerscher, D. Lohse and F. Toschi, J. Fluid Mech. 607, 13-24 (2008).
• Baum & Street  (1971) M. M. Baum and P. J. Street, Combustion Science and Technology 3, 231-243 (1971).
• Bec et al.  (2007) J. Bec, L. Biferale, M. Cencini, A. Lanotte, S. Musacchio and F. Toschi, Physical Review Letter 98, 084502 (2007).
• Squires & Eaton  (1991) K. D. Squires and J. K. Eaton, Journal of Fluid Mechanics 226, 1-35 (1991).
• Wood et al.  (2005) A. M. Wood, W. Hwang and J. K. Eaton, International Journal of Multiphase Flow 31, 1220-1230 (2005).
• Eaton & Fessler  (1994) J. K. Eaton and J. R. Fessler, International Journal of Multiphase Flow 20, 169-209 (1994).
• Toschi & Bodenschatz  (2009) F. Toschi and E. Bodenschatz, Annual Review of Fluid Mechanics 41, 375-404 (2009).
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# Relative magnitudes of intermolecular forces from phase diagrams

Is it possible to say whether or not a system A or B has stronger intermolecular forces from a phase diagram according to which:

1. The $T_{\text{b}}$ of A is lower than the $T_{\text{b}}$ of B.
2. The $T_{\text{melting}}$ of A is higher than the $T_{\text{m}}$ of B.
3. The triple points of A and B are the same.

Or is it not possible to compare A and B's intermolecular forces from a phase diagram?

I was sure that B has stronger intermolecular forces, as its boiling point is higher than for A at the same pressure.

## 1 Answer

This is an interesting situation. It would be easier to answer if the mp and bp of A were both either higher or lower than for B.

But your measure of the strength of intermolecular forces will be melting point - and boiling point. And they give different answers, so how do we make a decision?

Consider the boiling point first: the higher boiling point of B suggests higher intermolecular forces, as you have concluded. Heat of vaporization is needed to put a molecule into the vapor phase, and it seems that B needs more energy to go gaseous.

But A melts at a higher temperature, suggesting that its molecules are held together in a solid phase by stronger bonds. Solid A needs more energy to turn it into a liquid; perhaps A can pack into a tighter solid structure. It would then need more heat to melt it, so it has a larger heat of fusion.

Then your answer must have two parts: A is more tightly bound in a solid framework, but after you put enough heat into both A and B to melt them, B has more intermolecular bonding capacity as a liquid than A does. So B boils higher and is more strongly bound as a liquid.

• James, first of all thank you for your answer. Though I'm still not sure how to answer this question. Do you think it would be appropriate to say, that as A and B are in different states of matter, therefore it's impossible to compare their intermolecular forces? – Daniella Jun 14 '18 at 15:51
• Rather, melting and boiling are two different types of strength measurements; each has a heat required, and they are not necessarily related. So you can compare one set of heats (fusion) and decide which of A or B has stronger bonds. Then you compare the heats of evaporation and decide which of A or B seems to show stronger intermolecular strength at the higher temperature. Perhaps strength is a concept that you can measure by the amount of heat required to "break". – James Gaidis Jun 15 '18 at 13:32


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• #### Accelerated Reduction in $SO_2$ Emissions from the U.S. Power Sector Triggered by Changing Prices of Natural Gas

(American Chemical Society, 2012)
Emissions of sulfur dioxide ($SO_2$) from the U.S. power sector decreased by 24% in 2009 relative to 2008. The Logarithmic Mean Divisia Index (LMDI) approach was applied to isolate the factors responsible for this decrease. ...


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# what does it mean to be a high frequency wave

Whatever frequency we are vibrating at, we attract like frequency. Frequency, the number of waves that pass a fixed point in unit time; also, the number of cycles undergone during one unit of time by a body in periodic motion. Do we have to be a âmonkâ to awake to âlove, compassion and kindness â or over the age of 30 is doable ð? Online jelentkezés az itt letölthető dokumentumon keresztül. You could have a high amplitude low frequency signal, which would have a long wavelength but high peaks, but since frequency is defined as the number of wave peaks But that doesn't mean that it's independent of the wave length and the period. high frequency: [noun] a radio frequency between very high frequency and medium frequency â see Radio Frequencies Table. The more cycles that occur per second, the higher the frequency. Adatvédelmi szabályzat Frequency generally refers to a voltage that is changing in the form of a sine wave. 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However, if we say that a wave has a frequency of 60 Each wave has a certain shape and length. The higher the frequency, the shorter the distance between each successive compression (or rarification) in the incoming sound wave. It â¦ Learn vocabulary, terms, and more with flashcards, games, and other study tools. This MATLAB function estimates the mean normalized frequency, freq, of the power spectrum of a time-domain signal, x. (Magyar nyelvű dokumentum - online_elojegyzes.pdf), Online jelentkezés az itt letölthető dokumentumon keresztül. For FFT to be fully fast, it required 2^n data points. Since a sound wave is a pressure wave, a detector could be used to detect oscillations in pressure from a high pressure to a low pressure and back to a high â¦ Just through the standard formula. Frequency is measured in units of hertz (Hz) which is equal to one occurrence of a repeating event per second. This doesn't mean they are 'louder' or have a Now if I've got a Frequency is measured in hertz. It is important to note that the amplitude is not the distance between the top and bottom of a wave. Frequency, wavelength, amplitude and wave speed You need to know about these quantities used in waves - what they mean, the symbols used for them and the units used to measure them. In the case of electrical current, frequency is the number of times a sine wave repeats, or completes, a positive-to-negative cycle. It is common for kilohertz (kHz), megahertz (MHz) and gigahertz (GHz) to be used when waves have very high frequencies. To get different images, the wave height/amplitude can be changed as well as the starting point of the wave, also called the The intermediate frequency is created by mixing the carrier signal with a local oscillator signal in a process called heterodyning , resulting in a signal at the difference or beat frequency . The note A which is above Middle C (more on this later) has a frequency of 440 Hz. Our tips from experts and exam survivors will help you through. Proper usage and audio pronunciation (plus IPA phonetic transcription) of the word high frequency. Éppen dolgozunk a weboldalon, de hamarosan visszatérünk. It is often easiest to measure this from the, of one wave to the trough of the next wave, or from the. Higher frequencies tend to be more 'directional' than lower frequencies. The distance between peaks (high points) is called wavelength. What does high frequency mean? High frequency (HF) is an electromagnetic frequency defined by the International Telecommunications Union (ITU) for radar and radio communications. rate of vibration and oscillation measured over a specific period of time (usually one second most people cannot hear a high-pitched sound above 20 kHz; radio stations broadcast radio waves with frequencies of about 100 MHz; most wireless computer networks operate at 2.4 GHz. (Német nyelvű dokumentum - online_anmeldung.pdf). If the amplitudes of the vibrations are large enough and if natural frequency is within the human frequency range , then the vibrating object will produce sound waves that are audible. How i â¦ It is also the number of waves that pass a certain point each second. This is a high â¦ Speed is equal to distance divided by time. The frequency of a wave is the number of wave crests passing a point per unit time. Define high frequency. Research by psychologists has shown that on any given day, 99% of our thoughts are exactly the same as the day before. In communications and electronic engineering, an intermediate frequency (IF) is a frequency to which a carrier wave is shifted as an intermediate step in transmission or reception. The back-and-forth vibrational motion of the particles of the medium would not be the only observable phenomenon occurring at a given frequency. BETA WAVEâthe beta wave is known in the spectrum as âhigh frequency low amplitudeâ and are normally present during a waking state. A body in periodic motion is said to have undergone one cycle after passing through a series of events or â¦ A dokumentumot kitöltve kérjük küldje vissza a következő email címre: info@osztrakmagyarovi.hu, Online jelentkezés az itt letölthető dokumentumon keresztül. Start studying Physics Chapter 12. Please update your bookmarks accordingly. Definition of wave frequency and how it is related to the energy of a wave. If a voltage swings from 1 volt to -1 volt and does that a million times in a second, thats a frequency of 1 megahertz. It is also known as the decameter band or decameter wave as its wavelengths range from one to ten decameters (ten to one hundred meters). We have moved all content for this concept to for better organization. When a wave contacts matter, there are several different interactions that can occur. It is often used as a reference frequency for tuning musical instruments. As waves travel, they set up patterns of disturbance. It operates in a range of three to thirty MHz and as the wavelengths spread over ten to one hundred meters, it is also known as decameter wave and designated as band 7. [1] It is also referred to as temporal frequency, which emphasizes the contrast to spatial frequency and angular frequency. They allow for logical thought processes, and therefore are instrumental in problem-solving For example: most people cannot hear a high â¦ The frequency of a wave can be calculated using the equation: $\text{frequency f =}~\frac{\text{number of waves to pass a point}}{\text{time taken in seconds}}$. frequency Symbol: f , Î½.The number of oscillations per unit time of a vibrating system. Learn about how waves are measured according to amplitude, wavelength and frequency. The amplitude of a wave is its maximum disturbance from its undisturbed position. Higher frequencies have a higher average power for any given section of the wave, than lower frequency waves. Power spectral density (PSD), specified as a vector or matrix. The symbol for wavelength is the Greek letter lambda, λ. A smoother frequency response is better One of those interactions is wave absorption. High-Frequency Trading - HFT: High-frequency trading (HFT) is a program trading platform that uses powerful computers to transact a large number of orders at very fast speeds. Online jelentkezés az itt letölthető dokumentumon keresztül. The wavelength of a wave is the distance between a point on one wave and the same point on the next wave. n. Abbr. But it doesn't matter where you measure it - as long as it is the same point on each wave. I have a data set with 15000 data points. It is common for kilohertz (kHz), megahertz (MHz) and gigahertz (GHz) to be used when waves have very high frequencies. Meaning of high frequency. The frequency of a wave is the number of waves produced by a source each second. The difference in wavelength is the way we tell different kinds of electromagnetic energy apart. Period: Period of a wave is the time required for a vibration of a wave to make a complete cycle. It is often easiest to measure this from the trough of one wave to the trough of the next wave, or from thecrest of one wave to the crest of the next wave. Wavelength and frequency are interdependent. Although frequency is a measurement of rate of movement, it is not identical to velocity. (Magyar nyelvű dokumentum - online_elojegyzes.pdf) Second Frequency: Each additional wave or frequency brings along more ripples and as such, more detail. We generally think of an electrical signal as a one-way phenomenon; it travels from the output of one component to the input of another componâ¦ To be more clear. I want to transform the data to frequency domain using FFT. 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## Physics: Principles with Applications (7th Edition)

a. Here, the force is the same. $F = k_{1}x_{1} = k_{2}x_{2}$. The work done on spring 1 is $$W_{1} = \frac{1}{2} k_{1}x_{1}^{2}$$ The work on spring 2 is $$W_{2} = \frac{1}{2} k_{2}x_{2}^{2} = \frac{1}{2} k_{2}\frac{k_{1}^{2} x_{1}^{2}}{ k_{2}^{2}}$$ $$W_{2} = W_{1} \frac{k_{1}}{k_{2}}$$ Because the first spring is stiffer, the work done on spring 2 is greater. b. Now, the distance stretched is the same. As before, the work done on spring 1 is $$W_{1} = \frac{1}{2} k_{1}x^{2}$$ The work on spring 2 is $$W_{2} = \frac{1}{2} k_{2}x^{2}$$ Because the first spring is stiffer, the work done on spring 1 is greater.


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How many different BCD numbers can be stored in 12 switches ? (Assume two position or on-off switches).

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ISRO Scientist CS 2014 Official Paper
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1. 212
2. 212 - 1
3. 1012
4. 103

Option 4 : 103
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ISRO Scientist CS 2020 Official Paper
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80 Questions 240 Marks 90 Mins

Detailed Solution

The correct answer is option 4.

Key Points

Binary Coded Decimal (BCD) encoding scheme each of the decimal numbers (0 - 9) is represented by its equivalent binary pattern. Which is generally 4-bits.

Step1: A Switch can store on or off which means it takes one bit(0 for off, 1 for on) and each digit of BCD takes 4 bits.

Step2: So there will be $$\frac{12}{4}$$ = 3 BCD digits. A BCD digit can be from 0 to 9  which is total 10 possibilities.

Step3: Therefore, diffrent possible BCD numbers in 12 switches are = 10 * 10 * 10 = 1000 = 103

∴ Hence the correct answer is 103.