Chapter 1 Introduction and Kinetics of Particles

Transcription

1 Chapter 1 Introduction and Kinetics of Particles 1.1 Introduction There are two ain approaches in siulating the transport equations (heat, ass, and oentu), continuu and discrete. In continuu approach, ordinary or partial differential equations can be achieved by applying conservation of energy, ass, and oentu for an infinitesial control volue. Since it is difficult to solve the governing differential equations for any reasons (nonlinearity, coplex boundary conditions, coplex geoetry, etc.), therefore finite difference, finite volue, finite eleent, etc., schees are used to convert the differential equations with a given boundary and initial conditions into a syste of algebraic equations. The algebraic equations can be solved iteratively until convergence is insured. Let us discuss the procedure in ore detail, first the governing equations are identified (ainly partial differential equation). The next step is to discretize the doain into volue, girds, or eleents depending on the ethod of solution. We can look at this step as each volue or node or eleent contains a collection of particles (huge nuber, order of ). The scale is acroscopic. The velocity, pressure, teperature of all those particles represented by a nodal value, or averaged over a finite volue, or siply assued linearly or bi-linearly varied fro one node to another. The phenoenological properties such as viscosity, theral conductivity, heat capacity, etc. are in general known paraeters (input paraeters, except for inverse probles). For inverse probles, one or ore therophysical properties ay be unknown. On the other extree, the ediu can be considered ade of sall particles (ato, olecule) and these particles collide with each other. This scale is icroscale. Hence, we need to identify the inter-particle (inter-olecular) forces and solve ordinary differential equation of Newton s second law (oentu conservation). At each tie step, we need to identify location and velocity of each particle, i.e, trajectory of the particles. At this level, there is no definition of teperature, pressure, and thero-physical properties, such as viscosity, theral conductivity, heat capacity, etc. For instance, teperature and pressure are related to the kinetic energy of the particles (ass and velocity) and frequency of particles A. A. Mohaad, Lattice Boltzann Method, DOI: / _1, Ó Springer-Verlag London Liited

2 2 1 Introduction and Kinetic of Particles bobardent on the boundaries, respectively. This ethod is called olecular dynaics (MD) siulations. To get an idea, a nuber of equations need to be solved; 10 3 c of air at the roo conditions contains about olecules. One ole of water (16 g, a sall cup of water) contains ore than 6: olecules. To visualize this nuber, if we assue that the diaeter of these olecules is 1 (tip of a pen, a dot) and these dots are arranged side by side, then the total area that can be covered by these dots is 6: k 2 : The total surface area of the earth is about 5: k 2 and area of the United States is 9: k 2 and area of Africa is 1: k 2 : This eans that we can cover the total area of the earth by dots ade fro nuber of olecules of 16 g of water if the olecule diaeter is 1. Furtherore, it is interesting to calculate how any years we need to coplete the project of doting the world by using pen tip. For a reasonable fast person, 6 dots per second is a good estiate. Hence, it needs at least about years to coplete the project. The question is do we really need to know the behavior of each olecule or ato? In bookkeeping process we need to identify location (x; y; z) and velocity (c x ; c y ; and c z are velocity coponents in x; y and z direction, respectively) of each particle. Also, the siulation tie step should be less than the particles collision tie, which is in the order of fero-seconds (10 12 s). Hence, it is ipossible to solve large size probles (order of c) by MD ethod. At this scale, there is no definition of viscosity, theral conductivity, teperature, pressure, and other phenoenological properties. Statistical echanics need to be used as a translator between the olecular world and the acroscopic world. The question is, is the velocity and location of each particle iportant for us? For instance, in this roo there are billions of olecules traveling at high speed order of 400 /s; like rockets, hitting us. But, we do not feel the, because their ass (oentu) is so sall. The resultant effect of such a chaotic otion is alost nil, where the air in the roo is alost stagnant (i.e., velocity in the roo is alost zero). Hence, the behavior of the individual particles is not an iportant issue on the acroscopic scale, the iportant thing is the resultant effects. Fundaentally, MD is siple and can handle phase change and coplex geoetries without any difficulties and without introducing extra ingredients. However, it is iportant to specify the appropriate inter-particle force function. The ain drawback or obstacle of using MD in siulation of a relatively large syste is the coputer resource and data reduction process, which will not be available for us in the seen future. What about a iddle an, sitting at the iddle of both entioned techniques, the lattice Boltzann ethod (LBM). The ain idea of Boltzann is to bridge the gap between icro-scale and acro-scale by not considering each particle behavior alone but behavior of a collection of particles as a unit, Fig The property of the collection of particles is represented by a distribution function. The keyword is the distribution function. The distribution function acts as a representative for collection of particles. This scale is called eso-scale.

3 1.1 Introduction 3 Continuu( Macroscopic scale), finite difference, finite volue, finite eleent, etc), Navier-Stokes Equations Lattice Boltzann Method (Mesoscopic scale), Boltzann Equation Molecular Dynaics (Microscopic scale), Hailton s Equation. Fig. 1.1 Techniques of siulations The entioned ethods are illustrated in Fig LBM enjoys advantages of both acroscopic and icroscopic approaches, with anageable coputer resources. LBM has any advantages. It is easy to apply for coplex doains, easy to treat ulti-phase and ulti-coponent flows without a need to trace the interfaces between different phases. Furtherore, it can be naturally adapted to parallel processes coputing. Moreover, there is no need to solve Laplace equation at each tie step to satisfy continuity equation of incopressible, unsteady flows, as it is in solving Navier Stokes (NS) equation. However, it needs ore coputer eory copared with NS solver, which is not a big constraint. Also, it can handle a proble in icro- and acro-scales with reliable accuracy. 1.2 Kinetic Theory It is necessary to be failiar with the concepts and terinology of kinetic theory before proceeding to LBM. The following sections are intended to introduce the reader to the basics and fundaentals of kinetic theory of particles. I tried to avoid the detail of atheatics; however ore ephasis is given to the physics.

4 4 1 Introduction and Kinetic of Particles Note that, the word particle and olecule are used interchangeably in the following paragraphs Particle Dynaics As far as we know, the ain building block of all the aterials in nature is the olecules and sub-olecules. These olecules can be visualized as solid spheres oving randoly in conservatory anner in a free space. The otion satisfies conservation of the ass, oentu, and energy. Hence, Newton s second law (oentu conservation) can be applied, which states that the rate of change of linear oentu is equal to the net applied force. F ¼ dðcþ ð1:1þ dt where F stands for the inter-olecular and external forces, is the ass of the particle, c is the velocity vector of the particle and t is the tie. For a constant ass, the equation can be siplified as, F ¼ dc dt ¼ a ð1:2þ where a is the acceleration vector. The position of the particle can be deterined fro definition of velocity, c ¼ dr ð1:3þ dt where r is the position vector of the particle relative to the origin, as shown in Fig. 1.2 In the MD siulation, the above equations are solved provided that F is a known function. Fig. 1.2 Position and velocity vectors

5 1.2 Kinetic Theory 5 If an external force, F, is applied to a particle of ass, the velocity of the particle will change fro c to c þ Fdt= and its position changes fro r to r þ cdt; see Fig In the absence of an external force, the particle streas (oves) freely fro one location to another location without changing its direction and speed, assuing no collision takes place. The agnitude of the particle velocity increases and interaction between the particles increases as the internal energy of the syste increases (for exaple, heating the syste). Increases in the kinetic energy of the olecules are referred as increases in teperature in the acroscopic world. The particles (olecules) are continuously bobarding the container walls. The force exerted by those actions per unit area is referred as pressure in the acroscopic easure. Fro this siple odel, we can see that there is a relationship between teperature and pressure, as the teperature increases, which eans the kinetic energy of the olecules increase, we expect that the probability of particles bobarding the container wall, increases. In the following section, the relationship between pressure, teperature, and kinetic energy will be explored Pressure and Teperature Let us assue that a single particle oving with a speed, c x (in x-direction), inside a tube of length L and bobarding the ends of the tube, continuously. The force exerted by the particle on an end is equal to the rate of change of the oentu (assuing that the collision is perfect elastic), then FDt ¼ c x ð c x Þ¼2c x ; ð1:4þ where Dt is tie between hits. Equation 1.4 is an integration of Newton s second law, Eq The tie between hits is equal to 2L=c x ; which is the tie needed for the particle to travel fro one end to another end and return to the sae location, Fig Hence, 2LF=c x ¼ 2c x ; which yields, F ¼ c 2 x =L ð1:5þ after colliding before colliding x -C C L Fig. 1.3 A particle is freely oving in a box

6 6 1 Introduction and Kinetic of Particles The results can be generalized for N particles. The total force exerted by N particles is proportional to Nc 2 =L: In general, c 2 ¼ c 2 x þ c2 y þ c2 z ; where c x; c y and c z are velocity coponent in x, y and z-directions, respectively. It is fair to assue that these coponents are equal, therefore c 2 ¼ 3c 2 x : Then, the total force can be written as, F ¼ Nc 2 =ð3lþ ð1:6þ The pressure is defined as a force per unit area, perpendicular to the force vector, P ¼ F=A: Then, the pressure exerted by N particle on the ends of the tube is equal to, P ¼ Nc 2 =ð3laþ ¼Nc 2 =ð3vþ ð1:7þ where V stands for volue, which is equal to LA. This siple picture of olecular otion, relates the pressure in acroscopic sense to the kinetic energy of the olecules, i.e., P ¼ðc 2 =2Þð2=3ÞN=V In other words, the pressure is related to kinetic energy (KE) as, P ¼ 2 3 nke ð1:8þ ð1:9þ where n is the nuber of olecules per unit volue. In this siple odel (ideal gas odel), we neglected the effect of olecular interaction and effect of the olecular size. However, for a gas at a roo teperature, the results are surprisingly true. In a real syste, the particle has volue and collision take places between the particles. Experientally, it is well-established that for gases far fro the critical conditions, the state equation can be expressed as, PV ¼ nrt ð1:10þ where n is nuber of oles = N=N A ; where N A is Avogadro s nuber and R is the gas constant. Equating Eqs. 1.8 and 1.10, yields N=N A RT ¼ðc 2 =2Þð2=3ÞN ð1:11þ Introducing Boltzann constant (k ¼ R=N A ¼ 1: J/K), we can deduce that kinetic energy, KE, of a gas is KE ¼ c 2 =2 ¼ð3=2ÞkT ð1:12þ It is interesting that the teperature and pressure in the acroscopic world are no ore than scale of the kinetic energy of the olecules in the icroscopic world.

7 1.3 Distribution Function Distribution Function In 1859, Maxwell ( ) recognized that dealing with a huge nuber of olecules is difficult to forulate, even though the governing equation (Newton s second law) is known. As entioned before, tracing the trajectory of each olecule is out of hand for a acroscopic syste. Then, the idea of averaging cae into picture. For illustration purposes, in a class of 500 students (extreely sall nuber, copared with the nuber of olecules in volue of 1 3 ), if all the students started asking a question siultaneously, the result is noise and chaos. However, the question can be addressed through a class representative, which can be handled easily and the result ay be acceptable by the ajority. The idea of Maxwell is that the knowledge of velocity and position of each olecule at every instant of tie is not iportant. The distribution function is the iportant paraeter to characterize the effect of the olecules; what percentage of the olecules in a certain location of a container have velocities within a certain range, at a given instant of tie. The olecules of a gas have a wide range of velocities colliding with each others, the fast olecules transfer oentu to the slow olecule. The result of the collision is that the oentu is conserved. For a gas in theral equilibriu, the distribution function is not a function of tie, where the gas is distributed uniforly in the container; the only unknown is the velocity distribution function. For a gas of N particles, the nuber of particles having velocities in the x- direction between c x and c x þ dc x is Nf ðc x Þdc x : The function f ðc x Þ is the fraction of the particles having velocities in the interval c x and c x þ dc x ; in the x-direction. Siilarly, for other directions, the probability distribution function can be defined as before. Then, the probability for the velocity to lie down between c x and c x þ dc x ; c y and c y þ dc y ; and c z and c z þ dc z will be Nf ðc x Þf ðc y Þf ðc z Þdc x dc y dc z : It is iportant to ention that if the above equation is integrated (sued) over all possible values of the velocities, yields the total nuber of particles to be N, i.e., ZZZ f ðc x Þf ðc y Þf ðc z Þ dc x dc y dc z ¼ 1: ð1:13þ Since any direction can be x, ory or z, the distribution function should not depend on the direction, but only on the speed of the particles. Therefore, f ðc x Þf ðc y Þf ðc z Þ¼Uðc 2 x þ c2 y þ c2 z Þ ð1:14þ where U is another unknown function, that need to be deterined. The value of distribution function should be positive (between zero and unity). Hence, in Eq. 1.14, velocity is squared to avoid negative agnitude. The possible function that has property of Eq is logarithic or exponential function, i.e.,

8 8 1 Introduction and Kinetic of Particles log A þ log B ¼ logðabþ ð1:15þ or e A e B ¼ e ðaþbþ ð1:16þ It can be shown that the appropriate for for the distribution function should be as, f ðc x Þ¼Ae Bc2 x ð1:17þ where A and B are constants. The exponential function iplies that the ultiplication of the functions can be added if each function is equal to the exponent of a function. For exaple: then FðxÞ ¼e Bx ; FðyÞ ¼e Cy ; FðxÞFðyÞ ¼e Bx e Cy ¼ e ðbxþcyþ : But if FðxÞ ¼Bx and FðyÞ ¼Cy; then FðxÞFðyÞ ¼BxCy; in this case the ultiplication of function is not equal to the addition of the functions. Accordingly, it can be assued that, f ðcþ ¼Ae Bc2 x Ae Bc 2 y Ae Bc 2 z ¼ A 3 e Bc2 ð1:18þ Multiplying together the probability distributions for the three directions, gives the distribution in ters of the particle speed c. In other words, the distribution function is that giving the nuber of particles having speed between c and c þ dc: It is iportant to think about the distribution of particles in velocity space, a three-diensional space (c x ; c y ; c z ), where each particle is represented by a point having coordinates corresponding to the particle s velocity. Thus, all points lying on a spherical surface centered at the origin correspond to the sae speed. Therefore, the nuber of particles having speed between c and c þ dc equals the nuber of points lying between two shells of the sphere, with radii c and c þ dc; Fig The volue of the spherical shell is 4pc 2 dc: Therefore, the probability distribution as a function of speed is: f ðcþdc ¼ 4pc 2 A 3 e Bc2 dc ð1:19þ Integration of above function for the given Fig. 1.4 yields eight particles. The constants A and B can be deterined by integrating the probability distribution over all possible speeds to find the total nuber of particles N, and their total energy E.

9 1.3 Distribution Function 9 Fig. 1.4 Phase diagra Since a particle oving at speed c has kinetic energy 1 2 c2 ; we can use the probability distribution function to find the average kinetic energy per particle, as: 1 2 c2 ¼ R R c2 f ðcþ dc f ðcþ dc ð1:20þ The nuerator is the total energy, the denoinator is the total nuber of the particles. Notice that the unknown constant A cancels between nuerator and denoinator. Substituting the value of f ðcþ in the integrals, yields 1 2 c2 ¼ 3 ð1:21þ 4B Substituting the value for the average kinetic energy in ters of the teperature of the gas (Eq. 1.12), Hence, B ¼ =2kT; so 1 2 c2 ¼ 3 kt ð1:22þ 2 f ðcþ /c 2 e c2 2kT ð1:23þ The constant of proportionality is given by integrating over all speeds and setting the result as equal to one (since we factored out the nuber of particles N in our definition of f ðcþ). The final result is: f ðcþ ¼4p 2pkT 3 2 c 2 e c2 2kT ð1:24þ Note that this function increases parabolically fro zero for low speeds, reaches a axiu value and then decreases exponentially. As the teperature increases, the position of the axiu shifts to the right. The total area under the curve is always one, by definition. This equation called Maxwell or Maxwell Boltzann distribution function.

10 10 1 Introduction and Kinetic of Particles The probability of finding a particle that has a specific velocity is zero, because the velocity of particles change continuously over a wide range. The eaningful question is to find the probability of a particle or particles within a range of velocity rather than at a specific velocity. Therefore, Eq. 1.24, need to be integrated in that range of velocity. Exaple For air olecules (say, nitrogen) at 0 and 100 C teperatures, calculate the distribution function. The ass of one olecule of N 2 ; which is olar ass (28 g/ol or kg/ ol) divided by Avogadro s nuber (6: ol 1 ) gives 4: kg; which is ass of one N 2 olecule. The Boltzann constant is 1: J/K ðkg 2 =ðs 2 KÞ: Then =ð2kþ ¼4: =ð2 1:38Þ ¼1: ð 2 =ðs 2 KÞÞ: h Figure 1.5 shows the probability of as a function of olecular velocity (c) for N 2 at T ¼ 273 and 373 K. The area under each curve is unity. As the teperature increases, the nuber of olecules with high velocity increases. The ost probable speed is equal to rffiffiffiffiffiffiffiffi 2kT ð1:25þ This can be obtained by setting the derivative of the distribution function with respect velocity to zero, and then solve for velocity. The average speed is equal to rffiffiffiffiffiffiffiffi 8kT hci ¼ ð1:26þ p which is the weighted average velocity. It can be obtained by integrating distribution function fro zero to infinity, as Fig. 1.5 The probability distribution function for nitrogen gas as a function of olecular velocity, c f(v) T=273 K Maxiu Average Speed T=373 K Root-Mean-Average Speed V

11 1.3 Distribution Function 11 hci ¼ Z 1 The root-ean-average speed is equal to hc 2 i¼ Z cf ðcþ dc c 2 f ðcþ dc ¼ 3kT ð1:27þ ð1:28þ The ean average speed is equal to ðc 2 x þ c2 y þ c2 z Þ1=2 and average speed is equal to ðc x þ c y þ c z Þ=3: The root ean squared speed of a olecule is c rs ¼ ffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi p hc 2 3K B T i ¼ ð1:29þ Lighter gases usually have higher olecular speeds than heavy gases (higher olecular weight). Figure 1.6 illustrates the probability function for H 2 (2 kg/kole), N 2 (28 kg/kole) and CO 2 (44 kg/kole). At roo teperature, ost N 2 olecules travel at speeds around 500 /s, while ost hydrogen olecules travel at speed around 1,600 /s (supersonic rockets). Exercise Calculate the average and rs speed of hydrogen olecule at roo teperature (25 C). Exercise For a Maxwellian distribution function, calculate the total nuber of olecules striking unit area of a wall per unit tie. What will be the pressure on the wall? Fig. 1.6 Distribution function for N 2 ; H 2 and CO 2 at roo teperature f(v) T=298 K N 2 H 2 CO V

12 12 1 Introduction and Kinetic of Particles Boltzann Distribution Boltzann generalized the Maxwell s distribution for arbitrary large systes. He was the first to realize the deep connection between the therodynaic concept of entropy and the statistical analysis of possible states of a large syste that the increase in entropy of a syste with tie is a change in acroscopic variables to those values corresponding to the largest possible nuber of icroscopic arrangeents. Boltzann showed that the nubers of available icroscopic states for a given energy are far greater for acroscopic values corresponding to theral equilibriu. For exaple, for a given energy there are far ore possible icroscopic arrangeents of gas olecules in which the gas is essentially uniforly distributed in a box than that of all the gas olecules being on the left-hand half of the box. Thus, if a liter of gas over the course of tie goes through all possible icroscopic arrangeents, in fact there is a negligible probability of it all being in the left-hand half in a tie the age of the universe. So if we arrange for all the particles to be in the left-hand half by using a piston to push the there, then reove the piston, they will rapidly tend to a unifor distribution spread evenly throughout the box. Boltzann proved that the therodynaic entropy S, of a syste (at a given energy E) is related to the nuber W, of icroscopic states available to it by S ¼ k logðwþ; k being Boltzann s constant. There were soe abiguities in counting the nuber of possible icroscopic arrangeents which were rather troublesoe, but not fatal to the progra. For exaple, how any different velocities can a particle in a box have? This atter was cleared up by the quantu echanics. Boltzann was then able to establish that for any syste large or sall in theral equilibriu at teperature T, the probability of being in a particular state at energy E is proportional to e E kt ; i.e. f ðeþ ¼Ae E=kT This is called the Boltzann distribution. Let us consider kinetic energy of olecules in x-direction, then E ¼ 1 2 c2 x ð1:30þ ð1:31þ For a noralized probability function, the probability function integrated for all values of velocity (fro inus to plus infinity) should be one. Hence, Therefore, Z 1 1 Ae c2 x 2kT dc ¼ 1 ð1:32þ

13 1.3 Distribution Function 13 rffiffiffiffiffiffiffiffiffiffi A ¼ 2pkT The probability of finding velocity c x is rffiffiffiffiffiffiffiffiffiffi f ðc x Þ¼ 2pkT e c2 x 2kT We are interested on probability of three diensional velocity (c) where ð1:33þ ð1:34þ c 2 ¼ c 2 x þ c2 y þ c2 z ð1:35þ The probability of (c) is ultiple of probability of each function, i.e., which leads to f ðcþ ¼f ðc x Þf ðc y Þf ðc z Þ rffiffiffiffiffiffiffiffiffiffi 3 f ðcþ ¼ e 2kT ðc2 x þc2 y þc2 z Þ 2pkT ð1:36þ ð1:37þ or 3=2e f ðcþ ¼ c2 2kT ð1:38þ 2pkT It should be noted that the above equation does not take into account the fact that there are ore ways to achieve a higher velocity. In aking the step fro this expression to the Maxwell speed distribution, this distribution function ust be ultiplied by the factor 4pc 2 (which is surface area of a sphere in the phase space) to account for the density of velocity states available to particles. Therefore, Maxwell distribution function (Eq. 1.24) is covered. In fact, integration of Maxwell distribution function (Eq. 1.24) over a surface of sphere in phase space yields Eq An ideal gas has a specific distribution function at equilibriu (Maxwell distribution function). But Maxwell did not ention, how the equilibriu is reached. This was one of the revolutionary contribution of Boltzann, which is the base of the LBM. In the next chapter, Boltzann transport equation, which is a ain concern of this book, will be discussed.