Appendix to the paper on Significance of the microfluidic concepts for the improvement of macroscopic models of transport phenomena

Transcription

1 ppenix to the paper on ignificance of the microfluiic concepts for the improement of macroscopic moels of transport phenomena Michael Krol *1, * Tel ; michael_krol48@yahoo.com 1: Chemical Engineering Institute, Berlin Uniersity of Technology, ermany : ruziazka zkola Wyzsza (W), Polan ubject of this appenix to the paper submitte to the 3 r Micro an Nano lows Conference at Thessaloniki, reece, -4 ugust, 11 (MN 11) is aitional information about: a simple mathematical moel applicable to the broaest possible range of micro- an macroscopic transport phenomena with help of a minimum of basic concepts of corpuscular physics [, 3]. The moel is the basis of: Dynamics of isperse systems [9, 7-13] as since a longer time eelope common backgroun of apparently unrelate phenomena. Main features of the propose metho in istinction to the most popular methos of flui mechanics [e.g. 1, 4, 5] are: - Explicit use of Newton s law as equation of motion instea of erie equations like Naier-tokes [4], Bernoulli equation etc. Thereby Newton s law an conseration laws are applie to nonrelatiistic motion of particle systems in the range from iniiual particles characterize by mass an elocity, such as iniiual atoms (molecules) or een electrons [15-17], oer to macroscopic sets of such particles builing either soli boies or flowing systems of traitional mechanics [1, 5], up to celestial boies of classical astro-physics [6]. Main application fiel are single- an multi-phase flows in engineering. - Departure from ifferential notation where possible, thus aiming explicitly at tangible, total effects characterizing conitions in a efine olume or on a surface instea of imaginary conitions at a point. Integral notation allows etermining of physical quantities at require scale by choice of the consiere reference olumes/surfaces an use of the mean alue theorem (MT) of integral calculus [18]. - The obsere changes of the state of motion efine (name) forces, the exact nature of which is regare as unknown. Thereby, Newton s secon law [7 13] an conseration laws [1] in ifferential-integral notation buil always the same starting point for physically coherent explanation of phenomena roote in motion of matter, unerstoo as isperse system [, 3, 5-1]. Basic form of Newton s law reas [11, 1]: (1) t Equation (1) is use to efine graity, similarly as electrostatic forces, or macroscopic results of multiple electrostatic molecular interactions such as pressure or friction. The results of the basic experiments respectiely obserations are written own in form of spherical laws [3, 15-17]: m m (r) 1 for graitational r attraction of masses m 1 an m separate by istance of r with uniersal graitation constant, or as in case of electrostatic interactions [15-18]: 1 q q (r) 1 for electrostatic E 4 r attraction of charges with opposite polarity q 1 an q separate by r with ε enoting the ielectric constant. Corresponingly RE (r) may enote electrostatic repulsion for charges of the same polarity. When ealing with single molecules equation (1) may be written as: - 1 -

2 (r) E (r) RE (r) t () In case of a limite number of molecules it is possible to use approximations of forces () e.g. to calculate the elocities of iniiual molecules an other particle parameters releant for macroscopic flows. ig. 1 below shows a plot of ifferent molecular interaction forces esigne for use with moel particles of unit charge. 1( t) 1( 4 t4) ( t) ( 4 t4) ( t) () i 1() i () i r H () i ig. 1 Interaction forces for moel particles with unit charge an hyrogen mass. ig. below shows the corresponing elocities of the moel molecules calculate for interaction forces (r) an 1(r) shown aboe using form () of eq. (1). Thereby forces (r) an (r) mark the here aopte approximation ranges. (r) was constructe for electrostatic repulsion an graitational attraction. (r) stresses the influence of electrostatic attraction in presence of graitation an electrostatic repulsion. Influence of graitation in the plot of 1(r) was stresse to inicate its existence in a single iagram together with much stronger electrostatic forces. ig. 1 shows onset of graitational attraction alreay at istance of the force centres of just ca m, which was obtaine by amplifying graitational attraction to show all interactions in a single iagram. Calculations were conucte using eq. an ata accoring to [15-17] r H () t ig. elocities of moel particles obtaine using interaction forces plotte in ig. 1. or applications in macroscopic flow situations it is conenient to use the following representation of forces : g p t η t (3) The forces on the right sie of (3) are ientifie as [11, 1]: graity g, pressure p, friction t etc. acting in an, respectiely, on its limiting surfaces. The ouble unerlining enotes the consiere term (here t) as tensor of secon orer. Thereby the so calle flui quantities are unerstoo as result of aerage application of Newton s law eq. (1) an momentum conseration law [1] to micro scale particle systems (flui). Thus the statistical flui quantities like ensity, pressure p an friction t (represente as function of the ynamical iscosity η), are all efine at the micro scale by molecular numbers, masses, olumes, aerage collision free paths etc. [, 3, 7-13]. Irrespectiely of its applicability to macroscopic isperse systems equation (3) oes not mix up the particle an the flui quantities. It uses them consciously together, within the framework of a physically coherent uniersal metho capable of ealing with arbitrary particle or flui systems characterize by mass an elocity. In - -

3 compliance with this flui quantities like pressures p (or iscosity η) an such macroscopic equialents as boy forces are specialize representations of the aerage molecule impact forces. The term pressure will be outline on example of gas molecules shown schematically below (ig. 3): z x ig. 3: lui moelle as a molecular isperse system [7-13]. chematically shown are gas molecules in stochastic thermal motion. rrows symbolize the thermal molecule elocities. Deriations of the statistical physical quantities follow from the analysis of molecular motions in respect to chosen control surfaces. Basic examples are: Thermal gas pressure: We consier the magnifie microscopic molecule motion through a macroscopic flat control surface e.g. (x--y), perpenicular to z-axis an containing the x an y axis ig. 4. below. y x ig. 4 Molecular exchange through the control surface x--y. is symbolic representation of (aerage) collision free path. is ector of thermal elocity of a molecule. x symbolizes the ector of superimpose aerage bulk flow elocity (if present). is the surface ector of the consiere part of the x - y surface. The surface ector of the control surface is parallel to the z-axis. as molecules moe stochastically in all irections, but in such a way that on the aerage there is no resultant long term transport across. The aerage macroscopic flow elocity is zero, thus macroscopic friction is not present. raity influence is negligible. t these conitions the resulting aerage pressure force is efine by: p p (4) t ince the aerage flow is steay, the local time eriatie anishes an the left han sie of the aboe equation, which efines the force of total momentum exchange (pressure force), reuces to conectie component alone: p t t (5) With representing the aerage collision free traelling istance we assume, that the aerage elocities of the gas molecules are in equal. In particular it is: (z ) (z) an x,y,z = (6) Conitions (6) for aerages of molecular elocities are substitute into the conectie component, uner assumption that half of the surface area is crosse by molecules traelling e.g. ownwar an half in reerse irection. orce of momentum exchange by molecule collisions follows as: P, e / - e (z ) (7) e (z ) / Terms z represent results of scalar ector prouct approximating the aerage iffusie molecule transport through the control surface. The + sign results for transport along the -axis (in ig. upwar), corresponingly results for transport reerse to the z-axis (in ig. ownwars)

4 The inex means that the obtaine components are irecte along the z-axis, perpenicular to. Multiplication with the unit ector e results in: P, e / e / / - / (z ) (z ) (z ) (z ) (z) ( ) / (z) / The total interaction force follows after simplifying the last expression an applying the mean alue theorem of integral calculus. It reas: P Pressure p follows as force component iie by the area: p (8) imilar calculation is performe to fin expressions relating e.g.: the ynamic gas iscosity η (flui parameter) to molecular (particle) transport parameters such as the collision free path, aerage stochastic molecule elocity an gas ensity. To this purpose the bounary conitions are change to moel the behaiour of the gas clou in ig. 3 uner linear, steay shear loa. Conitions concerning the molecular elocities remain unchange; the main assumption this time concerns the aerage elocity of the bulk flow i.e. the x-component of the aerage gas elocity x oes not equal zero as before but changes linearly with z: (z ) (z) x (9) x x z or otherwise steay aerage flow elocity fiel x superpose on the stochastic molecule elocity an, with other assumptions unchange it follows, that: molecules crossing the control surface an colliing with molecules with ifferent local mean flow elocity x will on the aerage obtain the local aerage flow elocity thus changing their original momentum. To fin the resulting force balance we substitute again the aerage elocity fiel approximation in the conectie component as aboe: e e T, e / (z ) (z ) / The x-component of the shear force follows from multiplication with a unit ector e x : T, e (z ) e (z ) / / (z ) (z ) / / / (z) z ( (z) z / pplying the mean alue theorem of integral calculus an simplifying the result, we obtain: T, z (11) z The prouct of the gas ensity, of the aerage stochastic molecule elocity an collision free path efines the macroscopic ynamic iscosity of the consiere system []: η (1) z Expressions (11, 1) may be generalize to express the iscous rag force in one imensional flows as: T ) k(re) η W or, for practical z - 4 -

5 Purposes: k(re) η rel T W (13) R H Thereby k(re) is a geometry an flow type (laminar, turbulent) epenent coefficient. W represents the wette surface of the consiere flow geometry. The elocity graient in the bounary layer may be represente for practical purposes by a quotient of the aequate relatie elocity an hyraulic raius R H =/ W. In particular it was shown by the author, that for one imensional flows the coefficient k(re) allows a ery conenient representation of the rag force. Thus for: Couette flow (aboe): k C (Re) = 1 Pipe with circular cross section: kp(re) +.1 Re.75 or: kp(re) +.1 Re f T (Re) low aroun a sphere: k (Re) Re.5 +.5Re low through fixe particle be: k B (Re) Re f T (Re)/f T,max(Re) may sere as turbulence intensity inicator [11, 1]. imilar calculations to fin expressions relating macroscopic flui quantities to particle quantities are gien in the submitte paper, e.g. Heat conuctiity etc.[]. lows through systems of macroscopic particles hae similarities with microscopic flows of iniiual molecules. Howeer, there are also many ifferences e.g. in a system of macroscopic particles pressure buils a boy force an this boy force is experience not only uring particle collisions, but also uring the free flights, that particles make between collisions. To show this an also exactly how to apply the presente metho to examples exhibiting similarities of micro- an macroscopic systems we consier: I. rchimees principle: ig. 5 (below). y ig. 5 Macroscopic particle of ensity > suspene in a container fille with on the aerage motionless flui ( flui ensity). To etermine is the force require to hol the sphere motionless. irst step of analysis buils choice of proper theoretical tools. ince in this example we eal with forces in a macroscopic system, the aequate analysis tool will be form (3) of eq. (1). Written for a flui it reas: g p t η t (14) or suspene soli sphere eq. of motion must be extene with external force - e : t g g p Q t ( e ) (15) econ step is implementation of aequate simplifying assumptions: - Time epenence an transport of matter beyon the bounaries of any macroscopic x z - 5 -

6 partial olume (e.g. olume containing more than 1 1 molecules) anish in the consiere system. - riction anishes because the flui is in macroscopic sense motionless. rom equation of motion (3) for the flui follows: g p (16) or submerge sphere, with = e : g p (17) or the hyrostatic pressure we hae: p g (18) The thir step of analysis in all problems allowing it, is use of the mean alue theorem (MT) of integral calculus. pplying it to the aboe equation, for any horizontal flat surface Q follows: p Q = g (19) The fourth step is frequently obtaining the scalar components of the resulting ector equation. To this purpose we multiply the aboe result with unit ector e, it follows: p Q = g ; with: = Q z an: z = h for the hyrostatic pressure we obtain: p = g z = g h () Direct etermination of the force suspening the macroscopic sphere from mean alue theorem applie to eq. (17) is not possible, because pressure uner the integral oer sphere surface aries with the epth z respectiely with h. In such cases it is most conenient to use the auss-ostrograski theorem. It follows: g p g p (1) p e e Y e g z g x y z () The resulting pressure graient p g is constant in the whole flui olume an thus also on all surface parts of the submerge sphere. pplying the MT, now we obtain: g p g g (3) This result may easily be generalize for an arbitrary shape of a submerge boy. We obtain: weight of a boy submerge in a flui (force ) is apparently reuce by the weight of the isplace flui, q.e.. Extening the aboe example to a system of multiple macroscopic particles we continue with analysis of: II. ixe an fluiize bes, igs. 6 an 7 below. L PB 1 ig. 6 chematic representation of particle be at rest. Particles on the siee are supporte with total force equal to total particle weight less weight of the isplace flui. The conenient first step of analysis of an iealize fluiize be starts with etermination of forces acting on particles. These result from equation (3) applie to particles. It reas: (5) t p Q PB = s p g p p p p t η Thereby p is the olume fille by the particles; p is the ensity of the particles; p is respectiely the total surface of all particles regare here as total wette surface. Thus: W = P ; enotes the support forces acting on particles resting on the supporting siee. or the upwar irection follows [7, 11, 1]: g - 6 -

7 (p ) g p kb(re) η W R rel HP (6) Thereby is the flui ensity. R HP is the hyraulic raius estimating the bounary layer thickness in the pores between the particles [8, 11, 1]: R HP = ε / W. The aboe simplifie form of eq. (5) is ali for particle phase for situations represente in ig. 5 uring all perios of time in which the aerage absolute elocity of the particles w g along the graity ector anishes i.e. : w g = an rel =. The first term on the right sie of the aboe eq. (6) represents weight of the particles reuce by weight of the isplace flui (rchimees law, aboe). The secon term escribes the iscous rag force. The interstitial flow elocity in spaces between the particles is controlle by the flow intensity Q an by the geometry of the interstitial flow: Q = ε (7) Where is the total cross-section of the column an ε represents the porosity of the consiere particle ensemble. low intensity Q efines seeral operation moes of the system: a) Particle be at rest is characterize by the conition of total particle weight less weight of the isplace flui being larger than the rag exerte by the flow i.e., in respect to eq. (6) when: (8) b) Unstable state of the be beginning when the flow rag approaches the total weight of the particle be iminishe by the weight of the isplace flui, so that the supporting force (eq. (6)) anishes (s. ig. 8 further below). The interstitial flui elocity in spaces between the particles is still smaller than the aerage absolute seimentation elocity of the particles w g. The porosity of the particles ε aries locally between porosity of fixe be ε B an 1. Then following conitions preail: = ; < w g ; ε B ε 1 (9) c) teay state begins when the interstitial flui elocity reaches the seimentation elocity of single particles w g throughout the whole particle be at porosities ε just starting to excee the porosity of a fixe be ε B. This operation moe is approximate by equilibrium of rag an weight of single particles ( =). Iniiual particles start moing freely in the flow as shown in ig. 7 below builing together a macroscopic moel of a liqui (compare ig. 3 an 7). L B w g 1 Q PB = ig. 7 luiize particle be. Particles are separate from each other by the flow an moe freely aboe the supporting siee. The aerage seimentation elocity of the particles (black arrow w g ) approximately equals the aerage upwar interstitial flui elocity (blue arrow). low intensity through the fluiize be Q B is larger than flow intensity through the particle be at rest Q B > Q PB. Length L B excees length of the particle be at rest L B > L PB. Porosity of the particles ε in steay state of a fluiize be is estimate by an approximately continuous function of the flow intensity Q an of characteristic imensions an elocities in the column as: Q ε (3) Wg Wg Porosity of the fluiize particle layer increases linearly with increasing flow intensity Q from porosity of the fixe be ε B till the alue of 1. In respect to the calculation of pressures in the column it is to be remarke: - orces releant for the behaiour of the particles remain constant uner steay - 7 -

8 operation conitions allowing simple analysis an representation of results (ig. 8 below) () i () i L () i Q() i ig. 8: (i): force exerte by the particles on the supporting siee (rchimees principle). (i): Drag force on the particles in the fixe be an in steay state of the fluiize be (green line in an after the grey block of unstable operation moe), L (i): iscous rag force exerte by the flow on the particles. L (i) rises till unstable moe (grey block region) is reache. In the unstable region anything may occur. Re interrupte line shows iscous rag in the unstable be increasing till be breakup occurs (abrupt fall of the interrupte L (i) line to (i) leel). - Calculation of pressures relates forces to the magnitues of the cross-sections aailable to the flow. ree cross sections epen on the porosity. Thus pressure is a function of porosity an flow intensity: p = f(ε(q)). It may also be calculate by relating forces to the column cross section. This introuces the possibility of multiple interpretations of principally ientical results (s. ig. 9 below) p.l ( i) p.l ( i) p. ( i) p. ( i) 5..4 Q ( i).57 ig. 9 Calculate pressures: two pressure alues (Δp L (i) an Δp Lε (i) result from a single force alue L (i) or (i) of ig. 8 aboe. Pa Pressure rop fluiize be luiization point luiization point after be break up ig. 1 Col test of a fluiize be esigne for processing of woo splitters for regeneratie energy generation. (By courtesy of Thomas iegenhein, c.o. rank Behrent) serious problem with measurement results in fluiize bes is that they frequently look ery much ifferent than theoretical preictions base on iealize assumptions. Causes of eiations from theoretical preictions may be manifol. Besies inhomogeneity of flow an particle parameters many problems are associate with electrostatic loas acquire by macroscopic particles uring processing. urther, other processes like e.g. coalescence, air cleaning etc. require taking uner account electrostatic forces, which besies etermining the motion of iniiual molecules (ig. 1) an parameters of macroscopic flows at micro scale are in similar way highly releant at macro scale for the behaiour of macroscopic particles, or for esign of engineering processes. Unerstaning of releant phenomena oft cannot be limite to their single scale: micro or macro. Conclusions: Require is a physically coherent, ali in a broa range of scales, yet aequate for interisciplinary eucation possibly simple escription of all phenomena relate to the motion of matter. Experimental eience gathere in corpuscular physics [3, 15 18], in kinetic gas theory [, 3], as well as in classical astrophysics [6], - 8 -

9 confirms Newton s law as the simplest tool suite to escribe the non-relatiistic motion of matter generally. On the basis of author s earlier experience [19] an application examples [7 13] the submitte paper [] proposes use of Newton s law in integral notation together with conseration laws as explicit reference stanar an starting point of analysis of all phenomena, which may be iscusse in terms of ynamics of isperse systems. References [1] Hughes, W..; Brighton, J..; lui Dynamics; chaum s Outline eries in Engineering; Mcraw-Hill Book Co.; 1967 [] Reif,.; tatistical Physics; Mcraw-Hill Book Company; 1967 [3] Resnick, R; Halliay, D.; Physics; John Wiley an ons, Inc.; 1966 [4] efferman; Ch.; L.; Existence an smoothness of the Naier-tokes equation; Princeton; NJ ; [5] Hirsh, C.; Numerical Computation of Internal an External lows; Elseier; 7 [6] Hoyle; ; stronomy an Cosmology; reeman an Co.; 1975 [7] Krol, M.; Preiction of relatie particle flui isplacements in flowing suspensions at meium particle olume fractions by means of a simple stochastic moel; 4 th Miami Int. Conf. on: Multiphase Transport an Particulate Phenomena; Miami, U; 1986; Multi-Phase Transport an Particulate Phenomena; ol. 4; Hemisphere Publishing Corporation an pringer Publishing House (1988). [8] Krol, M.: Pressure rop in porous meia at arbitrary Reynols numbers; European erosol Conference; Helsinki; ept. 1995; [9] Krol, M.: luiynamik Dynamik isperser ysteme; 1997/98 [1] Król, M. :Equation of Motion of a compressible lui erie from the Kinetic Theory of ases an Newton's econ Law IMP PN Rep.No. 3195/3 an IMP PN Rep.No. 3197/3 [11] Król, M.; Physics of solis, liquis an gases; Lectures (W); Polan; 5-1 [1] Krol, M.; Physikalische runlagen er Ein- un Mehrphasen-trömungen; Lectures at Technical Uniersity of Berlin; ermany 6-11 [13] Krol, M.: Momentum Exchange as common physical backgroun of a transparent an physically coherent escription of Transport Phenomena; Begell House Inc. Proceeings of the International ymposium Turbulence, Heat an Mass Transfer 6 ;9. [14] iegenhein, Th.; ergasung on Holz in einer Wirbelschicht mit anschließener katalytischer Reinigung; Term tuie; achgebiet Energieerfahrenstechnik un Umwanlungstechniken regeneratier Energien; c/o rank Behrent; Technische Uniersität Berlin; 1. [15] Wichmann; H.,Eyin; Quantum Physics; Berkeley Physics Course olume 4; Mcraw Hill Book Company; 1971 [16] Purcell; M. Ewar; Electricity an Magnetism; Berkeley Physics Course olume ; Mcraw Hill Book Co.; 1965 [17] Pauling; Linus; Pauling; Peter; Chemistry; W. H. reeman an Co.: 1975 [18] Marsen,J.E.; Tromba,.J.; ector Calculus; reeman an Co.; 1976 [19] Krol, M.; Experimentelle Untersuchung er Partikelbewegung bei hohen eststoffkonzentrationen in turbulenten Mehrphasenströmungen; Ph. Thesis; Uniersity of Kaiserslautern; 1984 [] Krol, M.; ignificance of the microfluiic concepts for the improement of macroscopic moels of transport phenomena; paper submitte to the 3 r Micro an Nano lows Conference, Thessaloniki, reece;