By completing this chapter, the reader will be able to:

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1 hater 4. Mechanics of Particles Particle mechanics governs many rinciles of article measurement instrumentation an air cleaning technologies. Therefore, this chater rovies the funamentals of article mechanics. Because article mechanics involves both the articles (which are treate as iscrete subjects) an their carrying fluis (which are treate as a continuous meium), article mechanics is inevitably associate with flui mechanics. Unlike flui mechanics, in which the rimary concerns are mass an energy transfer carrie by the flui itself, article mechanics concerns the mechanical behaviors of the articles relative to the carrying flui. These relative behaviors inclue the relative velocity, article Reynols number, terminal velocity, relaxation time an stoing istance. Two unique stanarize arameters are introuce an will be wiely use in this book: article Reynols number an aeroynamic iameter. The former stanarizes the flui conitions aroun the article of concern, an the latter stanarizes the article sizes base on their aeroynamic behavior. By comleting this chater, the reaer will be able to: Determine Reynols numbers for flui (Re) an for articles (Re ), exlain their imlications to flow conitions an rags on articles. Aly Newton s resistance law to articles at any flow conitions, i.e., a wie range of Re, Stokes region an beyon. Aly Stokes law to article mechanics, an unerstan that Stokes law is just a articular case of Newton s resistance law. The majority of inoor air articular matter transortation falls into Stokes region. Determine the following mechanical roerties of articles by means of either analytical or exerimental methos, or a combination of the two, eening on the availability of variables; o unningham sli correction factor, c, o Terminal settling velocity, V TS, o Mechanical mobility, B, o Dynamic shae factor, χ, o Aeroynamic iameter, a, o Relaxation time, τ, o Stoing istance, s. Of the above roerties, aeroynamic iameter an terminal settling velocity are erhas the most imortant an commonly use, because a is a stanarize escrition of the article, an the V TS can be relatively easily measure. The rest of the mechanical roerties then can be erive. 4. Reynols Numbers for Fluis an Particles General flui ynamic equations (continuity, energy conservation an momentum) inclue the shear stress, which is a function of the viscosity. These general equations are comlicate nonlinear artial ifferential equations, an usually there are no general solutions. To best All right reserve

2 unerstan the ynamic roerties an the behavior of a flui, Osborne Reynols (883) investigate the ynamic similarity of ifferent flows. Two flow cases can be consiere to be ynamically similar when they are geometrically similar an have a similar attern of streamlines. Geometric similarity refers to the corresoning linear imensions that have a constant ratio. Streamline similarity refers to the ressures at corresoning oints that have a constant ratio. Reynols foun that the imensionless grou, UL/η, must be the same for two similar flows, where is the ensity of the flui, U is a characteristic velocity, L is a characteristic length an η is the viscosity of the flui. This imensionless grou is now calle the Reynols number: U L Re (4-) η Of the four variables in etermining the Reynols number, the characteristic length, L, is the most ifficult variable to efine for a oorly efine flow such as most inoor air flows. However, the other three variables,, U an η can be irectly measure. For most flows with hysical bounaries, the characteristic length can be estimate using the following equation: L 4 A w (4-) P w where A w an P w are the wette area an wette erimeter, resectively. The ratio of A w /P w is calle the hyraulic raius. For a circular uct, the hyraulic raius is only half of its geometric raius. From Eq. 4-, the characteristic length of a circular uct is the uct iameter D. Since the initial force (F I ) of an element is roortional to its mass (which is roortional to the ensity,, an its imensions) an velocity, an the shear or frictional force (F τ ) is roortional to the viscosity, the Reynols number can be viewe as the ratio of inertial force to shear or frictional force acting on an element of the flow in steay state. F I Re (4-3) F τ The inertial force, accoring to the Newton s secon law, is equal to the change of momentum of the flui element, ( mvt ) m FI Vt + t t V m t t (4-4) where m is the mass of the flui element, an V t /t is the total acceleration of the flui element. The total acceleration, V t /t contains two comonents: an acceleration ue to the change in the total flow system, U/t, an an acceleration resulting from the change of ositions of that flui element along the moving irection, UU/x. The UU/x is roortional to the flui velocity an the change of velocity with osition. For examle, when the flui flows aroun an elbow All right reserve

3 tube, the acceleration of a flui element is relate to its velocity an the location in the elbow. Thus, the inertial force can be exresse as F m U U Vt + m ( U ) (4-5) t t x I + For incomressible flow at steay state, m/t0, an U/t 0. The mass of the flui element is the ensity () times its volume, which is roortional to its L 3, where L is a characteristic length to reresent any linear length such as the iameter of a tube, a istance of travel, or the iameter of a shere. Thus, the inertia force can be written as U L U (4-6) L F I 3 The frictional force, F τ, is roortional to the viscosity, η, the surface area, A, an the velocity change erenicular to the moving irection, U/y, where y is a irection erenicular to the moving irection, x. Noting that the characteristic length, L, reresents any linear length, the area A L, an y L. Thus, the frictional force can be exresse as U Fτ η L (4-7) L Substituting Eqs. 4-6 an 4.7 into Eq. 4- yiels Eq. 4-. The nature of a given flow of an incomressible flui is characterize by its Reynols number. Reynols numbers for flui flow are commonly ivie into three regimes: Re <,000 for laminar flow;,000 < Re < 4,000 for transient between laminar an turbulent flow; an Re > 4,000 for turbulent flow. For large values of Re, one or all of the terms in the numerator are large comare with the enominator, i.e, large volume or exansion of the flui, high velocity, great ensity an small viscosity of the flui, or a combination of all of the above. A large Re (>4,000) imlies that the inertial force reominates the flow while the frictional force is often negligible. A large Re inicates a highly turbulent flow with losses roortional to the square of the velocity. The turbulence may be large scale, such as an air jet that encounters an obstacle an swirls in a room. Such large scale turbulence creates large eies which carry most of the mechanical energy. The large eies generate many smaller eies that raily convert the mechanical energy into irreversibilities via their viscous action. The turbulent flow may also be small scale, such as the air flow near the exit of a iffuser. A small scale turbulent flow has small eies an small fluctuations in velocity with high frequency. Generally, the turbulence intensity increases as the Reynols number increases. For intermeiate values of Re (,000 < Re < 4,000), both inertial an frictional effects on the flow are imortant. The changes in viscosity result in changes in velocity istribution an the All right reserve 3

4 resistance to the flow. Examles of such a transient flow are the room air at some bounaries such as along a wall or aroun a human boy. For small values of Re (Re <,000), the flow is sai to be laminar. Because the gases, incluing air, usually have very small viscosities (the enominator of the Reynols number), the Re values are usually large. For inoor air movement, the characteristic length, L, is large (in orer of meters). Even at very low air velocities such as 0. m/s, the room airflow is still highly turbulent. On the other han, some inoor air alications can be laminar flow. For examle, an air iffuser with an oening one centimeter wie an ischarge air velocity of two m/s has a Re value of,36, which is a laminar flow uner stanar air conitions. There are many Reynols Numbers in use toay eening on the situations of flow an the roblems of concern. For examle, the motion of water in a ie may be characterize by UD/μ, where is the ensity of water, U is the velocity of water, D is the iameter of the ie an η is the viscosity of the water. Another examle is the Reynols number of a shere moving in a flui, UD/η, where is the ensity of the flui, U is the velocity of the shere, D is the iameter of the shere an η is the viscosity of the flui. Although the exressions of the Reynols numbers are the same in both examles, the hysical arameters an the values of Re may be quite ifferent. The ifference is a result of the flow conitions an the roblems of concern. In this textbook, there are two ifferent Reynols Numbers: one for flui (usually air) flow, Re, an one for airborne articles, Re. The subscrit istinguishes the Reynols Number of articles from that of flui (Figure 4-). U U Figure 4-. Streamlines of a flui flow between two arallel walls with a shere in the flui. The flui velocity is U an the shere has a relative velocity, V r, to the flui flow. The Reynols Number for flui is escribe in Eq. 4-, where is the ensity of the flui, U is the velocity of the flui, L is a characteristic length of the flow such as the iameter of an air uct, an η is the viscosity of the flui. The Reynols Number for a article moving in a flui is exresse as Vr Re (4-8) η All right reserve 4

5 where is the ensity of the flui, V r is the relative velocity of the article to the flui, is the iameter of the article an η is the viscosity of the flui. The relative velocity of a article, V r, relative to the flui flow velocity, U, can be calculate as V r r U r U ( U U ) + ( U U ) + ( U U ) ) ( x x y y z z (4-9) where U r an U r are velocity vectors for the article an the carrying flui, resectively. Subscrits x, y an z are comonents for a carinal coorinate. When the motions of the article an the carrying flui are on the same line, but not necessarily in the same irection, the relative velocity becomes V r U U (4-0) If V r is a ositive value, it is the same irection as the U ; an if V r is negative, the irection of V r is oosite to the irection of U. There is often some confusion about the calculations of Re for the flui flow an the Re for a article in the flui flow. The following items may be helful to clarify this confusion: The ensity () an viscosity (η) of the flui are the same for flui flow Reynols number an article Reynols number. For flui flow Reynols number, the characteristic velocity is the flui velocity U, an the characteristic length L is the characteristic length of the flow fiel such as the iameter of the tube containing the flui flow. For article Reynols number, the characteristic velocity is the relative velocity of the article to the surrouning flui (V r ), an the characteristic length L is the iameter of the article ( ). Particle Reynols number characterizes the mechanical roerties of a article in a flui flow. These article mechanical roerties are iscusse in etail in later sections of this chater. Particle Reynols numbers have four regions: Laminar (which is more commonly calle Stokes region), transient, turbulent an Newton. Reynols numbers an tyical characteristics of each region for article motion are escribe below. Stokes region: Re <. Flui flow aroun the article is laminar. Frictional force exerte on the article is reominant, an the inertia force is negligible. Transient region: < Re < 5. Turbulence starts to occur aroun the article. Both inertial force an frictional force are imortant to the behavior of the article. Turbulent region: 5< Re <,000. Flui flow aroun the article is turbulent. Drag coefficient of the article ecreases as the Re increases. Newton s region: Re >,000. Flui flow aroun the article is highly turbulent. Drag coefficient of the article remains aroximately constant. In summary, the Reynols number has the following roerties: First, it is an inex of the flow regime that serves as the benchmark to inicate if the flow is laminar or turbulent, or how the All right reserve 5

6 resistance to a article changes. Secon, because it can be viewe as the ratio of inertial force to frictional force, its value is imortant for etermining which arameters are more imortant to the flow. Thir, it rovies a means to stuy a similar flow using a geometrically similar exerimental aroach. Examle 4-. The cross section of the hatch cabinet is one meter wie an 0.5 meters high. Air velocity through the cabinet (erenicular to the cabinet cross section) is 0. m/s. A newborn chicken walks own stream (the same irection as the air flow in the cabinet) along the length of the hatch cabinet at a see of 0.05 m/s. The chicken can be aroximate as a shere with a iameter of 5 cm. The ensity of air is. kg/m 3 an the viscosity is.8x0-5 N s/m. Determine: (a) Reynols number for the airflow in the hatch cabin. (b) Reynols number for the chicken. (c) What woul the Reynols number be if the chicken were a 50 μm article? Solution: (a) The characteristic length, L, is calculate from the with an height of the hatch, L 4A P w w 4 ( 0.5) ( + 0.5) 0.67 ( m) U L Re 5 η.8 0 8,884 The airflow in the hatch cabin is turbulent because Re 8,884 >,000. (b) Since the chicken is walking with the same irection of the airflow in the cabinet, the relative velocity of the chicken with the airflow is V r 0.05( m / s) 0. ( m / s) 0.5( m / s) The - sign of V r inicates that the irection of V r is oosite to the chicken s moving irection. Vr Re 5 η The airflow aroun the chicken is turbulent because Re 497 >. (c). If the chicken was a 50 μm article, the Reynols number for the article Vr Re η.8 0 The airflow aroun the article becomes laminar because Re 0.497<. All right reserve 6

7 4. Newton s Resistance Law One of the most imortant mechanical roerties of articles is the resistance of the carrying gas in which the article is traveling. In analyzing the resistance to the article motion, steay-state an streamline motion is assume. This assumtion is esecially vali for small articles such as those that are airborne. The rationale will be justifie in this chater by the short relaxation time an stoing istance of airborne articles. For airborne articles tyically smaller than 00 μm, the article motion in the air almost instantaneously becomes steay-state, regarless of its initial conitions. Newton s resistance law alies to article motion with high article Reynols numbers (Re > 000). In such a high Re region, the reominant force that governs the article motion is the inertial force. Other forces such as frictional force an gravity become negligible. Originally, Newton reasone that the resistance encountere by a cannonball travelling through air is a result of the acceleration of the air that has to be ushe asie to allow the shere to ass through (Figure 4-). At time t+t V r F D A A t time t Figure 4-. The resistance encountere by the shere is the momentum changes of the gas being ushe asie by the shere. Alying Newton s secon law, the resistance force or the rag force is roortional to the change of momentum of the relace gas: F D K ( m g t V r ) K V r m t g + m g V t r (4-) where K is a constant of roortionality, m g is the mass of gas that has been islace by the motion of the shere, an V r is the velocity of the shere. Newton initially thought that the K was a constant that i not vary with the velocity. The change of mass with resect to time, m g /t, is equal to the volume of gas that has been ushe asie in one secon, i.e., All right reserve 7

8 m t g π g AVr g Vr (4-) 4 where A is the cross-sectional area of the shere (Figure 4-). Because the article motion is steay state, V r /t 0. Substituting Eq. 4- into 4- gives F D mg π K Vr K g Vr t (4-3) 4 Eq. 4-3 is the restricte form of Newton s resistance equation, which is only vali for article motion with very high values of article Reynols number (Re >,000). This is true because the equation is erive base on the inertial force without consiering the viscous effect. At lower values of Re, the frictional force becomes more imortant. Thus, the roortional constant, K, is a constant only for high Re. A moifie Newton s resistance equation alicable for entire Re ranges can be written as: F D π 8 D g Vr (4-4) Where D is the rag coefficient that is eenent uon the article Reynols number. The rag coefficient, D, is the most imortant roerty in the mechanics of article motion an flui ynamics, as most alications in real life are emloye either to overcome or utilize the rag cause by the flui. Some examles of these alications are esigning airlanes, shis, an air cleaning evices to searate airborne articles, which will minimize the rag. Parachutes, on the other han, are esigne to increase the rag. Drag force will also vary with the article shaes. In this textbook, the articles we eal with are either sheres or have been normalize (geometrically or aeroynamically) as equivalent sheres. Figure 4-3 shows the rag coefficient of sheres with resect to the shere Reynols number. Drag coefficient for other shaes can be foun from many flui ynamics references. Drag coefficients of sheres in Figure 4-3 vary with Reynols numbers in a highly nonlinear fashion. However, for most engineering alications, the rag coefficient can be calculate base on four regions of article Reynols number: low Reynols number region ( Re < ); transient region ( < Re < 5); turbulent region (5 < Re <,000); an high turbulent region (Re >,000). Of the four, rag coefficient in the transient region is the most comlicate an several equations were roose base on regressions of exerimental ata. 3, 4 Most of these equations are sufficiently accurate an generally within 4% for Re < 800 an 7% for Re <,000. All right reserve 8

9 Drag coefficient, D D Re Stokes law Re Re 6 Transition region D 4 D + Re D 0.44 Newton s law ( 0.5 Re ) Turbulent region Reynols number, Re Figure 4-3. Drag coefficient of sheres versus the article Reynols number. D 4 (for Re < ) (4-5) Re D Re 6 Re (for < Re <5) (4-6) D ( Re ) (for 5 < Re <,000) (4-7) Re 0.44 (for Re >,000) (4-8) D For airborne articles, Re values rarely excee. Particles in some extreme situations, such as in a ust storm or an erution of a volcano, may fall in the lower range of transitional Reynols numbers. In inoor air quality alications, although the room air is almost always highly turbulent (Re >,000), the article Reynols numbers are almost always smaller than, which is well within the Stokes Region. 4.3 Stokes Law The general Navier-Stokes equation for incomressible flow is an alication of Newton s secon law, i.e., the change of momentum of a flui element is equal to the total external forces exerte on the flui element of concern. For airborne articles, the gravity an the buoyancy are All right reserve 9

10 negligible comare with other external forces such as the frictional force an static ressure (hea ressure). Thus, the general Navier-Stokes equation for each of the x, y an z irections can be simlifie as ( U U V W P U U U + U + U + U ) + η ( + + ) (4-9) t x y z x x y z ( V U V W P V V V + V + V + V ) + η ( + + ) (4-0) t x y z y x y z ( W U V W P W W W + W + W + W ) + η ( + + ) (4-) t x y z z x y z where U, V an W are velocity comonents corresoning to the x, y an z irections, resectively. P is the ressure. - P/ x inicates that the ressure force is oosite to the irection of the motion. Eqs. 4-9 to 4- are generally unsolvable because they are nonlinear artial ifferential equations. Stokes law is a solution to Eqs. 4-9 to 4- base on the following assumtions: ) The inertial force is negligible comare to the frictional force. This assumtion is true for article Reynols numbers smaller than. Thus, the term UU/x 0, or the carrying flui is incomressible. ) The article motion is in steay state, i.e., U/t 0. 3) The article is rigi an there is no other article nearby. 4) The velocity of flui at the article surface is zero. The above assumtions are accurate for most inoor air situations. First, the incomressible assumtion (UU/x 0) refers to the fact that the air near the articles oes not comress significantly when the articles flow through it; it oes not imly that the air is incomressible. A moving article can only significantly comress the air when it reaches the see of soun, which is usually not the case in an inoor environment. Secon, the article motion in the air can achieve the steay state almost instantaneously. This will be iscusse in etail in later sections (relax time an sto istance). Thir, for articles that have a much larger ensity than the air, the articles can be consiere rigi. For examle, the settling velocity of a water rolet in air is only 0.6% faster than reicte by Stokes law, inicating that the eformation of a article in the air flow an the effect on the Stokes law is negligible. Fourth, only a wall within ten times of the article iameter will affect the rag force of the article. Generally, only a small ortion of articles can get near that istance in an inoor environment. Thus, the nearby wall effect can be neglecte. For examle, consiering an extremely high article concentration of,000 articles/ml with a mean article iameter of 0 μm, the mean istance between two articles is,000 μm 00 times its mean iameter. The assumtion of zero-velocity at the article surface can cause significant error for small articles. This error can be correcte by the sli correction factor, which will be iscusse in the next section. With the above assumtions, an consiering the article motion in the x-irection only, Eq. All right reserve 0

11 4-9 can be reuce to only two terms: ressure an shear stress cause by viscosity, both oosite to the irection of the article motion (or the flui motion relative to the article): U U U P η ( + + ) 0 (4-) x y z x Eq. 4- is linear an can be solve if the bounary conitions are known. The rag force of the flui exerte on the shere is the sum of the ressure an frictional forces. Integrating the ressure stress an frictional force over the entire surface of the shere gives the rag force exerte on the shere: U U U P FD η ( + + ) S x y z x S S S τ sin θ S P conθ S (4-3) S S y r sinθ r sinθ φ r θ S θ θ x φ φ z Figure 4-4. A sherical coorinate system where an element surface area efine by the raius, r, an two angles, θ an φ. All right reserve

12 The first term at the right sie of Eq. 8-5 is cause by friction (or shear or viscous force), an the secon term is cause by ressure. If the shear stress an ressure istribution are known, Eq. 8-5 can be solve. 5, 6 Stokes foun the following shear stress an ressure istribution exerte on a shere surface with a iameter in a sherical coorinate system, r, θ an φ (Figure 4-4): τ 3η V r r 4 sinθ (4-4) 3η V 0 r P P r cosθ (4-5) where P 0 is the ambient ressure. Aarently, when r >> /, P P 0, which means that the ressure cause by flui is zero. In a sherical coorinate as shown in Figure 4-4, S r sinθ θ φ (4-6) where 0 < r /, 0 θ π, 0 φ π. Substituting Eqs. 4-4 to 4-6 into Eq. 4-3, noting that r /, an integrating, yiels: F 3η Vr 3η Vr sinθ S ( P0 cosθ ) S D S S π π 3 V r 3 ( sin θ )( ) θ φ η π π 3η V ( r cos θ P0 )( ) sinθ θ φ π ηv + π ηv r r 3π ηv r (for Re < ) (4-7) Eq. 4-7 is the Stokes law, stating that the rag force exerte on a article is roortional to the viscosity of the flui (η), the relative velocity of the article to the flui (V r ) an the iameter of the article ( ), for a article Reynols number smaller than unity. The first term at the right sie is cause by friction an the secon term is cause by ressure. In Stokes region, the ressure contributes one thir of the total rag, an the frictional force contributes to two thirs of the total rag force. Now let s revisit the Newton s resistance law, exresse by Eq Because the Newton s general resistance equation (Eq. 4-4) alies to the entire range of Re values, it also hols for Re <. Equating Eqs. 4-4 an 4-7, gives All right reserve

13 D π 8 V g r 3πηV (4-8) r Solving for D, noting that Re g V r /η, verifies Eq. 4-5, D 4 (4-5) Re Eq. 4-5 reresents the straight-line ortion in Figure 4-3 where the rag coefficient is inversely roortional to the article Reynols number. From Newton s to Stokes law, there is a change of relationshis between the rag force an its affecting variable, V r,, g an η. Newton s resistance equation contains the roerties of inertia, g, an the rag force is roortional to the squares of velocity an the shere iameter. While Stokes law contains the viscosity, η, the rag force is roortional to the velocity an shere iameter. The eenent variable of the rag force graually changes from V r to V r an from to as the Re ecreases. This change is reresente by the straight line ortion in Figure Sli orrection Factor One of the assumtions mae uring the erivation of Stokes law was that the velocity of flui at the surface of the article was zero. This assumtion is true for large articles. For small articles, the actual rag force is smaller than the value reicte in Eq It aears that very small articles sli in the flui. In fact, very small articles o sli with resect to the flui. This occurs because when the article is sufficiently small, esecially aroaching the size of the mean free ath of the gas, the robability of imacting with the gas molecules is smaller; hence, the rag force is smaller. To take this sli effect into account, the Stokes law can be moifie by a sli correction factor, c. Thus, the Stokes law becomes F D 3πηVr (for Re < ) (4-9) c The sli correction factor was first erive by unningham in 90 for articles larger than 0. μm. Thus the sli correction factor is often calle the unningham correction factor. Allen an Raabe roose a sli correction factor equation for oil rolets an for soli articles for all article sizes within.% of accuracy. 7, 8 In summary, the sli correction factor can be calculate using the following equations: c.5 λ + (for 0. μm) (4-30) All right reserve 3

14 λ c ex (for < 0. μm) (4-3) λ where λ is the mean free ath of the carrying flui. For air at stanar conitions (P 0.35 kpa an T 0 o ), λ μm. Mean free ath for ifferent gases can be calculate an is iscusse in the next chater. Eq. 4-3 can be alie to all sizes of articles. Because of its simlicity, Eq is recommenefor articles larger than 0. μm. For articles smaller than 0. μm, Eq. 4-3 must be use. The sli correction factor increases raily as the article iameter ecreases. For examle, uner stanar conitions, c.067 for a 0 μm article in air, the sli effect is only.67%. The sli effect in air increases to 6.7% for a μm article an 67% for a 0. μm article. Therefore, sli correction factor must be use for small articles, tyically smaller than 3 μm. The sli correction factor for a 3 μm article is.055, reresenting a 5.5% error if it is not being consiere. Substituting the c.055 into the Stokes law, the rag force for the 3 μm articles is 95% of that without a sli effect. For most inoor air roblems, sli effect is usually not a concern for articles larger than 3 μm. Tyical values for c versus article iameters carrie by air streams are lotte in Figure μm 5 Sli correction factor (c) μm -0 μm Particle iameter (μm) Figure 4-5. Sli correction factor at stanar atmosheric conitions, lotte using Eqs an 4-3. All right reserve 4

15 Generally, inoor air is at or near the stanar conitions, i.e., near the sea level ressure of 0.35 kpa with a temerature of 0 o. Uner such stanar conitions, Eqs an 4-3 are sufficiently accurate for most alications without consiering the ressure effect. When ambient conitions change substantially, such as when the air is uner vacuuming or ressurizing, the effect of the ressure an/or temerature has to be consiere. For a given iameter of a article, the sli correction factor is only eenent uon the mean free ath, λ. From the Boyal s ieal gas law an gas kinetics, the mean free ath of a gas is inversely roortional to the molecule concentration, n. The molecular concentration, n, is roortional to the absolute ressure an inversely roortional to the absolute temerature. Denoting λ 0, n 0, P 0 an T 0 as the mean free ath, gas molecule concentration, absolute ressure an temerature at stanar conitions (P kpa an T 0 93 o K), the mean free ath, λ, an the new molecule concentration, n, at a given absolute ressure, P an temerature, T, have the following relationshis with those of stanar conitions, λ0 λ n n 0 PT P T 0 0 (4-3) The mean free ath of the gas molecules from Eq. 4-3 is P T λ (4-33) PT 0 λ0 0 Substituting Eq.4-33 in to Eqs an 4-3, noting that P kpa an T 0 93 o K, gives the sli correction factor incluing the effects of ambient ressure an temerature. c λ0 T (for 0. μm) (4-34) P c λ 0 T ex.8 P λ0 P T (for < 0. μm) (4-35) where λ 0 is the mean free ath of the carrying gas molecules at stanar conitions, P is in kpa an T is in K. For air, λ μm. All right reserve 5

16 Table 4-. Terminal settling velocity an mobility for tyical sizes articles with stanar ensity uner stanar atmosheric conitions. (μm) Sli correction coefficient, c Mobility, B (m/n s) Terminal settling velocity, V TS (m/s) Examle 4-. At the to of a mountain, the atmosheric ressure is measure as 70 kpa, an temerature as 0 o. What woul the error of the sli correction factor be for a 0.3 μm article if the effect of the ressure an temerature were ignore? Solution: Without consiering the effect of ressure an temerature, λ When the effect of ressure an temerature is taken into account, λ T P The error of sli correction factor, ε, cause by ignoring the effect of temerature an ressure: ε c c c % 00% 9%.69 Settling velocity an Mechanical Mobility Settling velocity of a article is efine as the terminal free-fall velocity of the article after it is release in still air. The terminal settling velocity is a constant for a given size of article. When a article is release into still air, the article quickly aroaches its terminal velocity an settles own at a constant see. Aarently, in Stokes region, the rag force exerte on the article is equal to the relative gravity forces of the article to the gas, i.e., All right reserve 6

17 F D F m g m g (4-36) G g 3πηV c r π ( ) 3 g g (4-37) 6 where F G is the relative gravity of the article to the gas, m is the mass of the article, m g is the mass of the gas with the same volume as the article, an g 9.8 N m/s. The mass of gas reresents the effect of buoyancy which in ractice science can often be neglecte because the is much greater than g. For examle, a water rolet has a ensity of 800 times greater than the air. Neglecting the buoyancy effect of the air only gives 0.% error when calculating the settling velocity. For inoor air quality roblems, the buoyancy effect can be neglecte. The settling velocity of an airborne article, V TS, which is the relative velocity of the article to the air, can be erive from Eq g c V TS (for Re <.0) (4-38) 8η Particle terminal settling velocity increases raily with the article size as it is roortional to the square of the article iameter. The ensity of the gas within which the article becomes airborne has no significant effect on the settling velocity. For articles of 3 μm in iameter, the effect of the sli is aroximately 5% on its terminal settling velocity, an for articles of μm in iameter, the error increases to 7%. In most cases, we will neglect the sli effect for articles larger than 3 μm in iameter an still have a 95% or better accuracy. However, c shoul be consiere for articles smaller than 3 μm because the errors become significant. An imortant factor affecting the terminal settling velocity is the gravitational acceleration, g, which is roortional to the external force, m g, which causes the settling. Similarly, when a article is subjecte to other kins of external forces such as centrifugal or electrical, the acceleration, a e, can be calculate as F e a e (4-39) m where F e is the external force in Newtons, an m is the mass of the article in kg. For examle, the centrifugal acceleration a e is V T /R, where V T is the tangential velocity an R is the raius of the circular motion. The terminal velocity of a article in a centrifugal force fiel is V T cae cvt (for Re <.0) (4-40) 8η 8η R Note that V T here is the tangential article velocity, not the relative velocity of the article to the flui, V r. The tangential article velocity is equal to the flui velocity of the circular motion in a All right reserve 7

18 steay flow. The terminal velocity cause by centrifugal force can be many times higher than the terminal settling velocity cause by gravity. For examle, if a article has a circular motion at a tangential velocity of 8 m/s with an R 0.3 m, the terminal velocity cause by centrifugal force is.7 times as high as the terminal settling velocity. In this case, the gravity effect may be neglecte. The terminal velocity can be alie to the article searation for article samling an air cleaning technologies an will be iscusse in etail in later chaters. Particle mobility, B, is efine as the ratio of the article velocity relative to the flui (V r ) to the rag force exerte on the article. B V r c (4-4) FD 3π η The article mobility is also calle mechanical mobility to istinguish it from other mobilities such as electrical mobility. The mobility has a unit of m/n s. The article mobility is an inicator how mobile the article is. A higher mobility value inicates that either the article has a high velocity relative to the flui, or a small rag force is exerte on the article, or a combination of both. The mobility of the article ecreases as the viscosity of the flui increases, an ecreases as the iameter of the articles increase. In analogy, an airlane is more mobile than a shi of similar size because the viscosity of air is lower than the viscosity of water. A motor boat is more mobile than an oil tanker because the size of the motor boat is smaller than the oil tanker. ombining Eqs. 4-38, 4-39 an 4-4, the terminal velocity of a article is simly the rouct of its mobility an the external force exerte on the article, i.e., VTS Fe B (4-4) where the external force F e m g, when the article is settling in the air; or F e m V T /R as the centrifugal force when the article is in circular motion. Examle 4-3. A grain of concrete ust article is falling own onto the floor through room air. The article iameter is μm an the article ensity is,500 kg/m 3. Assuming the room air is still, etermine the terminal settling velocity, rag force an mobility of the articles. The room air is at stanar conitions. Solution: In orer to calculate the settling velocity, we first etermine the sli correction factor for the μm article. At stanar conitions, air viscosity η Pa s (N s/m ), air mean free ath λ μ m an air ensity. kg/m 3. λ Substituting c values to Eq. 4-30, gives All right reserve 8

19 V g 8η 6,000 ( 0 ) (.8 0 ) c TS ( m / s) Before alying the Stokes law to calculate the rag force, article Reynols number nees to be examine to ensure the article motion is in Stokes region. Re VTS η ( ) Since Re <<, Stokes law alies. F D 3πηV TS c π (.8 0 N 8. 0 Since the article is free falling, 5 9 )( yn 4 )( 0 6 ) F e π m g 6 3 π ( 0 g 6 6 ) 3, N The article mobility: B V F 4 TS e 8. 0 m / N s 4.5 Nonsherical Particles an Dynamic Shae Factor Most articles in ractice, esecially in an inoor environment, are nonsherical. Particle ynamic shae factor is efine as the ratio of the actual resistance force of a nonsherical article to the resistance force of a sherical article that has the same equivalent volume iameter an the same settling velocity as the nonsherical article. χ F D (4-43) 3πηVTS e where F D is the actual rag force exerte on the nonsherical article, V TS is the terminal velocity of the nonsherical article, e is the equivalent volume iameter of the nonsherical article, an c is the sli correction factor for e. Note that the enominator at the right sie of Eq uses V TS of the nonsherical article an thus it is not the actual rag force of the sherical article with e. The actual resistance force of the article with e woul be 3πη e V r, where V r > V TS. The ynamic shae factor is always greater than.0 excet for certain streamline shaes. All right reserve 9

20 Dynamic shae factors for some tyical nonsherical articles can be foun in references (Table 4-). The calculation or irect measurement of ynamic shae factors for irregular articles is very ifficult. Such irregularly shae articles inclue flakes (such as aner) an fibers (such as hair). Fibers have a relatively consistent ynamic shae factor of.06 (Table 4-) with a variation of less than 5%. This constancy occurs because the fiber tens to kee its long axis in alignment with the moving irection, esecially at low Reynols numbers. For flake shae articles, ynamic shae factors are much larger than.0. This is because it is ifficult to kee the long axes of the article in alignment with the moving irection. An examle of a flake shae article with a large ynamic shae factor is a arachute, which falls in air much more slowly than other shaes, even those with the same equivalent volume iameter. Table 4-. Dynamic shae factors for tyical articles. Shae Dynamic shae factor f, χ Axial ratio 5 0 Geometric shaes Shere.00 ube a.08 yliner a Vertical axis Horizontal axis Orientation average Fiber (L/ > 0) b,c.06 lustere sheres b,c chain. 3 chain.7 5 chain.35 0 chain.68 3 comact.5 4 comact.7 Dust Bituminous coal.08 Quartz.36 San.57 Talc e.88 Sources: a Johnson, D.L., D. Leith an P.. Reis, Drag on non-sherical, orthotroic aerosol articles, J. Aerosol Sci., Vol 8, 87-97, 987. b Dahneke, B.A., Sli correction factors for sherical boies III: The form of general law, J. Aerosol Sci, Vol 4,63-70, 973. c Dahneke, B.A., 979. Davis,.N., Particle Flui Interaction, J. Aerosol Sci., Vol 0, , 979. e heng, Y.S., H.. Yeh an M.D. Allen, Dynamic shae factors of late-like articles, Aerosol Sci. Tech, Vol 8, 09-3, 988. f Average over all orientations unless secifie. However, the ynamic shae factor for a secific article can be measure inirectly. Substituting Eq into Eq. 4-36, gives All right reserve 0

21 χ e gc (4-44) 8ηV TS The ynamic shae factor in Eq is a function of its terminal settling velocity, V TS. The equivalent volume iameter ( e ) an article ensity ( ), an the terminal velocity of an irregularly shae article can be measure in a calm air chamber, by recoing its falling istance an the traveling time, an thus the χ can be etermine. Such measurements require secial instrumentation an stringent roceures. 4.6 Aeroynamic Diameter The most imortant hysical roerty of a article is erhas its aeroynamic iameter. In reality, it is extremely ifficult to measure or calculate the equivalent volume iameters an shae factors because of the comlexity of the geometric shaes of articles. Thus, we nee an equivalent iameter that can be hysically etermine to characterize the article roerty an behavior. This equivalent iameter is calle the aeroynamic iameter, a. Aeroynamic iameter is efine as the iameter of a shere with a unit ensity an the same settling velocity as the article of concern. From this efinition an Eq. 4-44, e gce 0 a gca V TS (4-45) 8η χ 8η where ca an ce are sli correction factors for a an e, resectively, 0,000 kg/m 3. Solving Eq for a, gives ce a e (for nonsherical articles) (4-46) ca 0 χ For sherical articles, e an χ, then c a (for sherical articles) (4-47) ca 0 where is the actual geometric iameter of the article, an c is the sli correction factor for. Aeroynamic iameter may be more easily unerstoo an remembere using the following analogy. Dro the article in question an a water rolet (sherical with a unity ensity) in still air from the same height. If the article an the water rolet reach the groun at the same time (so that they have the same settling velocity), the aeroynamic iameter of the article in question is equal to the iameter of the water rolet. In this analogy we assume that the water All right reserve

22 rolet is a shere. In reality this is not true as the water rolet eforms when it is subjecte to the air resistance, thus it is not a true shere. A more accurate escrition of the aeroynamic iameter is shown in Figure 4-6, which emonstrates the equivalent volume an aeroynamic iameter of the same article. For articles larger than 3 μm, the sli effect can be neglecte, thus c for e, a an are equal unity, an Eqs an 4-47 can be written as: a e (for nonsherical articles, e 3 μm) (4-48) 0χ a (for sherical articles, 3 μm) (4-49) 0 An irregular article an its equivalent volume shere e 0 μm 3,000 kg/m 3, χ.3 The aeroynamic equivalent shere ea 5. μm,000 kg/m 3, Aeroynamically equivalent V TS m/s V TS m/s Figure 4-6. An irregularly shae article will behave similarly to the shere with the same aeroynamic iameter of the article. For articles smaller than 3 μm, the sli effect has to be consiere. In Eqs an 4-47, each contains a sli correction factor, ca, which is a function of the aeroynamic iameter, which is also in question. Aeroynamic iameter can be solve exlicitly for articles smaller than 3 μm but larger than 0. μm. Substituting Eq into Eqs an 4-47 an solving for a, gives a 6.35λ + 4 e ce. 5λ 0χ (for nonsherical articles an 0. μm < e < 3 μm) (4-50) All right reserve

23 a 6.35λ + 4 c. 5λ 0 (for sherical articles an 0. μm < < 3 μm) (4-5) Aeroynamic iameter may be quite ifferent from its equivalent volume or geometric iameters. For small articles, sli effect can also significantly alter its aeroynamic behavior, an thus the aeroynamic iameter. These ifferences can significantly alter the oututs of aeroynamically base article measurement instruments. The following examle gives a ersective on how the sli effect affects the aeroynamic iameter. Examle 4-4. Determine the ratio of the aeroynamic iameter to its actual geometric iameter. The article is a 0. μm iameter shere with a ensity of,500 kg/m 3. If the sli effect were ignore, what woul the aeroynamic iameter be? Solution: Because the article is smaller than 3μm, the sli effect must be consiere, c.5λ Substituting into Eq. 4-5, gives a 6.35λ + 4 c. 5λ 0, , μm The ratio of the aeroynamic iameter to the article iameter is a If the sli effect were ignore, the aeroynamic iameter woul be a 0,500 0., ( μm) All right reserve 3

24 omare with the a incluing the sli correction factor, ignoring the sli effect woul introuce a % error in its aeroynamic iameter. Aeroynamic iameter for articles smaller than 0. μm can be solve using iteration methos by substituting Eq. 4-3 into Eq or 4-5. In general, articles smaller than 0. μm ten to behave like gases, for which other article transort mechanisms an behavior, such as iffusion, becomes more imortant than the aeroynamic behavior. Gas roerties are iscusse in next chater. From Eq. 4-45, the aeroynamic iameter stanarizes both the ensity (to be the same as the water ensity) an the shae (to be a shere) in accorance with the terminal settling velocity. Because the V TS is relatively easy to obtain, the aeroynamic iameter becomes a very useful article arameter. In this textbook, article sizes are given by aeroynamic iameters unless secifie. 4.7 Relaxation Time It has been assume in the revious sections that a article reaches its terminal velocity very quickly when it is release into a flui or subjecte to an external force. The quickness is sai to be almost instantaneous. Thus, the terminal settling velocity can be exerimentally etermine by measuring the free fall istance an the time of travel. Now is the time to examine how fast a article accelerates in the air, an how long it takes a article to achieve its terminal velocity. onsier article motion in still air along the vertical irection. The article has an initial velocity, V 0, when t 0, an it is falling own in the air. A free boy iagram of the article is shown in Figure 4-7. There are two forces exerte on the article, gravity, m g, an rag force, F D. Alying Newton s secon law an Stokes law, gives m V ( t) 3πη V ( t) m g (4-5) t c where V(t) is the article velocity at any time. Substituting the mobility, B c /3πη (Eq. 4-4) an V TS m gb (Eq.4-4) into Eq. 4-5, gives m V ( t) B VTS V ( t) (4-53) t Searating variables, Eq can be written as ( V ( t) V ) V ( t) V TS TS t m B (4-54) Integrating Eq with resective range of V(t) an t, All right reserve 4

25 V ( t) V0 V ( t) ( V ( t) V ) V TS TS t 0 t m B (4-55) yiels where V ( t) VTS ( VTS V0 ) ex ( t m ) B t VTS ( VTS V0 )ex( ) (4-56) τ 3 π c c 0 a ca τ m B (4-57) 6 3πη 8η 8η The τ is referre to as the relaxation time of the article. The term of the relaxation time imlies the time erio require for a article to relax from a transient (acceleration or eceleration) state to a steay state (with a constant terminal velocity). Figures 4-7 an 4-8 show the relationshi between the terminal settling velocity an the time. Initial osition t 0 V(t) V 0 F D m V(t) m g Figure 4-7. A article with an initial velocity, V 0, is falling own in air. Before it reaches its terminal velocity, the article has an acceleration V(t)/t. From Eq. 4-57, τ is the time constant of the article neee to reach its terminal velocity when it is subjecte to an external force. When t 0, V(t) V 0, When t, V(t) V TS. Theoretically, V(t) never reaches its terminal velocity. Practically, V(t) reaches 63.% of (V TS V 0 ) when t τ, 95% of (V TS V 0 ) when t 3τ, an 99.3% of (V TS V 0 ) when t 5τ. All right reserve 5

26 % (V TS -V 0 ) V(t)/VTS V 0 /V TS t/τ Figure 4-8. Relationshi between the ercentage of terminal settling velocity an the normalize time t/τ. The initial velocity is assume to be zero. Examle 4-5. What is the relaxation time for a article with an aeroynamic iameter of 3 μm? If the article is injecte from a nozzle to air an has an initial velocity of 5 m/s towars the groun, an then allows the article free fall, what is the article velocity after 0.00 secon? Solution: The sli correction effect can be neglecte since a 3 μm. The relaxation time of the article: τ m B 0 a 8η ca,000 ( ) ( s) Since the relaxation time is so short, the article almost instantaneously reaches its terminal velocity. To answer the secon question, first etermine the terminal settling velocity V TS 0 a g 8η ca 6,000 (3 0 ) ( m / s) Alying Eq gives V t ) ex ( ) V ( V V 0.00 TS t s TS 0 τ ( ) ex ( ) ( m / s) Which is the same as the terminal settling velocity. This inicates that even with a high initial velocity of 5 m/s, the article reaches its terminal velocity in less than one millisecon. All right reserve 6

27 Although the relaxation time is erive using the gravity force, it can be erive when it is subjecte to other tyes of external forces such as centrifugal or electrical, simly by substituting B V T /F e into Eq. 4-57, where V T is the terminal velocity of the article, an F e is the external force exerte on the article. 4.8 Stoing Distance In aition to the terminal velocity of articles, the istance that a article can travel is another imortant characteristic of article mechanics. For examle, how far will a article generate from a grining wheel travel in still air? The answer to this question involves the concet of stoing istance. Stoing istance is efine as the maximum istance a article can travel with an initial velocity V 0 in still air in the absence of external forces. Such external forces inclue gravity, centrifugal, an electrical forces. Drag force is not consiere as an external force in article mechanics. onsiering the linear istance that a article travels along one irection in still air as S(t), then V(t) S(t)/t. From Eq. 4-56, the istance function, S(t), can be obtaine by integrating the S(t)/t over the time t, S( t) t t V TS ( VTS V0 )ex( ) t 0 τ t VTS t τ ( VTS V0 ) ex( ) (4-58) τ By the efinition of the stoing istance, two factors must be consiere. First, the maximum istance can only be achieve when the time. Seconly, there is no external force exerte on the article, i.e., V TS 0 when t. Tyically, we can consier V TS 0 when t > 5τ with a % accuracy. Thus, when t >> τ, V TS 0, an Eq yiels the stoing istance S c V0 τ Bm V0 V0 (Re 0 ) (4-59) 8η Similar to the terminal velocity, τ, the relaxation time is the time constant of the article neee to reach its stoing istance when there is no existence of external forces. Theoretically, S(t) never reaches its stoing istance. Practically, S(t) reaches 63.% of its stoing istance when t τ, 95% of the stoing istance when t 3τ an 99.3% of the stoing istance when t 5τ. Figure 4-9 shows the relationshi between the stoing istance an the time. All right reserve 7

28 00 Percent of stoing istance t/τ Figure 4-9. Relationshi between the ercentage of stoing istance an the normalize time, t/τ. Stoing istances calculate from Eq are for articles moving in Stokes region only, i.e., when Re <. When Re >, the stoing istance of the article is shorter than it is reicte by Eq This is because the rag force increases as the Re increases. Recalling Eqs. 4-6 an 4-7, the rag coefficient D increases as the Re increases. The increase of the rag force attenuates the velocity of the article, an hence reuces the stoing istance. However, since the Re is roortional to the article velocity an the rag force is roortional to the square of velocity, it is extremely ifficult to obtain an analytical equation for stoing istance outsie of Stokes region. Mercer 9 roose an emirical equation to calculate the stoing istance for articles with Re <,500: S g Re tan 3 Re ( 6 ) ( < Re 0,500) (4-60) where Re 0 is the article Reynols number at the initial article velocity, V 0, an the term 3 tan Re / 6 is in raian. 9 The accuracy of this estimation is within 3% of the calculate value. Stoing istances for tyical sizes of articles are liste in Table 4-3. The values are calculate using Eqs an 4-60 accoring to the article Reynols number at the initial velocity. When Eq.4-59 was use, sli correction factors were calculate using Eqs an 4-3 accoring to the article size. All right reserve 8