SRUBBERS FOR PARTIE OETION Backgroun To unerstan how scrubbers work, we must first efine some terms. Single roplet efficiency, η, is similar to single fiber efficiency. It is the fraction of particles in the gas upstream of a spherical roplet that collect on that roplet as the roplet moves through the gas. Although the same mechanisms contribute to single roplet efficiency that contribute to single fiber efficiency, an inclue impaction, interception, an iffusion, impaction alone usually ominates. For particle collection ue to impaction on a roplet, the following equation is often use: 2 Stk η = Stk.35 (1) + where Stk is the Stokes number for the rop moving through gas that contains particles. The particles are assume to be embee in the gas; that is, they o not move relative to the gas except to cross streamlines by impaction. Inspection of Eq (1) shows that η is zero if Stk is zero, an approaches a value of unity as Stk becomes large. Stk 2 ρp V / g =. (2) 18µ Here, is particle iameter, ρ p is particle ensity, V /g is the velocity of the roplet relative to the gas (see below) is the unningham slip correction factor, µ is gas viscosity, an is roplet iameter. Because roplets are liqui, particle collection on liqui roplets will epen on some mechanisms that o not operate for collection on ry spheres or on ry fibers. For example, iffusiophoresis an Stefan flow can be important. If the roplets are water, Stefan flow will enhance particle collection uner conitions when water vapor conenses on the roplets, but will reuce particle collection for evaporating roplets. Diffusiophoresis for conensing water vapor will ten to rive particles away from the roplet, an towar an evaporating roplet; however, this effect is usually less important than Stefan flow. iqui holup, H, is the fraction of the scrubber volume that is fille with roplets. It is analogous to soliity for a filter. If, for example, all the roplets in a scrubber were mae of ice, then the ice all melte, then H woul be the volume of the water from the ice ivie by the volume of the scrubber. -1-
A ifference between holup an soliity is that holup is a ynamic situation whereas soliity is a stationary situation. The value for scrubber holup epens on the rate that liqui is fe into the scrubber an the velocity with which the roplets move through the scrubber. volume of roplets in scrubber V / w H = = =, (3) volume of scrubber A A V where / w H is holup, or volume fraction comprise of roplets, is liqui volumetric flow into the scrubber, V /w is the velocity of the roplets relative to the scrubber wall (see below) A is the cross sectional area of the scrubber, an is scrubber length, measure perpenicular to A. Droplet velocities within the scrubber can be expresse in several ways. Sometimes the important concept is the velocity of the roplets relative to the gas, V /g. This velocity woul be important, for example, if we are intereste in the impaction of particles that are in the gas on roplets that move through the gas. At the same time that roplets move through the gas, the gas itself is moving through the scrubber with a certain velocity that shoul be taken relative to the scrubber wall,. This velocity is important, for example, if we are intereste in the resience time of the gas in the scrubber. A final velocity of interest is the velocity of the roplets relative to the wall of the scrubber, V /w. This velocity is the vector sum of the velocity of the rops relative to the gas, an the velocity of the gas relative to the wall r V / w r r = V + V. (4) / g g / w onsier a spray tower in which the gas enters at the bottom an leaves at the top. Water is spraye ownwar into the top of the rising gas. If, for example, the roplets fall through the gas with a ownwar terminal settling velocity of 2 cm/s relative to the gas, an the gas itself flows upwar through the scrubber with a velocity of 5 cm/s relative to the scrubber wall, then the ownwar velocity of the roplets as seen through a winow in the sie of the scrubber, or V /w, is - 2 + 5 = -15 cm/s V /g V /w or 15 cm/s in the ownwar irection. If the gas happene to flow upwar at the exact same velocity that the roplets fell ownwar through the gas, then the roplets woul appear stationary when seen through that winow, or V /w =. If the upwar gas velocity ha a higher numerical value than the ownwar roplet velocity, -2-
then the rops woul be blown out the top of the scrubber, even though they continue to settle ownwar through the rising gas. Relative velocity is important as we consier particle collection in scrubbers. Bear in min that the important velocity for particle collection is the velocity of the roplets relative to the gas, or V /g whereas the important velocity for roplet motion through the scrubber is the velocity of the roplets relative to the wall, or V /w. iqui Evaporation will occur if the gas stream is not saturate with vapor. iqui that evaporates is not available to collect particles, so evaporation must be consiere as we investigate scrubber performance. This stuy can be one using the psychrometric chart; see a link to such a chart on the course website. Similar charts are available in many books an at many websites. Use of the psychrometric chart can be shown through an example. onsier a case where 1, cfm of air at 15 F, an relative humiity of 2%, is to be scrubbe. Assuming that the air becomes saturate, etermine the exit gas temperature an the flow of water require to replace the water lost ue to evaporation. First, locate the point on the psychrometric chart that correspons to 15 F, 2% RH. Rea to the left to fin absolute humiity of about.33 lb water/lb ry air. At this point (15 F, 2% humiity) rea humi volume to be about 16.3 ft 3 /lb ry air. From this initial point, follow the aiabatic saturation line up an to the left to etermine the ew point or wet bulb temperature, T wb. For these conitions, T wb = 13 F;.45 lb water/lb ry air; 15.2 ft 3 /lb ry air. With these ata we can solve the problem. The key is to express air conitions on a basis of pouns of ry air, as that quantity oes not change as we a (or subtract) water vapor. a. Determine pouns of ry air. 1, ft min 3 lb ry air 3 16.3ft lb ry air = 6134 min b. Determine flow after saturation: lb ry air 15.2 ft ft 6134 = 93,3 min lbryair min c. Determine water consumption: lb ry air lb water 6134 = min lb ry air 3 3 lb water (.45.33) 73.6 or 8.8 gallons / minute Thus, before we can have water roplets in the gas stream at all, we must supply enough water to saturate the gas. That process causes the gas stream to ecrease in temperature from 15 F to 13 F, an to ecrease in volume from 1, cfm to 93,3 cfm. min -3-
Spray Tower The approach we will use to escribe particle collection in a spray tower is similar to the one we use for fiber filters, an to what we use for gravitational settling chambers. The approach is base on the rawing shown below. Particle-laen gas enters at the bottom of the scrubber an moves vertically upwars. iqui sprays into the top of the scrubber an falls ownwar as roplets, each of which falls at its terminal settling velocity. out, As always, we nee to make some assumptions: g 1. Particle concentration,, is uniform in any plane perpenicular to the irection of gas flow, 2. Gas velocity, V g, is uniform throughout, ( ) 3. Droplets are sprea evenly across the gas stream cross section, which has area A, A 4. Droplets all have the same iameter,, 5. No roplets collect on the walls, V /g in, g 6. No roplets agglomerate, The last four assumptions represent ieal roplet behavior. If they are true (an they are usually not true) then each roplet acts in a perfect way for particle collection. Departure from ieal behavior will cause less particle collection than what we etermine using an equation base on the ieal roplet assumptions. A balance for the rate at which particles pass through the ifferential slice yiels the equation given on the next page. As for the equation we evelope for particle collection in a filter, the term on the left sie of the equation represents the rate that particles enter the slice from the air. The first term on the right sie of the equation is for particles that leave in the air that flows from the back sie of the slice. The secon term on the right sie is for particle accumulation within the slice, where accumulation is cause by particle collection on roplets. within the slice. The sign on the accumulation term is negative because particles are remove from the air in the slice; that is, they o not accumulate there. When thinking about the mass rate balance equation, one must be careful to recognize the role of the several relative velocities that are important in scrubbers. -4-
Rate ust flows into the slice Rate ust flows out of the slice Rate ust accumulates in the slice g 2 π = ( ) A H 4 g V / g η 3 (5) π 6 roplet volume collection area/ volume collection area volume of gas swept/time volume of gas swept clean/time mass of particles swept clean/time Here, g is upwar gas volumetric flow thorough scrubber, is ust concentration in the slice A is scrubber cross-sectional area is scrubber height, H is holup, the fraction of scrubber volume mae up of roplets, see Eq (3), is roplet iameter, η is single roplet efficiency, see Eq (1). After substituting Eq (3) for holup, Equation (5) becomes out 3 V / g = η (6) 2 V or in G / w -5-
( ) out 3 V / g 3 V / g Pt = = exp η = exp η, (7) in 2 G V / w 2 G V / g Vg / w or in terms of efficiency, η 3 V / g = 1 exp η (8) 2 G ( V ) / g Vg / w Note that efficiency is high when the ratio of water use to gas treate, / G, becomes high. Efficiency also is high when the settling velocity of the roplets, V /g, approaches the upwar velocity of the gas,. In this case, holup becomes high; that is, the tower contains more suspene rops. Efficiency is high if the tower is tall, an if single roplet efficiency is high. The role of roplet iameter is complex. Eq (8) shows that efficiency increases as roplet iameter ecreases, because for the same amount of liqui use smaller roplets present greater surface area for particle collection. Smaller roplets have lower settling velocity, V /g, which ecreases the gas sweep rate an ecreases efficiency, but increases holup which increases efficiency. Smaller roplets have a lower Stokes number, which ecreases single roplet efficiency, η. Analysis suggests that single roplet efficiency, η, is maximize when roplet iameter is about 1 µm (1 mm) for particles of all sizes. Efficiency of the entire spray tower as given in Eq (8) is maximize when roplet iameter becomes so small that V /g, approaches the upwar velocity of the gas,. However, because roplets that are too small are likely to be blown out the top of the tower, an because spray nozzles prouce roplets with a istribution of iameters, practical consierations suggest that roplets with meian iameter of about 1 µm may be most effective. Eq (8) must be use with iscretion. It was evelope using assumptions that the roplets behave in an ieal way: that none evaporate, that all are istribute evenly over the scrubber cross section, that none collie with scrubber walls or with each other, etc. In an actual scrubber, none of these assumptions is likely to hol, with the result that scrubber efficiency is likely to be lower than that preicte. Preictions from Eq (8) shoul be regare as the highest efficiencies reasonably possible. -6-
Venturi Scrubber Analysis of particle collection in a venturi scrubber is similar to that use to escribe collection in a spray tower except that flow of gas an roplets is co-current rather than countercurrent. Droplets enter the venturi at the throat, then accelerate in the gas stream ue to rag force from the passing gas. Eventually, if the throat is long enough, the roplets will reach the full gas velocity. Note: Positive is towar the irection of gas flow. V /w As before, we can unerstan roplet motion using the concept of relative velocities. is the velocity of the gas relative to the venturi wall, an is taken to be constant through the venturi throat where the cross-sectional area is constant. V /w is the velocity of the rops relative to the scrubber wall; that is, the apparent velocity of the rops as seen through a winow in the sie of the venturi. This velocity will change along the venturi length. At the point the rops are introuce through the wall of the venturi, V /w will be zero. After the rops accelerate to the full gas velocity, V /w =. V /g is the velocity of the accelerating roplets relative to the gas. At the point where the roplets are introuce, this relative velocity is highest. Once the roplets fully accelerate, they have no velocity relative to the gas an V /g is zero. These an other relationships are shown schematically in the plots below, which apply if a positive irection is taken as towar the right; that is, if gas flows from left to right. Each plot shows a parameter such as velocity on the vertical axis, an position along the scrubber on the horizontal axis. The left figure in each plot applies to conitions within a venturi scrubber, whereas the corresponing conition for a spray tower is on the right. onitions in the spray tower are constant along the length of the tower, but velocities within the venturi change with length as the roplets accelerate. Recall that V /w = V /g + as given in Eq (4). In the venturi throat, is high an has a constant value. V /g starts at -, then increases to approach zero. Thus, V /w must start at zero, then increase to reach. From the stanpoint of a particle embee in the gas stream, roplets initially have a high, negative velocity; that is, they seem to the particle to move -7-
towar the left. After the roplets accelerate fully, the particles in the gas an the roplets have no relative velocity, as both move together with the gas. Venturi Spray Tower V /g : Velocity of the roplets relative to the gas. V /g V /g = terminal settling velocity H - - V /w : Velocity of the roplets relative to the scrubber wall. V /w H V /w - - H : Holup of liqui in the scrubber (volume fraction comprise of roplets) H A V / w H A V / w H -8-
η : Single roplet efficiency 1 1 η η H The equations that escribe roplet motion own the venturi throat are complicate, because roplet motion is in the turbulent regime; that is, the Reynols number for roplet motion through the gas in the venturi throat is initially much greater than unity. Several ifferent authors have evelope approaches to this problem, as escribe in the article by Runick et al. given on the link in our course website. Two approaches, that of alvert, an that of Yung et al., are given in the course spreasheet. The relevant equations for these approaches, taken from the Runick article, are given below. Droplet Diameter The iameter of roplets generate by pneumatic atomization has been stuie for many years. Although several equations to preict mean roplet size are available, one by Nukiyama an Tanasawa has been wiely use. This equation gives the Sauter iameter; that is, the surface-to-volume mean iameter. g / w i 3/ 2.5 = +.918 V V (9) G where is given in meters per secon, an V i is the axial velocity of the roplets at the time of atomization. Droplet iameter,, is given in meters in Eq (9). alvert Approach Here, efficiency epens on parameters that govern roplet motion through the scrubber, plus a factor f, use to fit experimental ata to the moel. ontroversy exists over the correct value of f to use, but in general a value of about.32 seems about right. The equation for penetration is given in Eq (1). -9-
Pt ρ Vg / w.35 + K i f.49 = exp 2.8ln + 1.4 4 K i f 55 g µ K i.35.35 + K i f (1) Here, K i is the value of the Stokes number uner conitions at the inlet to the venturi throat: K i 2 c ρp Vg / w = (11) 18µ Yung et al. Approach The approach of Yung et al. is more complex. Here, the acceleration of roplets in the venturi throat is moele using more realistic assumptions than use by alvert. The equation for penetration, copie from the Runick et al. article, is: ρ g ρg Pt = exp Di 1.5.5 4K (1 V* e ) + 2.1(1 V* e ).5 3.55K (1 V* e + 1.35/ K) tan (K (1 V* e) /.35).35+ K (1 V* e ) 2.1+ 4K.5 1 3.55K ( 1+.35/ K) tan (K /.35).35+ K.5.5 (12) Note that the inverse tangents use in the above equations must be expresse in raians, not in egrees. The other terms in Eq (12) are: V* e 2 2 = 2 (1 X + X X 1 ) (13) 3 t Di ρg X = 1+ (14) 16 ρ Here, t is the length of the throat an Di is the rag coefficient that applies for the roplets at the point of liqui injection, the beginning of the venturi throat, where V /g =. The value of Di is etermine from the stanar curve of D versus Re for the roplets at the point where they are injecte. If roplets are injecte with a non-zero velocity, then the appropriate relative velocity between the gas an roplets at the point of injection must be use. Eqs (13) an (14) are also use to etermine venturi pressure rop as escribe below. The course spreasheet inclues equations for particle collection using both the alvert approach an the approach of Yung et al. The article by Runick et al. compares the preictive power of the alvert approach, the Yung et al. approach, an several other methos to preict venturi efficiency that are even more complex. -1-
Pressure Drop The pressure rop through a venturi scrubber is high by comparison to pressure rop across other particle collection evices. In a venturi, water rops are accelerate to the high velocity that exists in the venturi throat. With the assumption that other sources of pressure loss are negligible compare to that require to accelerate the liqui roplets, the following equation can be evelope. onsier the principle from physics that the impulse imparte to an object equals the momentum it attains. Impulse is force x time, whereas momentum is mass times velocity. Therefore, F t = m V / w (15) If we ivie both sies of Eq (15) by contact time, t, an by the cross-sectional area of the venturi throat, A, then we have an expression for pressure rop ue to accelerating liqui rops. F t m V / w ρ m V ρ V / w g / w P = = = (16) A t A t ρ t ρ A Vg / w or V w ρ V [ ] / g / w P = = ρ V / w Vg / w (17) g g The highest velocity the roplets can attain relative to the wall is if the roplets fully accelerate to the gas velocity by the time they reach the en of the venturi throat. In practice, the roplets probably o not fully accelerate. If we efine V* e such that V / w V * e = (18) Vg / w at the en of the venturi throat, where V* e < 1, then Eq (17) becomes V 2 P = V* e ρ g / w. (19) g As a practical matter, a value of V* e.85 can be use. Note that this assumption is that the rops attain 85% of the gas velocity by the time they reach the en of the venturi throat. Alternatively, V* e can be calculate from a force balance on the roplets uring their acceleration with the result given in Eq (13) above. A further refinement comes from recognizing that roplets accelerate in the throat at the cost of a certain pressure rop will return some of that pressure rop to the gas stream when the roplets ecelerate in the iverging section of the venturi that follows the venturi throat. If the gas velocity in the throat is an the gas velocity at the en of the venturi iverging section is V g/ then a correction can be applie to Eq (19) for the regain in pressure rop ue to roplet eceleration with the following, final result. -11-
V = ρ g / V 2 + P g / V g / w V* e 1 (2) g Vg / w Vg / w 2 The course spreasheet inclues pressure rops from Eqs (19) with V* e = β, which can be set by the user, with V* e from Eq (13), an from Eq (2) with V* e from Eq (13). References alvert, S. in Air Pollution, 3 r e., Vol IV. A.. Stern, e., Acaemic Press, New York 1977. eith, D., D. W. ooper an S.N. Runick, Aerosol Science an Technology, 4: 239 (1985). Runick, S. N., J.. Koehler, K.P. Martin, D. eith, an D.W. ooper, Env. Sci. Tech., 2: 237 (1986). Yung, S., S. alvert, H.F. Barbarika,.E. Sparks, Env. Sci. Tech. 12: 456-459 (1978). -12-