4.5 Evaporation and Diffusion Evaporation and Diffusion through Quiescent Air (page 286) bulk motion of air and j. y a,2, y j,2 or P a,2, P j,2

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1 4.5 Evaporation and Diffuion Evaporation and Diffuion through Quiecent Air (page 86) z bul otion of air and j z diffuion of air (a) diffuion of containant (j) y a,, y j, or P a,, P j, z 1 volatile liquid (pecie j) y a,1, y j,1 or P a,1, P j,1 Figure 4.10 Evaporation of a volatile liquid (pecie j) fro a container open at the top, partially filled with the volatile liquid. The evaporation a flow rate (a per tie) i given in all cae by = N M A (4-46) evap,j j j A log ean ol fraction (y a ) and a log ean partial preure ratio (P a ) are defined a and y P a a ya, ya,1 = ya, ln y a,1 = ( Pa, Pa,1 ) P ln P a, a,1 (4-53) (4-54) The a tranfer coefficient i defined a G. For pure diffuion, G i or, The olar flux (N j ) can be expreed a G G Dja = R T z z y ( ) u 1 a P Dja = R T z z P ( ) u 1 a j G j,1 j, Alternatively, the olar flux (N j ) can be written intead a (4-59) (4-60) N = P y y (4-61) N = R T c c = P P (4-6) j G u olar,j,1 olar,j, G j,1 j, Uually, point 1 and correpond to the liquid-ga interface (ubcript i) and the top of the container or tan (ubcript in Figure 4.10) repectively.

2 Exaple Evaporation of PERC fro a Nearly Epty Dru Given: A dry cleaning etablihent ue perchloroethylene, coonly called PERC or per (Cl C=CCl, CAS ), a the fluid to clean clothe. PERC i a colorle liquid with a chlorofor-lie odor and a PEL of 100 PPM. The liquid arrive in a 55-gallon dru and i tranferred to the cleaning equipent. A nearly epty dru with an open top i iproperly tored inide the facility and a all pool of PERC in the botto of the dru evaporate into the worplace. The dru ha a cro-ectional area of 0.5 and a height, (z z 1 ), of Fro the MSDS and Appendice A.8, A.9, and A.0, PERC ha the following propertie: - M = 165.8, ρ = 1.6 g/c 3 - odor recognition threhold (high value) = 69. PPM - boiling point (BP) = 50. o F - vapor preure, P v (at 0 C) = 14.0 Hg - D ja = 0.74 x 10-5 / To do: Etiate the rate at which PERC evaporate. Solution: In the far-field, P a, = P, ince P j, 0. At the liquid interface, the partial preure of the PERC i equal to the vapor preure at tandard atopheric condition, P j,1 = 14.0 Hg. Thu, Fro Eq. (4-54), P a a,1 j,1 P = P P = Hg = 746. Hg ( a, a,1) ( ) P P Hg = = = Hg P 760. a, ln ln P 746. a,1 The a tranfer coefficient ( G ) can be obtained fro Eq. (4-60), 5 ( 760. Hg) P D J N ja G = = RuT( z z1) P J a N Pa K Hg K 9 G = The olar flux i then found fro Eq. (4-6), N = P P = j G j,1 j, Pa Pa 9 = ( ) Hg = Pa Hg Finally, the a flow rate of the PERC i obtained fro Eq. (4-46), 9 g -7 g evap,j = N jm ja = =.9 10 which i approxiately 1.03 g/hr.

3 4.5.5 Evaporation of Single Coponent Liquid (page 91) P j, U U P j (z) U(z) N j z air and vapor liquid Figure 4.1 Ma tranfer through a ingle-fil; concentration and velocity profile in air paing over a volatile ingle olecular pecie liquid. We till ue the ae equation for the evaporation a flow rate, = N M A (4-46) evap,j j j And we till ue the ae expreion for olar flux (N j ), i.e., N = P P (4-67) j G j,i j, where the variable are defined (along with typical unit) a - N j = olar flux [/( )] - G = ga phae a tranfer coefficient [/( Pa)] - P j,i = partial preure of pecie j at the liquid-ga interface ( Hg or Pa) - P j, = partial preure of pecie j in the far-field ( Hg or Pa) Now, however, the expreion for G becoe ore coplicated, and Reynold analogy i ued. We introduce LU ρ LU Re (Reynold nuber) = = (4-70) µ ν µ ν Sc (Schidt nuber) = = D ρ D ja ja (4-71) Characteritic length (L) i choen by the uer or defined in a particular epirical equation. It i the practice in heat tranfer to alo expre the fil coefficient in dienionle for a a Nu = C Re Pr b (4-7) where C, a, and b are dienionle contant that depend on the geoetry of the urface and the range of Reynold nuber under conideration. The two new paraeter, Nuelt nuber and Prandtl nuber, are alo dienionle and are defined a Lh Nu (Nuelt nuber) = (4-73) µ c Pr (Prandtl nuber) = P (4-74)

4 where h i the heat tranfer coefficient. In all of the above equation, the dynaic vicoity (µ), ineatic vicoity (ν), denity (ρ), and theral conductivity () pertain to the air, not the liquid. Since the containant concentration i uually very all, the propertie of pure air can be ued. The ratio of the Schidt and Prandtl nuber i called the Lewi nuber (Le), We alo introduce the Sherwood nuber (Sh), Sc Le = Pr = ρ c (4-77) D P Sc Sh = Nu Pr Finally, an epirical expreion for the ga-phae a tranfer coefficient G can be written a b D 1 ja Sc P 1 G = Nu L Pr P a R u T b 1 ja (4-79) (4-80)

5 Exaple Etiating Evaporation Rate fro Fundaental Principle Given: Ethyl ercaptan (CH 3 CH SH, CAS ) i a liquid with an unpleaant un-lie odor. The aterial i a trong oxidizer with a PEL of 0.5 PPM. A 55-gallon dru of the aterial i handled roughly and a all lea develop in a ea. A circular pool, approxiately 10. in diaeter develop. The wind peed over the pool i etiated to be around 3.0 /. To do: Etiate the evaporation rate fro the pill (S, in g/hr), uing firt principle rather than epirical equation uch a Eq. (4-14). Solution: It i aued that the partial preure of the containant at the liquid-ga interface i equal to the vapor preure of the containant, P v,j the value of which can be found in Appendix A.8. It i alo aued that the characteritic length i the diaeter of the pool, i.e. L = D. The teperature i 17.7 C (90.85 K) and the total preure i 1.00 at (760. Hg). ethyl ercaptan air P j,i = P v,j = 400. Hg ν = x 10-6 / D ja = 0.9 x 10-5 / Pr = M j = 6.1 U = 3.0 / P a,i = P j,i = 360. Hg The preence of ethyl ercaptan in air jut above the interface influence the Prandtl (Pr) and Schidt (Sc) nuber. For a firt approxiation, thi influence i neglected becaue the concentration are not large. The following paraeter are coputed: ( ) Hg U 3.0 ( 10.0 ) P D a = = Hg Re = = = ν -5 ln The criteria in Table 4.3 how that the flow i turbulent. Table 4.3 indicate that the Nuelt nuber can be calculated a Nu = 0.037( Re) 871 ( Pr) = 0.037( ) 871 ( 0.707) = 685. The Schidt nuber i µ ν Sc = = = = Dja ρ Dja The a tranfer coefficient ( G ) can be coputed fro Eq. (4-80), where b 1 = 0.33, 033. Dja Sc P 1 G = Nu D Pr P a R u T Hg 1 J = 685 J ( K) K which yield Hg Pa 6 G = Pa The olar flux (N j ) can be coputed fro Eq. (4-67), auing that the ethyl ercaptan partial preure in the far field i zero (P j, = 0), 6

6 Pa 4 N j = G ( Pj,i Pj, ) = ( 400. Hg 0) = Pa 760. Hg Finally, the total evaporation rate fro the pool i obtained fro Eq. (4-46), 4 g π g evap,j = N jm ja = = hr hr or approxiately 1800 g/hr. Dicuion: If the epirical equation, Eq. (4-14), were ued intead to etiate the evaporation rate, a value of 1670 g/hr would have been obtained. The evaporation rate coputed fro fundaental principle i about 5% larger than the value coputed fro the epirical equation. Thi agreeent i excellent, conidering the nuerou approxiation and auption! An explanation for the dicrepancy ay lie in the fact that the equation ued to copute the Nuelt nuber aue that the velocity and concentration boundary layer begin at the leading edge of the pool. In the actual cae, air paing over the ground ha a fully etablihed boundary layer at the leading edge of the pool, and the equation for the Nuelt nuber ay not apply to the actual cae.

7 4.5.6 Single Fil Theory for Multi-Coponent Liquid (page 97) We till ue the ae equation for the evaporation a flow rate, = N M A (4-46) evap,j j j And we till ue the ae expreion for olar flux (N j ), i.e., Nj G Pj,i Pj, = (4-67) But now we need to be careful how to deterine the partial preure of pecie j at the interface. We ue either Raoult law, Pj = y jp Pj,i = y j,ip = x j,ipv,j (4-8) where P v,j i the vapor preure (ee Appendix A.8) of pure pecie j at the teperature of the liquid, and x j,i i the ol fraction at the interface in the liquid phae, or we ue Henry law, which can be expreed in everal for depending on the unit choen to repreent the olute concentration. where P = Hc = Hx c = Hx (4-97) j olar,j j olar,l j H = colar,lh (4-98) See text for equation for the ga-phae a tranfer coefficient G. For exaple, for large pill of petroleu on bodie of water, pill of the order of hundred of eter in diaeter, Driva (198) recoend the expreion for all the evaporating pecie, where U G RuT D0 Sc = (4-95) - G = a tranfer coefficient [ol/( at hr)] - U = air velocity (/hr) - D 0 = diaeter of the pill () - Sc = ga-phae Schidt nuber, uing a a weighted average for the liquid ixture (unitle) - R u = univeral ga contant [8.06 x 10-5 (at 3 )/(ol K)] - T = air teperature (K)

8 4.5.7 Two-Fil Evaporation of Multi-Coponent Liquid (page 30) P j, U air and containant P j (z) N j z P j,i liquid and containant x j,i x j (z) x j, Figure 4.13 Concentration profile for two-fil a tranfer; x j i the ol fraction in the liquid phae, and P j i the partial preure in the vapor phae. We till ue the ae equation for the evaporation a flow rate, = N M A (4-46) evap,j j j But ee text for epirical expreion for olar flux N j.

9 Exaple Evaporation of Volatile Copound fro a Stagnant Wate Lagoon Given: A wate lagoon i 5. x 40. x 3.5 deep. It contain 100. g/l of benzene (M b = 78.0) and 100. g/l of chlorofor (M c = 119.0) in water. The air and liquid teperature are 5.0 C and the wind peed i 1.70 / at z = 10.. To do: Etiate the evaporation rate (g/hr) of benzene and chlorofor. Solution: Firt, the Schidt nuber are lited: Sc (benzene-water) = 1,000 Sc (benzene-air) = 1.76 Sc (chlorofor-water) = 1,100 Sc (chlorofor-air) =.14 The total olar denity (c olar,l ) i g ρ 3 total ρwater ol 4 ol colar,l = = 5 g 3 Mtotal M water 18.0 = Henry law contant for the two cheical in the water can be looed up: - benzene: H b = 3.05 x 10 7 N/ H b = H b /c olar,l = 5.5 x 10-3 (at 3 )/gol - chlorofor: H c =.66 x 10 7 N/ H c = H c /c olar,l = 3.39 x 10-3 (at 3 )/gol The overall a tranfer coefficient K L,j i given by Eq. (4-106), and the a tranfer coefficient for the ga and liquid phae are calculated fro Eq. (4-109) through (4-111). The friction velocity U* can be found fro Eq. (4-113), * U = U U10 = ( 1.7) = 4.55 Since U * i greater than 0.3 /, Eq. (4-110) i relevant for thi exaple proble. Suarizing the a tranfer coefficient for the two cheical, Benzene (ubcript b): G,b = 4.1 x (1.9)(4.55)/ = ol/( at) L,b = x 10-4 (4.55)/1, = x 10-6 = 4.9 x 10-4 / K L,b = 4.85 x 10-4 / Chlorofor (ubcript c): G,c = 4.1 x (1.9)(4.55)/ = ol/( at) L,c = x 10-4 (4.55)/1, = x 10-6 = 4.69 x 10-4 / K L,c = 4.57 x 10-4 / The olar flux (N j ) for pecie j i given by Eq. (4-107) where c olar,j, i the olar concentration of the pecie in the liquid far-field and P j, i the partial preure of pecie j in the ga far-field. In the liquid far-field (near the botto of the lagoon), g 1 g 1 colar,b, = 100. colar,c, = 100. L M L M b but in the ga far-field (ditance far above the liquid-air interface) there i only pure air, o Eq. (4-107) then iplifie to P b, = Pc, = 0 c

10 P N K c K c H j, j = L,j olar,j, L,j olar,j, The a tranfer rate of benzene (j = b) and that of chlorofor (j = c) fro the lagoon (in g/hr) for urface area A = 1,000 are thu = NMA= K c MA and b b b L,b olar,b, b 4 g 1 g L g = Mb L M 6 = 3 b 10 g hr hr = NMA= K c MA c c c L,c olar,c, c 4 g 1 g L g = Mc L M 6 = 3 c 10 g hr hr Dicuion: Coparing the reitance of the liquid and ga fil to a tranfer, it i clear that the liquid fil reitance i nearly one hundred tie larger than the ga fil reitance (for either cheical).