Gas Film Liquid Film Bulk Gas

Transcription

1 Diagram of Two Film Theory showing gas adsorption (solid arrows) and gas stripping (dashed arrows). Dissolved gas is rought to interfacial film surface y mixing (convection) and diffuses to interface where a chemical reaction from liquid (C i ) to gas (p i ) phase. Then, the gas diffuses to film surface and convects to ulk atmosphere. Reverse occurs for stripping. Bulk liquid concentration is convected to surface, diffuses to interface, chemical reaction from liquid to gas phase occurs, gas is convected to ulk and stripping is complete. Bulk Gas Gas Film iquid Film Bulk Gas Mixing Interface Diffusion Mixing p a p GAS STRIPPING C i GAS ADSORPTION C Film Theory Two Theories - Penetration and Film Penetration Theory - Eddies originate in the ulk and migrate to the g/l interface and rief exposure results in gas asorance efore other eddies move the asorance eddies from the surface due to turulence. Gas transfer is a function of diffusivity, ******* gradient and surface renewal (eddy transfer). Two Film Theory - two films exist at surface, one gas/one liquid. They are stagnant ut furnish the resistance to g/l transfer. Three phases - ) transport to surface of film (usually controlled y mixing) )diffusion through the gas and then the liquid film and 3)transport to the liquid ulk. Transport is slow y diffusion ****** to mixing so that (transport) controls gas transfer. Molecular Diffusion moles gas A through quiescent gas B: where dna/dt moles gas A diffusing dn dt a DAp v ( Pa -P a) RTaS P t time (hr) D v vol. diff. (ft /hr) A area for diffusion p pressure (atm) p a - p a partial pressure diffusing gas (atm) P log mean pressure B S diff. Distance R 0.79 ft 3 *atm/l*mol* o R T o R

2 Molecular Diffusion Through iquids dn dt DAC ( Ca -C a) SCm a v a + Ca /Ca molar Conc. Diff. gas C a+ molar A+B / vol Cm log mean concentration Mass Diffusivity NH 3, CO, + O in Air 5 o C, in Water 0 o C AIR WATER Dv, (ft/hr) Ns c (u/pdv) Dv (ft /hr) Ns c (u/pdv) NH * CO * O * Assuming a steady state with no interface accumulation N a dna/dt k g A(ρ-ρ i )k i A(C i -C ) N a mol/time k g, k l mass transfer coeff. (mol/hr*ft *atm) for k g and (ft/hr) for k l ρ,ρ I partial pressure (atm) C,C conc. interface, ulk We have derived Henry s aw from G G o + RT ln Prod/React CO g + H 0 CO + H 0 G 0; ln P R x o G k RT CO gas is expressed as pressure K CO () /CO (g) (0 o C, atm) CO (g) atm CO (l) KCO (g) K(P CO(g) ) CO (l) 0-5 M/l (44,000mg/mol) 0.44 mg/l note CO (l) is independent of ph as predicted y Henry s aw P (/K)C HC

3 Example : Predict sat. concentration of O (mg/l) in H0, at STP H 4.0*0-4 atm/mol frac ρo 0. atm. O (g) + H O O (l) CO (l) kρ ρ/n 0. / 4.0*0 4 CO (l) 5.4*0-6 mol frac. n H0 000/ n O 5.4*0-6 (55.5).9*0-4 gm-mol 9.3 mg/l 0 P e Do not know interfacial concentrations - P i so use ulk concentrations P C e equilrium concentration to P P e equilirium pressure to C C C i C e So N a K A(C e -C ) K g AC(P -P e ) Defining P c H C i P ρ ρ m molar density liquid phase from interface equation m H C e P ρ m e H C ρ m KHA g N a K g AC( P P c ) ( C e C ) ρm Driving Force into the film C e - C (C i - C ) + (C e - C i ) Na KA Na + ka l PN m a HkgA Pm + overall liquid resistance K kl Hkg

4 can also derive H + overall liquid resistance Kg kg ρ mkl for large H, gas of low soluility this term is very small + /K l /k l + **** major resistance is in liquid film, reverse is true if H is small /K g /k g For high soluility gases, large K H, the overall transfer coefficient is equal to the k g. If K N is low then, ie. N, O, CO, K H is usually controlled y transfer across the liquid side. If we reduce the thickness of the liquid film y agitation then the overall gas transfer rate is increased. It is possile for oth of these K f + K g, gas transfer factors to e controlling for gases of intermediate soluility. Gas is controlled y concentration gradients in several effluent y hypothetical media s. These medias will limit the gas transfer process thus controlling the overall rate. The limiting soluility is a function of Henry s aw of the saturation limit for any gas. Equipment:. Gas Dispersers a)small orifice )hydraulic shear c)mechanical shear. Packed Columns 3. Tray Columns 4. Spray Units Gas Dispersers: Depth < 5ft Contact time < 30 seconds Bule must e small Can t measure ule surface area easily so use volumetric mass transfer coefficient are used which measure interfacial area/vol + mass transfer coefficient. da/dt N K A (C e - C ) K ga (P - P e ) K A, K ga hr -, mol/hr(ft 3 )(atm) all other units the same C e C t conc. C Time

5 Integrate dn/dt K A (C e - C ) dn dt dc Ce C dc KA( Ce C) dt C t Co 0 KA dt e ln C C e C t C KAt also solve ln P t P Po P e e KgaRTt Plot log C vs. time is straight line Asorption (O uptake) log C t - C eq Desorption (CO removal) Time Mass Transfer related to many variales K fcd, N re, N sc D v volumetric Diffusivity N re Reynolds No. N sc Schmitt No Dimensionless No., Derived from Pockingham Theory N sh m K N Re N Sc / (air ules/ H 0) KD Dv P K Dpu m tρ) u u ' ρdv / K Mass Transfer Coefficient, (ft/hr) D p ule diameter, (ft) D v olume Diffusivity, (ft) K Constant u t ule velocity, (ft/s) ρ density u asolute viscosity, (l mass/sec ft)

6 for depth > 5ft A q 6 Du p t Then K A surface area (ft ) volume (ft 3 ) q air flow (ft 3 /sec) K K A A A Du p t 6q m m K' ' q K' ' q solving for K A: K A / / D ( u / ρd ) D N Penetration Theory In + generation out + accumulation AO oz + A Z r o AN oz + Z + A Z δc/δt no chemical reaction δc δ D C δt δz consider 3 oundaries t0 Z>0 CC t>0 Z0 CC * t>0 Z CC average flux N o p D * ( C C ) t c v Shorter contact time yields higher transfer. Want to form many interfaces that are large in numer, small in size, and with a short t. Maximum O flux:. Decrease Contact time y increasing mixing energy. Shallow tanks, hi vol. A ir 3. Increase turulence y process geometry p sc Surface Renewal Theory

7 Consider liquid elements for finite time periods Random distriution Assume a steady production of new area rate r s independent of time or age. Area of Surface A(t) t at age etween t+ t Then A(t) t A(t- t) t-{a(t- t) t} r s t so t 0 da() t dt ra() t integrate and Ke -rst A(t) s Area considered Ke 0 s dt K r s rt At () re s rt s * earlier N ( C C ) o D t * N ( C C ) D t re o s 0 r s t dt r s t B N ( C C ) r * / o s D / 0 e B db N ( C C ) r * / o s * N ( C C ) Dr o s rd s / / Surface removal assumes turulent flow, Penetration does not. Bule aeration Most common through porous plate or tue Release is in a cloud - coalesce and form larger ules Oxygen transfer is proportional to mean ule diameter N o d α Q g n Q g flow α Factor for ule shape d ule diameter 0. < n <.0 Re < common dimensionless parameters for mass transfer studies Sherwood N o sphere 300 < Re < 4000 ellipsoidal Re > 4000 elliptical caps

8 K d /D molar transfer rate/unit area for spec. conc. Reynolds No: Re U d ρ w /u internal forces / viscous forces Schmidt No u/ρ w D kinematic viscosity/ diffusivity EckenFelder Proposal ) K d D duρw u B u ρd / ) B B' 3 / H B constant H /3 tank depth Usually estimate K l a ecause can t measure K est y: Qg a d d 3 d t c Average ule interface/tank unit volume QH g a 6 d u a area of interface/unit tank vol. t c gas/liquid contact time t c H / tank depth/ule velocity This term neglects the free surface ut the term is small and can e neglected. Comine interfacial area (a), time of concentration (c), and then solving for K and using B /H /3 gives: K a / 3 / / / 3 6B' H Qg Pw D 6B' H Qg sec / / d v( u) d vn sc Oxygen Transfer Rate: M O K a(c * -C ) 4. Example Design Aeration System

9 Tank olume 0 6 liters Depth 00 cm Diameter 30 cm O Concentration (mg/l) QC QC QC Average Bule Diameter (cm) Time (s) Estimate K l a for a tank 366cm deep and 458cm deep O s 7.8mg/l D. * 0-5 cm/s K a K a K a 3 3 / / 5 / 6B'( 00) ( 5)( ) (. * 0 ) 47. * 0 B 5 ' / 0.(. 4* 0 )(. 00) 37. * 0 B ' 33. * 0 B ' 3 Finding K a from plots of time concentration data: K a 0.005s - B 0.06 K a 0.00s - B K a s - B variation B is small while Q q is large. B is safe.

10 Temperature Effects: Ka K a Tu Tu u viscosity T o K Ka K a 0 (. 0) T 0 T o C Surface Active Agents: Film thickness Surface Tension f (S.A.A.) ie. detergents K a db S.A. Conc. S.A. Conc. Minimum Oxygen transfer Rate is unexplained ut thought to e a function of the sufaces of ules as affected y the S.A.A. concentration. K l a values in laoratory are generally higher than what is found in practice. Agitation from Aeration Important in Bio-systems ecause of clumping action of activated sludge. Bule Diameter of 0..0 cm diameter will have rise velocities of 0 to 50cm/s and R e of 300 to 5000 and Newton s aw controls. Addition Turulence from turulence: P d g π 3 ρ ω u 6 No. of Bules at any time n Qt q c QH q u Total Power n * P Q q H P w g Have to assume a distriution:

11 P πzgρ ω ( Z/ 4 + H ) u 4 t Z tank diameter ut average liquid velocity The fanning friction factor: f' ( P P ) Z 4 H ρ P P solve 4QH g Z Z π + H 4 Mixing and Transfer Eff. R e uzρ u 4QHρ g ω πuz Z + H 4 convective mixing Oxygen transfer/oxygen Del M K a C * C * ( ) K ac ( C) n Q M mass of O trans/ule g Eff. of O transfer E O m / P 0 E O * K E ( C C ) QP g O Mass transfer coefficient is α to gas flow rate E O 3 / / / * ω / du ρo 6BH ' ρ D ( C C ) controlling variales: liquid depth ulk O concentration Diameter will vary with Q g Note: C approaches C* quickly and unless O is removed from liquid phase y reaction, efficiency of transfer will drop to zero.

12 Example: O consumption rate 0.07 mg/s/l Depth 4.58 m B ol..439*03 m3 u 0.0 mg/cm-s ule diameter 0.cm D.*0-5 cm /s O sat 7.8 mg/l O density in ule 3.85*0-4 g/cm 3 Bulk O(avg) mg/l final O transfer eff. rate and air input rate required to keep 0.05*** part suspended 6(. 0 06)( 458) () (.* 0 ) (.* ) 0.(. 0 0) (. 385* 0 ) 3 / / 5 / 6 E O / or 8.7% Q O ( O consumption) eff.* ρ m 3 /s O 3 (. 439 * 0 )(. 0 07) ( 385. * 0 ) Qair 3 QO m / s 04. Air input to maintain scour force alance: m dv π 3 p dg( ρp ρ) CDApρ dt 6 terminal velocity d/dt 0 ρp.05 (trial and error) Re 3 Stoke s aw No Good C D **** 06. R e Terminal elocity D (. 0 07) d g( ρp ρ) (. 0 07)(. 0 05) 9. 8(. 0 05) cm / s u ρ (. 00) ()

13 64(. 0 05) Re Q q( mix) Z Z ( + H ) (. )(. )( ) + π H 4( 458) 563,406 cm 3 /s m 3 /s Aeration y Mechanical Mixers Devices are specified with respect to volume, depth, and area. Commercial literature gives Kla volumes of transfer rates. Elken Felder reported O transfer rates with surface turine reactors 3.6 Kg O /Kw-hr when D.O. init 0 Conventional activated sludge systems.5 kg O /Kw-hr to.4 kg O /kw-hr K a CH α N t B Z γ Nt impeller top speed Zi impeller diameter These circumstances are tank specific K a for fermentation industry: Elkenfelder: P Ka N 05. t nominal gas velocity K a a B Qg Z N γ. < α < < β < < γ < 9 Rememer that O transfer is a f(system equilirium, and the rate of O *****). ow O utilization rates do not necessarily mean that O transport can e increased. Most manufacturing data is repeated from a optimum condition - zero initial D.O., low dissolved salts. Data gathered as l. O transported/unit energy - input can e quite high.

14 Factors Affecting Gas Soluility Salinity: can approximate salinity effect y multiplying the soluility value for fresh water * ( - S*0-5) where S is the salinity in mg/l chlorides. Temperature: as o C increases the D.O. or N or CO decrease. The comined effects of dissolved salts and temperature on O soluility in clean water in contact with wet air: Co C T o ( T 73 ) T o K Since gas, O is introduced at depths of 0-5 feet, it is necessary to compute a mean saturation value at mid-depth, O transfer decreases as a function of ρp O in ule: C sm, ρ [ σ] t Cs Cs saturation ρ as. pressure in psi and point of introduction [σ] % O concentration in ule ***** tank Film Transfer Se have stated that: )for agitated systems diffusion across the film (liquid/gas) is limiting. )for gases of low soluility the diffusion through the liquid side of the film is limiting. (ie) if gas concentration is small, film thickness is small relative to ule The flux of gas in a direction normal to the interface is: F x o D dc dx Film Flow Oxygen Transfer case ) reactor only in semi-solid phase (a)(c) case ) reactor in /solid phase () case 3) mass transport controlling case 4) reaction rate controlling case 5) laminar flow case 6) turulent flow

15 Film/iqiud Film iquid Film iquid (a) () Simulation of slime/liquid gas transfer (c) Determination of Condition of Flow Media in trickling filter - rock (0.9 m /m ) or plastic (0.99 m /m ) H.G..R 5*0-5 m 3 /m -s to 4.5*0-4 to 4.5*0-4 m 3 /m Assume: (uniform distance) Bernoulli s equation: αp α U x ρg αx αy Assume: δ is constant αp αx 0 For accumulation in liquid phase: Acc input across film y diffusion x dc dt A (*** ) 0 n x n is unit vector F o D C C s x x 0 S D /S K f ) K KA f (hr - ) ( cs c) ( c c ) dc c c ln s ln &&& D D o s Kt ) where D i C s -C i o K A t Kt ) f

16 A/ Interfacial transfer area So max surface area for ule (smaller ules) will increase oxygen transfer as will more turulence however larger ules increase turulence so there is an optimum. iquid phase transport (quiescent conditions) : laminar film: max Xδ αc X D α C α αy length of film in Z δ film thickness X section of liquid film The continuity equiation for gas within the film αc αt j+ r at steady state with no chemical reaction or in one dimension j 0 d dx D dc dx 0 This equation may e integrated and dc/dx approximated y : c c s at x 0, c e, x δ F xo D c c δ [ s ] oviously K f D /δ For a gas that reacts in solution: D dc Kc dx 0

17 05. ( / ) ( K D ) F o DC K D s x x 0 δ tanh / Now reactive gas: F o 05. DK C x x 0 s So we see that the molecular diffusion coefficient is proportional to mass transfer coefficient to a power of 0.5 to