An Alternative Mathematical Model for Oxygen Transfer Evaluation in Clean Water

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1 An Alernaive Mahemaical Model for Oxygen Transfer Evaluaion in Clean Waer Yanjun (John) He 1, PE, BCEE 1 Kruger Inc., 41 Weson Parkway, Cary, NC ABSTRACT Energy consumpion from aeraion sysem is a bigges par of he oal energy cos in wasewaer reamen plan and accoun for as much as 6% of he energy consumpion for he acivaed sludge process. Therefore, i is very imporan o know how effecive he aeraion sysem and significan aenion has been paid o developmen and upgrade of sandard mehod for quanifying oxygen ransfer efficiency of he aeraion sysem. In order o evaluae he performance of differen ypes of aeraion sysems, he American Sociey of Civil Engineers (ASCE) and US EPA joinly developed a sandard for he measuremen of oxygen ransfer in Clean Waer in The Sandard was subsequenly improved, updaed, and republished in 1991 and 26. The focus of his paper has been o develop an alernaive model which includes more parameers han he sandard mehod. A new model would probably be a more accurae descripion of he aeraion process and give more reliable oxygen ransfer performance evaluaion resuls. KEYWORDS Aeraion, ASCE, SOTR, oxygen ransfer, ransfer coefficien, surface ransfer, Model INTRODUCTION The ASCE Sandard Mehod for he measuremen of oxygen ransfer in clean waer, which has also been called he Nonlinear Regression Mehod, is based on he unseady-sae removal of dissolved oxygen (DO) by sodium sulfie followed by re-oxygenaion back o sauraion or seady-sae condiions wih accurae experimenal measuremen of he DO level over ime. The resuls of he es are convered ino he values a he sandard condiions and he final resuls are expressed as sandard oxygen ransfer rae (SOTR) or sandard oxygen ransfer efficiency (SOTE), a hypoheical oxygen mass ransferred per uni of ime or fracion of oxygen in an injeced gas sream dissolved in clean waer when DO concenraion is zero in he waer volume, he waer emperaure is 2 C, and he baromeric pressure is 1. am (11 kpa). The following sandard simplified mass ransfer model is used o analyze DO measuremen daa o esimae volumeric mass ransfer coefficien: dc = K α (C - C ) (1) d Where K ɑ is volumeric mass ransfer coefficien (T -1 ), C is he sauraion or near sauraion DO concenraion a infinie ime (mg/), and C is he dissolved oxygen a ime. 1

2 Inegraion and re-arrangemen of Eq. 1 gives: ( C - C) 1 ( C - C ) ( - ) K α = In (2) Once K ɑ value a sie condiions is deermined by Eq. 2, he SOTR can be readily calculaed by he following equaion: ( 2-) SOTR = K α θ CS, 2 V (3) Where Ѳ is he empirical emperaure correcion facor and V is liquid volume in he es ank wih aeraor. Examinaion of Eq. 1 reveals ha he sandard model simply assumes he following: (1) waer in he ank is compleely mixed; (2) he overall oxygen ransfer is only from air bubbles, oxygen ransfer from waer surface is no aken ino consideraion; (3) he equilibrium DO concenraions are he same everywhere in he ank, he DO equilibrium concenraions variaion wih waer deph is also no aken ino consideraion. Since BIDs are won and los by as lile as 2-5% difference in oxygen ransfers raes, i is imporan o know how accurae he sandard mehod for quanifying oxygen ransfer rae. The paper repors how big he impac of oxygen ransfer from he air is a he waer surface in SOTR calculaions. The paper also repors he effec of he equilibrium DO concenraions, which is also depending on he raio of oxygen in he air bubble which varies wih waer deph in SOTR calculaions. METHODOOGY Modified Sandard Mehod Model 1 When a submerged diffuser is operaed, here are wo main inerfaces over which oxygen ransfer occurs. Oxygen ransfer occurs across he bubble inerfaces as he bubbles rise hrough he waer column. Oxygen ransfer also occurs across he waer surface where oxygen in he air dissolved ino waer. A he waer surface, oxygen ransfer is driven by he DO concenraion gradien beween gas-liquid films. The oxygen ransfer across bubbles is driven by he DO concenraion gradien beween he equilibrium DO concenraion and acual DO concenraion in liquid phase. In order o describe he oxygen ransfers from he waer surface and bubbles, we separae he oal mass ransfer ino wo differen pars make i possible o evaluae he oxygen ransfer from he air bubbles and a he waer surface. The mass ransfer coefficien for he air bubbles, K ɑ, is assumed o be presen in he whole ank and he waer is assumed o be compleely mixed. K ɑ s, which is he mass ransfer coefficien a he waer surface, is only presen a he waer surface. K ɑ s value depends on level of urbulence a waer surface. The more urbulence a he waer surface he more oxygen is ransferred and K ɑ s increases (DeMoyer e al, 23). How urbulen he waer surface is can depend on he ype of aeraion sysem and he air flow rae. 2

3 dc d = K α (C - C ) + K s α s ( C - C ) SC (4) Where K ɑ s is volumeric mass ransfer coefficien (T -1 ) a waer surface, C SC is he sauraion concenraion a sie emperaure and amospheric, which is equal o C, and C is he dissolved oxygen a ime. Inegraion and re-arrangemen of Eq. 4 becomes: ( C λ + ξ) 1 ( ξ + λ C ) ( - ) ln = λ (5) Where ξ= K a C SC + K s a s C SC, λ= - K a - K s a s, and C is DO concenraion a ime zero. If he iniial dissolved oxygen (D.O) concenraion is mg/ a ime zero, Eq. 5 becomes: ( C λ + ξ) 1 ln = λ (6) ξ ( - ) When sysem is a he seady sae condiion, C () C SC, Eq. 6 can be simplified as: 2 K a s ln = K a + K a s s ( - ) λ (7) Because C SC is DO sauraion concenraion a sie condiions and can be calculaed based on Henry s aw or can be found in he chemical engineering exbook or elsewhere (ewis, 26). K a s is he only parameer ha mus be evaluaed. Once K a s is known, i is a simple maer o calculae K a a sie condiions and calculae he SOTR using Sandard Mehod as discussed in preceding paragraph. a s is raio of oal waer surface area (As) o oal ank volume (V), K s is air-waer ransfer coefficien and i has been deermined by a number of researchers (McWhirer and Huer, DeMoyer, C.D. e al) under various es condiions. K a s could be also measured. The sandard simplified mass ransfer model as discussed in preceding paragraph is used o analyze DO measuremen daa o esimae volumeric mass ransfer coefficien K a s. Modified Sandard Mehod Model 2 Because he liquid phase equilibrium concenraion of a given bubble is no only a funcion of emperaure and amospheric pressure, bu also hydrosaic pressure and gas phase oxygen composiion, oxygen ransfer across bubbles as bubbles rise hrough he waer column mus include some addiional consideraions. Firs is ha bubble-gas ransfer of oxygen, nirogen and some oher race gases occurs due o absorpion process in he liquid phase. As a resul, gas phase oxygen composiion changes over he deph in he gas bubble. Second even ha complicaes he modeling of oxygen ransfer is ha DO equilibrium concenraion also varies wih waer deph. According he Henry s law, he higher he pressure, he more oxygen can be dissolved in he waer. 3

4 In order o describe he oxygen ransfer behavior wih consideraions of oxygen ransfer a waer surface and he difference in equilibrium concenraions wih deph, he following simplified mass ransfer model ha combine Sandard Model and Modified Model 1is used o analyze DO measuremen daa o esimae volumeric mass ransfer coefficien: dc d = K α (C d () z - C )dz + K α ( C - C ) s SC (8) Where d is side waer deph o aeraion sysem (), is side waer deph (), and C o is dissolved equilibrium concenraion (mg/). The DO equilibrium concenraion, C o can be esimaed by following equaion (McWhirer and Huer, 1989): ( - ) d P - PWV y C () z = CSC [ ] (9) 1- P.266 WV where y (kmol O 2 /kmol/n 2 ) is gas-phase oxygen composiion, which is he molar raio of oxygen o nirogen gas in he gas phase, P WV (am) is waer vapor pressure, P (am) is amospheric pressure,.266 (kmol O 2 /kmol N 2 ) a z = when he bubbles released from he diffusers. Inegraion of Eq. 8 yields: dc d = K α d C () z - C )dz + K αs ( CSC - C ) s (1) In order o solve Eq. 1, C o mus be compued. Because y mus be deermined o be able o calculae C in Equaion 8, and y is also deph dependen, i makes he compuaion of Eq. 1 exremely complicaed. However, because a he seady sae condiion, C C and dc, Eq. 1 can be simplified as: d K α (C () z - C )dz = (11) d Re-arrangemen of Eq. 11 gives: () z - C )dz d K α = (C (12) The mass ransfer model as discussed in preceding paragraph is used o deermine K a s. K a can be easily calculaed by Eq. 12. Once K a s and K a are known, i is a simple maer o calculae he SOTR using Sandard Mehod as discussed in Insrucion paragraph. () SC 4

5 RESUTS AND DISCUSSION Fundamenals of Ka Predicaion Using ASCE Sandard Mehod ASCE Sandard Mehod (Figure 1) is based on a very simple model by assuming oxygen ransfer only occurs across he bubbles and liquid waer. The mehod requires measuremen of DO over ime and esimaion of K a by using nonlinear regression. Once K a is deermined, SOTR is calculaed using he Eq. 3. Figure 1. ASCE Sandard Mehod Model Schemaic Showing Oxygen Transfer Only Occurs across Air Bubbles and Bulk iquid Because ASCE Sandard Mehod requires coninuously recording DO values unil DO concenraion in he ank reaches a leas 98% of he presumed seady sae concenraion, all measured DO concenraions over ime are given in Table 1. Table 1 also liss he bes esimaes of hree major parameers (K a, DO sauraion concenraion, and iniial DO concenraion) ha mus be deermined in he sandard model. C is he DO concenraion a ime zero, which is.4 mg/. C is he sauraion or near sauraion DO concenraion a infinie ime (mg/), his can measured or calculaed based on he equaion in he ASCE sandard. K a values are calculaed based on Eq. 2. Figure 2 shows re-aeraion daa and % of DO sauraion concenraion over ime, ploed o conform o ASCE sandard mehod. Table 2 shows esimaed SOTR and SOTE values using K a, ank volume, air flow rae, and diffuser submerged deph. 5

6 Table 1. Measured DO Daa and Esimaed Parameers for Sandard Model Time DO DO Sauraion Iniial DO Conc. % Cs og Defici K a (minues) (mg/) (mg/) (mg/) % NA hr %.1 NA % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %

7 Figure 2. Measured DO Concenraions and % of DO Sauaraion over Time Table 2: Sandard Oxygen Transfer Rae and Efficiency Tes Resuls Using ASCE Sandard Model Air Flow Tank Volume Diffuser Deph K Sie K a (2 C) SOTR SOTE SOTE (SCFM) (MG) (f) (hr -1 ) (hr -1 ) (lbs/hr) (%) %/f 3, Fundamenals of K a Predicaion Using Modified Sandard Model 1 As discussed in Mehodology Secion, here exis wo difference mass ransfer zones for oxygen ransfer (Figure 3) in a diffused aeraion sysem where he air is discharged ino liquid on he boom of he aeraion ank and rise hrough liquid o he waer surface. Because oxygen ransfer rae consiss of wo erms: one from he air bubbles and he oher from he waer surface, K a s mus be deermined o be able calculae K a. 7

8 Figure 3. Modified Model 1 Schemaic Showing Oxygen Transfer Only Occurs across Air Bubbles and Waer Surface Because a s is raio of oal waer surface area (As) o oal ank volume (V), K s is air-waer ransfer coefficien and i has been deermined by a number of researchers (McWhirer and Huer, DeMoyer, C.D. e al) under various es condiions, K a s calculaion becomes very simple. Once K a s is calculaed, K a can be calculaed using he following equaion: [ K α (C - C ) K α ( C - C ) ] SOTR = V + (13) s s SC In order o use Eq. 7 o deermine K a, an ieraive approach mus be used by assuming an arbirary K a value as an iniial guess and repea same calculaion ill values in wo sides are equal. All esimaed parameers using Modified Sandard Model 1 are summarized in Table 3. As shown in Table 3, K s a s value is relaively small in comparison o he oal K a, his can be explained by bubble oxygen ransfer is sill he primary mode of oxygen ransfer for aeraion sysem in he deep aeraion ank (17f) in comparison wih surface oxygen ransfer. Table 3. Sandard Oxygen Transfer Rae and Efficiency Tes Resuls Using Modified Sandard Model 1 Air Flow Tank Volume Diffuser Deph K a (2 C) K a s (2 C) SOTR SOTE SOTE (SCFM) (MG) (f) (hr -1 ) (hr -1 ) (lbs/hr) (%) %/f 3, Fundamenals of K a Predicaion Using Modified Sandard Model 2 8

9 In his mass ranspor model, he air phase oxygen composiion in he bubbles varies wih deph. Air released from he diffuser conains an iniial air phase oxygen composiion (mole raio) of.266 (.21 O 2 /.79 N 2 ). A he same ime, alhough solubiliy of nirogen gas is almos wice as much as oxygen, he concenraion gradien of nirogen is significanly lower (DeMoyer e al). As a resul, gas phase oxygen composiion decreases over deph. K a s can be deermined by he same mehod as discussed in preceding paragraph. C can be deermine d by Eq. 9. Once K a s and C are known, K a can be deermined by Eq. 12. All esimaed parameers using Modified Sandard Model 2 are summarized in Table 4. As shown in Table 4, he impac of oxygen ransfer from he air is a he waer surface in SOTR calculaions can be significan (~1%). Meanwhile, he equilibrium DO concenraions and oxygen composiion in gas bubble, which vary wih waer deph have a significan impac in SOTR calculaions. Table 4: Sandard Oxygen Transfer Rae and Efficiency Tes Resuls Mehod K a (2) K s a s SOTR SOTE SOTE/f K a Difference hr -1 hr -1 lbs/hr % % % Sandard Modified % Modified % CONCUSIONS Field oxygen ransfer resuls presened in his paper suppored he conclusion ha sandard mehod for oxygen ransfer measuremens oversimplified he oxygen ransfer process by assuming surface oxygen ransfer is negligible Field oxygen ransfer resuls presened in his paper also suppored he conclusion ha sandard mehod for oxygen ransfer measuremens oversimplified he oxygen ransfer process by assuming he equilibrium DO concenraions are he same everywhere in he ank. Field oxygen ransfer presened in his paper indicaed oxygen ransfer from waer surface is around 1% of bubble ransfer oxygen. Field oxygen ransfer presened in his paper shown oxygen mass ransfer efficien and SOTR measuremen error could be up o 2% due o surface oxygen ransfer and liquidwaer DO equilibrium concenraions and gas phase oxygen composiion variaion over he deph. Sandard mehod for oxygen ransfer measuremens dose no describe he whole aeraion process, in order o ge comparable resuls, all invesigaes mus he mehod a he same condiions. 9

10 REFERENCE ASCE, 27. Measuremen of Oxygen Transfer in Clean Waer. American Sociey of Civil Engineers, Virginia McWhie, J.R. & Huner, J.C., Improved Oxygen Mass Transfer Modeling for Diffused/Subsurface Aeraion Sysem. American Insiue of Chemical Engineers Journal, Vol. 35, no. 9, pp DeMoyer, C.D., Schierholz, E.., Gulliver, J.S., and Wilhelms, S.C. 23. Impac of Bubble and Free Surface Oxygen Transfer on Diffused Aeraion Sysems. Waer Research. Vol. 37, Sensrom, M.K, eu, S.Y., and Jiang, P. 26. Theory o Pracice: Oxygen Transfer and he New ASCE Sandard. Proceedings of WEFTC 26, pp Ingrid Fandriks Alernaive Mehods for Evluaion of Oxygen Transfer Performance in Clean Waer. Maser Thesis. Deparmen of Informaion Technology, Uppsala Universiy, Uppsala. ewis, M.E. 26. Dissolved Oxygen (version 2.1) Chaper A6, Secion 6.2. hp://waer.usgs.gov/owq/fieldmanual/chaper6/6.2_ver3.pdf 1

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