Development of a model to determine mass transfer coefficient and oxygen solubility in bioreactors

Development of a model to determine mass transfer coefficient and oxygen solubility in bioreactors Development of a model to determine mass transfer coefficient and oxygen solubility in bioreactors Jo[.]

Development of a model to determine mass transfer coefficient and oxygen solubility in bioreactors

Johnny Lee*

Kitchener, Waterloo, Ontario, Canada

* Corresponding author at: 317 Pine Valley Drive, Kitchener, Ontario, N2P 2V5, Canada.

E-mail address: fearlessflyingman@gmail.com (J Lee).

Abstract

The objective of this paper is to present an experimentally validated mechanisticmodel to predict the oxygen transfer rate coefficient (Kla) in aeration tanks fordifferent water temperatures Using experimental data created by Hunter andVogelaar, the formula precisely reproduces experimental results for thestandardized Kla at 20 °C, comparatively better than the current model used byASCE 2–06 based on the equation Kla20 = Kla (θ)(20−T) where T is in °C.

Currently, reported values for θ range from 1.008 to 1.047 Because it is ageometric function, large error can result if an incorrect value of θ is used.Establishment of such value for an aeration system can only be made by means ofseries of full scale testing over a range of temperatures required The new modelpredicts oxygen transfer coefficients to within 1% error compared to observedmeasurements This is a breakthrough since the correct prediction of the volumetricmass transfer coefficient (Kla) is a crucial step in the design, operation and scale up

of bioreactors including wastewater treatment plant aeration tanks, and theequation developed allows doing so without resorting to multiple full scale testingfor each individual tank under the same testing condition for differenttemperatures The effect of temperature on the transfer rate coefficient Kla isexplored in this paper, and it is recommended to replace the current model by thisnew model given by:

Kla20¼ Klað E
σ Þ 20

E
σ

ð Þ T

T20T

5

where T is in degree Kelvin, and the subscripts refer to degree Celsius; E,ρ, σ areproperties of water Furthermore, using data from published data on oxygensolubility in water, it was found that solubility bears a linear and inverserelationship with the mass transfer coefficient

Keywords: Physics methods, Physical chemistry, Energy, Chemical engineering,Civil engineering

1 Introduction

The main objective is to develop a mechanistic model (based on experimentalresults of two researchers, Hunter [1] and Vogelaar [2]) to replace the currentempirical model in the evaluation of the standardized mass transfer coefficient(Kla20) being used by the ASCE Standard 2–06[3] The topic is about gas transfer

in water, (how much and how fast), in response to changes in water temperature.This topic is important in wastewater treatment, fermentation, and other types ofbioreactors The capacity to absorb gas into liquid is usually expressed assolubility, Cs; whereas the mass transfer coefficient represents the speed oftransfer, Kla, (in addition to the concentration gradient between the gas phase andthe liquid phase which is not discussed here) These two factors, capacity, andspeed, are related and the manuscript advocates the hypothesis that they areinversely proportional to each other, i.e., the higher the water temperature, thefaster the transfer rate, but at the same time less gas will be transferred

This hypothesis was difficult to prove because there is not enough literature orexperimental data to support it (Some data do support it, but they are approximate,because some other factors skew the relationship, for example, concentrationgradient; and the hypothesis is only correct if these other factors are normalized orheld constant)[4]

This hypothesis may or may not be proved by theoretical principles, such as bymeans of thermodynamic principles to find a relationship between equilibrium-concentration and mass transfer coefficient, but such proof is beyond the expertise

of the author

However, the hypothesis can in fact be verified indirectly by means ofexperimental data that were originally used to find the effects of temperature onthese two parameters, solubility (Cs) and mass transfer coefficient (Kla).Temperature affects both equilibrium values for oxygen concentration and therate at which transfer occurs Equilibrium concentration values (Cs) have beenestablished for water over a range of temperature and salinity values, but similarwork for the rate coefficient is less abundant

This paper uses the limited data available in the literature, to formulate a practicalmodel for calculating the standardized mass transfer coefficient at 20 °C The workproceeds with general formulation of the model and its model validation using thereported experimental data It is hoped that this new model can give a betterestimate of Kla20 than the current method.

2 Model 2.1 The temperature correction model for Kla 2.1.1 Basis for model development

Using the experimental data collected by two investigators [1, 2], datainterpretation and analyses allowed the development of a mathematical modelthat related Kla to temperature, advanced in this paper as a temperature correctionmodel for Kla The new model is given as:

KlaT ¼ K  T5 E
σ

where Kla = overall mass transfer coefficient (min−1); T = absolute temperature ofliquid under testing (°K); the subscript T in the first term indicates Kla at thetemperature of the liquid at testing; and K = proportionality constant E = modulus

of elasticity of water at temperature T, (kNm−2); ρ = density of water attemperature T, (kg m−3);σ = interfacial surface tension of water at temperature T,(N m−1); Ps is the saturation pressure at the equilibrium position (atm) Thederivation is based on the following findings as described in Section3

The model was based on the two film theory by Dr Lewis and Dr Whitman[5],and the subsequent experimental data by Professor Haslam[6], whose finding wasthat the transfer coefficient is proportional to the 4th power of temperature Furtherstudies by the subsequent predecessors [1,2,7] unveiled more relationships, whichwhen further analyzed by the author, resulted in a logical mathematical model thatrelated the transfer coefficient (how fast the gas is transferring when air is injectedinto the water) to the 5th order of temperature Perhaps this is also a hypothesis, but

it matches all the published data sourced from literature

Similarly, using the experimental data already published for saturation dissolvedoxygen concentrations, such as the USGS (United States Geological Survey) tables[8], Benson and Krause’s stochastic model [9], etc., it was found that solubilityalso bears a 5th order relationship with temperature

So, there are actually three hypotheses But are they hypotheses or are they in factphysical laws that are beyond proof? For example, how does one prove Newton'slaw? How does one prove Boyle's law, Charles' law, or the Gay-Lussac's law?They can be verified of course, but not lend themselves easily to mathematical

derivation using basic principles As mentioned, Prof Haslam found that the liquidfilm transfer coefficient varies with the 4th power of temperature, but how doesone prove it by first principles? The model just fits all the data that one can findalthough it would be great if it can be proven theoretically However, thecorrelation coefficients for (Eq.(1)) are excellent as can be seen in the followingsections.

The paper is not a theory/modelling paper in the sense that a theory was not derivedbased on first principles Nor in fact is it an experimental/empirical paper since theauthor did not perform any experiments However, the research workers who didthe experiments did not recognize the correlation, and so they have missed theconnection This paper revealed that these data can in fact support a new model thatrelates gas transfer rate to temperature that they missed They used their data forother purposes, and drew conclusions for their purposes

Further tests may therefore be required to justify these hypotheses Although otherpeople's data are accurate since they come from reputable sources, they aredifferent from experiments specifically designed for this model developmentpurpose only The novelty of the proposed model is that it does not depend on apre-determined value of theta (θ) to apply a temperature correction to a test data forKla, if all other conditions affecting its value are held constant or convertible tostandard conditions

The current model adopted by ASCE 2–06 is based on historical data and is given

by the following expression:

Kla20KlaT

N¼Kla20KlaT

factor is also dependent on turbulence, as well as the other properties as shown in(Eq (1)) Current wisdom is to assign different values of theta (θ) to suitdifferent experimental testing While adjusting the theta value for differenttemperatures may eventually fit all the data, this may lead to controversies.Furthermore, it is necessarily limited to a prescribed small range of testingtemperatures.

2.1.2 Description of proposed model

The purpose of the manuscript is to improve the temperature correction method forKla (the mass transfer coefficient) used on ASCE Standard 2–06[3]and to replacethe current standard model by (Eq.(1))

The proposed model can also be expressed in terms of viscosity as describedbelow Viscosity can be correlated to solubility When a plot of oxygen solubility

in water is made against viscosity of water, a straight-line plot through the origin isobtained[10] When the inverse of viscosity (fluidity) is plotted against the fourthpower of temperature, the linear curve as shown inFig 1below was obtained.Therefore, viscosity happens to have a 4th order relationship with temperature, sothat (Eq (1)) can be expressed in terms of viscosity and a first order oftemperature, instead of using the 5th order term The concept of molecularattraction between molecules of water and the oxygen molecule is important sincechanges in the degree of attraction would influence the equilibrium state of oxygensaturation in the water system as well as its gas transfer rate Although the aboveplot (Fig 1) shows that the reciprocal of viscosity (fluidity) is linearly proportional

to the 4th order of absolute temperature, the line does not pass through the origin

Fig 1 Reciprocal of viscosity plotted against 4th power of temperature [10] [4].

As viscosity is closely correlated to solubility, it is obvious that the molecularattraction between water molecules that influences viscosity and the molecularattraction between water and oxygen molecules are interrelated This correlationdoes not establish that an alteration of water viscosity, such as changes in thecharacteristics of the liquid, will have an impact on oxygen solubility However, itwill certainly affect the mass transfer coefficient Viscosity due to changes intemperature is therefore an intensive property of the system, whereas viscosity due

to changes in the quality of water characteristics is an extensive property Theequation relating viscosity to temperature is given byFig 1as:

1

μ ¼ 0:2409  103

T1000

whereμ = viscosity of water at temperature T, (mPa.s)

Rearranging the above equation, T4 can be expressed in terms of viscosity andtherefore,

T4¼ K′ 1μþ 0:7815

(6)

where K’ is a proportionality constant

Substitute (Eq.(6)) into (Eq.(1)), therefore,

where K" is another proportionality constant

Therefore, Kla can be expressed as either (Eq.(8)) or as (Eq.(1)) For the sake ofeasy referencing to this model, this model shall be called the 5th power model

2.1.3 Background

The universal understanding is that the mass transfer coefficient is more related todiffusivity and its temperature dependence at a fundamental level on a microscopicscale Although Lewis and Whitman long ago advanced the two-film theory[5]and subsequent research postulated that the liquid film thickness is related to thefourth power of temperature in °K [6], it was not thought that this relationshipcould be applied on a macro scale In a laboratory scale, Professor Haslam [6]conducted an experiment to examine the transfer coefficients in an apparatus, usingsulphur dioxide and ammonia as the test solute Based on Lewis and Whitman’s

finding[5] that the molecular diffusivities of all solutes are identical, he derivedfour general equations that link the various parameters affecting the transfercoefficients which are dependent upon gas velocity, temperature, and the solutegas He found that the absolute temperature has a vastly different effect upon thetwo individual film coefficients The gas film coefficient decreases as the 1.4thpower of absolute temperature, whereas the liquid film coefficient increases as thefourth power of temperature The discovery that the power relationship betweenthe liquid film coefficient and temperature can be applied to an even highermacroscopic level where Cs is a function of depth, is based on a combination ofseemingly unrelated events as follows:

i Lee and Baillod [12, 13] derived by theoretical and mathematicaldevelopment, the mass transfer coefficient (Kla) on a macro scale for a bulkliquid treating the saturation concentration Cs as a dependent variable;

ii The derived Kla mathematically relates to the“apparent Kla”[3]as defined inASCE 2-06[3];

iii It was thought that KL(the overall liquid film coefficient) might perhaps berelated to the fourth power of temperature on a bulk scale similar to the samefinding by Professor Haslam on a laboratory scale, as described above;

iv John Hunter[1] related Kla to viscosity via a turbulence index G;

v It was then thought that viscosity might be related to the fourth powertemperature and a plot of the inverse of absolute viscosity against the fourthpower of temperature up to near the boiling point of water gives a straightline;

vi The interfacial area of bubbles per unit volume of bulk liquid under aeration is

a function of the gas supply volumetric flow rate which is in turn a function oftemperature;

vii It was then thought that Kla might be directly proportional to the 5th power ofabsolute temperature and indeed so, as verified by Hunter’s data described inSection4 (Fig 2); the relationship, however, was not exact because the dataplot deviates from a straight line at the lower temperature region;

viii Adjustment of the initial equation based on observations of the behavior ofcertain other intensive properties of water in relation to temperature improvedthe linear correlation with a correlation coefficient of R2= 0.9991 (Fig 3);

ix The relationship is based on fixing all the extensive factors affecting the masstransfer mechanism Specifically, Kla is dependent of the gas mass flow rate.Since Hunter’s data has slight variations in the gas mass flow rate over thetests, normalization to a fixed gas flow rate improves the accuracy to R2=0.9994 (Fig 4), with the straight line passing through the origin

Based on the above reasoning, data analysis as described in detail in the followingsections confirmed the validity of (Eq.(1)), but only for the special case where Ps

is at or close to atmospheric pressure (i.e Ps = 1 atm) The experiments described

in this Paper have not proved that Kla is inversely related to Ps The authoradvances a hypothesis that Kla is inversely proportional to equilibriumconcentration (Cs), which can be related to pressure which therefore in turn isrelated to the depth of a column of water Since saturation concentration is directlyproportional to pressure (Henry’s Law), therefore Kla must be inverselyproportional to pressure, if the reciprocity relationship between Kla and Cs is true.Furthermore, the concept of equilibrium pressure Ps and how to calculate Ps must

be clarified for a bulk column of liquid (The details for the pressure adjustment aregiven in ASCE 2–06 Section 5 and ANNEX G) [3] Insofar as the currenttemperature correction model has not accounted for any changes in Ps due totemperature, this manuscript has assumed that Ps is not a function of temperaturefor a fixed column height and therefore does not affect the application of (Eq.(1))for temperature correction

2.1.4 Theory

The Liquid Film Coefficient (kl) can be related to the Overall Mass TransferCoefficient (KL) for a slightly soluble gas such as oxygen For any gas-liquidinterphase, Lewis and Whitman’s two-film concept[5] proved to be adequate toderive a relationship between the total flux across the interface and theconcentration gradient, given by:

Fig 4 Kla vs temperature, modulus of elasticity, density, surface tension, gas flow rate.

When the liquid film controls, such as for the case of oxygen transfer or other gastransfer that has low solubility in the liquid, the above equation is simplified to

This means that the gas transfer rate on a macro scale is the same as in a microscale when the liquid film is controlling the rate of transfer due to the fact that theliquid film resistance is considerably greater than the gas film resistance The fourequations Prof Haslam[6] developed are given below:

Eq (15) Because the interphase concentrations are impossible to determineexperimentally, only the overall mass transfer coefficient KLcan be observed in hisapparatus However, by substituting the values of the film coefficients calculatedusing the above equations into Eq.(10), excellent agreement was found betweenthe observed values of the overall coefficients and those calculated Because ofEqs.(11)and(14), it can be concluded that the overall mass transfer coefficient in

a bulk liquid is proportional to the fourth power of temperature, given by:

where k’ is a proportionality constant

For spherical bubbles, the interfacial area (a) is given by:

The contact time is dependent upon the path of the bubble through the liquid and can

be expressed in terms of the average bubble velocity vband the liquid depth Zd:

tc¼Zd

vb

(18)

where, vb= average bubble velocity, (m s−1)

The area of bubble interface per unit of tank volume V is then

a¼ 6  Qa

This shows that for a given tank depth, and a fixed aeration system, ‘a’ isproportional to the gas flow rate Qa The mass transfer coefficient is dependent onthe volumetric gas flow rate which changes with temperature and pressure–– thehigher the gas flow the faster is the transfer rate The average gas flow rate isdependent on the test temperature of the bulk liquid With this in mind, Qacan bedetermined as follows:

Combining Eq A-1b in Section A.5.1 of the ASCE standard 2–06[3]Annex A and

Eq A-2b where they were written as:

Q1= gas flow at the gas supply system

QP = gas flow at the point of flow measurement (at the diffuser depth)

Ps = standard air pressure, 1.00 atm (101.3 kPa)

P1= ambient (gas supply inlet) atmospheric pressure

PP= gas pressure at the point of flow measurement

Ts = standard air temperature (293 K for U.S practice)

T1= ambient (gas supply inlet) temperature, °K (= °C + 273)

TP= gas temperature at the point of flow measurementSubstituting (Eq.(20)) into (Eq.(21)), we have

Qa¼

QSPSTP2

As stated above, the response of Kla to temperature is affected by the behaviour ofthe water properties that are the other variables that affect the 5th order temperaturerelationship As the temperature drops, the density of water (ρ) increases, and themaximum density is at about 4 °C Similarly, the surface tension (σ) also increaseswith the decrease of temperature However, the modulus of elasticity (E) decreases

as the temperature decreases This is because the modulus of elasticity isproportional to the inverse of compressibility, which increases as the waterapproaches the solid state Compressibility of water is at a minimum at around 50

°C Combining all the three variables in response to temperature with the 5th orderrelationship would result in a curve that resembles the error structure in Hunter’sexperiment as described in Section 4 below These changes in water propertieswith respect to temperature are shown inFig 5,Fig 6, andFig 7

The variability of the compound parameter (Eρσ) with temperature is also shown inFig 7for the elasticity curve Taking into account the changes in water properties

in response to temperature, (Eq.(27)) can be simplified to:

KlaT¼ K  T5E
σ

Ps

(28)where the symbols are as defined in (Eq.(1)) The inverse relationship betweenKlaTand Psis a hypothesis, based on the assumption that KlaTand Csthe solubilityare inversely related (See Section4.3below.)

Fig 5 Density vs temperature °C.

3 Materials and methods

To derive a temperature correction model, there are two ways One is to use thesolubility law derived from the solubility table for water, (Section 4.3), and theknowledge that Kla is inversely proportional to Cs, under a reasonable temperatureboundary range The other method is by use of examination and interpretation of

data performed by numerous investigators, such as Hunter’s data [1], on therelationship between Kla and temperature.

The new model for the correction number N as defined by (Eq.(3)), is based on the5th power proportionality Numerous investigators have performed experiments ofKla determination at different test water temperature, ranging from 0 °C to 55 °C.These data appear to support the hypothesis that Kla is proportional to the 5thpower of absolute temperature for a range of temperatures close to 20 °C andhigher For temperatures close to 0 °C, however, the water properties begin tochange in anticipation of a change of physical state This change from a liquid state

to a solid state at this low temperature is unique to water However, byincorporating these changes of the relevant properties into the Kla equation, asdescribed previously, it becomes possible to find a high degree of correlation forthe data interpretation

The following paragraphs describe the derivation method to arrive at the proposedtemperature correction model by use of experimental data This derivation is purelybased on data interpretation and data analysis using linear graphical verification,and is not derived theoretically

3.1 Hunter ’s experiment

Hunter [1] performed an experiment for the case of laboratory-scale submergedturbine aeration systems Results shown inTable 1 He derived an equation thatrelates Kla to the various extensive properties of the system and to viscosity, andcorrelated his data for a temperature range of 0–40 °C His method is described inthe Paper cited in the manuscript and in his dissertation: Hunter, John S.“A Basisfor Aeration Design” Doctor of Philosophy Dissertation, Department of CivilEngineering, Colorado State University, Fort Collins CO, 1977

3.2 Vogelaar et al ’s experiment

The experiments performed by J.C.T Vogelaar et al.[2]consist of determining Klausing tapwater for a temperature range of 20–55 °C using a cylindrical bubblecolumn with an effective volume of 3 litres and subject to aeration flow rates of0.15, 0.3, 0.45, and 0.56 vvm (volume air volume liquid−1min−1) The results forone particular volumetric air flow rate (0.3 vvm) among all the data are presented

in Section4.1.2below

The following section describes how the data from these two research workershave been used to develop the temperature correction equation for determiningKla20for any clean water test carried out in accordance with ASCE 2–06, and it isproposed that this new equation is to be used to replace the current equation asstated in ASCE 2–06 Section5and the relevant sections concerning the use of (θ)

in the calculation of this important parameter Kla20–the standardized Kla atstandard conditions as defined in the ASCE Standard.

4 Results and discussion

Hunter[1]has suggested that turbulence can be related to viscosity as well as theaeration intensity that created the turbulence In surface aeration, aeration intensitycan be the power input to the water being aerated, while in subsurface diffusedaeration, it is likely to be the air bubbles flow rate Therefore, for certain fixedpower intensity, Hunter surmised that Kla is only a function of viscosity which inturn is a function of temperature He created a mathematical model that relatedKlaTto viscosity at different temperatures from 0 °C to 40 °C His results are given

in Table 1, where Kla(G) are his modelled results The model he used wasexpressed as:

KlaðGÞ ¼ ð 4:04 þ 0:00255G2 ðD

where D/T is a geometric function [Note that T in his equation is NOTtemperature], G2= P/V/μ where μ is viscosity, P is the power level (total powerinput into the water being aerated in ergs/s, and V is the volume of tank in cm3).The term G was defined as the turbulence index However, just as in solubility, it iserroneous to consider G as a function of viscosity because viscosity is an intensiveproperty not extensive Changing the viscosity would not increase turbulence, inthe same way turbulence does not affect viscosity for a fixed temperature.However, in his paper’s attachment, he has theoretically derived a relationshipbetween r, the rate of gas-liquid interfacial surface renewal, and the turbulence

Table 1 Hunter’s experimental data (*Note: The air flow rate Q is back calculatedfrom Hunter’s equation at D/T = 0.35, P/V = 2000)

T(°C) Viscosity(poise) T(°K) (T/1000)5× 104 Kla(h−1) Kla(G)(h−1) Q(SCFH)*

index G, that they are equal Since KLthe liquid film coefficient is related to r, itcan be concluded that turbulence affects the mass transfer coefficient, but this isnot due to the apparent correlation between G andμ.

In this table, the observed Kla results are given in column 5 His modelled resultsare given in column 6 As one can see, his predicted results match up quite wellwith the true results for those tests carried out at 20 °C and above At the lowertemperature range, however, his errors increase progressively as the temperaturedrops to the water melting point (freezing point) His results can be seen from thefollowing plot inFig 8

Hunter did not explain why the errors in terms of percent difference become morepronounced toward the lower end of the temperature spectrum, since the turbulenceindex G has already accounted for the increase of viscosity due to temperature, and

so if turbulence was only a function of viscosity, the changes due to viscosity to themass transfer coefficient should have been taken care of in his equation However,

in his attachment, he did derive an equation that relates Kla not only to G, but also

to other system variables which he had not defined (Note: Hunter’s formula didinclude the extensive properties as system variables in his experiment: geometry,power level, volume, gas flow rate But while the extensive properties areimportant factors affecting Kla, it is found in this study that the relationshipbetween Kla and the intensive properties is always linear, and this linearrelationship is independent on the extensive properties The intensive properties areall temperature dependent.) Hunter did not know of the 5th power model Had heplotted his Kla(G) values against the 5th power of absolute temperature, he wouldhave been astonished to see a perfect straight line as shown inFig 2

Fig 8 Hunter ’s data of Kla plotted vs temperature °C.