Theoretical analysis and experimental study of oxygen transfer under regular and non-breaking waves

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2013,25(5):718-724 DOI: 10.1016/S1001-6058(13)60417-5

Theoretical analysis and experimental study of oxygen transfer under regular and non-breaking waves*

YIN Ze-gao (尹则高) Shandong Province Key Laboratory of Ocean Engineering and Engineering College, Ocean University of China, Qingdao, China Department of Civil and Environmental Engineering, University of Alberta, Edmonton, Canada, E-mail: [email protected] DAVID Z. Zhu Department of Civil and Environmental Engineering, University of Alberta, Edmonton, Canada LIANG Bing-chen (梁丙臣) Shandong Province Key Laboratory of Ocean Engineering and Engineering College, Ocean University of China, Qingdao, China WANG Le (王乐) Engineering College, Ocean University of China, Qingdao, China (Received September 18, 2012, Revised November 20, 2012) Abstract: The dissolved oxygen concentration is an important index of water quality, and the atmosphere is one of the important sources of the dissolved oxygen. In this paper, the mass conservation law and the dimensional analysis method are employed to study the oxygen transfer under regular and non-breaking waves, and a unified oxygen transfer coefficient equation is obtained with consi- deration of the effect of kinetic energy and wave period. An oxygen transfer experiment for the intermediate depth water wave is per- formed to measure the wave parameters and the dissolved oxygen concentration. The experimental data and the least squares method are used to determine the constant in the oxygen transfer coefficient equation. The experimental data and the previous reported data are also used to further validate the oxygen transfer coefficient, and the agreement is satisfactory. The unified equation shows that the oxygen transfer coefficient increases with the increase of a parameter coupled with the wave height and the wave length, but it decreases with the increase of the wave period, which has a much greater influence on the oxygen transfer coefficient than the coupled parameter. Key words: dissolved oxygen, regular and non-breaking waves, oxygen transfer coefficient

Introduction The dissolved oxygen concentration is an impo-

rtant index of the water quality. Generally, the dissolved oxygen in a water body originates from the atmosphere and the photosynthesis of the water-inhabiting

* Project supported by the National Natural Science Foun- dation of China (Grant Nos. 51009123, 50809065), the Natural Science Foundation of Shandong Province (Grant No. ZR2009FQ003). Biography: YIN Ze-gao (1977-), Male, Ph. D., Associate Professor

plants, where the atmosphere is one of the important sources of the dissolved oxygen in water[1,2].

Classical film-based theories, such as two films, penetration, and surface renewal theories, were deve- loped to explain the dependency of the oxygen transfer coefficient on the hydrodynamic and aerodynamic factors and the molecular transport properties of gases[3,4]. Although these film-based theories give simple physical pictures of the gas absorption processes, the transfer coefficient cannot be predicted by these theories alone because of existence of unknown parameters, including the film thicknesses and the surface renewal rates[5]. As far as the real ocean is conce- rned, the oxygen transfer coefficient is usually influe- nced by the wind speed. The wind tunnel experiment

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and the field work show that a strong correlation exists between the mass transfer coefficient and the wind speed. Under high wind conditions, this coefficient is found to be proportional to the wind speed at the sta- ndard height of 10 m above the wave surface[6-8].

These studies improved our understanding of the interfacial gas transfer processes, however, one does not have enough knowledge to predict the oxygen transfer coefficient for waves with a low wind speed (less than 3 m/s) and the zero wind speed as ship-generated waves[5]. Without the wind speed above the wave surface, the wave geometry parameters should be used rationally to express the oxygen transfer capacity. The wave slope, the drag force coefficient, the wave height and fetch were introduced to compute the mass transfer coefficient[9,10]. But they can be used only for a specific type of waves, e.g., the shallow water wave or the deep water wave, and their applica- tions are somewhat limited.

Though there are several theoretical and field studies concerning the oxygen transfer coefficient for shallow water waves, intermediate depth water waves and deep water waves, a unified equation is not avai- lable for the mass transfer coefficient covering all above cases. Because of the limitations of the existing equations, one often has to choose a suitable equation among them to predict the transfer coefficient, depe- nding on the wave pattern parameters. However, these equations do not predict similar transfer rates. Espe- cially when under a condition near the transition condition between different wave types, a great discrepancy occurs between them[8]. Therefore, it is difficult to select an appropriate equation without detailed knowledge of how and under what conditions they were formulated and validated.

This paper proposes a unified oxygen transfer coefficient equation which can be used for all regular and non-breaking waves without the additional consi- deration of the water depth effect. Theoretical analysis and an experiment are carried out to study the oxygen transfer coefficient of regular and non-breaking waves, with considerations of the effect of the kinetic energy and the wave period. A unified equation for the oxygen transfer coefficient is proposed and validated by experimental data of intermediate depth water waves. 1. Theoretical analysis 1.1 Oxygen transfer coefficient deduction for regular

and non-breaking waves The oxygen transfer flux at wave surface can be

written as follows

= ( ) =L s LF K C C K D (1)

where F is the amount of the oxygen mass transfe- rred over unit area and time, LK is the oxygen transfer

coefficient, sC is the saturation dissolved oxygen con-

centration, C is the Dissolved Oxygen Concentration (DOC), D is the oxygen deficit and = sD C C .

The oxygen transfer into a stagnant water comes only through the molecular diffusion between the water and the atmosphere, but the oxygen transfer of a wave comes not only through the molecular diffusion, but also through the turbulent diffusion. Generally, the intensity of the molecular diffusion is much less than that of the turbulent diffusion, so the molecular diffusion here can be neglected to simplify the study.

According to the law of the mass conservation, the dissolved oxygen mass variation in the control volume per unit time is regarded as the net mass di- fference between those entering and leaving the control volume per unit time, which can be described as follows

1 1

0 00 0d d = d d

A t V C

t CF t A c V (2)

where A is the wave surface area of the control volume,

2 2/ 2 2

20

2= 2 1+ cos d

L HA x x

L L

,

L is the wave length, H is the wave height, t is time,

0t and 1t are the initial time and the end time, respe-

ctively, V is the volume of the control volume and

0C and 1C are the average DOC of the control

volume at the initial time and the end time, respectively.

Substituting Eq.(1) into Eq.(2) and integrating over time and space, we obtain

0 1

=0 =0

( )=

( )L I N

ns i n i

i n

C C VK

C C t A

(3)

where niC is the DOC at i point and n time step,

= 0,1i I , = 0,1n N .

1.2 Oxygen transfer coefficient dimensional analysis

of regular and non-breaking waves With regards to the oxygen transfer coefficient of

regular and non-breaking waves, the wave height, the wave length, the wave period and the water depth influence the process of the oxygen transfer.

The stronger the turbulence intensity, the larger the wave kinetic energy, and the oxygen transfer coe-

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fficient will be. The wave kinetic energy can be denoted by a parameter coupled with the wave height and the wave length of regular and non-breaking waves

21 1= =

16 16E H L E

where E is the wave kinetic energy per unit width,

is the water specific weight, E is a parameter coupled with the wave height and the wave length and

2= /E H L . So the coupled parameter in the wave kinetic

energy is taken as an independent variable in the dimensional analysis integrated with the water depth and the wave period. Here the coupled parameter expresses the horizontal space characteristics of the wave geometrical property, the water depth expresses the vertical space characteristics of the wave, and the wave period expresses the time characteristics.

LK can be denoted as follows

= ( , , )LK f E d T (4)

where d is the still water depth and T is the wave period.

In general, the oxygen transfer rate increases with the increase of the wave height, but decreases with the increase of the wave length, the wave period and the water depth. In fact, the oxygen transfer between the water and the atmosphere occurs only through the wave surface. The wave height, the wave length and the wave period influence directly the wave surface area and the occurrence frequency. For a shallow water wave, the water depth influences the oxygen transfer coefficient through the related wave geometry parameters (wave height, wave length and wave period). For a deep water wave, the wave parameters are not related with the water depth; the water depth does not influence the oxygen transfer coefficient, but influences the oxygen transfer mass and the DOC distribution. For an intermediate depth water wave, the water depth influences not only the oxygen transfer coefficient through the previously mentioned wave geometry parameters, but also the oxygen transfer mass amount and the DOC distribution. It is not suitable to express the oxygen transfer coefficient by the water depth especially in a deep water wave, and maybe one can use the related wave parameters, ex- cept for the water depth[11]. So the water depth is neglected here to deduce a unified equation for the oxygen transfer coefficient.

With the fundamental variables of length and time, we have [ ] = [ ] [ ]x y

LK B E T (5)

where B is an undetermined constant. After the dimensional analysis, we have = 1/ 3x ,

= 1y . So

1/ 3 1=LK BE T (6)

The next step is to determine LK with Eq.(3) and

experimental data. Then B in Eq.(6) can be calcula- ted.

Fig.1 Schematic diagram of experiment (#1 and #2 indicate lo-

cations of 2 dissolved oxygen probes) 2. Experiments 2.1 Experiment setup

An experiment was conducted in the wave channel in the hydraulic lab of Ocean University of China. The length, the width and the height of the wave channel are 30 m, 0.6 m and 1 m, respectively. The regular and non-breaking waves were generated by a pis- ton type wave generator. The operating frequency and the stoke length of the wave generator were controlled to adjust the wave parameters as required. The tap water was used in the experiment. The water temperature T was controlled by a heater to make sure that it was close to 20oC. The measured water temperature varied from 19.9oC to 20.1oC, and was considered as 20oC without further corrections. The measured area lies in the middle of the wave channel. Two dissolved oxygen probes of JPBJ608 type were used to measure the DOC after the calibration with Winkler technique[12]. The distance between 2 dissolved oxygen probes is 0.8 m, and the #1 dissolved oxygen probe was put in the wave trough position to measure the DOC near the wave surface. The #2 dissolved oxygen probe was put at the position of half the still water depth to measure the DOC as the average value of the control volume. This assumption deserves a comment here. It is justified based on the references[13,14] , where it was found that the DOC decreases from a higher value near the surface to a lower value near the bottom al- most linearly. Thus a depth averaged dissolved oxygen can be found at the mid-depth. In addition, the gravity center of the regular wave is the same as the still water, so the assumption could be used without

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causing major errors for the average DOC of the control volume. A wave height meter was put at the 0.3 m position after the #2 dissolved oxygen probe to measure the wave height, see Fig.1.

It was indicated that an oxygen scavenger method could be used to obtain the DOC, and the an- hydrous sodium sulfite (Na2SO3) is capable of scave- nging the oxygen effectively and has long been used in the presence of a catalyst (CoCl2) to scavenge the oxygen in water[15]. According to the reaction of Eq.(7), a suitable amount of Na2SO3 was added into the whole channel to lower the DOC so as to reach the desired value without a complete removal of the dissolved oxygen in the experiment. CoCl2 was also used to accelerate the reaction speed to ensure that the reaction is completed before the measurement. The amount of CoCl2 was suggested to be about 0.1% of Na2SO3 by weight[15].

2CoCl2 3 2 2 42Na SO + O 2Na SO (7)

With the water depth varying from 0.2 m to

0.33 m, the wave height from 0.04 m to 0.13 m, and the wave period from 0.8 s to 2 s (see Table 1), a total of 125 experimental conditions were obtained and in- vestigated, the DOC, the wave heights, the water length and the wave period were measured, respectively. Table 1 Parameters of experiments

Water depth, d (m)

Wave height, H (m)

Wave period, T(s)

0.20 0.04 0.8

0.23 0.06 1.0

0.28 0.08 1.4

0.30 0.10 1.8

0.33 0.13 2.0

2.2 Experiment results and analysis

Figure 2 shows the DOC of #1 and #2 dissolved oxygen probes against time for a water depth of 0.3 m. Here 1stC and 2ndC represent the measured DOC of #1

and #2 dissolved oxygen probes, respectively. Figure 2 shows that the DOC of #1 and #2 disso-

lved oxygen probes increase faster at the initial stage because of the great oxygen deficit in the control volume. With the increase of time, more and more oxygen is dissolved into the water, the oxygen deficit decreases to make the DOC increase more slowly and close to the saturation condition ultimately. When the DOC of the control volume reaches the saturation concentration, the mass of the oxygen entering per unit time is equal to that leaving per unit time through the wave surface, and the dynamic balance of the dis-

solved oxygen in the water body is reached. Figure 2 also shows that the DOC of the #1 dissolved oxygen probe increases faster than that of the #2 dissolved oxygen probe at the initial stage, and they approach each other in the end.

Fig.2 DOC time history of #1 and 2# dissolved oxygen probes

Figures 2 (a) and 2(b) show that the DOC increases fast with a larger E under the same T . With =T 1.4 s, E increases from 0.0041 m3 to 0.0190 m3, the saturation time decreases from 140 min to 65 min. Figures 2(c) and 2(d) show that if T decreases to 44%

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Table 2 20LK comparison between Eq.(8) and experimental data

Experiment No. H (m) L (m) T (s) 20LK of experiment (10–5m/s) 20LK of Eq.(8) (10–5 m/s)

1 0.04 0.96 0.80 2.32 2.51

2 0.08 0.96 0.80 4.21 3.96

3 0.12 0.96 0.80 6.17 5.17

4 0.04 1.37 1.00 1.98 2.25

5 0.08 1.37 1.00 3.55 3.56

6 0.12 1.37 1.00 5.13 4.65

7 0.04 2.15 1.40 1.57 1.87

8 0.08 2.15 1.40 2.74 2.95

9 0.12 2.15 1.40 3.81 3.86

10 0.04 2.89 1.80 1.29 1.60

11 0.08 2.89 1.80 2.27 2.53

12 0.12 2.89 1.80 2.99 3.31

13 0.04 3.26 2.00 1.21 1.50

14 0.08 3.26 2.00 2.15 2.37

15 0.12 3.26 2.00 2.87 3.10

(0.8 /1.8 0.44) and E decreases to 33% (0.0058 /

0.0174 0.33) , the oxygen transfer coefficient will

increase to 158% according to Eq.(6). Assume that E and T are doubled, respectively, and the oxygen transfer coefficient will decrease to 63%, showing that T has a much greater influence on the oxygen transfer coefficient than E according to Eq.(6). 3. Determination and validation of oxygen transfer

coefficient 3.1 Determination of oxygen transfer coefficient

Here, the experimental results of previously mentioned 125 conditions are used with regression analysis to determine and validate B in Eq.(6), where 110 conditions are adopted randomly to determine it. The statistical software package SPSS v17.0 is employed to accelerate the analytical process with least squares method. Equation (8) is proposed to represent 20LK

from Eq.(6).

1/ 3 120 = 0.00017LK E T (8)

where 20LK is the oxygen transfer coefficient when

the water temperature is 20oC. 3.2 Validation of oxygen transfer coefficient

Equation (8) is validated by the data from other 15 working conditions with water depth ( )d of 0.3 m,

see Table 2, here the wave type is all the intermediate depth water wave. Table 2 shows the 20LK compari-

son between Eq.(8) and experimental data, and it is shown that the computed oxygen transfer coefficient based on Eq.(8) agrees with the experimental results approximately.

In order to further validate Eq.(8), the previous reported data[11] are also used, see Table 3, here the wave type again is all the intermediate depth water wave. The water temperatures in their experiments were not always kept at 20oC, and 20LK was determi-

ned by the transformation as follows[16]

0.5 0.50.5

20

20 20 20

293= =

+ 273L

L

K Sc

K Sc T

(9)

where Sc is the Schmidt number, = / mSc D , with

being the kinematic viscosity of the water body,

mD being the molecular diffusivity, 20Sc is the

Schmidt number at 20oC, 20 is the kinematic visco-

sity of the water body at 20oC, and 20 is the water

density at 20oC. Table 3 shows that the results of 20LK based on

Eq.(8) agree with the transformed experimental data of 5 conditions (No. 16, No. 17, No. 18, No. 20 and No. 21) approximately, but some greater discrepancy exists in No.19 condition. The reason is that an incon- sistency maybe exists between No. 19 and No. 20, because they have the same wave length, water depth

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Table 3 20LK comparison between Eq.(8) and previous reported data[11]

Experiment No. of previous

reported data

H (m)

L (m)

d (m)

T (s)

T (oC)

LK of

previous reported data

(10-5m/s)

20LK of previous

reported data (10-5m/s)

20LK of Eq.(8)

(10-5m/s)

16 0.109 1.86 0.72 1.10 17.9 4.80 5.10 4.39

17 0.203 3.30 0.72 1.55 17.9 6.20 6.50 5.68

18 0.125 2.50 0.72 1.55 17.9 3.00 3.10 3.76

19 0.283 4.37 0.72 1.90 12.7 3.40 4.20 6.33

20 0.211 4.37 0.72 1.90 13.2 3.50 4.20 5.21

21 0.202 3.30 0.72 1.55 13.9 3.90 4.60 5.66

and wave period but different wave height and water temperature. LK of No. 19 should be larger than that

of No. 20 theoretically, but their experimental data are just the opposite. So, it is unsuitable to use the data of No. 19 to validate Eq. (8). Tables 2-3 show that it is reliable and valid to compute the oxygen transfer coefficient with Eq.(8) at 20oC of the water temperature.

The limitations of these experiments should be pointed out. All the previously mentioned conditions are for the intermediate depth water wave due to the limitation of experimental facilities, and Eq.(8) is validated only for intermediate depth water waves. More field work and experiment data for shallow water wave and deep water wave are still expected to derive the B value of Eq.(6) and to compare with Eq.(8) in the future work. 3.3 Discussions

Compared with the oxygen transfer coefficient in an open channel flow[17] (for example, =LK

0.5 1.50.000045U d , where U is the average velocity),

the result of Eq.(8) is much smaller in general. The reason is that the average velocity magnitude in an open channel flow is often larger than that of a regular wave, which dissolves much more oxygen into the water. And the vertical mixing capacity also increases with the increase of the average velocity, contributing to the increase of the oxygen transfer coefficient. 4. Conclusions

In this paper, the theoretical analysis and experiments are used to deduce the oxygen transfer coefficient equation for regular and non-breaking waves. The results show that the oxygen transfer coefficient increases with the increase of the parameter coupled with the wave height and the wave length, but it decreases with the increase of the wave period, which has a much greater influence than the coupled parameter. The limitation of research is that all the previously

mentioned experiments are for the intermediate depth water wave, and more data for shallow water wave and deep water wave are still expected to validate the oxygen transfer coefficient equation.

Generally speaking, the natural waves behave in much more complex way than those in the lab. For example, if a breaking wave occurs, it will bring a lot of bubbles into the water. Due to the interaction between the water and the bubbles through the bubble’s surface, the DOC of the water body will change much more greatly than that on the wave surface, which is an interesting topic that requires further investigation. If wave groups occur, the wave geometry differs greatly from the regular wave, resulting in a more complicated oxygen transfer process. In future work, attention should be paid on the oxygen transfer of the breaking wave and the irregular wave to solve the practical environment problem. References [1] SCHMID B. H., KOSKIAHO J. Artificial neural net-

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[3] JIRKA G. H., HERLINA H. and NIEPELT A. Gas transfer at the air–water interface: experiments with different turbulence forcing mechanisms[J]. Experiments in Fluids, 2010, 49(1): 319-327.

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Theoretical analysis and experimental study of oxygen transfer under regular and non-breaking waves