The data from the steady-state tank experiments was all processed using the computer program "Stanford Graphics". A multilayer spreadsheet and associated graphs were set up for processing all the tank data, with modifications for each experiment as appropriate. The section f this chapter follow approximately the order in which calculations were made in this spreadsheet. Firstly, Section 6.2 lists the formulae for the thermodynamic and kinetic constants based on temperature and salinity. Section 6.3 explains the carbonate speciation based on the pCO_{2} and alkalinity, and also discusses the relative contributions of the various species, and the internal consistency of the carbonate system. Section 6.4 gives the formulae of the predicted CO_{2} transfer velocities due to both diffusion and reaction, and Section 6.6 explains how the actual CO_{2} transfer velocities and their associated error ranges were calculated. Finally mass-balance estimates of total and biological carbon are explained

Notes to table 6-1 :

TCO_{2}, alkalinity, pH, chlorophyll, and SF_{6} and O_{2} transfer velocities were only measured occasionally, and were calculated as described in Section 5.9 , Section 5.10 , Section 5.10 , Section 5.4 , Section 5.11 and Section 5.12 respectively. The carbonate system calculations were therefore based on pCO_{2} and a constant total alkalinity (except when it was known to change). The measured TCO_{2} and pH have been plotted on the "Carbon budget" graphs in chapter 8 for comparison with the values calculated from the pCO2 measurements.

The baseline for predicted CO_{2} transfer velocities due to diffusion only was derived from the temperature and rpm calibration experiments (see Section 6.4 , Section 7.2 and Section 7.3 ). Therefore SF_{6} / O_{2} gas exchange measurements are plotted only for comparison.

Day and time were combined into one variable "time" = day + time / 24 which forms the x-axis of many graphs, the axis ticks representing midnight. Since many measurements were made in the evening, the data may be closer to the label for the following "day" than the date of measurement.

The typical values given are illustrative and not intended to be internally consistent. For more detail on measurement errors see the relevant section f chapter 4 and chapter 5. Factors which might increase the error of a particular CO_{2} transfer velocity measurement were noted and incorporated into the calculations as described in

k_{1} H_{2}CO_{3} <==> HCO_{3}^{-} + H^{+}

k_{2} HCO_{3}^{-} <==> CO_{3}^{2- }+ H^{+
}

k_{w} H_{2}O <==> H^{+ }+ OH^{-
}

kb B(OH)_{3} + H_{2}O <==> H^{+} + B(OH)_{4}^{-
}

kp_{1} H_{3}PO_{4} <==> H^{+} + H_{2}PO_{4}^{-}

kp_{2} H_{2}PO_{4}^{-} <==> H^{+} + HPO_{4}^{2-
}

kp_{3} HPO_{4}^{2- }<==> H^{+} + PO_{4}^{3-
}

Each constant k is defined as [H] [A] / [HA], where HA and A are the acid and base forms of each pair respectively. They are *apparent* equilibrium constants based on *concentrations* rather than *activities*, valid only for one specific temperature and salinity. In other words, the *activity* coefficients, which are determined predominantly by the bulk seawater medium rather than the species reacting, are built into the constant. The units are mol kg^{-1} (solution) rather than mol l^{-1}. For more detail on this topic refer to papers by Hansson (1973) and Dickson (1993, 1994). All the thermodynamic constants used here can be found in Dickson (1994), where the chemistry of the carbonate system is discussed extensively.

Notes on table 6-2 :

1. The Oxygen and SF_{6} constants were not used routinely, but are included here for convenience

2. The rate constants for k_{CO2} and k_{OH}K_{w} are taken directly from the paper of Johnson (1982). -Emerson (1995) used a much lower value for k_{OH}K_{w}, which I believe to be incorrect, for reasons already explained in Section 1.5.2 . Moreover, the experimental results reported in Section 7.5 suggest that the value may need to be higher, rather than lower.

3. The Schmidt number formulae given by Wanninkhof (1992) are derived from Jahne (1987b) for CO_{2} and from Wilke and Chang (1955) for SF_{6} and Oxygen. King and Saltzman (1995) gave a slightly different formula for the SF_{6} Schmidt number.

However, an iterative process is required to calculate this speciation exactly from these parameters. This is because the inclusion of borate, phosphate, and hydroxyl ions in the total alkalinity results in a set of seventh order differential equations, which cannot be solved algebraically. On the other hand, iteration is inconvenient in a spreadsheet with many data points.

A compromise was found, whereby the hydrogen ion concentration was estimated first assuming that the carbonate alkalinity was equal to the total alkalinity. The borate and phosphate were then speciated using this hydrogen ion concentration. The carbonate alkalinity was then recalculated by deducting the contributions of the borate, phosphate and hydroxyl ions from the total alkalinity. From this second carbonate alkalinity, a second hydrogen ion concentration and carbonate speciation were calculated. The method should become clearer from a study of table 6-3 .

It was found that a third iteration (i.e. recalculating the borate and phosphate speciation using the second hydrogen ion concentration, and from that recalculating a third carbonate alkalinity and hydrogen ion concentration) made no significant difference to the TCO_{2} and hydroxyl ion concentrations, which are the parameters of most interest.

Figure 6-1 shows the measured TCO_{2} as a function of the TCO_{2} calculated by this method. Generally, the internal consistency seems sufficient for the purposes of this investigation

It would be confusing to include the concentrations of every chemical species on all the carbon budget graphs, and as they are in equilibrium this would not convey independent information. However, it is interesting to see how the speciation changes as a function of TCO_{2}. This is shown in figure 6-2 for a typical algal bloom culure.

The data in figure 6-2 is taken from the fourth *Dunaliella* bloom. Note that the phosphate concentration, similar to that in f/2 medium, is much higher than is typical of seawater, and therefore makes a significant contribution to the total alkalinity especially at low pCO_{2} (or TCO_{2}). Before the phosphate was included in the calculations as described above, it was not possible to reconcile the predicted and measured TCO_{2}, nor to explain the unusually high measured alkalinity.

Once the unenhanced transfer velocity has been calculated, the predicted enhancement due to reaction can be added by employing the formula of Hoover and Berkshire (1969). This uses the carbonate speciation as calculated in table 6-3 , and also the reaction rate constants from table 6-2 . The limitations of the Hoover and Berkshire equation have already been discussed in Sections 1.5.3-4 and will be considered further in Section 7.7.

The formula has been simplified by the use of a parameter "b" in the spreadsheet. "t" is a parameter used in Hoover and Berkshire's formulation. This should be clear from table 6-4

As the CO_{2} transfer velocity is the key quantity of interest in this study, it is important to consider the error associated with its measurement. The errors in the various measured parameters have already been noted in the previous chapters. However, the total error lies not in any one measurement but chiefly in the ratio of the two pCO_{2} differences (pCO_{2} into headspace - pCO_{2} in headspace) / (pCO_{2} in headspace - pCO_{2} in water). It is very difficult to express this error as a standard deviation, even if the standard deviation of the individual measurements are known. This is because the top and bottom parts of the ratio are not statistically independent, and so it is not possible to use any simple statistical formula to calculate the error in this ratio.

On the other hand, it is relatively easy to express an error range, by combining the errors in the individual measurements to determine the most extreme cases. If the errors in the measurements are assumed to be standard deviations, the error range of these extreme cases will be considerably greater than the standard deviation. However, this quantity is useful to indicate the variation in the error between data points, i.e. showing which are the more reliable.

The errors in the individual measurements also vary. The error in measuring the flow rate was taken to be inversely proportional to the flow rate (it's easier to measure a slow flow). The error in the pCO_{2} entering the headspace was normally taken to be the same as the LiCOR error (1ppm), because the constant pressure flowing from the pump ensured that the LiCOR cell was well flushed. On the other hand, when the pure CO_{2} cylinder was used to deliver a high concentration of CO_{2} then this pCO_{2} varied more (as described in Section 5.7 ) and the error due to this variation was noted at the time. Generally, for such high-CO_{2} flow, the error was proportional to pCO_{2}. The error in pCO_{2} leaving the headspace was taken to be higher (usually 3ppm) due to the lower pressure drop across the LiCOR and hence slower flushing. The error in water pCO_{2} was usually taken as 2ppm, but higher values were used when pCO_{2} was changing rapidly, as during intense algal growth (see plots in chapter 8).

An additional error might be caused by the headspace not having reached steady-state. An exponential formula describing the gradual approach towards steady-state has already been derived in Section 4.4 . Table 6-6 shows the calculation of non-steady-state error, based on this exponential formula. These non-steady-state errors were usually so small (<0.1ppm), that they were not incorporated into the error bars plotted on the graphs. However, these calculations gave a useful indication of whether the headspaces ought to be in steady state, and generally implied that discrepancies must be due to other factors.

Therefore to make these composite plots less confusing, I also calculated "weighted average" transfer velocities for each datapoint. Whenever two headspaces were used simultaneously, usually one for "influx" and one for "efflux", the formula was:

weighted average transfer velocity=

__ (flow / area) _{1} (pCO_{2} in - pCO_{2} out)_{1 } - (flow / area)_{2} (pCO_{2} in - pCO_{2} out)_{2
}__

dimensionless solubility * (pCO

Where the subscripts indicate the two different headspaces.

The weight is thus apportioned according to the difference between the pCO_{2} flowing into and out of each headspace, which is in most cases in inverse proportion to the maximum expected error. The water pCO_{2} is not used. This simpler formula was more convenient to calculate in the spreadsheet than one incorporating the full "expected error" calculation (as in table 6-5 ). Where only one headspace was used the transfer velocity was calculated in the normal way.

The method of calculating the total carbon is simple but only approximate. Basically, the steady-state of each headspace at the time of each measurement is assumed to have persisted since the previous measurement. The CO_{2} flux leaving the tank is subtracted from the CO_{2} flux entering the tank, and this difference is multiplied by the time interval, and added to the previous value for total carbon. The formulas are shown in table 6-7 .

Notes on table 6-7 :

1. The time interval was normally calculated from the data automatically. However, on various occasions this was overridden following changes in the set-up of the gas flow system.

2. As this is only an approximate calculation, the air temperature was assumed constant at 20C. Note that the air temperature is not a factor in the transfer velocity calculation.

3. the factor of 10^{-12} comes from the conversions: ml --> m^{3} (in calculating gas volume) and ppm --> atm

4. The first value for totc was set equal to TCO_{2}, i.e. bioc = 0

This method makes several simplifying assumptions, which are only reasonable for large blooms of algae in highly-buffered seawater. Chiefly, these are:

1. The headspaces are always in steady state. This means that the total CO_{2} in the headspace does not change during the time interval between measurements. Obviously this is not true, as it implies no change in the transfer velocity, and also an instant adjustment after any perturbation in the flow rate or pCO_{2} flowing into the headspace at the time of the previous measurement. However, it is reasonable, partly because the timescale for the headspace to reach steady state (see Section 4.4 ) is typically about 6 hours, less than the interval between most sets of measurements. Also, a large change in total CO_{2} in the headspace would only correspond to a small change in the total CO_{2} in the water, because of the buffering capacity of seawater (dpCO_{2} / dTCO_{2} = approximately 10). Therefore, the CO_{2} flux through the headspace during the time interval can be assumed to be much larger than the change in the headspace.

2. The initial total carbon is assumed to be all inorganic, i.e. equal to TCO_{2}. This implies that there is negligible organic carbon in the initial filtered seawater, and in the starter culture. Again, the latter is obviously untrue, however if the initial organic carbon were measured, the only effect would be to add a fixed offset to the biological carbon figures.

3. The assumption that the total organic carbon is a good indicator of algal biomass, implies that there is not a large pool of dissolved organic carbon, bacteria, or dead cells etc. A comparison with the chlorophyll measurements (see Section 8.2 ) should help to indicate whether this is reasonable. Note that when the seawater was filtered, the bacteria should initially be low. At the end of each bloom the algae were observed to flocculate and settle out on the base of the tank, but before this occurred the base appeared clean and the water homogenous. Notes on such observations are included with the graphs.

With a dynamic computer model of the tank, combined with initial organic carbon measurements, it would be possible to calculate the total carbon more accurately. However, since we are only using the total carbon to give an indication of the algal biomass, such accuracy is not critical for the purposes of this study.

CO

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