4.1 A holistic experiment achieved by a steady-state method?

However, such an experiment had never been done before because it is difficult to measure the air-water CO₂ flux whilst CO₂ is simultaneously being taken up or released by algae. If I tried to estimate the CO₂ flux (and hence the transfer velocity) based on measurements of changing water pCO₂, the signal to noise ratio would be small due to chemical buffering by the alkaline seawater, and it would probably not be possible to determine the small variations in transfer velocity which might be caused by chemical enhancement and biological catalysis.

Therefore it would be better to measure changes in the air pCO₂ in a closed headspace. The simplest way to do this would be to perturb the air pCO₂ and then observe its rate of return to "equilibrium" with the water. However, the water pCO₂ would also be changing during this reequilibration, both due to gas exchange and due to photosynthesis and respiration, so many measurements would still be needed from each phase. Moreover, if the water pCO₂ is a major factor influencing the gas exchange rate (as predicted by theory and borne out by the results of this work), this would add a further complication to analysis of the results.

An alternative approach is to construct a tank with several separate gas-tight headspaces, all exchanging gases with the same body of water such that there is a net flux of CO₂ into the water from one headspace and out of it from another. Thus it should be possible to balance the air-water fluxes and the biology (on a short timescale) such that the change in pCO₂ of the water is negligible during the course of the transfer velocity measurement. A third "equilibrium" headspace could be used to measure the pCO₂ of the water, and thus no water-phase pCO₂ measurements would be needed.

A plan view of such a steady-state tank with four headspaces is shown in figure 4-1 . All the headspaces are in contact with the same body of water, which must be well mixed. The pCO₂ in one headspace is greater than that of the water, while the pCO₂ in another headspace is less than that of the water, such that the two air-water fluxes roughly balance.

To measure the gas exchange, you need to know only the flow rate of air entering each headspace, the pCO₂ of air entering and leaving each headspace in steady state, and the pCO₂ of the water. The latter should not change significantly during the timescale of achieving the steady state, so it can be measured from another "equilibrium" headspace.

4.2 Advantages and disadvantages of the steady -state method.

• The system can be balanced such that the change in water pCO₂ is negligible, even during an actively photosynthesising algal bloom.
• All pCO₂ measurements are made in the air phase by the same instrument.
• No perturbation is required between gas exchange measurements.
• All the measurements needed to calculate the CO₂ transfer velocity can be made at the same time. This is very convenient, avoiding the need for regular sampling throughout the day.
• The transfer velocity can be calculated from a simple formula immediately after measurement, allowing a rapid response to unexpected changes.
• Two independent transfer velocity measurements are made concurrently (influx and eflux), thus providing a check on the accuracy and an immediate indication of any problem.
• Calibration errors in both the zero and span of the pCO₂ measurements cancel entirely as long as the response remains linear.
• Atmospheric pressure variation also cancels as demonstrated in the equations below ( Section 4.3 ).

The chief drawback of the steady-state method is that it is more complicated to design and construct such a tank with multiple headspaces. The lid had to be removable to allow access, but a gas-tight seal had to be made with all of the walls dividing the headspaces. Each headspace also had to be stirred independently. Therefore it took a long time (about five months) to have the tank designed and built by the workshop. It also took several more months to develop the temperature control, appropriate air and water stirring systems ( Section 4.7 ), and a reliable supply of continuously flowing air with a constant but adjustable pCO₂ ( Section 5.7 ). However, at the early design stage we did not know how long it would take to get the whole system going! Another problem is that the steady state method is not appropriate for low- solubility inert trace gases, as will be demonstrated below ( Section 4.5 ).

In retrospect, it might have been easier to use a simpler tank with only one headspace, making gas exchange measurements by following reequilibration after perturbation of air pCO₂, combined with continuous automated pCO₂ measurements saved automatically onto the computer attached to the LiCOR pCO₂ analyser. A sophisticated computer programme could then have been written to analyse such large datasets and compute the gas exchange compensating for various estimated biological fluxes and chemical buffering effects. However, as we did not have the LiCOR analyser until after the tank was complete, the full potential of such an automated system was not recognised at the early design stage.

4.3 Principle of steady-state CO₂ gas exchange calculation

Figure 4-2 shows the basic principle of such a measurement for any one headspace.

The formula for calculating the transfer velocity is derived as follows. Firstly, symbols and units are given in the table below.

Quantity	Symbol	Units
Transfer Velocity	k	m s-1
Water Surface Area	A	m2
Flux	F	mol s-1
CO2 concentration (water)	[CO2]	mol m-3
Flow rate	f	m3 s-1
Atmospheric Pressure	p	Pa (=kg m-1 s-2 )
Partial Pressure of CO2	pCO2	parts per million (dimensionless)
Temperature	T	K
Gas Constant	R	kg m2 s-2 K-1 mol-1
dimensionless solubility	a	dimensionless
(mol l-1water / mol l-1 air)
solubility	K0	mol kg-1 atm-1

In Steady-State the Flux (F) into the water must be the difference between the amount of CO₂ entering and leaving the headspace. Applying the gas law pV = nRT gives :

F = f * p * ( pCO_2in - pCO_2out ) / RT

As explained in Section 1.2.1 , the transfer velocity (k) can be defined as:

k = (F / A) / ([CO₂]_air - [CO₂]_water)

[CO₂]_air is the concentration of CO₂ in the water at the top of the water-surface microlayer, whereas [CO₂]_water is the concentration in the bulk water. Both of these can be derived from the pCO₂ measured in the gas-exchange and equilibrium headspaces respectively, using the solubility K₀ given by Weiss (1974) (see Section 1.2.2 , Section 6.2 ).

[CO₂]_air = ( 1 / 103.5 ) * K₀ * p * pCO_2out

[CO₂]_water = ( 1 / 103.5 )* K₀ * p * pCO_2eq

(the factor 1 / 103.5) converts mol kg^-1 atm^-1 to mol m^-3 Pa^-1 assuming salinity = 35)

Altogether we have:

k = [f * p * ( pCO_2in - pCO_2out )] / [A* RT * ( 1 / 103.5 ) K₀ p ( pCO_2out - pCO_2eq) ]

It can be seen that the pressure p now cancels.

K₀ RT (1 / 103.5) is the dimensionless solubility "a" .

Substituting this in gives:

k = (f / a A) * ( pCO_2in - pCO_2out ) / ( pCO_2out - pCO_2eq)

which is the steady-state equation used for calculating transfer velocities.

4.4 Timescale for approach to steady state

4.4.1 Equations describing approach to steady state in the headspace for CO₂

We can now write the rate of change of pCO₂ in the headspace:

d(pCO_2out - pCO_2ss) / dt = dpCO_2out / dt = [kaA (pCO_2eq -pCO_2out) +f (pCO_2in -pCO_2out)]/ vol_air
where vol_air is the volume of the headspace (m³) and other symbols are as in the table above.

This can be rearranged to:

d(pCO_2out - pCO_2ss) / dt = [(kaA pCO_2eq + f pCO_2in)- (kaA pCO_2out +f pCO_2out) ]/ vol_air

Also, rearranging the steady state equation gives:

kaA pCO_2eq + f pCO_2in = kaA pCO_2ss + f pCO_2ss

Substituting this in and rearranging gives:

d(pCO_2out - pCO_2ss) / (pCO_2out - pCO_2ss ) = - [ (kaA + f ) / vol_air ] dt

Integrating is now straightforward:

(pCO_2out - pCO_2ss) / (pCO_2initial - pCO_2ss) = exp -t [ (kaA + f ) / vol_air ]
where pCO_2intial = pCO_2out at time t = 0

The quantity [ (kaA + f ) / vol_air ] can be thought of as the number of times the headspace is flushed per unit time. This quantity is used in Section 6.5 to calculate the error in each measurement due to the headspace not being in steady state. Generally it was very small.

Introducing some typical numbers (at the slow end of the range),

k = 3 cm hr^-1, a = 1, A = 1130 cm², f = 60 cm³ min^-1 (=3600 cm³ hr^-1 ), vol_air = 11300 cm³

gives 0.619 "flushes per hour", or an e-folding time of 1.616 hours.

The corresponding "half life" for approach to steady state is 1.12 hours, and after 8 hours the error is less than 1%. Typically, gas flows were set up in the evening and measured assuming steady-state the following morning, or vice versa, allowing sufficient time for the headspace to reach steady state.

The rate of change of pCO_2eq in the water is more difficult to derive algebraically as it depends on the chemical buffering factor ¶ TCO₂ / ¶ pCO₂ which varies with pCO₂. However, due to this chemical buffering, it is reasonable to assume for the purpose of the calculation above that pCO_2eq (pCO₂ of the water) is constant over the timescale that the headspace takes to approach steady-state. This is confirmed by the computer model below.

The buffering is greatest when pCO₂ is low, although this also corresponded to the periods of intense algal uptake. The "Carbon Budget" plots in chapter 8 indicate the rate of change of pCO₂ during the algal blooms. When there were few or no algae, and there was no reason to require a change in pCO_2eq, then the air inflows were normally set up to balance such that pCO_2eq should remain constant.

4.4.2 Computer model of approach to steady-state

The numbers in the figure indicate the situation in the tank headspaces and water at the moment of the "snapshot". This corresponds to the right hand end of the coloured curves, which show how each variable changes as a function of time. The left hand end of the curves corresponds to the disequilibrium state defined by the initial variables listed in the table. By changing these key variables, many different scenarios for running the tank were investigated with the computer model. The carbonate speciation, solubilities and transfer velocities were calculated from these variables using temperature dependent equilibrium constants as described in detail in

4.5 Low-solubility inert gases

4.5.1 Problem with steady-state measurement of transfer velocity

The problem is caused by the low solubility of SF₆ in seawater, whose dimensionless solubility a [mol l^-1 (water) / mol l^-1 (air)] is about 1/300 of that of CO₂. Since the gas exchange rate is dominated by transfer across the water surface microlayer rather than the air boundary layer, then the rate of change of concentration in the air phase due to air-water exchange is proportional to this dimensionless solubility. Therefore the air-water fluxes for SF₆ are very small compared to the fluxes of SF₆ in the air flowing in and out of the headspaces, and so the difference between the SF₆ concentration in the air entering and leaving the headspaces is tiny - as shown by the blue curves on the figure which almost meet the green lines at the top and the bottom (the lower green line is obscured by the blue SF₆ one). Hence the difference between pSF_6out and pSF_6in is greater than the precision of measurement, and the error is very large.

To achieve an accurate steady-state measurement of the SF₆ transfer velocity, the quantities kaA and f (from the equations above) would have to be similar in size, in order to minimise the error in the calculation of k (see also Section 6.5 ). Therefore when the dimensionless solubility "a" is very small the corresponding flow rate "f" (of air into or out of the headspace) would also have to be small. The rate of approaching steady state, determined by [ (kaA + f) / vol_air ] as above, would likewise be very slow, and the experiment would take far too long.

Nevertheless it is critical to compare the gas exchange for CO₂ with that of inert gases, both to show the effect of chemical enhancement and to calculate the "Schmidt number dependence" of gas exchange in this tank (see introduction Section 1.2.4 ). For the former a gas whose diffusivity is close to that of CO₂ (such as oxygen) is most suitable, whereas for the latter a pair of gases are needed whose diffusivities differ as much as possible (the diffusivity of SF₆ is much lower than that of CO₂ or O₂). These inert gases would also show whether there is any physical impact of a surface organic "film" reducing the transfer velocity, as discussed in Section 1.4.5 .

By changing the Schmidt number and solubility within the program, the computer model was also used to investigate other gases (O₂, N₂O, CCl₄) instead of SF₆. Although the dimensionless solubility of O₂ is about 1/30th of that of CO₂ (10x greater than SF₆), it was still too low for satisfactory steady-state measurement of the transfer velocity. The solubility of N₂O is similar to that of CO₂, but unlike CO₂ it is not stabilised by chemical buffering and it might still be affected by biological activity in the water. CCl₄ was ruled out because it is highly toxic.

Therefore alternative methods had to be developed for measuring the transfer velocity of inert gases for comparison with CO₂.

4.5.2 Principle for measuring SF₆ transfer velocity

However it is easier to follow algebraically if we assume that all headspaces are closed, and in practice this was usually the case as several headspaces could then be sampled to check consistency.

If "a" is the dimensionless solubility as above, "t" is the total amount of SF₆ (in moles) added to the water, and e_w , v_w , e_a and v_a are the SF₆ equilibrium concentrations (mol m^-3) and the volumes (m³) of the water and of all four headspaces respectively, then at equilibrium the SF₆ will be partitioned such that
e_w = a e_a.
e_w v_w + e_a v_a = t

If c_w and c_a are the respective concentrations before equilibrium, A the total water surface area (m²) , and k the transfer velocity (m s^-1 ), then gas exchange can be represented by:

d(e_a - c_a) / dt = - dc_a / dt = - kA (c_w - a c_a) / v_a

Also mass balance requires that at all times: c_w v_w + c_a v_a = t = e_w v_w + e_a v_a

Substituting c_w = (e_w v_w + e_a v_a - c_a v_a) / v_w gives:

d(e_a - c_a) / dt = - kA (e_w v_w + e_a v_a - c_a v_a - a c_a v_w ) / v_a v_w

Substituting for e_w and rearranging gives:

d(e_a - c_a) / (e_a - c_a) = [ kA (a v_w + v_a ) / v_a v_w ] dt

As the part in brackets is a time constant (units s^-1), integration is now straightforward.

If i_a is the initial air concentration at time t=0, then:

ln [(e_a - c_a)/ (e_a - i_a)] = t kA[ (a v_w + v_a) / v_a v_w]

This expression can be used to estimate k from a series of measurements of c_a, i_a and e_a, and known tank dimensions.

In practice a is very small and i_a = 0 if the spike is added to the water, so the expression simplifies to:

ln [ 1 - c_a / e_a ] = t k / d*

Where d* = v_w / A, which is the "effective depth" calculated in figure 4-5 .

Note that because the gas exchange is dominated by the water phase concentration, the solubility a disappears altogether in this approximation. Also e_a should be equal to t / v_a, which can be checked if the amount of SF₆ added is known.

4.5.3 Principle for measuring Oxygen transfer velocity

The same symbols will be used as above, but this time we are interested in the change in the water phase, so:

d (e_w -c_w ) / dt = - d c_w /dt = -kA (a c_a - c_w ) / v_w

as before we can use the mass balance and equilibrium to give:

c_a = ( e_w v_w + e_a v_a - c_w v_w ) / v_a , and e_a = e_w / a

substituting these two and rearranging gives:

d (e_w - c_w) / (e_w - c_w) = - [ kA (a v_w + v_a ) / v_a v_w ] dt

which has the same time constant as above (for the air phase). Integrating gives:

ln [(e_w - c_w)/ (e_w - i_w)] = t kA[(a v_w + v_a) / v_a v_w]

In this case i_w is not zero and a is larger than for SF₆, although still small, so it is better to retain the full expression.

Details of the measurement and calculation procedures for SF₆ and oxygen are given later in Section 5.11 and Section 5.12 .

4.6 Design and Construction of the Steady-State Tank

4.6.1 Shape and Dimensions

The overall diameter of the tank was constrained to 80cm by the size of UEA's lathe. The height of the headspace - 10cm, was constrained by the time required for it to reach steady-state, as modelled by the computer program described above. For the water depth the main consideration was the volume of seawater we could easily obtain and develop as an algal culture - a depth of 10cm corresponded to a volume of about 50 litres. All measured dimensions are shown in figure 4-5 . From these the water volume and surface areas can be calculated, the volume being corrected for immersed objects, and also checked by filling from volumetric flasks. The tank was always filled to the same depth, 10cm below the lid, marked on the inner wall. The volume/surface area ratio, termed "effective depth", is used for the oxygen and SF₆ transfer velocity calculations, whereas for CO₂ only the surface area is needed, the CO₂ transfer calculation is independent of depth.

4.6.2 Materials

4.6.3 Gas tight headspaces

Two methods were used to check for leaks. During development, the tank was filled with CO₂, which could be detected by a commercial leak-detector, which measures the thermal conductivity of air entering the probe tip. This indicated a small escape of gas around the perimeter, which was prevented by the addition of silicone sealant to widen the top of the O-ring.

A more routine method for leak testing made use of the multiple headspaces. The tank was 2/3 filled with water, and air was pumped into one headspace while the others were left open to the atmosphere. This creates an imbalance in water levels between the different headspaces, due to different air-pressures (about 8cm water or 8millibar). The stoppers were all closed and the tank left overnight. If air could leak over the top of the dividing walls, the water levels would slowly reequilibrate. By a systematic repetition of this procedure, all the section f wall could be tested and the seal improved, again with silicone sealant. This check was made every couple of months.

The seal was never perfect, for if left long enough, the water level would always adjust slightly (a few mm / day). Headspace B, which also had the water cooling pipes, and oxygen electrode and temperature probe cables entering through the wall, was less pressure-tight than the others. However, it should be noted that during normal operation, the pressure difference between the headspaces is extremely small (<1mm water) and therefore there is minimal driving force for a leak. Also, as there is a continuous flow out to the atmosphere from the steady-state headspaces, any leak should also be in that direction. As the gas exchange calculation is based upon the measured flow rate of gas entering the headspace, this would not be affected by such a leak.

4.6.4 Temperature Control

Temperature was initially monitored with a thermistor probe immersed in the tank water, connected to a "Squirrel" logger by a cable through the tank wall. This consistently read 1^oC lower than thermometers against which it was checked, so a correction was made. In later experiments only thermometers were used.

4.7 Stirring

4.7.1 Water Stirring

However, changing the paddles was inconvenient because to do this the lid of the tank had to be removed, and the process of attaching a paddle underwater might contaminate the water sample.

Therefore a separate variable-speed power supply was later set up for the water-stirring motor. This was used with a fixed paddle of radius 24cm and height 1cm, producing a typical transfer velocity of about 3cm/hr for oxygen at 13 revolutions per minute. However, as the air-temperature in the lab increased, the motor tended to slow slightly, and wear on the gearbox and brushes also caused minor variation. Therefore the motor speed was checked during every gas exchange measurement. The motor speed was measured simply by timing 5 (or as appropriate) full revolutions of the paddle (typically 23 seconds), and repeating at least three times until consistent to within 1%, the short-term variability. The variability was much less after June 1996, when a better motor was found.

Later, many measurements were made to investigate the effect of varying the paddle speed on the enhanced and unenhanced CO₂ transfer velocity -these are reported in

4.7.2 Air Stirring

The left hand plot shows an experiment in which pure oxygen flowed through one of the four headspaces, which was stirred by a variable-speed fan. The other headspaces were initially in equilibrium with the water and were not stirred. The circles show measurements of oxygen in the water, and the line shows the speed of the rotating fan tip. Clearly the speed of the fan has no significant effect on the rate of O₂ invasion into the water. A similar experiment in which the flux of oxygen was from the water to the air (which was continuously flushed by N₂) showed the same result.

The right hand plot shows another experiment, in which oxygen was initially bubbled through the water and then gradually returns to equilibrium with the air pCO₂ (hence the slight curve). A large fan was positioned above all four headspaces. Again, switching the fan on and off seems to make no difference to the transfer velocity.

Since the air stirring would not dominate the gas exchange, serving merely to ensure that the headspaces are well-mixed, it did not need to be too vigorous, and a minor variation between the headspaces would not be critical to the results. Therefore, we decided to use four separate but identical DC motors, connected to simple fan blades (8x 45^o angle blades 5cm radius 2cm high) by a shaft passing through the lid (as in figure 4-6 ) and turning at about 300rpm when all went well. Unfortunately these motors did not turn out to be very reliable, and occasionally one or two were not operating. Even so, these headspaces were still useful for equilibration, lagging only a little behind stirred headspaces.

4.7.3 Discussion of relative importance of air and water stirring

Continue to Chapter 5:
Operation of steady-state gas exchange tank

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