# Deriving the flux equation

The LI-8250 Multiplexer can be used to measure fluxes of many trace gases that can be reliably detected with compatible gas analyzers. The flux equation remains the same for all gases.

At constant pressure, the total rate at which water evaporates into the chamber `sf _{w}` (mol s

^{-1}) is balanced by a small flow rate of air out of the chamber

`u`(mol s

^{-1}). The gas mole fraction of the air outside the chamber is

`c`, inside the chamber is

_{a}`c`, and in the soil is

_{c}`c`, all in mol mol

_{s}^{-1}. The chamber air water vapor mole fraction is

`w`(mol mol

_{c}^{-1}). The rate constant

`k`(s

^{-1}) characterizes leaks (if any) due to diffusion of gas between the soil chamber and outside air. The chamber volume

`v`includes the volume of the pump and measurement loop.

A chamber of volume `v` (m^{3}) and surface area `s` (m^{2}) sitting over the soil, which has a gas efflux rate `f _{c}` (mol m

^{‑2}s

^{‑1}) and water evaporation flux rate

`f`(mol m

_{w}^{‑2}s

^{‑1}).

The mass balance equations for the gas of interest, water vapor, and air take the form

A‑2`storage = flux in - flux out`

We neglect the effects of leaks for now, but we will consider them later.

### Gas mass balance

### H_{2}O mass balance

### Air mass balance

is the number density of gas in the chamber, is the number density of water vapor in the chamber, and is the total number density of air in the chamber (all in mol m^{-3}); , where is the number density of dry air in the chamber.

The number density of air is given by the ideal gas law, , where `R` is the gas constant (8.314 Pa m^{3} K^{-1} mol^{-1}), and `T _{K}` is the absolute temperature (K). From equation A‑5, with

`ρ`and

`T`constant, and

_{K}`sf`,

_{w}>> sf_{c}Combining equations A‑4 and A‑6, and noting that , we find

Combining equations A‑3, A‑6 and A‑7 gives

Equation A‑8 can be simplified by defining , which is the chamber dry mixing ratio, corrected for water vapor dilution, reported in units of µmol/mol. This is different from `C _{dry}`, the dry mole fraction, reported in units of parts-per-million (ppm) in the data output.

Differentiating we find,

A‑9

Substituting this into equation A‑8 gives

Equation A‑10 has an important advantage over equation A‑8 because it is not necessary to estimate the rate of increase in water vapor mole fraction. In most measurements, the water vapor mole fraction increases in a highly non-linear fashion, and the rate is estimated with a linear function. Thus, in effect, equation A‑8 forces us to use average values for and . But with equation A‑10, the dilution correction is made point-by-point, and estimates of the initial values at time zero are used to estimate `f _{c}` at the instant the chamber closed. This is both easier and more accurate than the procedure required to implement equation A‑8.

In order to use equation A‑10 the initial values must be known for `ρ` and `T _{K}` (to compute

`ρ`), as well as the initial values for

_{c}`w`and . After the chamber closes, the LI-8250 Multiplexer performs a linear regression with time on the first 10 values of each measured variable. The initial values of

_{c}`ρ`,

`T`and

_{K}`w`are obtained from the time zero intercepts of these regressions; however, finding the initial value for requires a little more work.

_{c}To do this, `f _{c}` is defined in terms of the mole fraction gradient across the soil- to-chamber interface and a transfer coefficient, to obtain

where `c _{s}` is the mole fraction in the soil surface layer communicating with the chamber (mol mol

^{-1}),

`g`is conductance to the gas of interest (m s

^{-1}), and

`ρ`is the density of air (mol m

_{c}^{-3}). The soil and chamber must be isothermal for equation A‑11 to hold.

Combining equation A‑11 with equation A‑10, considering all variables except `c _{c}'` to be constant, and rearranging, gives

where . When `w _{c} = w_{s}`,

`c`, gives the water vapor dilution-corrected mole fraction in the soil layer communicating with the chamber. We do not expect

_{s}'`w`to equal

_{c}`w`exactly, but most of the time they will differ by less than 0.02 mol mol

_{s}^{-1}or so, which introduces only a small uncertainty in

`c`. If

_{s}'`c`is taken as a constant, then equation A‑12 can be integrated to give

_{s}'where `α = sg v ^{-1}` is a rate constant (s

^{-1}) and

`c`is the initial value of the dilution- corrected mole fraction when the chamber closes. The rate of change in

_{c}'(0)`c`at any time can be computed from the derivative of equation A‑13.

_{c}'(t)A‑14

## Calculating the flux from measured data

In the LI-8250 Multiplexer, equations A‑10, A‑12 and A‑13 are implemented in a form that presents the variables in more familiar and intuitive units. Equation A‑10 is computed as

where `F _{c}` is the soil gas efflux rate (μmol m

^{-2}s

^{-1}),

`V`is volume (cm

^{3}),

`P`is the initial pressure (kPa),

_{0}`W`is the initial water vapor mole fraction (mmol mol

_{0}^{-1}),

`S`is soil surface area (cm

^{2}),

`T`is initial air temperature (°C), and is the initial rate of change in water vapor dilution corrected mole fraction (μmol mol

_{0}^{-1}s

^{-1}) of the gas of interest.

Examine Figure A‑2 to see `C'(t)` vs `t` data that were obtained from a soil CO_{2} flux measurement with two observations. The data are marked to show when the chamber closed and when it opened.

The deadband is the time until steady chamber mixing is established, and typically lasts 10s to 30s. After mixing is stable, the data are fit with an empirical equation that has a form similar to equation A‑13:

where `C'(t) `is the instantaneous water-corrected chamber mole fraction of the gas of interest, `C _{0}'` is the value of

`C'(t)`when the chamber closed, and

`C`is a parameter that defines the asymptote, all in μmol of gas per mol dry air (µmol mol

_{x}'^{-1});

`a`is a parameter that defines the curvature of the fit (s

^{-1}).

The initial value of `C'(t)`, called `C _{0}'` in equation A‑16, is computed from the intercept of a linear regression of the first 10 points after the chamber closes. This is used as a parameter in the non-linear regression that fits equation A‑10 to the

`C'(t)`vs

`t`data between the end of the deadband and the end of the observation. This regression yields values for the parameters

`C`, a and

_{x}'`t`.

_{0}`t = t`represents the time when

_{0}`C'(t)`in equation A‑16 equals its initial value when the chamber closes, or

`C'(t`. The delay between the instant the chamber closed and

_{0}) = C_{0}'`t`gives the time required to establish steady mixing. Gas offsets or time delays can occur when the chamber closes, and these events can cause

_{0}`t`to be positive or negative in value.

_{0}All the initial values needed to obtain the soil gas efflux rate, `F _{c}`, in equation A‑10 can now be computed. The initial values

`P`,

_{0}`T`and

_{0}`W`are all obtained from the intercepts of linear regressions of the first 10 measurements of

_{0}`P`,

`T`and

`W`after the chamber closes. The rate of change of dilution-corrected chamber mole fraction can be computed at any time from

When `t = t _{0}`,

Equation A‑18 gives an estimate of the rate of change in `C'` at the instant the chamber closed. This value must be estimated mathematically. It cannot be measured directly at any time during the measurement because imperfect mixing prevents an accurate estimate early in the measurement cycle, and later in the cycle, the increasing chamber gas concentration continuously reduces the gradient between soil and chamber. This suppresses the rate, as can be seen from equation A‑17 and also in Figure A‑2.

### Relationship between the model and the empirical equation

The diffusion model provides an equation with a form that allows correction for the effect of changing gradients on the rate, which in turn, makes it possible to estimate the initial rate. It is worthwhile to distinguish between the model function given in equation A‑13 and the empirical function in equation A‑16. As just described, the units are different in the two expressions; but more important, for the parameters `c _{s}` and

`A`in equation A‑13 to have their defined meaning, the assumptions underlying the derivation must be true. By contrast, equations A‑16 through A‑18 are treated as empirical functions and are used only to estimate the rate of change,

`dC'/dt`. The parameters

`C`,

_{x}'`a`and

`t`do not depend upon a specific theoretical interpretation, and may or may not provide reliable estimates of soil parameters.

_{0}### Evaluating other methods for computing soil gas flux

Other approaches have been used for computing gas flux in transient measurements. One commonly used method is to fit a linear function to what is sometimes referred to as "the linear portion" of the curve. Unfortunately, there is no linear portion, as can be seen from careful inspection of Figure A‑2. The slope is meaningless in the initial phase before steady mixing is established, and after steady mixing is established, the extent of the non-linearity depends upon the soil surface-to-chamber volume ratio and the flux rate. During this time, gas vs time curves are always concave in a downward direction, meaning that linear regression over this portion of the dataset will give an underestimate of the rate of change. In every case we have tested so far, the average rate measured by linear regression is less than the initial rate measured by non-linear regression. Nevertheless, linear regression is a robust numerical approach. We recommend you use these only for comparison to the initial values, which are obtained by fitting equation A‑16 to the data using a non-linear regression method.

Another approach that has been used to estimate the initial rate is to fit a polynomial to the gas concentration vs time data. This approach is theoretically sound inasmuch as a power series can be generated from a Taylor series approximation to equation A‑16. Usually, the data are fit with a quadratic equation. We tested this approach and found that while it can be justified on theoretical grounds, it does not work very well in practice. The shape of even a second order polynomial is sensitive to small perturbations in the data. This makes initial rate calculations subject to much larger variations than when the same data are analyzed by nonlinear regression using equation A‑16.

### Effects of high chamber gas concentrations

Finally, we consider the importance of choosing appropriate observation times and prepurge times. We do not have experience on all soil types, and cannot give absolute recommendations for the best observation length in all situations.

Nevertheless, our experience so far suggests that 60 to 120 seconds will often work well, though flux measurements of some trace gases may require longer measurements. A measurement of 60 to 120 seconds prevents large build-ups in chamber gas concentration at typical change rates such as 0.5 ppm s^{-1}. We have found that optimal deadband length can vary from about 10 to 60 seconds, with 30 seconds being a good value to use as a first estimate.

Deadbands and observation times can be adjusted after the fact, using SoilFluxPro Software. This program provides curve fit analysis tools that can be used to find the optimal deadband lengths and observation lengths. Therefore, it is not critical to choose the right deadband and observation length in the field; as long as the observation lengths are long enough, they can be optimized later, if necessary.

Long observations can have the effect of capping the soil and causing the gas concentration to build up in the soil under the chamber. This phenomenon can be observed by performing a sequence of observations in which the chamber concentration is allowed to increase several hundred ppm during each observation. When the prepurge is set to be just long enough to allow the chamber atmosphere to come back to the ambient gas concentration, the initial rates in sequential observations can often be observed to increase as the soil gas concentration increases. This is expected according to equation A‑18, `dC _{c}'/dt = a(C_{x}' - C_{ambient}'`), if it is assumed that

`C`. Thus, long observation lengths may perturb the very process we wish to measure.

_{x}' = C_{soil}'Another effect of high chamber gas concentrations is to promote leaks between the chamber and atmosphere. Leaks can be ignored when the gradient between the chamber atmosphere and ambient atmosphere are small. But when the gradient is not small, leaks cannot be neglected and it can be shown that the parameters in equation A‑13 are altered to become

A‑19

where `k` and `c _{a}'` are the leak rate time constant and water-corrected ambient gas concentration, respectively, and the expression for

`c`replaces

_{x}'`c`in equation A‑13. Thus, when chamber gas concentrations are high, the rate constant and asymptote will reflect leaks from the system.

_{s}'