Equation Summary

The LI-840A computes CO₂ and H₂O concentrations using an equation of the form

B‑1

where c is concentration, f() is the calibration function, α is the absorptance, g (α,P) is the pressure correction, S(α) is the span, and T is the temperature (°C) of the gas in the cell, typically 51.5 °C. Absorptance is computed from

B‑2

where V and V_o are the raw detector sample and reference readings, and Z is the zeroing parameter.

Span is a linear function of absorptance.

B‑3

H₂O Equations

Absorptance α_w for water vapor is computed from

B‑4

where V_w and V_wo are the sample and reference raw detector readings and Z_w is the zero parameter. The pressure correction for water vapor is an empirical function g_w() of absorptance and pressure P:

B‑5

The value of P_o is 99 kPa. When the pressure correction is not enabled, g_w() is simply 1.0. Values for g_w() can be viewed on the diagnostics window (see Operation).

Water vapor concentration W (mmol mol^-1) is computed from

B‑6

where f_w(x) is a third order polynomial whose coefficients are given on the calibration sheet.

B‑7

CO₂ Equations

The measurement of CO₂ is a bit more complicated than for H₂O because of the influence of water vapor. There is a slight direct cross sensitivity in the CO₂ signal to H₂O. This is measured at the factory and accounted for in the computation of absorptance (equation B‑8). There is also a band broadening effect that is accounted for in the computation of concentration (equation B‑12).

CO₂ absorptance α_c is computed from

B‑8

where V_c and V_co are the raw detector signals for sample and reference, Z_c is the CO₂ zero parameter, and X_wc is a cross sensitivity parameter for the effect of water vapor on CO₂. Its value is reported on the calibration sheet as “XS=”.

The empirical pressure correction function g_c() depends on CO₂ absorptance and pressure:

When P = P_o, g_c() = 1.

When P < P_o

B‑9

where a = 1.10158, b = -6.1217 * 10^-3, c = -0.266278, d = 3.69895, and z is the asymptotic value of absorptance, obtained from the calibration coefficients (equation B‑13).

B‑10

When P > P_o

B‑11

where X, A, and B are computed as in equation B‑9. The variable g_c() is viewable on the diagnostics window (see Operation).

CO₂ concentration C (µmol mol^-1) is computed from

B‑12

where f_c(x) is a function whose inverse is a double rectangular hyperbola, and whose coefficients (a1…a4) are given on the calibration sheet.

B‑13

Solving equation B‑13 for C yields the calibration function

B‑14

Where

B‑15

ψ(W) accounts for band broadening by water vapor, and is viewable on the Diagnostic window.

B‑16

The band broadening coefficient h(α_c) has been determined to be 1.45 for the LI-840A for CO₂ concentrations near ambient. At higher concentrations, the value decreases. We capture this behavior with an empirical relationship (equation B‑17).

B‑17

Where z is from equation B‑10, and b_w is the low concentration band broadening coefficient: 1.45. This is the value shown on the calibration sheet as BB=1.45. The present value of h(α_c) is viewable on the Diagnostics screen (Section 3). The typical relationship between h(α_c) and CO₂ concentration is illustrated below. (‘Typical’ because the exact relationship depends on the relationship between absorptance and CO₂, which is the calibration curve.)

Implementation Note: We formulated B‑17 with 0.64b_w – 0.64 instead of the simple equivalent (0.29) because this allows band broadening corrections to be turned off by setting b_w to 1. When b_w =1, h(α_c) = 1 everywhere. Also, to avoid computational problems (underflows, overflows, and division by zero) we constrain the argument α_c when computing h(α_c) to be

B‑18

is typically equivalent to about 600 ppm.

Calibration Equations

Zeroing H₂O

When the command for zeroing water is received, the LI-840A computes the water zero from equation B‑19, where and are averaged for 5 seconds.

B‑19

Zeroing CO₂

When the command for zeroing CO₂ is received, the LI-840A computes the CO₂ zero term from equation B‑20, where , , , and are averaged for 5 seconds.

B‑20

Spanning H₂O

When the command for setting the span for H₂O is received, along with the target concentration W_T, the LI-840A computes S_w0 from equation B‑21 , where is averaged over five seconds.

B‑21

where

B‑22

The instrument retains the following values, which are used for subsequent secondary spans:

B‑23

B‑24

Secondary Span H₂O

When the secondary span command for H₂O is received, the instrument computes new values for both S_w0 and S_w1. First, it measures a new and computes a new from equation B‑22. Then, it uses these plus the retained values (and from the previous normal span) to compute

B‑25

Given the new span slope S_w1, update the span offset S_w0 by equation B‑21.

Spanning CO₂

When the command for setting the span for CO₂ is received, along with the target concentration C_T, the LI-840A computes S_c0 from equation B‑26, where and are averaged for 5 seconds.

B‑26

where

B‑27

Note that

B‑28

The instrument retains the following values which are used for subsequent secondary spans, if necessary:

B‑29

B‑30

Secondary Span CO₂

When the secondary span command for CO₂ is received, the instrument computes new values for both S_c0 and S_c1. First, it measures a new and computes a new β_c from equation B‑27. Then it uses these, plus the retained values (α_c1 and β_c1 from the previous normal span) to compute

B‑31

Given the new span slope S_c1, update the span offset S_c0 by equation B‑26.

Symbol Summary

Many of the quantities described in the above section are available from the LI-840A. These are summarized in Table B‑1.

Table B‑1. Symbol summary

Symbol	XML Tag: <LI840>+	Description
αw	<data><h2oabs>	H2O absorptance
gW(αw,P)	<AUXdata><pHA>	Pressure corrected absorptance of H2O
W	<data><h2o>	H2O (mmol mol-1)
P	<data><cellpress>	Pressure (kPa)
T	<data><celltemp>	Temp (C)
αc	<data><co2abs	CO2 absorptance
gc(αc,P)	<AUXdata><pcA>	Pressure corrected absorptance of CO2
C	<data><co2>	CO2 (µmol mol-1)
Vw	<data><raw><h2o>	Raw detector readings
Vwo	<data><raw><h2oref>
Vc	<data><raw><co2>
Vco	<data><raw><co2ref>
aw1...aw3	<poly><h2o>	Calibration coefficients
ac1...ac4	<poly><co2>	Calibration coefficients
Zw	<cal><h2okzero>
Sw0	<cal><h2okspan>
Sw1	<cal><h2okspan2>
Zc	<cal><co2kzero>	Zero and Span values
Sc0	<cal><co2kspan>
Sc1	<cal><co2kspan2>
xwc	<poly><xs>	Cross sensitivity coefficient
bw	<poly><bb>	Band broadening coefficient for low CO2
h(αc)	<AUXdata><bb_EFF>	Computed band broadening coeff for the current absorptance
ѱ(W)	<AUXdata><psi>	Band broadening correction factor for water vapor