Equation summary
The LI-6800 measures and computes the following parameters related to leaf-level gas exchange measurements. Additional details are given in Theory and equation summary
Light
The photosynthetic photon flux density (PPFD) readings Q_{in} and Q_{out} are computed from
9‑19
9‑20
where S_{qin} and S_{qout} are the calibration factors, and I_{qin} and I_{qout} are the raw counts from the A/D converter.
Leaf temperature
Leaf temperatures T_{l1}, T_{l2} (°C) for the two thermocouples are computed from signals V_{l1} and V_{l2} (mV) by
9‑21
9‑22
where T_{j1} and T_{j2} are junction temperatures (°C).
Pressure
Atmospheric pressure P_{a} (kPa) is reported directly from the on-board sensor.
Chamber over-pressure ΔP_{c} (kPa) is computed from
9‑23
where V_{p} is the signal from the differential pressure sensor (in V, but taken to be kPa) V_{po} is the calibration zero for the sensor, G is fan speed (rpm), F is flow rate, and p_{ca} and p_{cb} are empirical coefficients that depend on chamber type (Table 9‑1).
Flow
The flow F (µmol s^{-1}) to the leaf chamber is computed from
9‑24
where af1...af7 are factory calibration constants, V_{f} is flow meter signal (V), V_{fo} is calibration zero, and T_{k} is IRGA block temperature.
Reference and sample IRGA cell flows F_{r} and F_{s} (µmol s^{-1}) are computed from
9‑25
9‑26
where V_{fr} and V_{fs} are cell exit flow sensor voltage, V_{fro} and V_{fso} are the offset value of that voltage, and ar1...ar4 and as1...as4 are factory calibration coefficients.
CO_{2}
Reference and sample CO_{2} concentrations C_{r} and C_{s} are given by
9‑27
where M_{c}() is the CO_{2} match correction function:
9‑28
9‑29
9‑30
where α_{cr} and α_{cs} are absorptances, Pa is atmospheric pressure, T_{r} and T_{s} are the cell inlet temperatures, f_{c} is a 5th order polynomial with coefficients (br1...br5 or bs1...bs5) determined at the factory, S_{cro}; S_{cso} and S_{cr1}; S_{cs1} are the span1 and span2 user calibration settings, W_{a} and W_{b} are water concentrations, X_{o} is the oxygen concentration in percent. Ѱ_{c} is the band broadening function for the effect of water and oxygen on CO_{2}, given by
9‑31
where B_{wc} and B_{oc} are the band broadening coefficients for H_{2}O and O_{2} respectively on CO_{2}.
Reference and sample absorptances α_{cr} and α_{cs} are corrected for zero drift with temperature, span drift with temperature, and cross sensitivity with water.
9‑32
9‑33
where cr1...cr3 and cs1...cs3 are empirical coefficients for CO_{2} absorptance span drift determined during calibration, I_{cr}, I_{cro}, I_{cs}, and I_{cso} are the raw IRGA detector absorbing and non-absorbing readings for CO_{2}, and I_{wr}, I_{wro}, I_{ws} and I_{wso} are those for water. X_{wcr} and X_{wcs} are empirical cross sensitivity coefficients for water on CO_{2} determined during calibration, Z_{cro} and Z_{cso} are the current user CO_{2} zero factors, and Z_{cr} and Z_{cs} are the factors for CO_{2} zero drift with temperature, determined during calibration.
Dry mole fractions C_{rd} and C_{sd} are computed from
9‑34
H_{2}O
Reference and sample H_{2}O concentrations H_{r} and H_{s} are given by
9‑35
where M_{w}() is the H_{2}O match correction function:
9‑36
9‑37
9‑38
where α_{wr} and α_{ws} are the reference and sample absorptances, f_{w} is a 3rd order polynomial with coefficients (dr1...dr3 and ds1...ds3) determined at the factory, S_{wro}, S_{wr1} and S_{wso}, S_{ws1} are the H_{2}O span1 and span2 user calibration settings for reference and sample, Ѱ_{o} is the band broadening function for the effect of oxygen on H_{2}O, given by
9‑39
where B_{ow1} and B_{ow2} are empirical coefficients.
Absorptances α_{wr} and α_{ws} are corrected for zero drift with temperature, span drift with temperature, and cross sensitivity with water.
9‑40
9‑41
where w_{r1}...w_{r3} and w_{s1}...w_{s3} are empirical coefficients for H_{2}O absorptance span drift determined during calibration. X_{cwr}, X_{cws} are empirical cross sensitivity coefficients for CO_{2} on H_{2}O determined during calibration, Z_{wro} and Z_{wso} are the current user H_{2}O zero factors, and Z_{wr} and Z_{ws} are the factors for H_{2}O zero drift with temperature, determined during calibration.
The vapor pressure (kPa) of the air in the reference and sample cells e_{r} and e_{s} is given by
9‑42
9‑43
Reference and sample cell dew point temperatures T_{dr} and T_{ds} are given by
9‑44
9‑45
Humidity indicators
The leaf chamber vapor pressure e_{c} (kPa) is given by
9‑46
The saturation vapor pressure e_{sc} in the leaf chamber is a function of chamber air temperature T_{a}:
9‑47
where e_{s}(T) is the saturation vapor pressure function:
9‑48
The relative humidity h_{c} (%) in the leaf chamber is given by
9‑49
The vapor pressure deficit of the leaf e_{Δl} is computed from
9‑50
where T_{l} is leaf temperature ().
Transpiration
The mass balance of water vapor in an open system is given by
where s is leaf area (m^{2}), E is transpiration rate (mol H_{2}O m^{-2} s^{-1}), u_{e} and u_{o} are incoming and outgoing flow rates (mol s^{-1}) from the chamber, and w_{e} and w_{o} are incoming and outgoing water mole factions (mol H_{2}O (mol air)^{-1}). Since
we can write
9‑53
which rearranges to
The relationships between the terms in equations 9‑51 through 9‑54 and what the LI-6800 measures are
where F is air flow rate (µmol s^{-1}), W_{s} and W_{r} are sample and reference water mole fractions (mmol H_{2}O (mol air)^{-1}), and S is leaf area (cm^{2}). The equation that the LI-6850 uses for transpiration is thus
9‑56
Stomatal conductance
The total conductance g_{tw} of the leaf to water vapor is
9‑57
where W_{l} is the molar concentration of the water vapor within the leaf (mmol H_{2}O (mol air)^{-1}), which is computed from the leaf temperature T_{l} (°C) and the total atmospheric pressure in the leaf chamber
9‑58
We assume that the total resistance for the upper r^{u} or lower r^{l} surface of a leaf is the sum of the stomatal resistance r_{s} and boundary layer resistance r_{b} of that surface
9‑59
9‑60
and that the upper and lower boundary layer resistances are the same
9‑61
and we define K to be the ratio of stomatal resistances of the two sides
Leaf stomatal resistance r_{s} is given by
9‑63
9‑64
9‑65
9‑66
Total conductance g (the inverse of the total resistance r) can thus be written
9‑67
For water vapor, the total conductance g_{tw} is related to stomatal conductance gsw and one sided boundary layer conductance g_{bw} by
9‑70
Solving equation 9‑68 for g_{sw} yields
Note that although we defined K to be a particular ratio of upper to lower stomata resistances (equation 9‑62), since K appears as , we get the same mathematical result if we had defined it the other way. In other words, K is equivalent to 1/K. Note also that the LI-6400 does not use equation 9‑71, but rather a simplified approximation:
9‑72
Boundary layer
The one sided boundary layer conductance to water vapor g_{bw} for a broadleaf is a function of fan speed G (rpm) and leaf area S (cm^{2}).
9‑73
where f is
9‑74
and s is forced to be
9‑75
The empirical coefficients c_{o}...c_{4}, reference pressure P_{o}, and leaf area limits S_{min} and S_{max} depend on chamber type.
Chamber | co | c1 | c2 | c3 | c4 | Po | Smin | Smax |
---|---|---|---|---|---|---|---|---|
6800-01 Flr | 0.250 | 0.35860 | -4.01816E-3 | 0.00451074 | -0.0044762 | 96.9 | 1 | 6 |
6800-01A 6 cm^{2} | 0.578 | 0.5229739 | 3.740252E-3 | -6.197961E-2 | -5.608586E-3 | 96.9 | 1 | 6 |
6800-01A 2 cm^{2} | 0.572 | 0.3872742 | -1.870584E-2 | 0.0 | -7.37389E-3 | 96.9 | 1 | 2 |
6800-12 3x3 | 0.500 | 0.44869569 | 1.9000035E-3 | -4.26088781E-2 | -3.456516E-3 | 96.7 | 2 | 9 |
6800-12A 9 cm^{2} | 0.579 | 0.3210639 | -1.109987E-3 | 5.106816E-3 | -3.283688E-3 | 96.7 | 2 | 9 |
6800-12A 6 cm^{2}FB | 0.345 | 0.552336 | -4.7985e-3 | 0.0 | -7.3557e-3 | 96.7 | 1 | 6 |
6800-12A 6 cm^{2}SS | 0.418 | 0.5145466 | -2.5106E-3 | 0.0 | -8.1206E-3 | 96.7 | 1 | 6 |
6800-12A 3 cm^{2}FB | 0.188 | 0.5795409 | -1.15295E-2 | 0.0 | -9.7259E-3 | 96.7 | 1 | 3 |
6800-12A 3 cm^{2}SS | 0.141 | 0.5263354 | -1.27376E-2 | 0.0 | -1.10157E-2 | 96.7 | 1 | 3 |
6800-13 6x6 | 0.430 | 0.267827 | -1.164018E-4 | 2.248202E-3 | -5.109462E-3 | 96.8 | 6 | 36 |
Net assimilation
The mass balance of CO_{2} in an open system is given by
9‑76
where a is assimilation rate (mol CO_{2} m^{-1} s^{-1}), c_{e} and c_{o} are entering and outgoing mole fractions (mol CO_{2} (mol air)^{-1}). Using equation 9‑52, we can write
9‑77
which rearranges to
To write equation 9‑78 in terms of what the LI-6800 measures, we use equations 9‑55 and
9‑79
where C_{r} and C_{s} are sample and reference CO_{2} concentrations (µmol mol^{-1}), and A is the net assimilation by the leaf (µmol m^{-2} s^{-1}). Substitution yields
9‑80
9‑81
9‑82
Intercellular CO_{2}
The intercellular CO_{2} concentration C_{i} (µmol CO_{2} (mol air)^{-1}) is given by
9‑83
where g_{tc} is the total conductance to CO_{2}. From equation 9‑69, we can write
where 1.6 is the ratio of the diffusivities of CO_{2} and water in air, and 1.37 is the same ratio for the boundary layer. This is another departure from the LI-6400, which does not use equation 9‑84, but a simplified approximation
9‑85
Energy balance
The LI-6800 provides several potential sources for leaf temperature: it can be directly measured with some combination of the two leaf thermocouples (T_{l1} and T_{l2}), or computed indirectly from a leaf energy balance (T_{eb}). The user can specify what combination to use via three weighting factors f_{T1}, f_{T2}, and f_{Teb}:
9‑86
The energy balance temperature T_{eb} assumes that the energy balance of a leaf in the chamber has three components: net radiation R (W m^{-2}), sensible heat flux H (W m^{-2}), and latent heat flux L (W m^{-2}), and that they all sum to zero:
We consider two components of net radiation: short wave (visible and near IR) and thermal.
where R_{abs} is absorbed short wave, and R_{nt} is net thermal. The absorbed short wave radiation is computed by
9‑89
where Q_{abs} is the absorbed irradiance by the leaf (µmol m^{-2} s^{-1}), and k is the conversion factor for transforming (µmol m^{-2} s^{-1}) to (W m^{-2}), based on the spectral characteristics of the light source.
The net thermal balance is based on the leaf temperature T_{l} and the surrounding chamber wall temperature T_{w}, so the total radiation balance R can be written as
where ϵ is the thermal emissivity of the leaf (usually assumed to be 0.95), and σ is the Stefan-Boltzmann constant (5.67 W m^{-2} K^{-1}). The 2 in equation 9‑90 accounts for both sides of the leaf. Wall temperature T_{w} is not measured, but depends on a user-specified offset ΔT_{w} from chamber air temperature.
9‑91
The latent heat flux L is the transpiration rate E converted to W m^{-2}.
9‑92
The sensible heat flux H is a function of the leaf - chamber air temperature difference T_{l}−T_{a}, the specific heat capacity of the air c_{p} (29.3 J mol^{-1} K^{-1}) and the one sided boundary layer conductance for heat transfer of the leaf g_{bh}, which is 0.92 times the boundary layer conductance for water vapor g_{bw}.
9‑93
9‑94
Equation 9‑87 becomes:
If we let ΔT = T_{l} − T_{a}, and note for small ΔT
Substituting equation 9‑96 into 9‑95 and solving for ΔT yields
9‑97
The energy balance leaf temperature T_{eb} is then
9‑98
Sensor head calibration coefficients
Symbol | Description | XML Locator (/licor/li6850/...) |
---|---|---|
a_{f1}...a_{f7} a_{r1}...a_{r4} a_{s1}...a_{s4} |
Main flow sensor Ref flow sensor Sample flow sensor |
../factory/flowmeter/a...g ../factory/irga_a/flow/a1...a4 ../factory/irga_b/flow/a1...a4 |
B_{wc} B_{oc} B_{ow1}, B_{ow2} |
Band broadening coefficient for water on CO_{2} Band broadening coefficient for oxygen on CO_{2} Band broadening correction for oxygen on water |
../factory/bb/ch ../factory/bb/cx ../factory/bb/hx0, hx1 |
b_{r1}...b_{r5} b_{s1}...b_{s5} |
Reference CO_{2} calibration coefficients Sample CO_{2} calibration coefficients |
../factory/irga_b/co2/a1...a5 ../factory/irga_a/co2/a1...a5 |
d_{r1}...d_{r3} d_{s1}...d_{s3} |
Reference H_{2}O calibration coefficients Sample H_{2}O calibration coefficients |
../factory/irga_b/h2o/a1...a3 ../factory/irga_b/h2o/a1...a3 |
m_{co}...m_{c3}
m_{wo}...m_{w3} |
CO_{2} match coefficients H_{2}O match coefficients |
../cfg/match/co2_adj ... co2 adj 3 ../cfg/match/h2o_adj ... h2o adj 3 |
p_{ca}
p_{cb} |
Pressure correction or fan speed and flow rate Pressure correction or fan speed and flow rate |
../factory/chamber/pca ../factory/chamber/pcb |
σ_{cr1}...σ_{cr3}
σ_{cs1}...σ_{cs3 } σ_{wr1}...σ_{wr3} σ_{ws1}...σ_{ws3} |
CO_{2} reference absorptance span drift with temp CO_{2} sample absorptance span drift with temp H_{2}O reference absorptance span drift with temp H_{2}O sample absorptance span drift with temp |
../factory/irga_b/co2/s1...s3 ../factory/irga_a/co2/s1...s3 ../factory/irga_b/h2o/s1...s3 ../factory/irga_a/h2o/s1...s3 |
S_{cro}
S_{cr1} S_{cso } S_{cs1} |
Span1 for reference CO_{2} Span2 for reference CO_{2} Span1 for sample CO_{2} Span2 for sample CO_{2} |
../cal/co2bspan1 ../cal/co2bspan2 ../cal/co2aspan1 ../cal/co2aspan2 |
S_{qin}
S_{qout} |
In-chamber light sensor cal External quantum sensor cal |
../cfg/ppfdin/mult ../cfg/ppfdout/mult |
S_{wro }
S_{wr1 } S_{wso } S_{ws1} |
Span1 for reference H_{2}O Span2 for reference H_{2}O Span1 for sample H_{2}O Span2 for sample H_{2}O |
../cal/h2obspan1 ../cal/h2obspan2 ../cal/h2oaspan1 ../cal/h2oaspan2 |
V_{fo}
V_{fro } V_{fso} |
Zero offset for main flow meter Zero offset for reference flow meter Zero offset for sample flow meter |
../cal/flowmeterzero ../cal/flowbzero ../cal/flowazero |
V_{po} | Zero parameter for differential pressure sensor | ../cal/chamberpressurezero |
X_{o} | Oxygen concentration, percent | ../factory/cal/oxygen |
X_{cwr }
X_{cws } X_{wcr } X_{wcs} |
Cross sensitivity, CO_{2} on H_{2}O, reference cell Cross sensitivity, CO_{2} on H_{2}O, sample cell Cross sensitivity, H_{2}O on CO_{2}, reference cell Cross sensitivity, H_{2}O on CO_{2}, sample cell |
../factory/irga_b/xhc ../factory/irga_b/xch ../factory/irga_a/xhc ../factory/irga_a/xch |
Z_{cr}
Z_{cro } Z_{cs } Z_{cso} |
Zero drift with temperature for reference CO_{2} Zero offset for reference CO_{2} Zero drift with temperature for sample CO_{2} Zero offset for sample CO_{2} |
../factory/irga_b/z ../cal/co2bzero ../factory/irga_a/z ../cal/co2azero |
Z_{wr}
Z_{wro } Z_{ws} Z_{wso} |
Zero drift with temperature for reference H_{2}O Zero offset for reference H_{2}O Zero drift with temperature for sample H_{2}O Zero offset for sample H_{2}O |
../factory/irga_b/z ../cal/h2obzero ../factory/irga_a/z ../cal/h2obzero |
Sensor measurements and computations
Symbol | Description (units) | Name (Label) | Group |
---|---|---|---|
α_{cr}
α_{cs} α_{wr} α_{ws} |
Reference cell CO_{2} absorptance Sample cell CO_{2} absorptance Reference cell H_{2}O absorptance Sample cell H_{2}O absorptance |
abs_c_b abs_c_a abs_h_b abs_h_a |
Raw Raw Raw Raw |
c_{o}...c_{4} | Boundary layer function coeffs | blc_a...blc_e | ChambConst |
C_{a}
C_{b} C_{r} C_{rd} C_{s} C_{sd} ∆P_{c} |
Sample cell CO_{2}, not adjusted for match Sample cell CO_{2} (µmol mol^{−}^{1}) Reference cell CO_{2} (µmol mol^{−}^{1}) Reference cell CO_{2}, dry mole fraction Sample cell CO_{2} (µmol mol^{−}^{1}) Sample cell CO_{2}, dry mole fraction Chamber over pressure (kPa) |
CO2_a CO2_b CO2_r CO2_r_d CO2_s CO2_s_d Pchamber (∆Pcham) |
Meas Meas2 Meas Meas Meas Meas Meas |
e_{r}
e_{s} I_{cr} I_{cro} I_{cs} I_{cso} |
Reference cell vapor pressure (kPa) Sample cell vapor pressure (kPa) Reference CO_{2} raw detector count Reference CO_{2} raw detector reference count Sample CO_{2}, raw detector count Sample CO_{2} raw detector reference count |
e_r e_s Wc_r Wco_r Wc_s Wco_s |
Meas2 Meas2 Raw Raw Raw Raw |
I_{qin}
I_{qout} |
In-chamber PPFD sensor raw counts External quantum sensor raw counts |
||
I_{wr}
I_{wro} I_{ws} I_{wso} |
Reference H_{2}O raw detector count Reference H_{2}O raw detector reference count Sample H_{2}O raw detector count Sample H_{2}O raw detector reference count |
Ww_r Wwo_r Ww_s Wwo_s |
Raw Raw Raw Raw |
P_{a} | Atmospheric pressure (kPa) | Press (Pa) | Meas |
F
F_{r} F_{s} |
Flow to chamber (µmol s^{−}^{1}) Flow from reference cell (µmol s^{−}^{1}) Flow from sample cell (µmol s^{−}^{1}) |
Flow Flow_r Flow_s |
Meas Status Status |
G | Chamber fan rotation rate (rpm) | Fan speed | Status |
Q_{in}
Q_{out} |
In-chamber PPFD External PPFD |
PPFD_in (Q_amb_in) PPFD_out (Q_amb_out) |
Meas Meas |
T_{a} | Leaf chamber air temperature (C) | Tchamber (Tair) | Meas |
T_{dr}
T_{ds} |
Dewpoint temperature reference cell (C) Dewpoint temperature sample cell (C) |
Td_r Td_s |
Meas2 Meas2 |
T_{j1}
T_{j2} T_{k} |
Leaf T/C 1 junction temperature (C) Leaf T/C 2 junction temperature (C) IRGA block temperature (C) |
Tleafjunction (Tleaf_j) Tleafjunction2 (Tleaf2_j) Tirga_block (Tirga) |
Status2 Status2 Status |
T_{l1}
T_{l2} T_{r} T_{s} |
Leaf temperature 1 (C) Leaf temperature 2 (C) Reference cell inlet temperature (C) Sample cell inlet temperature (C) |
Tleaf Tleaf2 Tb (Tr) Ta (Ts) |
Meas Meas Status Status |
V_{f}
V_{fr} V_{fs} |
Signal (V) from main flow sensor Signal (V) from reference flow sensor Signal (V) from sample flow sensor |
Flow Flow_b_v (Flow_r_v) Flow_a_v (Flow_s_v) |
Raw Raw Raw |
V_{l1}
V_{l2} |
Leaf temperature 1 signal (mV) Leaf temperature 2 signal (mV) |
leaf_t_mv (Tleaf_mv) leaf2_t_mv (Tleaf2_mv) |
Raw Raw |
V_{p} | Differential pressure signal (V or kPa) | VPchamber | Raw |
W_{a}
W_{b} W_{r} W_{s} |
Sample cell H_{2}O not corrected for match Reference cell H_{2}O Reference cell H_{2}O (mmol mol^{−}^{1}) Sample cell H_{2}O (mmol mol^{−}^{1}) |
H2O_a H2O_a H2O_r H2O_s |
Meas Meas2 Meas Meas |