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).

Table 9‑1. Values for pressure correction coefficients for each chamber type.

Chamber	pca	pcb
6800-01 Fluorometer	0	0
6800-01A Fluorometer	4.3321E-10	1.0215E-5
6800-12 3x3	0	0
6800-12A 3x3	2.6520E-10	1.3064E-5
6800-13 6x6	4.3998E-11	1.4792E-5
6800-17 Small Plant	0	0
6800-19 Custom	0	0

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.

CO₂

Reference and sample CO₂ concentrations C_r and C_s are given by

9‑27

where M_c() is the CO₂ 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 CO₂, given by

9‑31

where B_wc and B_oc are the band broadening coefficients for H₂O and O₂ respectively on CO₂.

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 CO₂ 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 CO₂, 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 CO₂ determined during calibration, Z_cro and Z_cso are the current user CO₂ zero factors, and Z_cr and Z_cs are the factors for CO₂ zero drift with temperature, determined during calibration.

Dry mole fractions C_rd and C_sd are computed from

9‑34

H₂O

Reference and sample H₂O concentrations H_r and H_s are given by

9‑35

where M_w() is the H₂O 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 H₂O span1 and span2 user calibration settings for reference and sample, Ѱ_o is the band broadening function for the effect of oxygen on H₂O, 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 H₂O absorptance span drift determined during calibration. X_cwr, X_cws are empirical cross sensitivity coefficients for CO₂ on H₂O determined during calibration, Z_wro and Z_wso are the current user H₂O zero factors, and Z_wr and Z_ws are the factors for H₂O 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

9‑51

where s is leaf area (m²), E is transpiration rate (mol H₂O 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 H₂O (mol air)^-1). Since

9‑52

we can write

9‑53

which rearranges to

9‑54

The relationships between the terms in equations 9‑51 through 9‑54 and what the LI-6800 measures are

9‑55

where F is air flow rate (µmol s^-1), W_s and W_r are sample and reference water mole fractions (mmol H₂O (mol air)^-1), and S is leaf area (cm²). 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 H₂O (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

9‑62

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

9‑68

9‑69

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

9‑71

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 (cm²).

9‑73

where f is

9‑74

and s is forced to be

9‑75

The empirical coefficients c_o...c₄, reference pressure P_o, and leaf area limits S_min and S_max depend on chamber type.

Table 9‑2. Values for boundary layer conductance for chamber types.

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 cm2	0.578	0.5229739	3.740252E-3	-6.197961E-2	-5.608586E-3	96.9	1	6
6800-01A 2 cm2	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 cm2	0.579	0.3210639	-1.109987E-3	5.106816E-3	-3.283688E-3	96.7	2	9
6800-12A 6 cm2FB	0.345	0.552336	-4.7985e-3	0.0	-7.3557e-3	96.7	1	6
6800-12A 6 cm2SS	0.418	0.5145466	-2.5106E-3	0.0	-8.1206E-3	96.7	1	6
6800-12A 3 cm2FB	0.188	0.5795409	-1.15295E-2	0.0	-9.7259E-3	96.7	1	3
6800-12A 3 cm2SS	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 CO₂ in an open system is given by

9‑76

where a is assimilation rate (mol CO₂ m^-1 s^-1), c_e and c_o are entering and outgoing mole fractions (mol CO₂ (mol air)^-1). Using equation 9‑52, we can write

9‑77

which rearranges to

9‑78

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 CO₂ 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 CO₂

The intercellular CO₂ concentration C_i (µmol CO₂ (mol air)^-1) is given by

9‑83

where g_tc is the total conductance to CO₂. From equation 9‑69, we can write

9‑84

where 1.6 is the ratio of the diffusivities of CO₂ 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:

9‑87

We consider two components of net radiation: short wave (visible and near IR) and thermal.

9‑88

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

9‑90

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:

9‑95

If we let ΔT = T_l − T_a, and note for small ΔT

9‑96

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

Table 9‑3. Sensor head calibration coefficients, with XML location.

Symbol	Description	XML Locator (/licor/li6850/...)
af1...af7 ar1...ar4 as1...as4	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
Bwc Boc Bow1, Bow2	Band broadening coefficient for water on CO2 Band broadening coefficient for oxygen on CO2 Band broadening correction for oxygen on water	../factory/bb/ch ../factory/bb/cx ../factory/bb/hx0, hx1
br1...br5 bs1...bs5	Reference CO2 calibration coefficients Sample CO2 calibration coefficients	../factory/irga_b/co2/a1...a5 ../factory/irga_a/co2/a1...a5
dr1...dr3 ds1...ds3	Reference H2O calibration coefficients Sample H2O calibration coefficients	../factory/irga_b/h2o/a1...a3 ../factory/irga_b/h2o/a1...a3
mco...mc3 mwo...mw3	CO2 match coefficients H2O match coefficients	../cfg/match/co2_adj ... co2 adj 3 ../cfg/match/h2o_adj ... h2o adj 3
pca pcb	Pressure correction or fan speed and flow rate Pressure correction or fan speed and flow rate	../factory/chamber/pca ../factory/chamber/pcb
σcr1...σcr3 σcs1...σcs3 σwr1...σwr3 σws1...σws3	CO2 reference absorptance span drift with temp CO2 sample absorptance span drift with temp H2O reference absorptance span drift with temp H2O 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
Scro Scr1 Scso Scs1	Span1 for reference CO2 Span2 for reference CO2 Span1 for sample CO2 Span2 for sample CO2	../cal/co2bspan1 ../cal/co2bspan2 ../cal/co2aspan1 ../cal/co2aspan2
Sqin Sqout	In-chamber light sensor cal External quantum sensor cal	../cfg/ppfdin/mult ../cfg/ppfdout/mult
Swro Swr1 Swso Sws1	Span1 for reference H2O Span2 for reference H2O Span1 for sample H2O Span2 for sample H2O	../cal/h2obspan1 ../cal/h2obspan2 ../cal/h2oaspan1 ../cal/h2oaspan2
Vfo Vfro Vfso	Zero offset for main flow meter Zero offset for reference flow meter Zero offset for sample flow meter	../cal/flowmeterzero ../cal/flowbzero ../cal/flowazero
Vpo	Zero parameter for differential pressure sensor	../cal/chamberpressurezero
Xo	Oxygen concentration, percent	../factory/cal/oxygen
Xcwr Xcws Xwcr Xwcs	Cross sensitivity, CO2 on H2O, reference cell Cross sensitivity, CO2 on H2O, sample cell Cross sensitivity, H2O on CO2, reference cell Cross sensitivity, H2O on CO2, sample cell	../factory/irga_b/xhc ../factory/irga_b/xch ../factory/irga_a/xhc ../factory/irga_a/xch
Zcr Zcro Zcs Zcso	Zero drift with temperature for reference CO2 Zero offset for reference CO2 Zero drift with temperature for sample CO2 Zero offset for sample CO2	../factory/irga_b/z ../cal/co2bzero ../factory/irga_a/z ../cal/co2azero
Zwr Zwro Zws Zwso	Zero drift with temperature for reference H2O Zero offset for reference H2O Zero drift with temperature for sample H2O Zero offset for sample H2O	../factory/irga_b/z ../cal/h2obzero ../factory/irga_a/z ../cal/h2obzero

Sensor measurements and computations

Table 9‑4. Sensor measurements and computations, and where to find them in the Data Dictionary.

Symbol	Description (units)	Name (Label)	Group
αcr αcs αwr αws	Reference cell CO2 absorptance Sample cell CO2 absorptance Reference cell H2O absorptance Sample cell H2O absorptance	abs_c_b abs_c_a abs_h_b abs_h_a	Raw Raw Raw Raw
co...c4	Boundary layer function coeffs	blc_a...blc_e	ChambConst
Ca Cb Cr Crd Cs Csd ∆Pc	Sample cell CO2, not adjusted for match Sample cell CO2 (µmol mol−1) Reference cell CO2 (µmol mol−1) Reference cell CO2, dry mole fraction Sample cell CO2 (µmol mol−1) Sample cell CO2, 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
er es Icr Icro Ics Icso	Reference cell vapor pressure (kPa) Sample cell vapor pressure (kPa) Reference CO2 raw detector count Reference CO2 raw detector reference count Sample CO2, raw detector count Sample CO2 raw detector reference count	e_r e_s Wc_r Wco_r Wc_s Wco_s	Meas2 Meas2 Raw Raw Raw Raw
Iqin Iqout	In-chamber PPFD sensor raw counts External quantum sensor raw counts
Iwr Iwro Iws Iwso	Reference H2O raw detector count Reference H2O raw detector reference count Sample H2O raw detector count Sample H2O raw detector reference count	Ww_r Wwo_r Ww_s Wwo_s	Raw Raw Raw Raw
Pa	Atmospheric pressure (kPa)	Press (Pa)	Meas
F Fr Fs	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
Qin Qout	In-chamber PPFD External PPFD	PPFD_in (Q_amb_in) PPFD_out (Q_amb_out)	Meas Meas
Ta	Leaf chamber air temperature (C)	Tchamber (Tair)	Meas
Tdr Tds	Dewpoint temperature reference cell (C) Dewpoint temperature sample cell (C)	Td_r Td_s	Meas2 Meas2
Tj1 Tj2 Tk	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
Tl1 Tl2 Tr Ts	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
Vf Vfr Vfs	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
Vl1 Vl2	Leaf temperature 1 signal (mV) Leaf temperature 2 signal (mV)	leaf_t_mv (Tleaf_mv) leaf2_t_mv (Tleaf2_mv)	Raw Raw
Vp	Differential pressure signal (V or kPa)	VPchamber	Raw
Wa Wb Wr Ws	Sample cell H2O not corrected for match Reference cell H2O Reference cell H2O (mmol mol−1) Sample cell H2O (mmol mol−1)	H2O_a H2O_a H2O_r H2O_s	Meas Meas2 Meas Meas