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Equations for respirometry without gas scrubbing
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Equations used for computing exchange rates are derived in part from
Depocas and Hart (1957; J. Appl. Physiol. 10:388-392),
Hill (1972, J. Appl. Physiol. 33:261-263) and Withers
(1977, J. Appl. Physiol. 42:120-123); others I derived
myself. If you want to see a (relatively) simple example of deriving an equation for VO2, look at this page -- but note that the example shown assumes scrubbing of both water and CO2 from excurrent gas.
The following symbols are used: FRadj =adjusted flow rate, V = exchange rate for the gas in question (oxygen, CO2, or water vapor), STP = factor for converting to standard conditions of temperature and pressure, Fi = input fractional concentration, Fe = excurrent fractional concentration, RQ = respiratory quotient.
The calculations for VO2 and VCO2 depend on the position of the flowmeter, although the effect is relatively small for VCO2 . Flow rate (FR) through the metabolism chamber is best measured upstream from the chamber in most cases -- this provides greater control (or understanding) of incurrent gas concentrations. To account for the presence of water vapor in the analyzed excurrent gas stream, two adjustments need to be made. First, the animal is assumed to have added water vapor, thereby increasing excurrent flowrate. To account for this, the program subtracts the fraction of excurrent flow comprised of water vapor (FeH2O) and uses the adjusted flow rate in calculations of VO2 and VCO2: FeH2O = water vapor pressure (kilopascals) / ambient pressure (kilopascals) Second, water vapor dilutes both excurrent oxygen concentration (FeO2 ) and excurrent CO2 concentration (FeCO2), so these values have to be adjusted upward, as follows:
'real' FeO2 = measured FeCO2 / (1 - FeH2O)
The algorithms used to compute vapor density are derived from Properties of Air, by Tracy, Welch, and Porter (1980; University of Wisconsin; you can find a pdf on the Web via Google Scholar). In turn, these are based on the Smithsonian Meteorological Tables. For those interested, the formulae used are as follows: • Vapor pressure (pw) at temperatures over liquid water (Smithsonian Tables, 1984, after Goff and Gratch, 1946):
Log10 pw = -7.90298 (373.16 T-1) • Vapor pressure (pi) at temperatures below 0 °C (over ice; Smithsonian Tables, 1984):
Log10 pi = -9.09718 (273.16/T - 1) The Goff-Gratch equation (for air over liquid water) covers a temperature range of -50 °C to about 100 °C, but is mostly theoretical for very low temperatures. Accuracy is probably ±0.5% or better at temperatures between -20 and 70 °C.
• CAUTION: regardless of the accuracy of the equations, technically it is very difficult to avoid condensation or freezing of water vapor when working at subzero temperatures.
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