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A dewpoint hygrometer for water potential measurement
Agricultural Meteorology, 12 (1973) 113-121 © Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
A DEWPOINT HYGROMETER FOR WATER POTENTIAL MEASUREMENT* E R I C C. C A M P B E L L I , G A Y L O N S. C A M P B E L L 2 and W A Y N E K. B A R L O W z
Wescor, Inc., Logan, Utah (U.S.A.} 2Department of Agronomy and Soils and Biophysics Program, Washington State University, Pullman, Wash. {U.S.A.J (Accepted for publication June 8, 1973)
ABSTRACT Campbell, E. C., Campbell, G. S. and Barlow, W. K., 1973. A dewpoint hygrometer for water potential measurement. Agric. Meteorol., 12: l 13-121.
An improved instrument for determining water potential by measuring dewpoint depression has been developed. Theoretical considerations show the instrument to have a sensitivity of 0.75 uV bar -~ and a change in sensitivity with temperature of 0.45% °C-~. Without compensation for changes in Peltier cooling, sensitivity, and heat dissipation with temperature, errors of +-6% result from changing the temperature of the sensor between 20 and 50°C. Performance of the instrument appears to agree well with the theory developed for the technique. INTRODUCTION A technique was introduced recently by Neumann and Thurtell (1972) for measuring the dewpoint temperature in a small sealed cavity adjacent a leaf surface. Leaf water potential was inferred from the dewpoint reading. Dewpoint temperature depression was measured using two thermocouples having a common measuring junction. A current was passed through one thermocouple cooling the measuring junction by the Peltier effect while the other thermocouple measured the junction temperature depression. A characteristic curve relating temperature depression to cooling current for the thermocouple in dry air was traced using an x - y plotter. When water vapor was introduced into the chamber, part of the electrical energy was used to condense water vapor if the junction was below the dewpoint temperature. The dewpoint depression was determined by cooling the thermocouple below the dewpoint temperature and condensing water on it, then manually decreasing the cooling current until the x - y recorder pen reached the dry curve. Neumann and Thurtell (1972) called their device a thermocouple hygrometer. The design and construction of an improved dewpoint meter based upon maintaining a thermocouple at dewpoint temperature, together with pertinent theory and results of experimental tests, is presented in this paper. * Presented at the Summer Meeting of the American Society of Agricultural Engineers at Hot Springs, Arkansas, June 1972 as paper No.72-436.
114
E.C. CAMPBELL, G.S. CAMPBELL AND W.K. BARLOW
DESIGN CONSIDERATIONS If held at the dewpoint temperature, a wet thermocouple junction will neither lose water through evaporation nor gain water through condensation. Consider a hypothetical thermocouple junction whose temperature is determined exclusively by the heat transferred to it or away from it by condensing or evaporating water. Assume also that the junction has an initial temperature T, and that it is covered with a film of water. If T is above the dewpoint, water will evaporate from the junction carrying with it the heat of vaporization until the temperature of the junction fails to the dewpoint. If T is below the dewpoint additional water will condense upon its surface, with the heat of condensation raising the temperature of the junction until it reaches the dewpoint. Therefore, given the aforesaid independence from other heat transfer mechanisms, the temperature of the wet junction will always converge upon the dewpoint. Under practical conditions, it is not possible for a thermocouple junction to be independent of heat transfer mechanisms which exist in ordinary environments. Nevertheless, by considering heat exchange conditions which prevail when a measurement of water potential is to be made, it is possible to simulate the above described hypothetical situation. During the measurement the wet junction temperature will always be below the temperature of its surroundings so heat will flow from the surroundings to the junction. Using Peltier cooling, a counter flow can be created and adjusted electrically to exactly balance the heat inflow for a net transfer of zero. If this condition is set up on a dry thermocouple to balance all heat transfer other than condensing or evaporating water, then when the junction is wet its temperature will converge on the dewpoint just as in the hypothetical example. The dewpoint readout must be designed to accomplish this objective. An additional consideration, that the dewpoint meter should operate with existing two and three wire psychrometers, requires that a chopping, or time sharing, system be worked out to cool and read on the same thermocouple. Consider a thermocouple with Peltier coefficient, P through which a current, i flows. The thermocouple will be time-shared such that the current will be applied to the junction for a fraction, L, of the total chopping cycle. The fraction of time the thermocouple is connected to the amplifier is 1 - L. The dry thermocouple temperature depression is given by (Peck, 1968): Odry-
- L II
a
-L/
. ( lh'/J - (2~2i2/TrrwZJT) (coth 71 - cosech 71)
~ + ( ~ ) ~ ~
"/
(1)
where 1/2
(2)
B = 27rka/ln(rc/rw)
(3)
3' = ( B @ r w 2 k w )
and where 11 is defined as a thermocouple cooling coefficient to be used in subsequent considerations, and c~converts temperature to thermocouple EMF. Other terms in eq.1, 2, and 3 are defined in the Notation.
DEWPOINT HYGROMETER FOR MEASURING WATER POTENTIAL
115
Radiative heat loss f r o m the j u n c t i o n is only about 1% of the other modes o f heat loss, so it is n o t included in eq.1. The m a x i m u m possible cooling for a given t h e r m o c o u p l e occurs w h e n L = 1 and is a f u n c t i o n only of t h e r m o c o u p l e and d e w p o i n t m e t e r design parameters. Thus for a given t h e r m o c o u p l e , environment, and cooling current, the t h e r m o c o u p l e NOTATION List of symbols B D Dv E Eo i J ka kw l L P rc rj rw R S Sd Sp t T a 13 e, eD 3' k I1 Ov 0 q, ~2
defined in eq.3 dewpoint gain, V -1 diffusion coefficient for water vapor in air, cm 2 sec thermocouple output voltage, Volts dewpoint meter offset voltage, Volts thermocouple cooling current, 7.5 • 10 -3 Amp. mechanical equivalent of heat, 4.2 Joules cal.-' thermal conductivity of air, cal. sec-~ cm -~ °C-1 thermal conductivity of wires, 0.05 cal. sec-' cm-' °C-' (Peck, 1968). length of thermocouple wires, 0.13 cm cooling current "on time" Peltier cooling coefficient given by aT, Volts radius of hygrometer chamber, 0.3 cm radius of thermocouple junction, 10 -2 cm radius of thermocouple wire, 1.3.10 -3 cm gas constant, 4.60 bar °C-' observed hygrometer sensitivity, V bar-' hygrometer sensitivity in the dewpoint mode, V bar-' hygrometer sensitivity in the psychrometer mode, V bar-' Celsius temperature Kelvin temperature thermocouple sensitivity, given by 58 + 0. It ~aV °C-' slope of the saturation vapor density-temperature function, g cm -3 °C-' offset error, Volts error caused by errors in D defined in eq.2 latent heat of vaporization for water, cal./g thermocouple cooling coefficient, Volts saturation water vapor density, g cm -s measuring junction temperature depression, °C water potential, bars resistivity of thermocouple wire, 6- 10 -s Ohm cm (Peck, 1968)
cooling coefficient, II, is constant. To exactly m a t c h the energy gains and losses for the t h e r m o c o u p l e , we use the t h e r m o c o u p l e o u t p u t to control the " o n t i m e " o f the cooling current by setting
L =otOD The d e w p o i n t gain, D determines the " o n t i m e " per m i c r o v o l t o f input signal. When D = 1/II, the d e w p o i n t m e t e r is properly set to read d e w p o i n t t e m p e r a t u r e depression.
(4)
116
E.C. CAMPBELL,G. S. CAMPBELLAND W. K. BARLOW
ANALYSIS OF ERRORS Errors in the dewpoint measurement may arise from zero offset errors, changes in sensitivity and II with temperature, and errors in setting D. To determine the magnitude of these errors we first find an expression for the temperature of a wet thermocouple during cooling. We then use this expression to determine the effect of various factors on thermocouple temperature. Peck's (1968) equation for thermocouple temperature d.epression during condensation can be modified slightly to give the temperature depression of a wet thermocouple being cooled by the dewpoint meter. The equation becomes:
f LPf
2~2i2L7 (coth 3'1 - cosech 3`I) - 47rrjXDvPVqJRT 7rrw2J
0wet = -
(2B/7) coth 3'l + 47rrjka + 47rrjXDv~3
(5)
Substituting eq.1 and 4 into 5, multiplying by a to obtain the thermocouple EMF, and dividing by water potential (q0 to express the result as a sensitivity (volts/bar), we obtain:
S-
47rarjhDvPv/R T
(6)
{1 -DII)(2B/3' coth 7l + 47rrjka) + 4zrrjXDv/3 When D -- 1/II (dewpoint gain is properly set for a given thermocouple), eq.6 gives the dewpoint sensitivity:
Sd = apv/R TI3
(7)
Since, from the Clausius-Clapeyron equation we may obtain/3 = dpv/d T = pvX/R T 2, eq.7 may also be written as:
Sd = aT/X
(8)
When D = 0, the thermocouple behaves as a conventional thermocouple psychrometer with sensitivity given by:
47rarjXDvPv/R T Sp - 2B/3` coth 7l + 47rrjka + 47rrjXDv/3
(9)
an equation given previously by Rawlins (1966) and Peck (1968). Substitution of eq.7 and 9 into 6 gives: S=
Sp Sd [Sd - DFI(Sd - Sp)]
(10)
The voltage error which results from zero offset is given by:
eo = Eo + otEodO/dE
(11)
since the reading will be changed both by the actual offset and by the change in control point due to the offset. The change in cooling with voltage can be expressed as the change
DEWPOINT HYGROMETERFOR MEASURINGWATER POTENTIAL
117
in cooling with L multiplied by the change in L with voltage. Since, in eq.4, aq = E; dL/dE = D. Eq.5 is linear in L passing through 0p, the psychrometer temperature depression, at L = 0. Thus the derivative of eq.5 with L is obtained simply as dO/dL = (0 -Op)/L. And since 0 = S xP/a, 0p = Sp ~/ol and L = ED = S ~ D , eq. 11 becomes: e0 = Eo(2 - Sp/S)
(12)
Typical Eo values would be around 0.1/aV and S -~ Sd. This would give an eo of 0.14/aV at 25°C. This is equivalent to about 0.18 bar water potential. Both II and Sd change with temperature. Ideally, one would set D to equal 1/II for the hygrometer temperature at which measurements are being made and then convert the dewpoint reading to water potential using the appropriate value of Sd from eq.8. In practice, it is time consuming to make such corrections, and some error is always present in dewpoint gain settings. We wish to know how much error results from ignoring temperature change~ or from missetting D. The error in water potential measurement is the difference between the measured and the actual water potentials divided by the actual water potential. This can be expressed in terms of EMF's and sensitivities, since SxI' = E, giving: e = (S d - S)/S d
(13)
For small changes in D, S d - S can be approximated by dS which is obtained by differentiating eq.10. This is substituted into eq.13 to give: eD
= II(Sd/Sp
-
1)dD
(14)
where it has been assumed that DII ~ 1. Since D -~ 1/If, we can substitute -dI1/II for H dD. The r a t i o Sd/S p at 25°C is about 1.7. Typical errors in setting D are equivalent to less than 2% error in H, so the error due to D setting would be less than 1.4%. Errors due to changes in II with temperature are determined using eq.1, 6, 7, 9, and 13. Using data from Rawlins' (1966) Table 1, and the values given in the list of symbols, the error due to temperature changes was calculated (assuming no readjustment of D from the 25°C value and no correction for change of S d with temperature). The result is shown in Fig.1. The error is less than +-4% between 20° and 35°C, and is less than +-6% from 20 ° to 50°C. Correct adjustment of D becomes relatively more important at low temperatures because Sp becomes so small at these temperatures. Error due to temperature effects on II and Sd can be made negligibly small by using the appropriate value of Sd (from eq.7 or 8) and adjusting D to compensate for temperature changes. The 11 value for the hygrometer cited here was 64/JV and changed at a rate of 0.62 pV C-1 . The temperature coefficient for any hygrometer is easily obtained by measuring II at two temperatures. An empirical equation obtained from eq.8 for Sd as a function of temperature is Sd = 0.63 + 0.0045 t.
l 18
E.C. CAMPBELL, G. S. CAMPBELL AND W. K. BARLOW
40 3O tr o
zo uJ Z
"' I0 L) W 0_
0
-'Co
,b
2'o
3b
TEMPERATURE
~'o
so
- C
Fig.1. Error in dewpoint measurement resulting from temperature effects on thermocouple cooling coefficient, lq. ELECTRONICS The electronic system of the dewpoint meter, when in the dewpoint operational mode, is depicted in the block diagram, Fig.2. The voltage signal from the thermocouple is processed through the microvoltmeter section of the instrument and is fed to the recorder output terminals, the panel meter, and into the non-inverting input of a level comparison
MICROVOLTMETER MOOULE
Ponel 1 Mefer T
-
-
-
[ . . . . . .
I
J
'
-
-
O Recorder
Output
°.,::::: - _ : P d - ' d o ~
t
-I I"---'1r----'~r "fo Iwl
I $owh)oth/PU/le
~ / ~ / ~
o Generator
Y DUTY CYCLE CONTROL
l:ig,2. Dewpoint meter electronic system, dewpoint mode.
DEWPOINT HYGROMETERFOR MEASURINGWATER POTENTIAL
1 19
circuit with bistable logic output either "high" (positive) or "low" (negative) depending upon whether the voltage at the + input terminal is higher or lower than the voltage at the input terminal. The microvoltmeter signal is compared to the sawtooth ramp voltage by this circuit. Since the ramp voltage starts at zero, and assuming that some finite signal is present from the thermocouple (thermocouple temperature depressed from ambient), the comparator output will initially be "high", and the thermocouple will receive cooling current through the electronic switching circuits. When the ramp voltage becomes higher than the signal voltage, the output of the comparator switches to "low", and the cooling current is discontinued. The microvoltmeter reads the thermocouple voltage during this interval, and the sample hold circuit maintains the signal level from one reading interval to the next. The negative pulses produced by the sawtooth/pulse generator circuit are added to the thermocouple signal at the + input of the comparator. These pulses represent a 95% duty cycle giving a minimum read interval of at least 5% to maintain the output signal in the sample hold circuit. The height of the sawtooth wave form is adjusted by the dewpoint gain such that for a given thermocouple the cooling duty cycle, as dictated by the thermocouple temperature depression signal, will remove precisely that amount of heat which is flowing into the junction from its surroundings. If the water on the junction then causes a movement of the junction temperature toward the dewpoint, the output signal will change, dictating a corresponding change in the cooling duty cycle, and thus maintaining a balance of thermal energy being conducted into and out of the junction. To further illustrate, Fig.3 depicts the temporal relationship between the various signal and control waveforms and levels as they will appear during initial cooling and as the -
MAX kT
~
AVERAGE JUNCTION TEMPERATURE DEPRESSION
DEW POINT \Mg{l!Xl"
OUTPUT FROM SAMPLE
/CONTROL
HOLD CIRCUIT ~
WAVEFORM
i S~V
0
TIME
.~ COOLING CURRENT PULSES
-
COOL ~ ~ Duty Cycle at 95~ I for'Haxirnura CoolCooling
DEW POINT But,,
Cycle
D. . . . . . . . . . .
lunction
~l Cooling
rlutv
i~'i t h I u n c t i o n Temnerature at De~, P o i n t
Fig.3. Typical control and signal waveformsvs. junction temperature for the dewpoint meter.
]
120
E.C. CAMPBELL, G. S. CAMPBELL AND W. K. BARLOW
junction temperature converges to the dewpoint. The height of the sawtooth waveform is set by the dewpoint gain to match the thermocouple. Once D is set, the value of II can be read directly from the meter. For another junction having a different cooling efficiency, a different sawtooth waveform height will accordingly be set into the instrument. RESULTS
The dewpoint meter is presently commercially available (Wescor, Inc., Logan, Utah). Several of the units have been used for measuring water potential of leaves and soil in situ as well as measuring water potential of samples. Cooling coefficients have been found to vary widely between thermocouples, but a predetermined setting for a given thermocouple appears to work well for all subsequent measurements using that thermocouple when appropriate temperature corrections are made. The agreement between theoretical and actual dewpoint meter output for one hygrometer is shown in Fig.4. The agreement is very good to about -20 bars but fails off somewhat by - 3 0 bars. The II for this thermocouple was 50/aV. Another thermocouple with 11 of 60/aV gave good agreement with the theoretical value to beyond - 6 0 bars. Preliminary results with a number of thermocouple hygrometers indicate that, once 11 is properly set, the theoretical curve fits all hygrometers unless the measuring junction is contaminated. A contaminated measuring junction causes the readings to be too low and to drift if the contaminant is soluble. Non-soluble contaminants apparently slow down the convergence of the thermocouple on the dewpoint.
% %%
3O ::1. 1--
20~ O n~ t.i tuJ
i0 =E O r,L9 >T
-50
-4
-30
-20
-I0
BARS
Fig.4. Comparison of eq.8 (dashed line) with measured (points) hygrometer output. The solid line represents the exponential relationship between water potential and vapor density rather than the straight line approximation used here and by Peck (1968).
DEWPOINT HYGROMETER FOR MEASURING WATER POTENTIAL
121
Some attempts have been made to track water potential over a period of time using the dewpoint meter. Success of such attempts has been limited by temperature drift of the reference junction. With reference junctions designed to give very little drift, such as those of Neumann and Thurtell (1972), tracking appears to be feasible. Drift in the amplifier itself is made very small by a special chopper circuit.
REFERENCES Neumann, H. H. and Thurtell, G. W., 1972. A Peltier cooled thermocouple dewpoint hygrometer for in situ measurement of water potentials. In: R. W. Brown and B. P. Van Haveren (Editors), Psychromerry in WaterRelations Research. Utah Agric. Expt. Sta., Logan, in press. Peck, A. J., 1968. Theory of the Spanner psychrometer, I. The thermocouple. Agric. Meteorol., 5: 433-447. Rawlins, S. L., 1966. Theory for thermocouple psychrometers used to measure water potential in soil and plant samples. Agric. Meteorol., 3: 293-310.
A DEWPOINT HYGROMETER FOR WATER POTENTIAL MEASUREMENT* E R I C C. C A M P B E L L I , G A Y L O N S. C A M P B E L L 2 and W A Y N E K. B A R L O W z
Wescor, Inc., Logan, Utah (U.S.A.} 2Department of Agronomy and Soils and Biophysics Program, Washington State University, Pullman, Wash. {U.S.A.J (Accepted for publication June 8, 1973)
ABSTRACT Campbell, E. C., Campbell, G. S. and Barlow, W. K., 1973. A dewpoint hygrometer for water potential measurement. Agric. Meteorol., 12: l 13-121.
An improved instrument for determining water potential by measuring dewpoint depression has been developed. Theoretical considerations show the instrument to have a sensitivity of 0.75 uV bar -~ and a change in sensitivity with temperature of 0.45% °C-~. Without compensation for changes in Peltier cooling, sensitivity, and heat dissipation with temperature, errors of +-6% result from changing the temperature of the sensor between 20 and 50°C. Performance of the instrument appears to agree well with the theory developed for the technique. INTRODUCTION A technique was introduced recently by Neumann and Thurtell (1972) for measuring the dewpoint temperature in a small sealed cavity adjacent a leaf surface. Leaf water potential was inferred from the dewpoint reading. Dewpoint temperature depression was measured using two thermocouples having a common measuring junction. A current was passed through one thermocouple cooling the measuring junction by the Peltier effect while the other thermocouple measured the junction temperature depression. A characteristic curve relating temperature depression to cooling current for the thermocouple in dry air was traced using an x - y plotter. When water vapor was introduced into the chamber, part of the electrical energy was used to condense water vapor if the junction was below the dewpoint temperature. The dewpoint depression was determined by cooling the thermocouple below the dewpoint temperature and condensing water on it, then manually decreasing the cooling current until the x - y recorder pen reached the dry curve. Neumann and Thurtell (1972) called their device a thermocouple hygrometer. The design and construction of an improved dewpoint meter based upon maintaining a thermocouple at dewpoint temperature, together with pertinent theory and results of experimental tests, is presented in this paper. * Presented at the Summer Meeting of the American Society of Agricultural Engineers at Hot Springs, Arkansas, June 1972 as paper No.72-436.
114
E.C. CAMPBELL, G.S. CAMPBELL AND W.K. BARLOW
DESIGN CONSIDERATIONS If held at the dewpoint temperature, a wet thermocouple junction will neither lose water through evaporation nor gain water through condensation. Consider a hypothetical thermocouple junction whose temperature is determined exclusively by the heat transferred to it or away from it by condensing or evaporating water. Assume also that the junction has an initial temperature T, and that it is covered with a film of water. If T is above the dewpoint, water will evaporate from the junction carrying with it the heat of vaporization until the temperature of the junction fails to the dewpoint. If T is below the dewpoint additional water will condense upon its surface, with the heat of condensation raising the temperature of the junction until it reaches the dewpoint. Therefore, given the aforesaid independence from other heat transfer mechanisms, the temperature of the wet junction will always converge upon the dewpoint. Under practical conditions, it is not possible for a thermocouple junction to be independent of heat transfer mechanisms which exist in ordinary environments. Nevertheless, by considering heat exchange conditions which prevail when a measurement of water potential is to be made, it is possible to simulate the above described hypothetical situation. During the measurement the wet junction temperature will always be below the temperature of its surroundings so heat will flow from the surroundings to the junction. Using Peltier cooling, a counter flow can be created and adjusted electrically to exactly balance the heat inflow for a net transfer of zero. If this condition is set up on a dry thermocouple to balance all heat transfer other than condensing or evaporating water, then when the junction is wet its temperature will converge on the dewpoint just as in the hypothetical example. The dewpoint readout must be designed to accomplish this objective. An additional consideration, that the dewpoint meter should operate with existing two and three wire psychrometers, requires that a chopping, or time sharing, system be worked out to cool and read on the same thermocouple. Consider a thermocouple with Peltier coefficient, P through which a current, i flows. The thermocouple will be time-shared such that the current will be applied to the junction for a fraction, L, of the total chopping cycle. The fraction of time the thermocouple is connected to the amplifier is 1 - L. The dry thermocouple temperature depression is given by (Peck, 1968): Odry-
- L II
a
-L/
. ( lh'/J - (2~2i2/TrrwZJT) (coth 71 - cosech 71)
~ + ( ~ ) ~ ~
"/
(1)
where 1/2
(2)
B = 27rka/ln(rc/rw)
(3)
3' = ( B @ r w 2 k w )
and where 11 is defined as a thermocouple cooling coefficient to be used in subsequent considerations, and c~converts temperature to thermocouple EMF. Other terms in eq.1, 2, and 3 are defined in the Notation.
DEWPOINT HYGROMETER FOR MEASURING WATER POTENTIAL
115
Radiative heat loss f r o m the j u n c t i o n is only about 1% of the other modes o f heat loss, so it is n o t included in eq.1. The m a x i m u m possible cooling for a given t h e r m o c o u p l e occurs w h e n L = 1 and is a f u n c t i o n only of t h e r m o c o u p l e and d e w p o i n t m e t e r design parameters. Thus for a given t h e r m o c o u p l e , environment, and cooling current, the t h e r m o c o u p l e NOTATION List of symbols B D Dv E Eo i J ka kw l L P rc rj rw R S Sd Sp t T a 13 e, eD 3' k I1 Ov 0 q, ~2
defined in eq.3 dewpoint gain, V -1 diffusion coefficient for water vapor in air, cm 2 sec thermocouple output voltage, Volts dewpoint meter offset voltage, Volts thermocouple cooling current, 7.5 • 10 -3 Amp. mechanical equivalent of heat, 4.2 Joules cal.-' thermal conductivity of air, cal. sec-~ cm -~ °C-1 thermal conductivity of wires, 0.05 cal. sec-' cm-' °C-' (Peck, 1968). length of thermocouple wires, 0.13 cm cooling current "on time" Peltier cooling coefficient given by aT, Volts radius of hygrometer chamber, 0.3 cm radius of thermocouple junction, 10 -2 cm radius of thermocouple wire, 1.3.10 -3 cm gas constant, 4.60 bar °C-' observed hygrometer sensitivity, V bar-' hygrometer sensitivity in the dewpoint mode, V bar-' hygrometer sensitivity in the psychrometer mode, V bar-' Celsius temperature Kelvin temperature thermocouple sensitivity, given by 58 + 0. It ~aV °C-' slope of the saturation vapor density-temperature function, g cm -3 °C-' offset error, Volts error caused by errors in D defined in eq.2 latent heat of vaporization for water, cal./g thermocouple cooling coefficient, Volts saturation water vapor density, g cm -s measuring junction temperature depression, °C water potential, bars resistivity of thermocouple wire, 6- 10 -s Ohm cm (Peck, 1968)
cooling coefficient, II, is constant. To exactly m a t c h the energy gains and losses for the t h e r m o c o u p l e , we use the t h e r m o c o u p l e o u t p u t to control the " o n t i m e " o f the cooling current by setting
L =otOD The d e w p o i n t gain, D determines the " o n t i m e " per m i c r o v o l t o f input signal. When D = 1/II, the d e w p o i n t m e t e r is properly set to read d e w p o i n t t e m p e r a t u r e depression.
(4)
116
E.C. CAMPBELL,G. S. CAMPBELLAND W. K. BARLOW
ANALYSIS OF ERRORS Errors in the dewpoint measurement may arise from zero offset errors, changes in sensitivity and II with temperature, and errors in setting D. To determine the magnitude of these errors we first find an expression for the temperature of a wet thermocouple during cooling. We then use this expression to determine the effect of various factors on thermocouple temperature. Peck's (1968) equation for thermocouple temperature d.epression during condensation can be modified slightly to give the temperature depression of a wet thermocouple being cooled by the dewpoint meter. The equation becomes:
f LPf
2~2i2L7 (coth 3'1 - cosech 3`I) - 47rrjXDvPVqJRT 7rrw2J
0wet = -
(2B/7) coth 3'l + 47rrjka + 47rrjXDv~3
(5)
Substituting eq.1 and 4 into 5, multiplying by a to obtain the thermocouple EMF, and dividing by water potential (q0 to express the result as a sensitivity (volts/bar), we obtain:
S-
47rarjhDvPv/R T
(6)
{1 -DII)(2B/3' coth 7l + 47rrjka) + 4zrrjXDv/3 When D -- 1/II (dewpoint gain is properly set for a given thermocouple), eq.6 gives the dewpoint sensitivity:
Sd = apv/R TI3
(7)
Since, from the Clausius-Clapeyron equation we may obtain/3 = dpv/d T = pvX/R T 2, eq.7 may also be written as:
Sd = aT/X
(8)
When D = 0, the thermocouple behaves as a conventional thermocouple psychrometer with sensitivity given by:
47rarjXDvPv/R T Sp - 2B/3` coth 7l + 47rrjka + 47rrjXDv/3
(9)
an equation given previously by Rawlins (1966) and Peck (1968). Substitution of eq.7 and 9 into 6 gives: S=
Sp Sd [Sd - DFI(Sd - Sp)]
(10)
The voltage error which results from zero offset is given by:
eo = Eo + otEodO/dE
(11)
since the reading will be changed both by the actual offset and by the change in control point due to the offset. The change in cooling with voltage can be expressed as the change
DEWPOINT HYGROMETERFOR MEASURINGWATER POTENTIAL
117
in cooling with L multiplied by the change in L with voltage. Since, in eq.4, aq = E; dL/dE = D. Eq.5 is linear in L passing through 0p, the psychrometer temperature depression, at L = 0. Thus the derivative of eq.5 with L is obtained simply as dO/dL = (0 -Op)/L. And since 0 = S xP/a, 0p = Sp ~/ol and L = ED = S ~ D , eq. 11 becomes: e0 = Eo(2 - Sp/S)
(12)
Typical Eo values would be around 0.1/aV and S -~ Sd. This would give an eo of 0.14/aV at 25°C. This is equivalent to about 0.18 bar water potential. Both II and Sd change with temperature. Ideally, one would set D to equal 1/II for the hygrometer temperature at which measurements are being made and then convert the dewpoint reading to water potential using the appropriate value of Sd from eq.8. In practice, it is time consuming to make such corrections, and some error is always present in dewpoint gain settings. We wish to know how much error results from ignoring temperature change~ or from missetting D. The error in water potential measurement is the difference between the measured and the actual water potentials divided by the actual water potential. This can be expressed in terms of EMF's and sensitivities, since SxI' = E, giving: e = (S d - S)/S d
(13)
For small changes in D, S d - S can be approximated by dS which is obtained by differentiating eq.10. This is substituted into eq.13 to give: eD
= II(Sd/Sp
-
1)dD
(14)
where it has been assumed that DII ~ 1. Since D -~ 1/If, we can substitute -dI1/II for H dD. The r a t i o Sd/S p at 25°C is about 1.7. Typical errors in setting D are equivalent to less than 2% error in H, so the error due to D setting would be less than 1.4%. Errors due to changes in II with temperature are determined using eq.1, 6, 7, 9, and 13. Using data from Rawlins' (1966) Table 1, and the values given in the list of symbols, the error due to temperature changes was calculated (assuming no readjustment of D from the 25°C value and no correction for change of S d with temperature). The result is shown in Fig.1. The error is less than +-4% between 20° and 35°C, and is less than +-6% from 20 ° to 50°C. Correct adjustment of D becomes relatively more important at low temperatures because Sp becomes so small at these temperatures. Error due to temperature effects on II and Sd can be made negligibly small by using the appropriate value of Sd (from eq.7 or 8) and adjusting D to compensate for temperature changes. The 11 value for the hygrometer cited here was 64/JV and changed at a rate of 0.62 pV C-1 . The temperature coefficient for any hygrometer is easily obtained by measuring II at two temperatures. An empirical equation obtained from eq.8 for Sd as a function of temperature is Sd = 0.63 + 0.0045 t.
l 18
E.C. CAMPBELL, G. S. CAMPBELL AND W. K. BARLOW
40 3O tr o
zo uJ Z
"' I0 L) W 0_
0
-'Co
,b
2'o
3b
TEMPERATURE
~'o
so
- C
Fig.1. Error in dewpoint measurement resulting from temperature effects on thermocouple cooling coefficient, lq. ELECTRONICS The electronic system of the dewpoint meter, when in the dewpoint operational mode, is depicted in the block diagram, Fig.2. The voltage signal from the thermocouple is processed through the microvoltmeter section of the instrument and is fed to the recorder output terminals, the panel meter, and into the non-inverting input of a level comparison
MICROVOLTMETER MOOULE
Ponel 1 Mefer T
-
-
-
[ . . . . . .
I
J
'
-
-
O Recorder
Output
°.,::::: - _ : P d - ' d o ~
t
-I I"---'1r----'~r "fo Iwl
I $owh)oth/PU/le
~ / ~ / ~
o Generator
Y DUTY CYCLE CONTROL
l:ig,2. Dewpoint meter electronic system, dewpoint mode.
DEWPOINT HYGROMETERFOR MEASURINGWATER POTENTIAL
1 19
circuit with bistable logic output either "high" (positive) or "low" (negative) depending upon whether the voltage at the + input terminal is higher or lower than the voltage at the input terminal. The microvoltmeter signal is compared to the sawtooth ramp voltage by this circuit. Since the ramp voltage starts at zero, and assuming that some finite signal is present from the thermocouple (thermocouple temperature depressed from ambient), the comparator output will initially be "high", and the thermocouple will receive cooling current through the electronic switching circuits. When the ramp voltage becomes higher than the signal voltage, the output of the comparator switches to "low", and the cooling current is discontinued. The microvoltmeter reads the thermocouple voltage during this interval, and the sample hold circuit maintains the signal level from one reading interval to the next. The negative pulses produced by the sawtooth/pulse generator circuit are added to the thermocouple signal at the + input of the comparator. These pulses represent a 95% duty cycle giving a minimum read interval of at least 5% to maintain the output signal in the sample hold circuit. The height of the sawtooth wave form is adjusted by the dewpoint gain such that for a given thermocouple the cooling duty cycle, as dictated by the thermocouple temperature depression signal, will remove precisely that amount of heat which is flowing into the junction from its surroundings. If the water on the junction then causes a movement of the junction temperature toward the dewpoint, the output signal will change, dictating a corresponding change in the cooling duty cycle, and thus maintaining a balance of thermal energy being conducted into and out of the junction. To further illustrate, Fig.3 depicts the temporal relationship between the various signal and control waveforms and levels as they will appear during initial cooling and as the -
MAX kT
~
AVERAGE JUNCTION TEMPERATURE DEPRESSION
DEW POINT \Mg{l!Xl"
OUTPUT FROM SAMPLE
/CONTROL
HOLD CIRCUIT ~
WAVEFORM
i S~V
0
TIME
.~ COOLING CURRENT PULSES
-
COOL ~ ~ Duty Cycle at 95~ I for'Haxirnura CoolCooling
DEW POINT But,,
Cycle
D. . . . . . . . . . .
lunction
~l Cooling
rlutv
i~'i t h I u n c t i o n Temnerature at De~, P o i n t
Fig.3. Typical control and signal waveformsvs. junction temperature for the dewpoint meter.
]
120
E.C. CAMPBELL, G. S. CAMPBELL AND W. K. BARLOW
junction temperature converges to the dewpoint. The height of the sawtooth waveform is set by the dewpoint gain to match the thermocouple. Once D is set, the value of II can be read directly from the meter. For another junction having a different cooling efficiency, a different sawtooth waveform height will accordingly be set into the instrument. RESULTS
The dewpoint meter is presently commercially available (Wescor, Inc., Logan, Utah). Several of the units have been used for measuring water potential of leaves and soil in situ as well as measuring water potential of samples. Cooling coefficients have been found to vary widely between thermocouples, but a predetermined setting for a given thermocouple appears to work well for all subsequent measurements using that thermocouple when appropriate temperature corrections are made. The agreement between theoretical and actual dewpoint meter output for one hygrometer is shown in Fig.4. The agreement is very good to about -20 bars but fails off somewhat by - 3 0 bars. The II for this thermocouple was 50/aV. Another thermocouple with 11 of 60/aV gave good agreement with the theoretical value to beyond - 6 0 bars. Preliminary results with a number of thermocouple hygrometers indicate that, once 11 is properly set, the theoretical curve fits all hygrometers unless the measuring junction is contaminated. A contaminated measuring junction causes the readings to be too low and to drift if the contaminant is soluble. Non-soluble contaminants apparently slow down the convergence of the thermocouple on the dewpoint.
% %%
3O ::1. 1--
20~ O n~ t.i tuJ
i0 =E O r,L9 >T
-50
-4
-30
-20
-I0
BARS
Fig.4. Comparison of eq.8 (dashed line) with measured (points) hygrometer output. The solid line represents the exponential relationship between water potential and vapor density rather than the straight line approximation used here and by Peck (1968).
DEWPOINT HYGROMETER FOR MEASURING WATER POTENTIAL
121
Some attempts have been made to track water potential over a period of time using the dewpoint meter. Success of such attempts has been limited by temperature drift of the reference junction. With reference junctions designed to give very little drift, such as those of Neumann and Thurtell (1972), tracking appears to be feasible. Drift in the amplifier itself is made very small by a special chopper circuit.
REFERENCES Neumann, H. H. and Thurtell, G. W., 1972. A Peltier cooled thermocouple dewpoint hygrometer for in situ measurement of water potentials. In: R. W. Brown and B. P. Van Haveren (Editors), Psychromerry in WaterRelations Research. Utah Agric. Expt. Sta., Logan, in press. Peck, A. J., 1968. Theory of the Spanner psychrometer, I. The thermocouple. Agric. Meteorol., 5: 433-447. Rawlins, S. L., 1966. Theory for thermocouple psychrometers used to measure water potential in soil and plant samples. Agric. Meteorol., 3: 293-310.