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DIVISION S-1—SOIL PHYSICS

Calibration and Temperature Correction of Heat Dissipation Matric Potential Sensors

^a U.S. Geological Survey, Placer Hall, 6000 J Street, Sacramento, CA 95189
^b Decagon, Inc., 950 NE Nelson Ct., Pullman, WA 99163
^c 1111 Myrtle Dr., Burlington, WA 98233

* Corresponding author (aflint{at}usgs.gov)

	ABSTRACT

TOP ABSTRACT INTRODUCTION NOTES THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
This paper describes how heat dissipation sensors, used to measure soil water matric potential, were analyzed to develop a normalized calibration equation and a temperature correction method. Inference of soil matric potential depends on a correlation between the variable thermal conductance of the sensor's porous ceramic and matric potential. Although this correlation varies among sensors, we demonstrate a normalizing procedure that produces a single calibration relationship. Using sensors from three sources and different calibration methods, the normalized calibration resulted in a mean absolute error of 23% over a matric potential range of -0.01 to -35 MPa. Because the thermal conductivity of variably saturated porous media is temperature dependent, a temperature correction is required for application of heat dissipation sensors in field soils. A temperature correction procedure is outlined that reduces temperature dependent errors by 10 times, which reduces the matric potential measurement errors by more than 30%. The temperature dependence is well described by a thermal conductivity model that allows for the correction of measurements at any temperature to measurements at the calibration temperature.

Abbreviations: CSI, Campbell Scientific, Inc. • USGS, United States Geological Survey • WSU, Washington State University

	INTRODUCTION

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HEAT DISSIPATION SENSORS provide an indirect method of measuring soil water matric potential. In general, a heat dissipation sensor consists of a heating element and thermocouple embedded within a porous ceramic matrix. Water moves between the porous ceramic matrix and surrounding soil to achieve equilibrium with the matric potential of the soil. This type of sensor was first described by Phene et al. (1971). Later, a smaller version, which uses a line heat source, was developed at Washington State University (WSU). The smaller sensors were initially produced by Soiltronics¹ (Burlington, WA), and currently are available from Campbell Scientific, Inc. (CSI, Logan, UT). These sensors have the ability to make matric potential measurements in the range of approximately -0.01 to -100 MPa with a sensitivity that is proportional to the matric potential. Resolution is approximately 0.001 MPa at matric potentials greater than -0.1 MPa and is capable of responding to changes in the matric potential of dry soils. The sensors are also easy to read and are relatively inexpensive (generally <$100 per sensor).

The heat transfer properties between the heater and the ceramic of the sensor varies enough among sensors to make individual calibrations necessary, and the sensor response is sensitive to temperature and pressure, as well as water content. The calibration requirement forces the user to invest either time in the calibration process, or invest money to have the calibration done by a commercial laboratory that will follow calibration guidelines presented in this paper. The purpose of this paper is to demonstrate a normalization procedure that simplifies calibration of the sensors and to present a temperature correction method. This normalization procedure and correction method can be employed by the user to save time and money.

	THEORY

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The time dependence of temperature change, T, in a line heat source buried in an infinite medium can be approximated using the following equation modified from Shiozawa and Campbell (1990),

[1]

where T_f and T_o are the final and offset temperatures, respectively, in degrees Celsius (°C), q is the heat input in watts per meter (W m^-1), k is the thermal conductivity of the medium (W m^-1 °C^-1), t_f is final time (s), and t_o is the time (s) since the initiation of heating, which is when T_o is taken. The model results support the assumption that an infinite medium is applicable for heating periods to about 75 s and for heat input of 3 W m^-1. For these sensors a current of 0.05 A applied through the 32 ohm heating element, 28 mm long, produces a heat flux (q) of approximately 2.9 W m^-1. The current is usually maintained for 20 to 30 s. For our calibration procedure the temperature (T_o) of the sensor is recorded 1 s (t_o) after initiation of heating, and T_f is measured 20 s (t_f) later. If the temperature correction is to be implemented the initial temperature (T_i) is also recorded prior to the initiation of heating at time zero (t_i). Waiting 1 s after the initiation of heating before measuring T_o reduces the noise in the measurement as the current in the heater ramps up to a constant value. The difference in temperature (T) between the early reading (T_o) and the final reading (T_f) is dependent on the thermal properties of the ceramic, including water content, and therefore can be calibrated and used to compute the matric potential.

Reece (1996) showed that k, which is inversely proportional to T, is a useful measure for correlation with matric potential. If Eq. [1] correctly describes the temperature response of the heater in the matric potential sensor, then k - 1 will be proportional to the temperature change, no matter what measurement times or sequences are used, as long as t_o and q do not vary from measurement to measurement.

Measurements presented by Reece (1996) confirm the work of others, and show that much of the sensor variation is due to heating methods (i.e., heating current and heating time) and that the variation can be removed by normalizing the thermal conductivity measurements. Normalization in this case is simply the inverse of k at any saturation divided by the inverse of k when the ceramic is dry (Reece's [1996] method uses oven-dry probes to determine the dry reference k but does not explicitly state the oven temperature). Sensor-to-sensor variation using the same heating current and heating time in similar ceramic is likely due to variation in heater-ceramic contact. If constant current is applied for a specified period of time (typically 20 to 30 s), normalization can be accomplished by computing a dimensionless temperature rise:

[2]

where T is the temperature rise of the line heat source in the ceramic because of heating for the specified time, and T_d and T_w are the temperature changes for a dry and fully saturated ceramic, respectively. The normalization procedure produces a result similar to the one proposed by Reece (1996) who normalized to the dry-end thermal conductivity This method, however, normalizes sensor output to a range between 0 and 1 by correcting for small differences between both the wet and dry conductances of the ceramic.

The thermal conductivity of wet, porous materials has been modeled in detail (Campbell et al., 1994). Heat is conducted through the solid, liquid, and gas phases. The changes in k with temperature and pressure are mainly the result of changes in the gas phase conductivity (presented later in Eq. [3]) because of changes in the latent heat of distillation of the moisture across the pores of the ceramic. When the ceramic is fully saturated or completely dry, there is no effect of pressure on the thermal conductivity, and the change in temperature is mainly from temperature dependence of the thermal conductivity of water or air.

	MATERIALS AND METHODS

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The heat dissipation sensors used for the measurements reported here came from three lots. The first lot consisted of sensors that were from those originally constructed at WSU. The second lot was produced by Soiltronics (Burlington, WA). The third lot was from CSI (model 229-L, Campbell Scientific, Inc., Logan, UT). The design of the sensors from all three lots is similar to that presented in Reece (1996); all three used the same proprietary ceramic.

The sensors were calibrated using several methods and at different laboratories (WSU, CSI, and the United States Geological Survey [USGS]). In general, sensors calibrated at matric potentials greater than -0.02 MPa were equilibrated on a tension table or in a Tempe cell. For matric potentials between -0.02 and -1 MPa, the sensors were placed in soil in a pressure-plate apparatus and equilibrated at selected pressures, as described by Reece (1996). The pressure-plate vessel was depressurized prior to making calibration measurements to preclude the effects of pressure on the calibration. For soils drier than -1 MPa, a moisture characteristic was established for a silt loam soil using thermocouple psychrometer or dew point methods (Gee et al., 1992). The matric potential sensors were equilibrated with the silt loam soil, and the matric potential was determined from the water content of the sample. For matric potentials less than -1 MPa, sensors were also equilibrated in the atmosphere above saturated salt solutions to determine their response to changes in atmospheric humidity. The USGS sensors calibrated at matric potentials drier than -1 MPa were equilibrated in a relative humidity oven set to a specific water potential (generally near -300 MPa) then sealed in vapor proof containers and equilibrated at 20°C.

Water Content Scaling Factors
Sensors were saturated under vacuum to obtain the T_w values. The sensors were immersed in water in a plastic desiccation chamber and then a vacuum was applied (using a standard laboratory vacuum pump) until all bubbling stopped (usually <60 min). The T_w values were substantially lower than the values obtained by immersing sensors for several weeks without a vacuum, indicating hysteretic effects near saturation. Initially, values of T_d were obtained by oven drying at 105°C and equilibrating the sensors in a closed container above freshly oven-dried silica gel (representing -1000 MPa). However, the manufacturer does not recommend oven drying at 105°C and warns that exposing the sensors to 105°C may damage them. The current procedure requires the sensors be air dried at ambient conditions with the temperature and relative humidity known. The T at ambient conditions can be converted to T_d (representing -1000 MPa) using an equation presented later in this paper. Calibration experience suggests that two methods of calibration may be appropriate for determining T_w: vacuum saturation for sensors that may be fully submerged under field conditions, or simple immersion for 24 to 48 h for sensors that will remain drier than -0.01 MPa during field deployment. The best policy would be to collect both saturation values and apply the one most appropriate for the field conditions.

Temperature Scaling Factors
In addition to obtaining saturated and dry T, the effect of ambient temperature on the sensor response also was required for this study. The temperature response of the sensors measured at WSU was determined by sealing them in polyethylene bags containing previously characterized soil that was prewet to different matric potentials. The bags of soil were placed in a constant temperature air bath and the temperature was regulated between 0 and 40°C. The samples were held at selected temperatures to equilibrate for approximately 12 h before readings were taken. Temperature response of the sensors measured at the USGS laboratory was determined by sealing them in a depressurized pressure chamber placed in a temperature controlled environmental chamber regulated between 10 and 40°C. (A water bath was used for the saturated sensors). The slope of a linear fit between the measured T over a range of specified initial temperatures T_i at each matric potential is used in the temperature correction procedure described later.

Thermal Conductivity Model
The response of the ceramic thermal conductivity to changes in water content and temperature was simulated using a thermal conductivity model similar to that described in Campbell et al. (1994). The Campbell model is a dielectric mixing model similar to that developed by de Vries (1963) and provides a theoretical basis for the need for temperature correction. The thermal conductivity model requires a somewhat arbitrary choice (at least for a sintered ceramic) of shape factors (e.g., the shape of the grains) and continuous phase (e.g., continuity of the gas or liquid phase, see Eq. [2] and [3] in Campbell et al., 1994). A simpler dielectric mixing model was developed for our calculations. The model for composite thermal conductivity is:

[3]

where x_g, x_w, and x_c are the gas, water, and ceramic volume fractions respectively, k_g, k_w, and k_c are the gas, water, and ceramic thermal conductivities respectively, and is a constant. The thermal conductivity of the gas phase (k_g) is the thermal conductivity of dry air plus a correction to account for latent heat transport, as described by Campbell et al. (1994) and is strongly temperature dependent. The variation of the latent heat term with temperature is the primary factor responsible for temperature dependence of thermal conductivity in soil (Campbell et al. 1994).

The water content of the ceramic for selected matric potentials was measured using disks of the ceramic, 10 cm in diameter and 1.25 cm thick, in a pressure-plate apparatus, equilibrated along with matric potential sensors and then weighed to determine their water content. Thermal conductivity was measured for the matric potential sensors using the same selected matric potentials. These measurements were then used to obtain the thermal conductivity versus water content relation of the ceramic sensor.

	RESULTS AND DISCUSSION

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The results from six CSI model 229-L sensors calibrated by the USGS, and one CSI model 229-L sensor calibrated by CSI and data from several of the early sensors calibrated at WSU are shown on Fig. 1 . Calibrations for the USGS sensors were done at room temperature, between 19 and 22°C. All temperature rise measurements were normalized using Eq. [2]. Detailed information on saturated and dry measurements and calibration temperatures are not available for the WSU and CSI sensors and therefore were not corrected to 20°C. The uncertainty in the saturated and the dry measurements of those sensors resulted in some of the scatter shown in Fig. 1. The data from the WSU sensors are included, however, to show the stability of the ceramic production method over time. In spite of the scatter, the measurements from all the sensors are in reasonably close agreement with each other when normalized using Eq. [2]. This result is similar to that obtained by Reece (1996).

View larger version (19K): [in this window] [in a new window]   Fig. 1. Matric potential data as a function of dimensionless temperature rise for seven Campbell Scientific Inc. (CSI) 229-L sensors and several early heat dissipation sensors. The dashed line is a nonlinear fit to a van Genuchten-type model and the solid line is the same model scaled to -1000 MPa, both described in the text.

[4]

where is matric potential, _o is the air-entry value of the ceramic, T* is the dimensionless temperature rise (Eq. [3]) and n and m are fitting parameters. The line labeled "unscaled model" in Fig. 1 was fit by minimizing the mean absolute deviation of measurements from prediction for the six USGS data sets to the simplified van Genuchten equation. Values of _o, n, and m were sought which minimize

[5]

where _i is the predicted value () using Eq. [4] and _i is the measured value. The values obtained are _o = -0.056 MPa, n = 1.0 and m = 0.45. The mean absolute deviation of measurements from predictions for data between -0.01 and -35 MPa is 22.8%, which is the mean percentage error over the matric potential range specified, and comes directly from the solution of Eq. [5]. A unique calibration equation for each sensor can also be calculated if the specific accuracy for an individual sensor is required.

Accurate saturated and dry readings are critical for producing the normalized calibration. Reece's (1996) analysis showed little sensor-to-sensor variation in the saturated measurement. The dry readings, however, show a considerable variation from sensor to sensor. Similar results were obtained in the USGS laboratory where 250 sensors where calibrated using the standard methods described here. The coefficient of variation was 6.0% for T_w (using vacuum saturation) and 11.6% for T_d, which resulted in coefficients of variation of 20.5, 9.5, and 29.5% for _o, n, and m respectively for individual calibrations. These results show that wet and dry end measurements should be made for every sensor and care should be taken when using a single calibration equation for all probes. To reduce the need for drying each sensor to determine T_d, we equilibrated a matric potential sensor over saturated salt solutions at several humidities at room temperature. Figure 2 is a graph of the ratio of temperature change for dry ceramic matrix (T_d) to measurements of temperature change T in equilibrium with different relative humidities (h). A quadratic fit to the data gives T_d T^-1 = 0.0224h² + 0.0034h + 1, r² = 0.964. A measurement at any humidity can be converted to the dry measurement by multiplying the reading by the appropriate factor, depending on the humidity at which the sensor is equilibrated. This equation is, therefore, used to determine T_d from air-dry probes.

View larger version (12K): [in this window] [in a new window]   Fig. 2. Ratio of temperature change for a dry ceramic matrix (Td) to measurements of temperature change T in equilibrium with different relative humidities (h). The line is a quadratic fit to the data and is described in the text.

Figure 3 shows the results of thermal conductivity measurements of the heat dissipation sensor ceramic as a function of saturation (Eq. [1] was rearranged to solve for k at different T's resulting from different known saturations). Saturation is dimensionless and is calculated as the volumetric water content divided by the porosity of the ceramic. The line produces a good prediction from the thermal conductivity model (Eq. [3]) with the above constraints and = 0.3. The ratio of the saturated to dry conductivity from the model is 4.26 and a typical matric potential sensor is 4.21 (the thermal conductivity model and a single probe are shown in Fig. 3).

View larger version (13K): [in this window] [in a new window]   Fig. 3. Thermal conductivity data from heat dissipation sensor ceramic as a function of saturation of the ceramic. The line is computed from the thermal conductivity model described in the text.

[6]

View larger version (15K): [in this window] [in a new window]   Fig. 4. Slope of T* versus Ti (s*) as a function of T*. Squares are measurements from the Washington State University (WSU) heat dissipation sensors and triangles are measurements from the USGS heat dissipation sensors. The line is the thermal conductivity model response described in the text.

Equation [6] is used in an iterative process to correct matric potential sensor readings to 20°C. The steps involved are as follows:

1. The temperature rise (T) of a single measurement is converted to an initial estimate of dimensionless temperature rise (T*) using Eq. [2] and is defined as T^*₀.
   
2. An initial estimate of s* is obtained by substituting the value of T^*₀ from Step 1 into Eq. [6].
   
3. A new estimate of T* is computed from the equation T* = T^*₀ - s* where T_i is the initial temperature measured before heating and 20 is the reference or calibration temperature (note: if T_i is initially 20°C then there is no correction for T^*₀).
   
4. A new estimate of s* is obtained using the new T* and Eq. [6].
   
5. Steps 3 and 4 are repeated until T* changes <10^-3 (or other specified accuracy).
   
6. T* is converted to matric potential using Eq. [4].
   

The temperature correction procedure corrects the dimensionless temperature rise, and therefore the matric potential, to the value it would have been at 20°C which is the probe calibration temperature. For example, using Eq. [4] and [6] without using the correction method for a soil at -0.6 MPa would erroneously indicate the soil matric potential as -0.76 MPa at 30°C and -0.41 MPa at 10°C and would only provide accurate results when the soil was at 20°C. Because the measurements shown in Fig. 4 are at different water contents, they reflect both changes in thermal conductivity with temperature and changes in the actual sample matric potential with temperature (Campbell and Gardner, 1971). The procedure corrects both the thermal conductivity temperature dependence of the ceramic, and any temperature dependence of the matric potential, to a value at 20°C. Because of the uncertainty in the correct relation between matric potential and temperature (Liu and Dane, 1993), and therefore difficulty in separating the thermal conductivity effects and actual change in matric potential in our data, we applied a method that corrects the matric potential to a standard temperature (20°C). If the relation between matric potential and temperature is known, that relation can be used to convert the matric potential at 20°C to a matric potential at any other temperature. (Note: Calibration equations seldom yield -1000 MPa data at T* of 0.0, even including T_d representing -1000 MPa in the dataset. To more accurately represent matric potential at the dry end, the calibration equation can be scaled to -1000 MPa at 0.0 T* [the solid line labeled scaled model in Fig. 1] as suggested by M. Th. van Genuchten [personal communication, 1998]. The scaled dimensionless temperature rise [T^*_s] is determined as where T^*_-1000 is the value of T* at -1000 MPa [which is usually >0.0]. T^*_s can be substituted into [Eq. 4] once the unscaled model is determined and T* is corrected for temperature. This step is usually done in postprocessing analysis and is not an integral part of the method presented in this paper. The procedure is used to postprocess the field data presented in Fig. 5 and 6 .)

View larger version (44K): [in this window] [in a new window]   Fig. 5. Soil temperature and corresponding matric potential values for a heat dissipation sensor, uncorrected and corrected for temperatures. Also show are uncorrected matric potential values for when soil temperatures were between 19 and 21°C.

View larger version (30K): [in this window] [in a new window]   Fig. 6. Matric potential data for heat dissipations sensors in a desert soil in the northern Mojave Desert, relative to the atmospheric water potential indicating near air-dry soil in the top 20 cm, indicating the need for the extended range in the calibration equation.

To test the range of the calibration equation, four calibrated heat dissipation sensors were installed at four depths in a desert soil in the northern Mojave Desert and monitored for almost 2 yr (Fig. 6). Atmospheric conditions of temperature and relative humidity were collected simultaneously to determine the atmospheric water potential (Fig. 6). During this period there were only four precipitation events that infiltrated deeper than 10 cm. It can be seen that the lack of precipitation for extended periods of time allowed the near-surface soil to reach equilibrium with the atmosphere. The calibration equation used here, which ranges from 0.0 to -1000 MPa and extends beyond the -0.01 to -1.2 MPa range proposed by Reece (1996), captures the full range of matric potential seen under natural conditions in this desert soil.

	CONCLUSIONS

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A normalization of T based on saturated and dry temperature responses appears to reduce the matric potential-temperature rise response of matric potential sensors with a line heat source to a single relationship, using the same heating current and measurement times. This relationship is fit well by a van Genuchten-type equation and requires, at a minimum, measurements of wet (using the two methods presented) and dry T for each sensor. Additional measurements would allow for a separate calibration equation for each sensor and can be used when greater accuracy is required. Sensor calibrations are temperature dependent, and this temperature dependence is described well by a thermal conductivity model which is used to develop a polynomial temperature correction model. This model allows for the correction of measurements at any temperature to values at the calibration temperature (20°C) or other specific temperature.

	ACKNOWLEDGMENTS

 
The authors thank John Hoffmann, Vic Heilweil, and three anonymous reviewers for helpful suggestions on the manuscript. Thanks also to Jim Bilski for data made available and for additional helpful suggestions on the manuscript.

	NOTES

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¹ Use of trade names is for reference only and does not constitue endorsement by the U.S. Government.

Received for publication April 19, 2001.

	REFERENCES

TOP ABSTRACT INTRODUCTION NOTES THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 

• Campbell, G.S., and W.H. Gardner. 1971. Psychrometric measurement of soil water potential: Temperature and bulk density effects. Soil Sci. Soc. Am. Proc. 35:8–12.
• Campbell, G.S., J.D. Jungbauer, Jr., W.R. Bidlake, and R.D. Hungerford. 1994. Predicting the effect of temperature on soil thermal conductivity. Soil Sci. 158:307–313.
• Campbell, G.S., and J.M. Norman. 1997. An introduction to environmental biophysics. Springer–Verlag, New York.
• de Vries, D.A. 1963. Thermal properties of soil. p. 210–235. In Physics of Plant Environments. W.R. van Wijk (ed.) North Holland Pub. Co., Amsterdam, The Netherlands.
• Gee, G.W., M.D. Campbell, G.S. Campbell, and J.H. Campbell. 1992. Rapid measurement of low soil water potentials using a water activity meter. Soil Sci. Soc. Am. J. 56:1068–1070.
• Liu, H.H., and J.H. Dane. 1993. Reconciliation between measured and theoretical temperature effects on soil water retention curves. Soil Sci. Soc. Am. J. 57:1202–1207.[Abstract/Free Full Text]
• Phene, C.J., G.J. Hoffman, and S.L. Rawlins. 1971. Measuring soil matric potential in situ by sensing heat dissipation within a porous body: I. Theory and sensor construction. Soil Sci. Soc. Am Proc. 35:27–33.
• Reece, C.F. 1996. Evaluation of a line heat dissipation sensor for measuring soil matric potential. Soil Sci. Soc. Am. J. 60:1022–1028.[Abstract/Free Full Text]
• Shiozawa, S., and G.S. Campbell. 1990. Soil thermal conductivity. Remote Sens. Rev. 5:301–310.
• van Genuchten, M.Th. 1980. A closed–form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44:892–898.[Abstract/Free Full Text]

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This Article Abstract Figures Only Full Text (PDF) Free Alert me when this article is cited Alert me if a correction is posted Services Similar articles in this journal Similar articles in Web of Science Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via HighWire Citing Articles via Web of Science (15) Citing Articles via Google Scholar Google Scholar Articles by Flint, A. L. Articles by Calissendorff, C. Search for Related Content PubMed Articles by Flint, A. L. Articles by Calissendorff, C. GeoRef GeoRef Citation Agricola Articles by Flint, A. L. Articles by Calissendorff, C. Related Collections Soil Methods/Instrumentation Structure and Properties Soil Physics