A general in situ probe spacing correction method for dual probe heat pulse sensor
Gang Liua,∗, Minmin Wena, Ruiqi Rena, Bing Si b, Robert Hortonc, Kelin Hua
a Department of Soil and Water, College of Resources and Environment, China Agricultural University, Beijing 100193, PR China
b Department of Soil Science, University of Saskatchewan, Saskatoon, SK, S7N5A8, Canada
c Department of Agronomy, IA State Univ. Ames, IA 50011,USA
The dual probe heat pulse (DPHP) method is gaining popularity for measuring soil thermal properties. However, because of the fact that the DPHP measured heat capacity (c) of soil is hyper-sensitive to probe spacing variation, probe deflection causes large error in the measured c. To deal with probe deflection, recently, Liu et al. (2013) has proposed an in situ probe spacing correction method, but, their method was based on the inline deflection assumption. Wen et al. (2015) found that for non-inline deflected DPHP sensors, the method of Liu et al. (2013) provided poor reduction of the error in c. To cope with the noninline deflection, in this study, we introduce a new DPHP sensor design by using three thermistors within the same temperature probe. A related probe spacing correction method for non-inline deflected probes was also presented. Both numerical simulation and experiment were conducted to test the new model. We define as the inclination angle of the probe deviation from the vertical direction. For experiment of the outward deflection (5.8◦ < <6.8◦), the error in c is in the range from −23% to −76%; after inline in situ correction, the error decreases to the range of 12% to 44%; after using our non-inline in situ correction, the error is between 1% and 8%. Our three-dimensional finite element numerical simulation also demonstrates that, compared with the method of Liu et al. (2013), the new sensor and correcting method can significantly eliminate errors in c caused by probe deflections. The new DPHP sensor design and the non-inline deflection model have the potential to replace the method of Liu et al. (2013), and become the standard method for correcting probe spacing in situ.
1.Introduction
For soil water content (Tarara and Ham, 1999), soil thermal diffusivity (), thermal conductivity (), heat capacity (c) (Bristow et al., 1994), and subsurface soil water evaporation measurements (Heitman et al., 2008), the dual probe heat pulse method (DPHP) (Campbell et al, 1991) is a promising method and is gaining popularity (Bristow, 1998; Mori et al., 2003; Liu et al., 2013). In this method, at least two parallel thin stainless made probes were utilized, with one as the heater and the others surrounding the heater as the temperature probes (Mori et al., 2003). The diameter of the probes is approximately 1.2 mm, the length of the DPHP probes varies from 28 mm to 50 mm, and the probe spacing, i.e., the distance between the heater and the temperature probe, is 6 mm. Traditionally, the probe spacing was calibrated in agar-stabilized water (Campbell et al., 1991) before performing measurements of the thermal properties. The thin probes of a typical DPHP sensor can be easily deflected and result in changed probe spacing (Kluitenberg et al., 2010; Liu et al., 2013). Liu et al. (2008) theoretically showed that for either the heating probe or the temperature probe, one degree of deflection will lead to ≥ 10% error in c. Recently, under the assumption of inline deflection, Liu et al. (2013) proposed a newly designed DPHP sensor that can be used to correct inline deflected probe spacing changes in situ. However, actual probe deflections could not always be approximated as inline deflection, as was demonstrated by classical mechanical engineering research (Beer et al., 2006) and DPHP experiments (Wen et al., 2015). Furthermore, Wen et al. (2015) found that for DPHP sensors with top end deflection, the inline correction method of Liu et al. (2013) provides poor reduction of the error in c. In this paper, we modified the DPHP sensor design of Liu et al. (2013) to enable correction of the probe spacing in situ for both inline-deflected probe sand for non-inline-deflected probes. Both finite element method-based simulation and theoretical anal ysis were conducted to test the new DPHP sensor design and the correcting method.
2.Theory
The following section presents the theoretical background knowledge that we will use in this study. In an infinite, homogeneous, and isotropic medium at a uniform initial temperature, when an infinite line source (ILS) releases a heat pulse of durationt0, the temperature increasemeasured by the temperature probe is
where r (probe spacing) is the distance between the heater probe and the sensor probe, −Ei(-x) is the exponential integral, is the thermal diffusivity (m2s−1), q’ is the heating strength (J m−1s−1), and is the soil bulk density (kg m−3). Bristow et al. (1994) derived the single point method (Young et al., 2008) for calculating the values of and c; this method relies on a single point of the temperature rise versus time curve, i.e., the point Tm(r, tm)
where Tm is the maximum temperature increase, tm is the time when Tm occurs and c = /. In this study, we use both the ILS assumption and the single point method. When there are two thermistors inside the same temperature probe, Liu et al.(2013) demonstrated thatthe deflected probe spacing can be corrected in situ. They defined 1 and 2 as:
For homogeneous samples, the in situ corrected probe spacing measured by thermistor 1 (r1) and thermistor 2 (r2) satisfy
Below are the summarized results of Liu et al. (2013) for calculating the corrected probe spacings.
where l1 and l2 are the distances from thermistor 1 and thermistor 2 to the probe body surface, respectively (Fig. 1). r10 and r20 are the initial distances (no deflection) from thermistor 1 and thermistor 2 to the heating probe, respectively. The displacements of thermistor 1 (r1)and thermistor 2 (r2) satisfy
Here, we introduce a new method for treating the non-inline deflection. According to the study of Wen et al. (2015) and the theory of structural mechanics (Beer et al., 2006), the deflection or deformation of the probes are nonlinear. Wen et al. (2015) found that, because ofthe deviation from the inline assumption, when the probes were deflected on the top end, the use of inline deflection will result in larger errors in c. Therefore, it is necessary to develop the model of Liu et al. (2013) to deal with non-inline deflected probes.We assume thatthemeasured sample ishomogeneous with a constant initial temperature. We also assume that the deflected DPHP probes can be approximated by a DPHP sensor with only the temperature probe being deflected;this assumption was supported by the finding of Liu et al. (2008), whose results showed that if the new probe spacing of the deflected probes is known, then we could estimate c accurately. Once we accepted the second assumption, the ILS model (Eqs. (1)–(3) can be used and can significantly simplify our result. Compared with the ILS model, the results of Liu et al. (2008) are too complicated to be used for in situ probe spacing correction. As a result, in this study, we use the ILS model. Unlike the design of Liu et al. (2013), who used two thermistors inside the same temperature probe for inline deflection, we use one additional thermistor to correct the probe spacing caused by more general non-inline deflection because three points are the minimum requirement for determining a nonlinear curve, whereas two points can determine a line. Similar to Liu et al. (2013),we use i to denote the expression in the curly brackets of Eq. (2).
where tm1, tm2 and tm3 are the time for the first, second and third thermistor to reach a maximum value, respectively. For a homogeneous sample, the thermal diffusivities (1, 2 and 3) measured by the three thermistors inside the same temperature probe should be the same; based on Eq. (2), we have
where ri0 is the initial probe spacing measured by the ith thermistor. Because of the three thermistors inside the same temperature probe, we can use a second-order polynomial instead of a linear relationship (Liuet al., 2013;Wenet al., 2015)to approximate the nonlinear function of ri(z)
where z denotes the direction normal to the base of the probe body (Fig. 1) and a and b are regression coefficients. For a given deflection profile in which both a and b are known, the displacement of the thermistors from their initial position (parallel with the heater probe) can be calculated by substituting z with l1, l2 and l3. l1, l2 and l3 are the three distances from the first, second and third thermistor to the probe body (Fig. 1). Unlike the inline deflections, we do not have to distinguish whether the probes were outwardly deflected or inwardly deflected. When the temperature probe was deflected outwards, ri was positive, and when the temperature probe was outwardly deflected, ri was negative. We can rewrite Eq. (7) in the form of a system of equations in the two variables a and b
where we define l1-l2P as 1, l1 2-l2 2P as 1, r10-r20P as 1, l2-l3P2 as 2, l2 2-l3 2P2 as 2, and r20-r30P2 as 2. When b = 0, the non-inline deflection (nonlinear deflection) reduces to the inline deflection (Liu et al., 2013). Therefore, the method of Liu et al. (2013) is a special case of the solution we derived here. Our method can be applied to both inline deflection and non-inline deflection.
The DPHP sensors we used were similar to those of Wen et al. (2015). Each consisted of one heating probe and one temperature probe held in a PVC plug. The exposed probe length was 5 cm. The probes were constructed from stainless-steel tubing (Small Parts Inc., Miami Lakes, FL) with 1.27-mmo.d. and 0.84-mmi.d. The original effective spacing between the temperature probe and heating probe was 6 mm. The heating wire inside the heating probe was made of nichrome wire (820 m−1). Three thermistors were inserted into the temperature probe(l1 = 15 mm, l2 = 25 mm and l3 = 35 mm). Detailed information about the fabrication of the sensor and the container for holding soil samples can be found in the work of Wen et al. (2015).
Wen et al. (2015) showed that compared with bottom deflections, top deflections result in larger errors in c. Therefore, in this
study, we will consider top deflection only (Fig. 2a). Both of the probes, rather than the temperature probe alone, were subjected to deflection treatments.We used a bamboo stick and heat shrink tubing to achieve the top deflection. Because the experimental results and numerical simulation of Wen et al. (2015) showed that the influence of bamboo stick and shrink tubing are sufficiently small, their existence is negligible.
A CR1000 datalogger (Campbell Scientific, Logan, UT) powered by a 12V battery was used to collect the heat pulse data. In this study, we use t0 = 15 s and qı´ ≈ 40Wm−1for all of the samples. The sampling frequency of the data logger was set to 1 Hz. The experiments were conducted in a laboratory with a temperature of 21 ± 1 ◦ Celsius. For all of the measurements, the background temperature fluctuation was within ± 0.02 ◦ Celsius in a 10-min period.
Before performing the DPHP experiments, we investigated whether or not the new method is better than the method of Liu et al. (2013). For a non-inline deflected DPHP probe, the three-dimensional analytical solution is still unavailable. As an alternative, we used the finite element method (FEM) to simulate the evolution oftemperature field. The three-dimensionaltransient heat conductionmodule ofCOMSOL (COMSOL,Inc.,Burlington,MA) was chosen to evaluate the influence of non-inline probe deflection on the DPHP measurements. A three-dimensional cylinder of 6 cm in diameter and 6 cm in height (Fig. 3) with zero initial temperature was created to model the DPHP system. The stainless steel heater was approximated by a cylindrical surface source with a diameter of 1.2 mm. To reduce the number of elements in our three-dimensional numerical simulation,the epoxy filling,the wire thermistor and the PVC plug for holding the probes were ignored and the temperature probes was assumed made of stainless steel. Liu and Si (2010) implemented the same simplification for numerical simulation. Both the temperature probe and the heating probe were deflected with the same displacement, as shown in Fig. 2a. All of the boundaries were set as adiabatic boundary conditions. The parameters used in the simulations are: t0 = 8 s, qı´ = 80Wm−1, = 1650 kgm−3, c = 750 J kg−1 K−1and k = 0.3Wm−1 K−1. The COMSOL simulated temperature vs. time curves were used to obtain c from the inline deflection model of Liu et al. (2013) (Eqs. (4)–(10)) and our non-inline deflection model (Eqs. (11)–(16)) and (17)).
After the COMSOL simulation, laboratory DPHP experiments were performed on fine sand and silica sand, both of which were oven-dried and packed at constant bulk densities (1.73 g cm−3 for fine sand and 1.60 g cm−3 for silica sand). At 20 ◦C, DSC (differential scanning calorimetry, Model Q2000, TA Instruments, New Castle, DE) was used to measure the heat capacities of silica sand and fine sand, which were 742 J kg−1 K−1 and 751 J kg−1 K−1, respectively. We used these two values as reference values of c. As in the procedure of Wen et al. (2015), in this study, dry fine sand was used to calibrate r10 and r20. Both r10 and r20 were the average of eight repetitions. After calibrating the initial probe spacing, we conducted the DPHP experiments. For each deflection treatment, wemeasured eight times (eight packings). Based on Eqs. (11)–(17), the probe spacings r1, r2 and r3 were corrected in situ. With estimated r1, r2 and r3, a nonlinear fitting method (Liu et al., 2013) was used to calculate c. We defined the error in c (measured by deflected probes) as the deviation of measured c values from the reference values of c. Liu et al. (2013) and Wen et al. (2015) provided additional details regarding the determination of Tm and tm, data processing, and error analysis.
4.Results and Discussion
In Fig. 2a, we digitalized one outward top-deflected DPHP probe; the filled red circles () denote the sampling points ofthe positions. The filled squares () and triangles ( ) represent the displacements of the temperature probe and the heating probe from their initial position (un-deflected parallel probes), respectively. The filled black circles () denote the equivalent displacement of the temperature probe if we assume that the heating probe was not deflected. Fig. 2b demonstrates thatthe displacement oftwo probes can be better modeled by the non-inline deflection model(Eq.(15)). The inline deflection assumption of Liu et al.(2013) results in a poor approximation of the deflection of Fig. 2a.
pproximation of the deflection of Fig. 2a. Next, we used the digitalized probe deflection data to conduct a COMSOL simulation. First, according to the displacements relationship of probes (r(z) ∼ z) in Fig. 2, we created both the deflected heating probe and the deflected temperature probe in the threedimensional transient heat conduction module of COMSOL. Based on the classification of Liu et al. (2013) and Wen et al. (2015), these two probes are outwardly deflected probes. Next, in the draw menu of COMSOL, we copied the temperature probe and rotated it 180 ◦
around the z-direction axis, which passes through the center of the initial heating probe. This newly generated temperature probe and the heating probe form a so-called inward deflected DPHP sensor. In this way, as shown in Fig. 3a,for each COMSOL simulation, we can simulate both the outward and inward deflection simultaneously.
The COMSOL simulated temperature versus time curves ofthree thermistors (l1 = 10 mm, l2 = 20 mm and l3 = 30 mm) were used for in situ probe spacing correction. The corrected probe spacings are presented in Fig. 4. The solid lines were calculated using our new model (Eqs. (11)–(17)). The three other lines were generated from the inline deflection model (Eqs. (4)–(10)) of Liu et al. (2013) by selecting any two of the three thermistors. The good agreement between the real DPHP probes displacements () and the theoretical prediction (solid line) imply that the new model that we introduced in this study has advantages over the model of Liu et al. (2013) for in situ correction of the probe spacing of noninline deflected DPHP probes. Meanwhile, Fig. 4 suggests that the ILS model is a good approximation for non-inline-deflected DPHP probes of Fig. 2a. Therefore, at least for the deflected probes in Fig. 2a (5.8◦ < < 8.5◦), we can assume that the heating probe was not deflected and that the temperature probe was the only probe that was deflected. For DPHP probes with larger deflection angles thanthose inFig. 2a,the ILSmodelmightnot work well.However,in typical DPHP experiments (Wen et al., 2015), the deflection angle of Fig. 2a is uncommonly high (Table.1).We will not expand our analysis here, but will leave the work to clarify the applicability of the ILS model to our future study.
The results of in situ probe spacing correction for COMSOL simulated inwardly and outwardly deflected probes are listed in Table 1. In this study, we will only consider the influence of probe deflection on the DPHP-measured c because Kluitenberg et al. (2010) verified that probe deflection will not cause noticeable error in . We define the error in c as the deviation of c values from the c that we defined in the physical properties of COMSOL.
As seen in Table 1, the error in c for outward deflection without in situ probe spacing correction is between 20% and 122%; after inline in situ correction, the error is between −16% and −28%; after non-inline in situ correction, the error is between −9% and −11%. Compared with our new method, the inline in situ correction method of Liu et al. (2013) underestimates c and yields a poor prediction in c, which agrees with the finding of Wen et al. (2015). On the contrary, the non-inline correction method significantly reduces the errors in c. Regarding the inward deflection,the error in c is in the range of −23% and −76% and shifts to the range of 12% to 44% after inline in situ correction, which is not an obvious improvement. However, after using the non-inline correction method, the error in c is narrowed to between 1% and 8%. Therefore, the new method is superior to the inline correction of Liu et al. (2013) for correcting the probe spacing.
If probes are outwardly deflected, the average values of errors in c for our model, the values of the model of Liu et al. (2013) and the uncorrected values are −10%, −20% and 74%, respectively; for inward deflected probes, the corresponding errors are 4%, 26% and 49%, respectively. The non-inline deflection model works the best among the above-mentioned three data analysis methods. However, unlike inward deflection, outward deflection is prone to large errors in c. The flat region around the peaks (Fig. 5) in the temperature versus time curves of outwardly deflected probes might be the reason. Similarly, Wen et al. (2015) found that the accuracy of tm is worse whentemperatures areflat aroundthepeak. Fig.5 also shows that for inward deflection, the slopes of the curve around the peaks are much steeper than those in the case of outward deflection, thus increasing the accuracy of tm. We may draw the conclusion that if the single point method was used for in situ probe spacing correction, for the same deflection amplitude, the error in c of the outwardly deflected probes is larger than the error in c of the inwardly deflected probes. For future DPHP sensor design, increasing the steepness ofthe temperature versus time curves around the peaks might be one solution to improve the accuracy of the DPHP sensor. Another possible reason for the better performance of the results of inward deflection is that the ILS model is more effective when the probe spacing is small. For a given DPHP sensor with fixed heating probe length, increasing the probe spacing will make the system deviate from the ILS assumption; the larger the probe spacing is, the larger the deviation from the ILS assumption is, and thus, the larger is the error in fitted c.
In Table.1, our FEM simulation shows that the new in situ probe spacing correction method can predict the non-inline deflection of DPHP sensor well.However,inourCOMSOL simulation,the existing of epoxy filling, heating wire and thermistors were all ignored. The existence ofthermal contact resistance was also notincluded in the FEM simulation. To evaluate our new method, we must conduct experiments.
The experimental results of the correction of the in situ probe spacings for the sand and silica under inward deflection and outward deflection are listed in Table.2. It is clear that the trend and amplitude of the error in c for sand and silica are similar. For the DPHP experimental results, we define the error in c as the deviation of c values from the non-deflection probes measured c, which is equal to the DSC-measured c. Table 2 indicates that the error in c for outward deflection without in situ probe spacing correction is between 2.3% and 68.8%; after inline in situ correction, the error is between −34.3% and 5.9%; after non-inline in situ correction, the error is between −1.6% and 10.4%. Similar to our COMSOL simulated results and the findings of Wen et al. (2015), for outward deflection, the inline in situ correction method of Liu et al. (2013) underestimates c. In addition, compared with our new method, the method of Liu et al. (2013) provides poor prediction for c. However, when we use the non-inline correction method, the errors in c were reduced significantly. Regarding the inward deflection, the error in c is between −3.8% and −18% and changes to between 2.6% and 13.6% after inline in situ correction, which is not an obvious improvement. However, after using the non-inline correction method, the error in c is narrowed to the range of 0.6% to 4.5%.
Therefore, as was demonstrated by the results of our COMSOL simulation, the new method is superior to the inline correction of Liu et al. (2013) for correcting the probe spacing.
5.Conclusion
By introducing three thermistors inside a temperature probe and using a new probe spacing correction model, we can correctthe probe spacing of non-inline deflected probes in situ. Both FEM simulations and experiments in the laboratory verified that compared with the inline deflection model of Liu et al. (2013) , the new DPHP sensor and the new probe spacing correction method could signifi- cantly reduce the error in c caused by both inward and outward deflection. This new DPHP design and the non-inline deflection model can replace the inline deflection model of Liu et al. (2013) and has the potential to become the standard method for in situ correction of the probe spacing.
This work was supported by Natural Science Foundation of China (Grant No. 41371231 (G. Liu)). The financial support of National Science andTechnology Support ProgramProjects ofChina (Grant No. 2012BAD05B02) to G. Liu is gratefully acknowledged.