**2. Theory and methodology of the thermo-TDR technique**

#### **2.1 Basic principles**

The heat pulse technique can measure soil volumetric heat capacity (*C*), thermal conductivity (λ) and thermal diffusivity (κ) by analyzing the propagation of a heat pulse at a known distance from a line heat source [2, 9]. The TDR method determines the dielectric and electrical conductivity properties by sending an electromagnetic pulse along a metal TDR waveguide embedded in soil [10]. The pulse travel time is related to soil dielectric constant (*K*a), and the attenuation of pulse amplitude is affected by soil σ [11]. The TDR sensor is widely used to measure soil θ from *K*<sup>a</sup> using the equation from [10]. Noborio et al. and Ren et al. noticed the similarities in sensor materials and configurations between the heat pulse and TDR sensors, and integrated the two systems into a single unit, which was named the thermo-TDR sensor [1, 12]. The unified sensor combines the functions of the heat pulse sensor and TDR sensor, which allows thermal and electromagnetic pulses to be applied concurrently into the soil, and soil temperature, water content, thermal properties, and electrical conductivity are then determined simultaneously [1, 13].

#### **2.2 Theories and calculations**

#### *2.2.1 Determination of soil thermal properties using the ILS theories*

Thermo-TDR technique estimates soil thermal properties from the temperature change-by-time data at the sensing probes (heat pulse signals) based on line-source heat transfer models. The most widely known model is based on the infinite line source (ILS) theory considering an instantaneous or pulsed heating scheme, which assumes the heating probe as a line heat source with zero diameter and infinite length [9, 14–16]. For an isothermal and homogeneous soil with a uniform initial temperature distribution, the solution of the Fourier radial equation for heat conduction of a short-duration heat-pulse away from an infinite line source was developed by [17] further analyzed by [15, 16]. The temperature distributions in a cylindrical system are as follows:

$$T(r,t) = \begin{cases} T\_1(r,t); & 0 < t \le t\_0 \\ T\_2(r,t); & t > t\_0 \end{cases} \tag{1}$$

where.

$$T\_1(r,t) = -\frac{q}{4\pi\kappa C} \text{Ei}\left(\frac{-r^2}{4\kappa t}\right) \tag{2}$$

$$T\_2(r,t) = \frac{q}{4\pi\kappa C} \left[ \text{Ei}\left(\frac{-r^2}{4\kappa(t-t\_0)}\right) - \text{Ei}\left(\frac{-r^2}{4\kappa t}\right) \right] \tag{3}$$

in which *T* is the temperature (°C) at a radial distance *r* (m) away from the line heat source and at time *t* (s). *t*<sup>0</sup> is the heat pulse duration (s), �Ei(�*x*) is the exponential integral, κ is soil thermal diffusivity (m<sup>2</sup> s �1 ) and *C* is volumetric heat capacity (MJ m�<sup>3</sup> K�<sup>1</sup> ). Soil thermal conductivity (λ,Wm�<sup>1</sup> K�<sup>1</sup> ) is calculated as the product of κ and *C*. The optimized κ and *C* values are derived by fitting Eq. (3) to the measured heat pulse signals. The variable *q* represents the quantity of heat liberated per unit length per unit time (J m�<sup>1</sup> ), which is calculated from the current (*I*, Amps) applied to the heater wire for a time of *t*0,

*Applications of Thermo-TDR Sensors for Soil Physical Measurements DOI: http://dx.doi.org/10.5772/intechopen.100285*

$$q = I^2 R t\_0 \tag{4}$$

where *R* is the resistance per unit length of the heating wire (Ω m�<sup>1</sup> ). The ILS model is widely used for heat pulse determined soil thermal properties, because of its simple form and computational efficiency.

#### *2.2.2 Determination of soil thermal properties using the CPC solution*

Ignoring the finite heat pulse probe properties (finite radius and finite heat capacity) can be a significant source of error when estimating soil thermal properties with the ILS theory, especially when there is a large contrast between the physical properties of probes and soil [18, 19]. Peng et al. [8] showed that finite probe effects on temperature rise with time curves were most significant in dry soils, and faded with increasing θ; the ILS theory can cause about 6% relative error in dry soil thermal property estimates [20]. Knight et al. proposed a semi-analytical solution of the cylindrical perfect conductors (CPC) theory, accounting for the finite probe radius and finite probe heat capacity [18]. The CPC theory was successfully applied in various studies [19, 21, 22]. This is especially true for the large sensor designs, in which the CPC theory reduces the error due to the finite probe effects. The theories and applications of CPC theory can be found in [20].

**Figure 1** shows typical heat pulse signals (temperature change-by-time data) in two sensing probes of a thermo-TDR measurement on a loamy sand soil with water content of 0.15 m<sup>3</sup> m�<sup>3</sup> . Generally, soil temperature starts to increase when the heat pulse is initiated and then decreases with time after the heat pulse ceases. The heating rate *q* equals 45.43 W m�<sup>1</sup> (with a *R* of 888 Ω m�<sup>1</sup> , and a *t*<sup>0</sup> of 25 s). The CPC solution is applied to fit the measured data with the built-in nonlinear curve fitting functions (*nlinfit*) in MATLAB software (The Math Works Inc., Natick, MA). The estimated *<sup>C</sup>* and <sup>κ</sup> values are 1.59 MJ m�<sup>3</sup> <sup>K</sup>�<sup>1</sup> and 7.48 � <sup>10</sup>�<sup>7</sup> <sup>m</sup><sup>2</sup> <sup>s</sup> �1 , respectively. Multiplying *C* and κ gives a λ value of 1.20 W m�<sup>1</sup> K�<sup>1</sup> .

#### **Figure 1.**

*The temperature change-by-time data (circles) measured by two sensing probes with the large thermo-TDR sensor on a loamy sand soil. The lines represent the nonlinear curve fitting results for the CPC solution to the measured data. The heating duration (t0) and the heating power (q) are listed.*

#### *2.2.3 Determination of soil water content and electrical conductivity*

Soil θ and σ measurements are determined from the TDR waveforms obtained with the reflectometer device. **Figure 2** presents a typical TDR waveform generated with the TDR200 device (Campbell Scientific Inc., Logan, UT). The TDR technique determines *K*<sup>a</sup> from the propagation time of an electromagnetic wave through a TDR wave guide. When an electromagnetic wave along the coaxial cable reaches the probe embedded in the soil, part of the signal is reflected back to the cable tester due to impedance change, which is shown as the first reflection point (*L*1) on the waveform. The remaining signal travels continuously through the wave guide, and the second reflection point (*L*2) is generated when the signal reaches the end of the probe due to impedance mismatch (**Figure 2**). Thus, *K*<sup>a</sup> is calculated from [23],

$$K\_{\mathbf{a}} = \left(\frac{L\_2 \cdot L\_1}{L\_{\mathbf{a}}}\right)^2 \tag{5}$$

where *L*<sup>a</sup> is the apparent probe length (m), which needs to be calibrated before the thermo-TDR measurement.

Typically, *L*<sup>1</sup> and *L*<sup>2</sup> are determined with the tangent line method. For the shortprobe thermo-TDR sensors, the tangent line-second-order bounded mean oscillation model (TL-BMO) method can be used to determine the reflections positions even when multi-reflections occur in short probes [24–26]. Both tangent line and TL-BMO methods are built-in algorithms in the TDR200 reflectometer device for calculating soil *K*a. To estimate soil θ, the Topp et al. equation or a specific calibration of the *K*a-θ relation can be used [10].

The magnitude of soil σ depends on the transmission line impedance *R*total (Ω), which can be calculated from the amplitude of the TDR signal at very long times [27, 28],

$$R\_{\text{total}} = Z\_{\text{c}} \frac{1 + \rho\_{\text{os}}}{1 - \rho\_{\text{os}}} \tag{6}$$

#### **Figure 2.**

*A TDR waveform from the thermo-TDR sensor immersed in distilled water. v0 is the amplitude of the incident voltage waveform generated by cable tester, and v*<sup>∞</sup> *is final voltage amplitude in the transmission line after all multiple reflections have ceased. Part of the waveform framed in gray is used for water content calculation. L1 and L2 are the first and second reflection points on a TDR waveform (from [8]).*

*Applications of Thermo-TDR Sensors for Soil Physical Measurements DOI: http://dx.doi.org/10.5772/intechopen.100285*

where *Z*<sup>c</sup> is the characteristic impedance of the cable (75 Ω); *ρ*<sup>∞</sup> is the voltage reflection coefficient at long times where multiple reflections have ceased with the TDR waveform reaching a stable level, which is defined as,

$$
\rho\_{\infty} = \frac{v\_{\infty} \cdot v\_{0}}{v\_{0}} \tag{7}
$$

where *v*<sup>0</sup> is the amplitude of the incident voltage waveform generated by the cable tester, and *v*<sup>∞</sup> is final voltage amplitude in the transmission line after all multiple reflections have ceased (**Figure 2**).

Following Heimovaara et al. *σ* can be obtained with the following equation [29],

$$
\sigma = \frac{K\_\mathrm{p}}{R\_{\mathrm{total}} - R\_\mathrm{c}} f\_\mathrm{T} \tag{8}
$$

where *<sup>K</sup>*<sup>p</sup> is the cell constant of probe (�8.77 m�<sup>1</sup> ) determined by using the method in [29] with different KCl solutions; *R*<sup>c</sup> is the combined series resistance of the cable, connectors, and cable tester, and *f*<sup>T</sup> is the temperature factor,

$$f\_T = \frac{1}{1 + \delta(T - 25)}\tag{9}$$

in which *δ* is the temperature coefficient of the soil sample (0.0191°C�<sup>1</sup> , [29]), and *T* (°C) is the temperature of soil sample at the measurement time. Previous studies showed that *R*<sup>c</sup> in Eq. (8) was only a small fraction of the *R*total, which could be neglected without serious errors [30]. Wang et al. incorporated a piece-wise model for electrical conductivity calculations into the TL-BMO model for an accurate determination of σ, θ and *K*<sup>a</sup> simultaneously [31]. The corresponding computer program is available at https://github.com/cauwzj.

#### **2.3 Sensor configuration and construction**

The design of the thermo-TDR sensor must meet several criteria to achieve the requirements of line-source heat-pulse theory to measure soil thermal properties and TDR principles to derive soil water content and electrical conductivity [1, 22]. The key parameters are probe diameter (*d*), probe length (*L*) and probe-to-probe spacing (*r*). For the heat pulse measurement, *L*/*d* > 25, *L*/2*r* > 2.2, and *d*/2*r* < 0.13 should be considered to minimize the effects of axial heat flow and finite probe properties on soil thermal property measurements [15, 16, 32]. A *r*/*d* value less than 10 is necessary for reliable TDR data [33].

Various configurations have been proposed for the thermo-TDR sensor. The original sensor design consisted of three parallel probes with 40-mm length, 1.3 mm diameter, and 6-mm probe-to-probe spacing [1] (**Figure 3**). The middle probe acted as a heater that introduced a heat pulse into soil, while the two outer needles acted as the sensing probes that measured the soil temperature at a known distance (e.g., �6 mm) from the heating probe.

Newer versions of thermo-TDR sensor designs, with various probe sizes and configurations (i.e., *L*, *r*, *d*) have been developed to enhance the strength and robustness of the sensor. Liu et al. presented a sensor design to obtain accurate soil thermal properties and ρ<sup>b</sup> values under field conditions, by using large-size probes (45-mm length, 2-mm in diameter, and 8-mm probe-to-probe spacing) and adding pointed tips at the probe ends [34]. A similar design, with pointed tips, 40.5-mm length, 2-mm diameter, and 6-mm probe-to-probe spacing, was used by Yu et al. in

**Figure 3.** *Schematic view of the thermo-TDR sensor configuration in [1]. (Figure originally published in [2]).*

geothermal applications [35]. Wen et al. introduced a thermo-TDR sensor with relatively thin (1.27-mm) and long (60 mm) probes, but was capable of in situ corrections of *r* changes due to probe deflection [36]. A sensor with curved heaters was tested, but it introduced potential errors due to soil compaction caused by the relatively large heaters [37].

The small sensing volume of the Ren et al. sensor design made it suitable for fine-scale measurements, but the short probes somewhat restricted the accuracy of TDR measurements [1, 38]. Recently Peng et al. introduced a large-size thermo-TDR with a probe length of 70 mm, and a probe-to-probe spacing of 10-mm, a diameter of 2.38 mm for the heater probe, and a diameter of 2 mm for the sensing probe (**Figure 4**) [22]. As a result, this sensing volume was three times larger than that of the Ren et al. [1] sensor, and greater accuracy was achieved with TDR θ measurement accuracy due to the reduction of the superimposed reflections. Peng et al. also integrated updated algorithms to determine soil thermal and dielectric properties in order to produce accurate θ, ρ<sup>b</sup> and porosity values [22].

Thermo-TDR sensors are not readily commercially available. One may be able to make special order sensors from some companies, but in most cases the sensors are constructed in soil physics research laboratories. As shown in **Figure 3**, a thermo-TDR sensor usually consists of three probes that house the heating wire and temperature sensors (thermocouples or thermistors), an epoxy base that fixes the probes in place, extension wires for the heater and temperature sensors, and a coaxial cable for TDR measurement. The stainless-steel tubes that serve as housings for heating and sensing probes, can be custom made or produced from hypodermic needles with the specified diameter and length.

The heating probe is constructed by threading an enameled resistance heater wire (e.g., 38-gauge Nichrome 80 Alley), through the heating needle two or four times for a total resistance of about 888 Ω m<sup>1</sup> . The sensing probes are typically constructed by positioning a thermocouple or a thermistor enclosed at the midpoint

#### **Figure 4.**

*Schematic view of the thermo-TDR sensor configuration from [22].*

of each probe (**Figure 3**). More than one thermocouple (Type E, chromelconstantan, 40 American wire gauge [AWG]) can be also used to detect soil temperatures at several locations along the probe to enable in situ corrections of *r* [22, 36]. In probes, the resistance wires and thermocouples are kept in place with high-thermal-conductivity epoxy.

For TDR measurements, a 75-Ω coaxial cable is connected to the sensor by soldering the inner conductor to the central probe and the shield to the outer probes. The thermocouple wires are extended by connecting them to longer extension wires of the same type (e.g., Type E, chromel-constantan, 36 American wire gauge [AWG]). The extension thermocouple and resistance wires are kept within 5 m to avoid signal losses in long wires. Finally, the three probes and wires are kept in place with a mold and casting resin.

**Table 1** lists the key materials and specifications used in [1, 22] for making the thermo-TDR sensors.

#### **2.4 Equipment and sensor operation**

The operation of a thermo-TDR sensor requires a setup to generate the heat pulses, a TDR device that generates a fast-rise-time electromagnetic pulse, samples and digitizes the resulting reflection waveform, and data acquisition and control systems (**Figure 3**). For the TDR part, a coaxial cable tester (e.g., model 1502B, Tektronix Inc., Beaverton, OR) or a TDR200 reflectometer system (Campbell Scientific Inc., Logan, UT) generates the reflection waveform for analysis or storage. Simultaneous and automatic collection of multiple TDR measurements can be achieved with compatible multiplexers connected to a datalogger (e.g., model CR1000x or CR3000, Campbell Scientific Inc., Logan, UT) that retrieves TDR waveforms or dielectric constants for further analysis of θ or σ.

The experiment setup commonly used for a heat pulse measurement, which consists of a datalogger, a circuit, and a DC power (**Figure 5**). The circuit consists of a relay and a 1-Ω precision resistor, which is controlled by the datalogger. A DC power supply or a 12-volt battery applies a constant current for a fixed time to the heater wires to generate the heat pulse. The extension wires of thermocouples/ thermistors are connected to a datalogger for temperature measurements. A switch to control the heat pulse is through a relay embedded in the circuit that can be


#### **Table 1.**

*Materials used for making thermo-TDR sensors.*

**Figure 5.** *Experiment setup for a typical thermo-TDR measurement.*

activated by the datalogger. The resistance wire is heated for a controlled amount of time (typically 8–20 s for small sensors and 15–30 s for large sensors). During the heat pulse process, the current in the heater wire is determined automatically by measuring the voltage drop across a 1-Ω precision resistor which is in series with the heater wire.

Once the measurement is initiated, the current in the resistance wire and soil temperatures of the sensing probes are recorded at a 1-s interval for about 100– 300 s with a datalogger (e.g., model CR1000x or CR3000, Campbell Scientific Inc., Logan, UT). The total measurement time can be set to be longer than 300 s, especially when the background soil temperature varies significantly with time under the field conditions. In this case, a linear temperature correction procedure is

#### *Applications of Thermo-TDR Sensors for Soil Physical Measurements DOI: http://dx.doi.org/10.5772/intechopen.100285*

needed for the soil thermal property calculations [39, 40]. The heating intensity should be carefully controlled to achieve a clear heat pulse signals at the sensing probe and to avoid potential heat induced moisture redistributions at the same time. Normally, the heat pulse duration is set to make sure that the temperature changes at the sensing probes typically fall in the range of 0.5–1.0°C.

The thermo-TDR sensor can be placed horizontally or vertically in a soil profile, depending on the application objectives. Special care is required to avoid needle deflection and to keep good soil-probe contact during installation. It is recommended to install the sensor under moist conditions when probe deflection is less likely to occur [2].

### **2.5 Sensor calibrations**

Accurate information about parameters *r*, *L* and *K*<sup>p</sup> are needed to determine soil thermal properties, water content, and electrical conductivity with the thermo-TDR technique. A 2% change in *r* value can induce 4% error in *C* estimates. The probeto-probe spacing *r* is frequently calibrated in a medium with a known *C* value at room temperature, such as agar-stabilized water (at a concentration of 5 g L�<sup>1</sup> ) with a *C* value equal to that of water (4.18 MJ m�<sup>3</sup> K�<sup>1</sup> , [9]). The *r* value is calculated by nonlinear curve fitting to the measured heat pulse data based on ILS or CPC theory.

Wen et al. designed a probe-spacing-correction thermo-TDR sensor with 6-cm long sensing probe, each enclosed with three thermistors at different distances away from the sensor base [36]. This enabled the calculation of probe deflection angles to estimate actual in situ *r* values by using linear and nonlinear models proposed by [41, 42]. In field applications, Zhang et al. proposed an on-site calibration method that determined the insitu *r* value by using the theoretical *C* values estimated from a one-time ρ<sup>b</sup> and θ calibration using an intact soil core collected near the sensor location [43].

For *K*<sup>a</sup> and θ measurements with the thermo-TDR sensor, the *L*<sup>a</sup> of the sensor is calibrated by analyzing the TDR waveform obtained in distilled water at room temperature, which is calculated as,

$$L\_a = \frac{L\_2 - L\_1}{V\_p \sqrt{K\_w}}\tag{10}$$

where *K*w, apparent dielectric constant of water (80.1 at 20°C, Haynes and Lide, 2010). *V*<sup>p</sup> is a user-selected propagation velocity, which is usually set as 0.99. *L*<sup>1</sup> can be determined by shorting the three needles in air with a razor blade at the needle base [44].

For TDR-σ measurements with the thermo-TDR sensor, *K*<sup>p</sup> of the thermo-TDR sensor can be estimated following the procedures of [29]. The sensor is immersed in KCl solutions with a series of concentrations (e.g., 0.0001, 0.0005, 0.001, 0.005, 0.01, 0.02, 0.1, and 1.0 mol L�<sup>1</sup> ), and the TDR waveforms are collected. The voltage reflection coefficient at long times is determined from the TDR waveforms, from which *R*total is calculated. Meanwhile, the solution σ is measured with a conductivity meter. The *K*<sup>p</sup> value of the improved thermo-TDR sensor and *R*<sup>c</sup> are then estimated by using regression analysis of σ vs. *R*total [1, 8].

## **3. Applications of the thermo-TDR technique**

#### **3.1 Determination of soil thermal property and electrical conductivity curves**

The thermo-TDR technique permits routine measurements of soil thermal properties, water content and electrical conductivity on repacked soil columns and in situ field measurements. **Figure 6** presents the results of soil thermal properties on a repacked sand soil, showing typical trends of *C*, λ, and κ in relation to θ. Generally, *C* is linearly related to θ, while κ and λ vary nonlinearly with θ. Both κ and λ show rapid increases at θ < 0.10 m<sup>3</sup> m<sup>3</sup> , and afterwards λ continuously increases while κ values decrease. These typical trends agree with published soil thermal property datasets, and earlier studies of *C*, λ, and κ models in relation to soil texture, water content, porosities [45–47].

**Figure 7** shows measured apparent σ values for sand wetted by various salt solution concentrations to θ ranging from 0.08 to 0.25 m3 m<sup>3</sup> . It is clear that the increases in salt concentrations lead to significant increases in σ, and σ also increases with θ. Soluble salt ions in soil solution can enhance the electric conductivity of bulk soil. For salt affected soils, the Peng et al. [8] thermo-TDR sensor can measure σ values as large as 22.5 dS m<sup>1</sup> . Thus, important observations of solute, heat and water properties in soil are possible with thermo-TDR sensors.

**Figure 6.**

*Thermo-TDR determined thermal properties of a sand at bulk density of 1.47 Mg m<sup>3</sup> as a function of water content.*

#### **Figure 7.**

*Thermo-TDR measured bulk electrical conductivity of a sand soil as a function of KCl salt concentrations used to wet the soil to four selected water contents.*
