Time Domain Reflectometry for Water Content Measurement of Municipal Solid Waste

Apparatus of TDR

The TDR system used in this article include TDR100^® tester, coaxial head, mold probe, and extension rods. Signal processing and acquisition was done with TDR100 tester from Campbell Scientific and its associated software PC-TDR^®. The coaxial head sits on the cylindrical mold probe, which is formed by a cylindrical metal mold as the outer conductor and a metallic rod as the inner conductor. The mold is similar to a standard compaction mold but with a nonmetallic end plate. In order to separate the EM wave reflections at coaxial head and the surface of test sample, four 300 mm long extension rods were used to increase the length of air gap. The schematic diagram of the system is given in Fig. 2. When using travel time method, the coaxial head sits on the mold probe directly. To use surface reflection method, the extension rods are added between coaxial head and mold probe to increase the length of air gap. Figure 3 shows the TDR waveform measured in deionized water with and without the extension rods.

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FIG. 2.

TDR system. 1, coaxial head; 2, studs; 3, stainless ring; 4, extension rods; 5, center rod; 6, metal mold; 7, Delrin as insulating material in coaxial head; 8, Delrin as base of mold; 9, coaxial cable; 10, TDR100 pulse generator; 11, data cable; 12, computer.

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FIG. 3.

TDR waveform measured in deionized water using the probe with extension rods (348 mm air gap, solid line) and without extension rods (48 mm air gap, dashed line). A, reflection at coaxial head and cable interface; B, start of reflection at interface of the air gap and test sample; C, end of the reflection at interface of air gap and test sample; D, first reflection from probe end. Subscript: 0, travel time method without extension rods; 1, surface reflection method with extension rods.

Travel time method for the measurement of apparent dielectric constant

The travel time method measures the propagation velocity of EM wave in medium, which is related to the apparent dielectric constant K_a of medium filled between the inner and outer conductor of the probe, and can be expressed as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align*} { v } = \frac { 2 { L } } { \Delta { t } } = \frac { { c } } { \sqrt { { K } _a } } \tag { 1 } \end{align*} \end{document}

where v is the velocity of EM wave in test medium; L is the effective probe length in test medium, which can be calibrated using deionized water as the test medium; Δt is the two-way travel time along the probe (time from B to D in Fig. 3); and c is the velocity of EM wave in free space (c=2.298×10⁸ m/s). The K_a of test medium is then \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align*} K_a = \left( \frac { c \Delta t } { 2L } \right) ^2 \tag { 2 } \end{align*} \end{document}

Surface reflection method for the measurement of apparent dielectric constant

The surface reflection method makes use of information in the EM wave reflection at the soil surface. The length of a standard probe specified in ASTM D 6780 was extended with four 300-mm extension rods. The relationship between the reflection coefficient at the soil surface and the K_a of the soil was established theoretically by Chen et al. (2009). Therefore, the K_a of the soil can be estimated from the surface reflection coefficients. Results indicate that the dielectric constant can be determined with reasonable accuracy with this new method even for soils with high electrical conductivities, whereas the conventional travel time analysis fails due to significant signal attenuation.

Chen et al. (2007) stated that the small Step C in Fig. 3 is the end of the reflection at the soil surface, and the difference of the reflection coefficients between B and C depends on the dielectric constant of the soil in the mold. The apparent dielectric constant obtained by surface reflection method can be expressed as (Chen et al., 2009) \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align*} K_a = k^2 \left( \frac { \psi + \Delta \rho } { \psi - \Delta \rho } \right) ^2 \tag { 3 } \end{align*} \end{document}

where Δρ=ρ_tII−ρ_tIII, which depends on the dielectric constant of the material in the mold. ρ_tII and ρ_tIII are reflection coefficients at point B and C, respectively. k and ψ are probe constants associated with the probe's material and geometry, which can be obtained by calibration experiment.

Unlike soils, the electrical conductivity of waste materials in landfills is extremely high. Equation (3) provides satisfactory accuracy in measuring water content of soils when E_c<500 mS/m. For MSW, Equation (3) should be revised, taking into account the considerably high E_c (2000 mS/m or more) of leachate commonly encountered in landfills. To investigate the effect of the pore fluid's extremely high conductivity on Δρ, solutions with different calcium chloride (CaCl₂) concentrations (E_c range from 16 to 2000 mS/m) were measured with the surface reflection method (see Fig. 4). The electrical conductivities of the solutions were measured using an E_c tester. As the E_c of the solution increases, the Δρ=ρ_tII−ρ_tIII also increases. The change of Δρ (defined as Δρ_ec) can be fitted by

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FIG. 4.

Influence of E_c on surface reflection of water.

Then, Equation (3) can be rewritten for the influence of E_c on Δρ, which leads to \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align*} K_a = k^2 \left( \frac { \psi + ( \Delta \rho - \Delta \rho_ { ec } ) } { \psi - ( \Delta \rho - \Delta \rho_ { ec } ) } \right) ^2 \tag { 5 } \end{align*} \end{document}

Relationship between apparent dielectric constant and volumetric water content

The apparent dielectric constant of test sample can be uniquely correlated with the volumetric water content because of the significant differences between the dielectric properties of water and other materials (see Table 1). The dielectric constants of waste materials are within the same range as soils (e.g., 3.0 for dry paper, 2.1–2.6 for plastic, 2.3–4.0 for rubber, and 2.0–4.0 for dry wood). The relationship between K_a and θ of soils has been studied by many researchers. Whalley (1993) suggests that K_a is related to the dielectric constants of air (K_o), water (K_w), and solid matter (K_s) by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align*} \sqrt{{ K}_a} = { f }_o \sqrt{{ K}_o} + { f }_w \sqrt{{ K}_w} + { f }_s \sqrt{{ K}_s} \tag{6} \end{align*} \end{document}

where f_o, f_w, and f_s are the volume fractions of air, water, and solid matter, respectively. Rearranging Equation (6), substituting test sample properties for f_o, f_w, and f_s yields: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align*} \sqrt{{ K}_a} = ( { n} - \theta ) \sqrt{{ K}_o} + \theta \sqrt{{ K}_w} + ( 1 - { n} ) \sqrt{{ K}_s} \tag{7} \end{align*} \end{document}

where n is the porosity of test sample. Topp et al. (1980) proposed an empirical polynomial equation to correlate K_a with θ: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align*} \begin{split}\theta & = 4.3 \times 10^{ - 6} {K_a}^3 - 5.5 \times 10^{ - 4} {K_a}^2 \\& + 2.92 \times 10^{ - 2} K_a - 5.3 \times 10^{ - 2} \end{split} \tag{8} \end{align*} \end{document}

Equation (8) has been confirmed by a variety of researchers and is quite broadly applicable for soils (Topp and Reynolds, 1998). However, the application of this model is limited when applying it to MSW. Then, different calibration equations have been proposed for converting measured K_a values into θ for MSW (Li and Zeiss, 2001; Masbruch and Ferré, 2003).

MSW and its components

The compositions of MSW in urban China are given in Table 2, which indicate that the main components are putrescible organics of animal or plant origin, paper, plastic, and soil. In this study, experiments were conducted on paper, plastic, and organics such as leaves and watermelon peel. Experiments of soils by surface reflection method had been carried out by Chen et al. (2009). Furthermore, the EM properties of mixtures of waste materials were also studied.

The waste materials were shredded before the test, and the densities of paper, plastic, leaves, and watermelon peel samples were 0.17, 0.12, 0.27, and 0.59 g/cm³, respectively. The corresponding total porosities were 0.78, 0.87, 0.94, and 0.95. Then, on the basis of the representative composition in Table 2, MSW samples were prepared in the laboratory for an experiment with the porosity of 0.90 (see Table 3).

The pore water is also an important component that should be considered. In landfill, the E_c of leachate can reach 2000 mS/m or more, which is much higher than the E_c of pore water generally encountered in common soils. The influences of high E_c should be accounted for in the determination of water content with TDR. The dielectric behavior of the leachate is very close to that of salt water, despite the former's suspended solids, chemical composition, and biological constitution (Staub et al., 2010). Therefore, salt water was preferred for all the tests with MSW to simulate a highly conductive leachate. For each type of waste material, a highly conductive fluid (CaCl₂ solution, E_c= 2000 ms/m) and deionized water (E_c=16 ms/m) were used as pore water, respectively.

Experiment methods

Before the test, each waste material was air dried except organics. To investigate the relationship between K_a and θ, predetermined volumes of CaCl₂ solution or deionized water was sprinkled on waste materials, which were then mixed well to ensure a homogeneous distribution of water. Then, the mixture was sealed in a plastic bag for 24 h. Afterward, the sample was packed into the mold with a metallic rod located at the middle of the mold. For each type of waste material, samples infiltrated with deionized water were measured using the coaxial head without the extension rods, and those infiltrated with salt water were measured using the coaxial head with the extension rods. Finally, water content of the test sample was determined by the oven-dry method.

When the pore water is deionized water, the relationship between K_a and θ of test sample can be studied by the travel time method. If the pore water is CaCl₂ solution, the sample's electrical conductivity increases with increasing water content, which was also verified by Ferré et al. (1998). The reflection waveform was attenuated rapidly as the sample's E_c increases, which prevents the measurement of a TDR waveform interpretable by travel time analysis. However, the reflection coefficient at the junction of the air gap and the test sample could still be determined clearly by surface reflection method under high electrical conductivity. Then, the K_a of the test sample can be calculated by Equation (5). The category and number of test samples are shown in Table 4.

K_a-θ relationships for paper, plastic, and organics

The results of volumetric water content and measured dielectric constant were plotted for tested materials (see Fig. 5) with deionized water and CaCl₂ solution as the pore water, respectively. Examination of Fig. 5 indicates that for waste materials with different values of E_c, two methods give similar results. Moreover, the surface reflection method still works satisfactorily for situations where the conventional travel time method fails due to weak or nonexistent second reflection. Further, unlike the method that insulates the TDR probes with a dielectric coating, the surface reflection method measures directly the dielectric constant of waste materials, not affected by the dielectric properties of the coating. Besides, the electrical conductivity can still be acquired by surface reflection method.

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FIG. 5.

Measured relationship between dielectric constant and water content for major MSW components. Suffixes: -L, test sample was infiltrated with deionized water (E_c=16 mS/m); -H, test sample was infiltrated with salt water (E_c=2000 mS/m). MSW, municipal solid waste.

For each waste material, the best-fit calibration curve was determined. The calibration coefficients (based on the three-order polynomial equation) and the R² values are shown in Table 5. The overall results show R² values >0.99 for all regressions performed on the calibration experiments, which indicates significant fits for all waste materials. The model proposed by Topp et al. (1980) was also included for comparison purpose. From Fig. 5, apart from water content and porosity, the effect of waste material type (different materials and their combinations) on dielectric constant seems to be negligible, which is consistent with the conclusion proposed by Li and Zeiss (2001).

It also indicates that the curve of Topp's equation is not equally in accordance with our measurements. The porosity of MSW is larger, and the contribution of the solid phase on dielectric constant is less compared with soils. Therefore, the dielectric constant of MSW is smaller than that of soil at the same volumetric water content, which can also be explained by Equation (7). So, the data points of waste materials are located above the curve of Topp's equation. However, with the increasing of volumetric water content (greater than 60%), the sample tends to be saturated, and the proportion of air is reduced significantly. The test results are consistent with the curve of Topp's equation. Therefore, the porosity has influences on measured K_a.

From Fig. 5, the oven-dry experiments showed that the intrinsic water of organics, for example, the leaves or watermelon peel, accounts for 95 percent of the total mass of organics. The K_a of a plant is a mixture of free water, bound water, and plant dry matter (Burke et al., 2005). Therefore, before being mixed with any water, the K_a of leaves or watermelon peel is much higher than paper and plastic. The corresponding initial θ of leaves and peel are 25.1% and 61.6%, respectively. As a result, the TDR technique can measure water content in organics in the absence of significant bound water effects.

Figure 6 shows the square root of the dielectric constant as a function of the volumetric water content for different waste materials. The data of Topp's equation and the linear relationship proposed by Ferré et al. (1996) that are suitable for soils have been added for comparison. For each material, the best-fit calibration linear equation was determined, and the coefficients and R² values (>0.97) are listed in Table 6. A calibration formed using all the data of MSW components gave a global calibration with a slope and intercept of 0.1219 and −0.0974, respectively, with an R² of 0.9872. The slope of the calibration equation is consistent with the value of 0.1181 determined by Ferré et al. (1996), but the intercept is smaller due to the large porosity of waste materials compared with soils. The highly linear relationship suggests that a two-point calibration could be performed using waste samples collected in landfill, which greatly simplifies the calibration procedure than the three-order polynomial equation.

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FIG. 6.

Measured relationship between square root of dielectric constant and water content for major MSW components.

The K_a-θ relationship for MSW

In addition, the relationship between K_a and θ of MSW is shown in Figs. 7 and 8. The results indicate that the dielectric property of MSW is similar to that of each component, and the water content plays a decisive role in the dielectric constant of MSW. The relationship between K_a and θ of MSW can be expressed by Equation (9). According to Fig. 8, it can be concluded that the equation provides good fit (R²=0.9985), and the fitting parameters are consistent with that of the square-root regression in Fig. 6.

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FIG. 7.

Measured relationship between dielectric constant and water content for MSW.

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FIG. 8.

Measured relationship between square root of dielectric constant and water content for MSW.

Comparison with some existing results

The comparison of the test results of this article with existing results is shown in Fig. 9. The local porosity in waste samples varies by the type of material and compaction. The higher the porosity of the sample, the lower the volume of solids per unit of total volume, and, therefore, the less the influence and contribution of the solid phase and its characteristics on the bulk dielectric constant will be resulted (Li and Zeiss, 2001). In dry state, the dielectric constant of MSW sample is close to 1 due to the major contribution of air. However, the dielectric constant of the soil sample is close to 3–5 under the same dry state because of the dominant contribution of soil particle under lower porosity.

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FIG. 9.

Comparison of research results with existing results. Data points M1 and Clay are test results of MSW and clay from this article, respectively. Data points M2–M7 are test results of MSW from Staub et al. (2010). Data points M8–M11 are test results of MSW from Masbruch and Ferré (2003).

It indicates that the results of clay agree well with the curve of Topp's equation. However, all the data of waste materials by surface reflection method, Masbruch and Ferré (2003), and Staub et al. (2010) are almost located above the curve of Topp's equation. The reason is that the porosity of waste materials is larger than that of soil. Parts of the data points of Staub et al. (2010) are located below the curve of Topp's equation.