HOME HELP FEEDBACK SUBSCRIPTIONS ARCHIVE SEARCH TABLE OF CONTENTS		QUICK SEARCH:   [advanced] Author: Keyword(s): Year:  Vol:  Page:

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 Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via HighWire Citing Articles via Google Scholar Google Scholar Articles by Liu, G. Articles by Horton, R. Search for Related Content PubMed Articles by Liu, G. Articles by Horton, R. Agricola Articles by Liu, G. Articles by Horton, R. Related Collections Heat Transport Analytical Solutions

SOIL PHYSICS

Analytical Solution of the Heat Pulse Method in a Parallelepiped Sample Space

^a Laboratory for Plant–Soil Interaction Processes, Ministry of Education, College of Resources and Environment, China Agricultural Univ., No. 2 Yuanmingyuan Xi Lu, Beijing 100094, P.R. China
^b Dep. of Agronomy, Iowa State Univ., Ames, IA 50011

^* Corresponding author (liug{at}cau.edu.cn).

	ABSTRACT

TOP NOTES ABSTRACT INTRODUCTION MATHEMATICAL MODEL RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

 
The heat pulse method enables estimation of the thermal diffusivity (k), volumetric heat capacity (C), and thermal conductivity of soils () and soil water content. In this study, analytical solutions were derived by the method of Green's function for a finite pulse cylinder source in a parallelepiped sample of size a by b by c with two kinds of boundary conditions. One is the zero surface temperature (ZST) boundary condition, and the other is the adiabatic boundary condition (ABC). The proposed solutions may be useful for evaluating the errors of a dual-probe heat pulse (DPHP) system introduced by approximating the finite heater with an infinite line source (ILS) model. Applications of the solution are presented in the context of an air-dried virtual soil sample to demonstrate how different factors (boundary conditions, soil sample size, heater needle length and radius, probe spacing, heating duration, and strength) can affect the error in k and C caused by using an ILS model. For a given parallelepiped (or soil column), the larger the ratio of lengths of the heater probe and the sample, the smaller the boundary influence on the temperature rise at the mid-needle temperature sensing location, and the smaller the errors introduced by using the ILS approximation. For various heating strengths, it was found that the errors in both k and C were relatively constant when all other parameters were fixed. These errors increased monotonically and slowly, however, as heat duration increased.

Abbreviations: ABC, adiabatic boundary condition • DPHP, dual-probe heat pulse • ILS, infinite line source • PVC, polyvinyl chloride • ZST, zero surface temperature

	INTRODUCTION

TOP NOTES ABSTRACT INTRODUCTION MATHEMATICAL MODEL RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

 
The heat pulse method is a promising technology for measuring soil thermal properties, soil water content, soil water flux, and soil temperature. Probes and sensors are relatively inexpensive to build and can be readily automated using data acquisition systems to obtain near-continuous data. Most probes have at least two needles, with one acting as a heater to release a short-duration heat pulse and the other acting as a temperature sensor (Campbell et al., 1991). An important application of the DPHP method is to monitor heat storage in a soil layer (Ham, 2001). Because volumetric heat capacity C is dependent on soil water content, DPHP sensors also can be used to monitor soil water content (Bristow et al., 1993; Tarara and Ham, 1997; Heitman et al., 2003) and plant water use (Song et al., 1998). Based on measuring the convective transport of a heat pulse introduced by a small heater, heat pulse methods can also be used to measure soil water fluxes (Ren et al., 2000; Ochsner et al., 2005).

Campbell et al. (1991) used an instantaneous heated infinite line source (IHILS) model to calculate the induced temperature distribution. The temperature vs. time measurements were used in conjunction with analytical theory to calculate volumetric heat capacity. Kluitenberg et al. (1993, 1995) conducted studies to examine possible errors in the use of the IHILS theory by comparing the IHILS model with three other models that account for the following characteristics of heat pulse probes: finite probe length, cylindrical heater geometry, and short-duration (non-instantaneous) heating. Two advantages make these models popular. One is the good agreement between model predictions and experimental observations, and the other is the relative ease of computation. Philip and Kluitenberg (1999) and Kluitenberg and Philip (1999) did extensive theoretical research on errors of heat pulse probes due to soil heterogeneity across a plane interface. Exact solutions were found for different probe and soil configurations with IHILS.

In spite of these advantages, all of the models use infinite space as an approximation for finite soil sample size. Thus far, others have not discussed the influence of finite soil sample size, nor have they considered different boundary conditions of the soil samples. In this study, we used rectangular parallelepiped container models for several reasons. First, some heat-pulse experiments (Bristow et al., 1993, 1994) were conducted in rectangular polyvinyl chloride (PVC) boxes. Second, the solutions in these cases are easy for computation; sines and cosines rather than more complex special functions such as Bessel functions are involved. Third, it may be possible to determine the influence of sample geometry, sample size, and probe needle spacing on temperature distribution. Such analysis was omitted in the earlier models.

The objective of our research was to derive transient analytical solutions for temperature distribution after a short-duration heat pulse from a finite cylinder source is released in a parallelepiped sample for two kinds of boundary conditions. Our models were then compared with other existing models. Finally, we analyzed the influence of factors such as sample size, probe spacing, and probe length on estimation errors of thermal conductivity, diffusivity, and heat capacity by comparing models for a short-duration ILS (Kluitenberg et al., 1993, 1995) with our newly derived models.

	MATHEMATICAL MODEL

TOP NOTES ABSTRACT INTRODUCTION MATHEMATICAL MODEL RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

 
Heat Conduction Equation
The heat conduction models that we developed assumed homogeneous and isotropic soil and no contact resistance between the soil and the needles. An exact temperature measurement in the sensing needle at distance r from the heater was also assumed (i.e., the needles were assumed to have infinitely small heat capacity and infinitely large thermal conductivity). Furthermore, we assumed that there is no heat exchange between the parallelepiped and the epoxy-filled sensor body. The heat conduction equation for a homogeneous and isotropic solid with thermal diffusivity k (m² s^–1) is

[1]

where T is temperature (K), t is time (s), and ² is the Laplacian (m^–2). The thermal conductivity (W m^–1 K^–1) and the volumetric heat capacity C (J m^–3 K^–1) are related to k by

[2]

The general surface condition usually arising in the mathematical theory of conduction of heat is linear heat transfer at the surface, for which the flux across the surface is proportional to the temperature difference between the surface and the surrounding medium (in most heat-pulse experiments, it is air). This is called the Newton–Richmann relation. The boundary condition is (Carslaw and Jaeger, 1959)

[3]

where /n_i denotes differentiation along the outward-drawn normal, T_s is the wall or boundary temperature, T_f is the characteristic fluid (air surrounding the soil columns) temperature, and h is related to the heat transfer coefficient (W m^–2 K^–1) and by h = /. As h 0 this tends to the ABC, and as h it tends to a prescribed surface temperature condition. It approaches a ZST condition for the special case where the prescribed surface temperature is zero. We studied both ABC and ZST conditions. Equation [3] can be used in calculations of convective (free convective or forced convective) heat transfer, and for solving problems of external heat exchange between a soil column and its surroundings. The heat transfer coefficient depends on the thermal properties of the medium, the hydrodynamic characteristics of its flow, and the hydrodynamic and thermal boundary conditions. Approximate values of are 5 to 37 W m^–2 K^–1 for gases in free convection (Hewitt et al., 1997). For turbulent flow of air, the values can be about 10 times larger.

Pulsed Infinite Line Source in an Infinite Solid
For a pulsed heat source of duration t₀, either of finite length or continuous, cylinder shapes or other shapes, the solution of Eq. [1] can be divided into two parts:

[4]

where r is the radial distance from the heater (m).

The solution for radial conduction of a short-duration heat pulse away from an ILS (Kluitenberg et al., 1993, 1995; Knight and Kluitenberg, 2004) in an infinite solid with zero initial temperature is

[5]

[6]

in which –Ei(–x) is the exponential integral and Q' is the source strength per unit length per unit time (m² K s^–1); Q' is equivalent to q'/C where q' is the quantity of heat liberated per unit length per unit time (J m^–1 s^–1).

Pulsed Finite Line Source in an Infinite Solid
The solution for radial conduction of a short-duration heat pulse in a plane normal to the source midway (z = 0) away from a line source (Kluitenberg et al., 1993) in an infinite solid with zero initial temperature is

[7]

[8]

where erf (x) represents the error function, u = r²/4k(t – t'), and l₀ (m) is the length of the heat source, which is along the z axis with –l₀ < z < l₀.

Pulsed Infinite Cylindrical Source in an Infinite Solid
The solution for radial conduction of a short-duration heat pulse in a plane normal to the source midway (z = 0) away from an infinite cylindrical source of radius r₀ (m) (Kluitenberg et al., 1993) in an infinite solid with zero initial temperature is

[9]

[10]

where I₀(x) represents a modified Bessel function of order zero.

Zero Surface Temperature Rectangular Parallelepiped Pulsed Finite Cylinder Source
The rectangular parallelepiped (0 < x < a, 0 < y < b, 0 < z < c) heat-pulse system consists of two parallel needles. One needle contains a heater and the other needle houses a thermocouple or thermistor (usually positioned halfway along the needle body). The DPHP has a PVC or epoxy body to hold the needles steady (Bristow et al., 1993; Ham and Benson, 2004) and the probe body is outside of the soil (Bristow et al., 1993; Mori et al., 2005). In this study, only the needles were inserted into the parallelepiped soil sample. A schematic of the parallelepiped and the heat-pulse needles used in our model is presented in Fig. 1 ; however, the probe body is not shown in the figure. We made the assumption that the effect of needles inside the PVC mounting (Bristow et al., 1993) or plug (Mori et al., 2005) can be ignored. The (x, y) plane is normal to the cylindrical heater probe, which is located at (x, y) = (a/2, b/2) and parallel with the z axis with a length of c₀ (Fig. 1a). We can also use polar coordinates (r, ) such that the heating probe is at r = 0 (Fig. 1b) and the position of the sensor probe is

[11]

The parallelepiped container is filled with soil.

View larger version (42K): [in this window] [in a new window]   Fig. 1. Schematic of the parallelepiped and the heat-pulse needles used in the model: (a) three-dimensional view [the heater needle is at (x, y) = (a/2, b/2), the sensor is at (x, y) = (a/2 + rcos[], b/2 + rsin[])] and (b) top view in polar coordinate [the heater needle is at (0, 0), the sensor is at (r, )]. For illustration purpose, the radius of heater probe and temperature probe are exaggerated and are not to scale.

[12]

and the boundary condition as

[13]

where denotes the boundaries of the parallelepiped. The temperature distribution in the rectangular parallelepiped with ZST boundary condition and zero initial temperature distribution (T₀ = 0) due to release of a heat pulse from a finite cylindrical source (Appendix A) is

[14]

[15]

where V is the volume of the parallelepiped (a x b x c); l, m, and n are indices of the Fourier series; and c₀ is the heater probe length.

The ZST is defined by Eq. [13], which is a special case of the constant-temperature boundary condition. The constant-temperature condition is a Dirichlet condition or boundary condition of the first kind. Typical soil thermal conductivity is in the range 0.2 1.8 W m^–1 K^–1 (Dane and Topp, 2002), while brass has 61 111 W m^–1 K^–1 and steel has 10 73 W m^–1 K^–1 (Holman, 1981). Compared with soil, saturated or not, the conduction of heat in the walls of brass or steel containers is relatively quick. As a result, brass or steel containers can be approximated as an idealized ZST system when placed in a controlled constant temperature room. Another possible example of ZST is the heat-pulse experiment of Bristow et al. (1993) for measuring changes in soil water content. They used a room fan to blow air across the surface of the soil to enhance the rate of soil drying during the measurements with the heat-pulse probes. In that case, the soil surface that was exposed to the fan lost heat into the atmosphere instantly (h ), which matched the definition of ZST.

Generality is not lost by taking T₀ = 0 in Eq. [12]. The principle of superposition (Carslaw and Jaeger, 1959) permits application of the solutions to cases with an arbitrary initial uniform temperature T₀. In our various solutions, T then becomes T = T – T₀, the temperature rise above the initial T₀. Since in our models the initial temperature was T₀ = 0, the temperature distribution T(x, y, z) was equal to temperature rise T(x, y, z) = T(x, y, z). When the surface temperature varies with time or is equal to a nonzero value, the solution may be deduced by Duhamel's theorem (Carslaw and Jaeger, 1959), from the case in which the boundary temperature is zero.

Adiabatic Rectangular Parallelepiped Pulsed Finite Cylinder Source
For a rectangular parallelepiped with zero initial temperature distribution and boundary condition,

[16]

The temperature distribution due to release of a heat pulse from a finite cylindrical needle parallel with the z axis (Appendix B) is

[17]

[18]

The explicit expressions for symbols such as , , ,, etc., are given in Appendix B. The boundary condition of Eq. [16] is called an adiabatic boundary condition or boundary condition of the second kind (or Neumann condition), and physically corresponds to no net heat flow into or out of the parallelepiped across the boundaries. Examples of processes proceeding under adiabatic conditions are those of a fluid medium in heat-insulated pipes and channels. Typical soil thermal conductivity is in the range 0.2 1.8 W m^–1 K^–1 (Dane and Topp, 2002), while plastic insulation materials have 0.03 W m^–1 K^–1 and PVC has 0.16 W m^–1 K^–1 (Hewitt et al., 1997). The conduction of heat in soil within PVC or plastic insulated containers can be approximated as an idealized adiabatic system.

In reality, most DPHP experiment systems have boundary conditions of neither ZST nor ABC. They actually have boundary conditions between ZST and ABC. The temperature of ABC and ZST boundary conditions give upper and lower bounds, respectively, for estimating the real temperature in experiments.

Model Error
Bristow et al. (1994) showed that differentiating Eq. [6] with respect to time and setting the results equal to zero give the following expression for k:

[19]

where we have inserted t_m for t to indicate that this expression holds at the time of the temperature maximum. For times t > t₀, an expression for C is obtained by rearranging Eq. [6] and then substituting temperature maximum, T_m, for T and t_m for t to yield

[20]

Equation [20] gives C when a value for k is substituted from Eq. [19]. The product of Eq. [19] and [20] yields . The method of Bristow et al. (1994), therefore, relies on the extraction of T_m and t_m from a temperature–time curve. To determine the model error that occurs when using Eq. [5–6] in the heat-pulse method, our newly derived models and the ILS model were compared.

If Eq. [5–6] are poor physical representatives of the experiment, error in the determination of C and k may result because Eq. [5–6] will not accurately describe the relationship between q', r, T_m, and C. The models of a parallelepiped with finite heat source (Eq. [14–15] and [17–18]) can be used to provide an assessment of the error incurred when the theory (Eq. [19–20]) of Bristow et al. (1994) is used to approximate a soil column of finite size.

The general approach that was used to evaluate model error was to generate a temperature–time curve using known thermal properties in Eq. [15] and [18]. The temperature maximum (T_m) and time to the maximum (t_m) of these generated curves were extracted and then substituted into Eq. [19–20] to calculate thermal constants based on the ILS theory of Bristow et al. (1994). Deviation between the original (known) thermal constants and those obtained from Eq. [19–20] yields the error in thermal properties attributable to model error.

To evaluate model error in k, the parameter t₀ was fixed, and Eq. [15] and [18] were solved for t_m with a golden section search technique (Press et al., 1986). Once t_m was obtained, t₀ and t_m were substituted into Eq. [19] to compute , where the hat () notation has been added to indicate that this diffusivity may differ from the fixed value that was introduced into Eq. [15] and [18] to get t_m. Relative error in k then was expressed as

[21]

Similarly, relative error in C was computed from

[22]

Comparisons of Different Models and Example Calculation
Based on Eq. [14–15] and [17–18], we can make comparisons among the various models. The integrals in Eq. [A8] and [B20–B22] are difficult to evaluate and make the calculations a lengthy process. This difficulty suggests the use of an approximation for heaters. Many probes use needles made from 1.27- or 1.65-mm-diameter (16 or 18 American wire gauge) hypodermic needle tubing with 6-mm spacing between the needles (Ham and Benson, 2004). The length of the needles is between 28 and 40 mm (Ham and Benson, 2004; Ren et al., 1999, 2000, 2003). As shown in Fig. 2 for a parallelepiped of size 5 by 5 by 5 cm³ with specific parameter values as given in Table 1 (the parameters of Fig. 2–15 are shown in Table 1), the line source model (Bristow et al., 1994) is not suitable for treating DPHP systems with a large ratio of r₀/r and, if used, might lead to significant errors. In the case of r₀ = 0.75 mm and r = 2 mm, the errors in C and k will be 36 and –9%, respectively, if the heater probe is treated as a line source of zero radius. When r₀ = 0.75 mm and r = 5 mm, the errors in C and k will be 2.2 and –0.2%. Hence, under the condition that r 5 mm, the influence of the probe needle radius (r₀ 0.75 mm) can be neglected. Most DPHP experimental needle diameters satisfy r₀ 0.5 mm; thus, we can simplify the calculation by setting r₀ = 0, i.e., we approximate the cylindrical source as a finite line source release of heat for the duration t₀ with strength q'. In the limit of r₀ 0, the integrals of Eq. [A8] and [B20–B22] can be simplified remarkably and can be evaluated analytically as given by Eq. [A9] and [B23–B25], respectively. All of the results reported here use the approximation of r₀ = 0. We assumed that the heating needle is at (x, y, z) = (a/2, b/2, 0 < z < c₀), and the coordinate for the point at which temperatures were simulated is (x, y, z) = (a/2 + r, b/2, c₀/2) or (r, ) = (r, 0°). All of the simulation results (except Fig. 7 and 14) use the point (x, y, z) = (a/2 + r, b/2, c₀/2) for the temperature. In Fig. 7 and 14, the z coordinate of the point for temperature simulation will distribute continuously in the range 0 z c. We used a = b = c = 5 cm, c₀ = 4 cm, C = 1.188 x 10⁶ J m^–3 K^–1, k = 1.91 x 10^–7 m² s^–1, q' = 16.8229 J m^–1 s^–1, and t₀ = 15 s for default parameters in most of the simulations (see Table 1). The specific values of r, c₀, t₀, q' were chosen to match the thermo-time domain reflectometry probe of Ren et al. (2003). The size of the parallelepiped used for simulation is similar to the size of the cylinder of Ren et al. (2003). The values of C = 1.188 x 10⁶ J m^–3 K^–1 and k = 1.91 x 10^–7 m² s^–1 correspond to air-dried sand (Dane and Topp, 2002). We focused our analysis on conduction heat transfer in soil, so we considered the condition of air-dried sand or soil to avoid soil conditions favorable to convective heat transfer. In moist soils, water movement in response to thermal gradients results in convective heat transfer (Bennacer et al., 2001; King et al., 2005).

View larger version (14K): [in this window] [in a new window]   Fig. 2. (a) Relative errors in volumetric heat capacity (C, filled symbols) and thermal diffusivity (k, hollow symbols) that occur when the model of Bristow et al. (1994) (Eq. [19–20]) is used to compute k and C rather than the model with zero surface temperature (ZST) boundary conditions (Eq. [15]). (b) Enlargement of the highlighted rectangular area of (a). Probe spacings are r = 2, 3, 4, 5, and 6 mm.

View this table: [in this window] [in a new window]   Table 1. Parameters used in the simulation results presented in Fig. 2 through 15.

View larger version (32K): [in this window] [in a new window]   Fig. 3. The relative errors of temperature change in a parallelepiped (5 by 5 by 5 cm3) introduced by approximating the infinite sum with a truncated series (which contains a finite number of terms, Nseries) at time t = (a, b, and c) 60 s and (d, e, and f) 16 s.

View larger version (20K): [in this window] [in a new window]   Fig. 4. Calculated long-time temperature (T) curves (a) at the thermocouple location for a pulsed infinite line source (dash-dot line), a pulsed infinite cylindrical source (long dash line), a pulsed finite line source (short dash line), and a pulsed finite line source in a parallelepiped (5 by 5 by 5 cm3) with zero surface temperature (ZST, solid line) and adiabatic boundary condition (ABC, dotted line); and (b) for a pulsed finite line source in the same parallelepiped.

View larger version (38K): [in this window] [in a new window]   Fig. 5. The relative errors of temperature distribution (Eq. [23]) as a function of angle in a parallelepiped (5 by 5 by 5 cm3) with specified needle spacing for (a, b, and c) an adiabatic boundary condition (ABC) and (d, e, and f) a zero surface temperature (ZST) boundary condition with time t = 50, 100, 200, and 300 s and probe spacing r = 6, 10, 15, and 20 mm.

View larger version (51K): [in this window] [in a new window]   Fig. 6. Three-dimensional surface graphs of temperature (T) distribution at time t = 300 s and z = 2 cm for (a1) zero surface temperature (ZST) and (b1) adiabatic boundary conditions (ABC); (a2) and (b2) are the corresponding isothermal plots of the temperature curves of (a1) and (b1), respectively.

View larger version (30K): [in this window] [in a new window]   Fig. 7. The temperature (T) distribution along the direction of the sensing needle in a parallelepiped (5 by 5 by 5 cm3) with time t = 60 s for zero surface temperature (ZST, dotted lines) and adiabatic boundary conditions (ABC, solid lines). The numbers on the curves are the values of (a) needle spacing with fixed probe length (c0 = 4 cm) and (b) probe length with fixed needle spacing (r = 6 mm). The point TILS on the curve denotes the temperature of an infinite line source.

View larger version (39K): [in this window] [in a new window]   Fig. 8. Relative errors in (a) volumetric heat capacity (C) and (b) thermal diffusivity (k) that occur when the model of Bristow et al. (1994) (Eq. [19–20]) is used to compute k and C rather than the model with zero surface temperature (ZST) boundary conditions (Eq. [15]). Relative errors are shown for sample heights of c = 4, 4.5, 5, 6, 8, 10, 12, 15, and 20 cm.

View larger version (35K): [in this window] [in a new window]   Fig. 9. Relative errors in (a) volumetric heat capacity (C) and (b) thermal diffusivity (k) that occur when the model of Bristow et al. (1994) (Eq. [19–20]) is used to compute k and C rather than the model with adiabatic boundary conditions (ABC, Eq. [18]). Relative errors are shown for sample heights of c = 4, 4.5, 5, 6, 8, 10, 12, 15, and 20 cm.

View larger version (18K): [in this window] [in a new window]   Fig. 10. Relative errors in (a) volumetric heat capacity (C) and (b) thermal diffusivity (k) that occur when the model of Bristow et al. (1994) (Eq. [19–20]) is used to compute k and C rather than the model with zero surface temperature (ZST) boundary conditions (Eq. [15]). Relative errors are shown for sample lengths (a) and widths (b) of 25, 28, 31, 35, 40, and 60 mm, probe length (c0) of 4 cm, and sample height (c) of 5 cm.

View larger version (20K): [in this window] [in a new window]   Fig. 11. Relative errors in (a) volumetric heat capacity (C) and (b) thermal diffusivity (k) that occur when the model of Bristow et al. (1994) (Eq. [19–20]) is used to compute k and C rather than the model with adiabatic boundary conditions (ABC, Eq. [18]). Relative errors are shown for sample lengths (a) and widths (b) of 25, 28, 31, 35, 40, and 60 mm, probe length (c0) of 4 cm, and sample height (c) of 5 cm.

View larger version (20K): [in this window] [in a new window]   Fig. 12. Relative errors in (a) volumetric heat capacity (C) and (b) thermal diffusivity (k) that occur when the model of Bristow et al. (1994) (Eq. [19–2]) is used to compute k and C rather than the model with zero surface temperature (ZST) boundary conditions (Eq. [15]). Relative errors are shown for sample heights (c) of 3, 3.2, 3.4, 12, and 21 cm and probe length (c0) of 3 cm.

View larger version (27K): [in this window] [in a new window]   Fig. 13. Relative errors in (a) volumetric heat capacity (C) and (b) thermal diffusivity (k) that occur when the model of Bristow et al. (1994) (Eq. [19] and [20]) is used to compute k and C rather than the model with ABC (Eq. [18]). Relative errors are shown for sample heights (c) of 3, 6, 12, 15, and 21 cm and probe length (c0) of 3 cm.

View larger version (20K): [in this window] [in a new window]   Fig. 14. The temperature (T) distribution along the axis of the sensing needle in a parallelepiped (length a = width b = 5 cm for lines, and a = b = 10 cm for circle symbol C) for (a) adiabatic boundary conditions (ABC) and probe length of 3 cm, and (b) zero surface temperature (ZST) boundary condition and probe length/sample height ratios of 0.1, 0.2, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1 at time t = 60 s. Needle spacing was fixed (r = 6 mm). Arrows indicate the midpoint position of the sensing needle, which is parallel along the z axis.

View larger version (37K): [in this window] [in a new window]   Fig. 15. Relative errors in volumetric heat capacity (C) and thermal diffusivity (k) that occur when the model of Bristow et al. (1994) (Eq. [19–20]) is used to compute k and C rather than the model with zero surface temperature (ZST, Eq. [15]) and adiabatic boundary conditions (ABC, Eq. [18]). Relative errors are shown for different (a) quantities of heat liberated (q') and (b) pulsed heat source duration (t0) and the enlargements of the highlighted rectangular areas of (a) and (b).

	RESULTS AND DISCUSSION

TOP NOTES ABSTRACT INTRODUCTION MATHEMATICAL MODEL RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

 
A graphical examination of the various models is helpful for a comparison of the solutions. The results (Fig. 4 ) show good agreement between the analytical solutions of the various models. This agreement is rather surprising, given the differences between the models. The robustness of the models (Eq. [5–10]) lies in the fact that, in all cases, the actual temperature sensors are positioned at the mid-length of the temperature needle. Models will not make accurate predictions if this condition is not satisfied, and might lead to large errors as shown in the following graphs. This can be further understood by considering the longitudinal temperature distribution of the sensor needle in the parallelepiped (Fig. 7).

When compared with actual temperature curves, the Dirichlet boundary condition (Fig. 4b) leads to underestimation of the overall temperature response, while the Neumann boundary condition leads to overestimation of the overall temperature response. The solution of Eq. [1] with the general boundary condition (Eq. [3]) might provide good agreement with the experimental data; however, the complex expression of the solution requires additional time for computations. Figure 4b also shows that for both ZST and ABC, the temperatures reach the same maximum T_m. On the other hand, the temperature for ZST with t 200 s is smaller than that of ABC. The temperature for ZST will approach zero at large time, and temperature approaches a constant value for ABC.

Relative Error in Temperature
The hypothetical probe used in our model was assumed to be positioned in a parallelepiped. It is thus of some interest to investigate the temperature variation with angle (Fig. 1b) for a fixed probe spacing of r. This was accomplished by evaluating the relative error in temperature T(r, z, ) at a given time t. Deviation between the temperature T(r, z, ) and the arithmetic mean of temperature T(r, z, ) yields the relative error in T(r, z, ), which is expressed as

[23]

Taking advantage of the symmetry of T(r, z, ) (Fig. 1b), we need only model one-quarter of the geometry (0 90°). Figure 5a shows the relative error of T(r, z, ) as a function of . Here we used the same parameters as that of Fig. 4 except r (Table 1). Results are shown for a heating duration of 15 s and for several values of t as well as r. The parallelepiped model was used at time t = 300 s and needle spacing r = 20 mm for both ABC and ZST boundary conditions, which corresponds to a –5 to 5% error in T(r, z, ) across the range 0 90. The corresponding error in T(r, z, ) for t 300 s and r 10 mm is –0.04 to 0.04% across this range. The relatively small errors in T(r, z, ), indicate that for a 5 by 5 by 5 cm³ parallelepiped, the temperature distribution with needle spacings r 10 mm could be assumed as isotropic in the radial coordinate. Note that for larger values of needle spacing r (r 15 mm) and t (t 300 s), however, this assumption can lead to highly inaccurate results (Fig. 5a and 5d). With all other parameters fixed, the larger the value of t, the larger the boundary influence on the temperature, and thus the larger the relative error in T(r, z, ). The same tendency also holds for the cases with different needle spacings. Figure 5 also shows that the torsion or the second curvature for ABC and ZST behaves differently. When the relative error curve of ABC is a right-handed curve, the corresponding curve for ZST will be a left-handed curve. Aside from the difference in torsion, ABC and ZST both have the same order of magnitude of the relative error in T(r, z, ).

Isothermal Plots of Temperature
With the same parameters as Fig. 3 and 4, Fig. 6 shows the parallelepiped (5 by 5 by 5 cm³) isothermal plots of temperature curves for ZST and ABC at time t = 300 s and z = 2 cm. As can be seen in Fig. 6a2 and 6b2, for r 20 mm, the isothermal plots can still be considered as perfect circles, which agrees with the results for the relative error in T(r, z, ). Figures 6a1 and 6a2 also show that, for ZST and ABC, the temperature approaches a maximum as r 0. The temperature for ZST approaches zero as the sensor needle approaches the boundary (r/a 0.5). The temperature for ABC is larger than that for ZST, and it approaches a nonzero small value at the boundary. In all cases, temperatures decrease rapidly as r increases. The ABC provides upper temperature bounds while ZST gives lower temperature bounds.

Probe Spacing and Probe Length
The effects of needle spacing are examined by using r = 2, 4, 6, 8, 10, and 12 mm. Figure 7 shows the variation with z of temperature distribution T(r, z, ) in a parallelepiped (5 by 5 by 5 cm³) at time t = 60 s when -dependent heterogeneity is ignored (Fig. 5). Figure 7a shows that the T(r, z) curves of ABC are sigmoidal and are almost constant for 0 z 3 cm. On the other hand, the temperature increase of ZST may reach zero for small values of z. The accumulation of heat near the surface for the ABC system and the loss of heat on the surface for the ZST system cause this difference. In all cases, the magnitude of the temperature increase is always less for ZST than for ABC. The temperature increases slow rapidly for z 3 cm and approach zero for z 5 cm. This is the result of c₀ < c, i.e., the T(r, z) for z c₀ was caused by longitudinal heat flow. For r = 2, 4, 6, 8, 10, and 12 mm, the temperature increases within the region of 1.5 cm z 3 cm are about 1.5, 1.1, 0.6, 0.3, 0.2, and 0.1 K, respectively. The rapid decrease of T(r, z) with increasing r is consistent with the results of Fig. 6.

Figure 7b shows the variation of the temperature increase T(r, z) with z, for corresponding heater probe lengths of 1, 1.5, 2, 2.5, 3, 3.5, 4, and 4.5 cm. Shortening the heater probes from 4.5 to 1 cm caused a significant deviation of temperature increase from the ILS model for ZST in the region with small z position (0 z 1 cm); the same tendency was not obvious for ABC. In other words, the temperature increase near the ZST boundary will deviate from the ILS more than will that of ABC. The fact that heat transfer near the top and the bottom of the parallelepiped is different than that of an infinite line source (T_ILS) is not accounted for in Eq. [5–6] and [9–10]. Heat transfer in either finite soil columns or our parallelepiped is also different from that of a finite source model in an infinite space (Eq. [7–8]), when the boundary conditions (both ABC and ZST) are ignored. For a given parallelepiped or soil column, with all other parameters (, C, k, a, b, c, q', and t₀) fixed, the longer the heater probe, the smaller the boundary influence on the temperature increase, and thus, the smaller the deviations of T(r, z) from linear infinite source predictions. Figure 7 confirms the report that the tip of the temperature-sensing needle does not rise to the same temperature as the rest of the needle after a heat pulse (Ham and Benson, 2004). Although a shorter temperature needle is less likely to deflect during insertion into the soil, Fig. 7b illustrates that the location of the temperature sensor should be far enough away from the surface (z = 0 or c) to warrant the relationship T(r, z) = T_ILS. For the case that we studied in Fig. 7, the temperature sensor should be put at least 1.5 cm away from ZST surfaces. For ABC, the sensor should not be placed too near to the surface z = c. By having the sensor away from the surfaces, deviations from the ILS model can be ignored. Similar to the analysis we presented here, with the help of the solutions that we derived in Eq. [14–15] and [17–18], the best location to measure temperature can be determined if thermal properties, probe geometry, and sample size are known. This is useful for designing a DPHP experimental system that can use the ILS model for good approximations.

The theory presented here permits an evaluation of the potential sources of error in determining C and k by the method of Bristow et al. (1994). They used the theory for the radial conduction of heat away from an ILS following a period of constant heat input. This theory is adequate if the actual heat source of finite length approximates an infinite source, and if the finite soil column with different boundary conditions approximates an infinite sample. Below, we will examine the validity of these two approximations by separately assessing the effect of finite heater length and finite sample size with different boundary conditions and then assessing how these effects combine.

Error Due to Different Probe Length/Sample Height Ratios
In the limit of c₀ and c , the heater will approach the ILS model. Since both c₀ and c are finite in reality, the effect of finite probe length, i.e., the deviation from ILS, can be examined by fixing the probe spacing (r = 6 mm), sample length (a = 5 cm) and width (b = 5 cm), and k and C of assumed air-dried sand, while varying the value of c₀/c for different sample heights c. In this case, Eq. [19–22] yield the curves shown in Fig. 8 (ZST) and Fig. 9 (ABC). Figures 8 and 9 show the relative errors in k and C that result from using Eq. [5–6] to compute k and C rather than the model for pulsed heating from a line heat source of finite length in a parallelepiped. Results are shown for samples with heights of c = 4, 4.5, 5, 6, 8, 10, 12, 15, and 20 cm and heating durations of 15 s. The reported values of c range from 3.9 to 27 cm. Although model error is shown for 0.05 < c₀/c < 1, values of c₀/c usually range from 0.1 to 0.8. Relative errors in k and C are always positive and, therefore, indicate that treating the finite probe as an infinite source consistently leads to overestimation of k, C, and . Figures 8 and 9 also show that relative errors in k and C decline with increasing c₀/c. Errors become <<1% in the limit as c₀/c approaches 1, even though Eq. [14–15] and [17–18] do not reduce to the ILS model. Relative errors in k and C also decline as the height (c) increases (Fig. 8 and 9). These results show that model error can be controlled by using a heater with relatively large c₀/c. Meanwhile, the model error for samples with a fixed value of c₀/c can also be controlled by increasing the height of the sample.

The heat-pulse device used by Bristow et al. (1993) consists of three-needle probes mounted in parallel to provide a heater, sensor, and reference probe. The needles were made from thin stainless steel tubing, 0.813 mm in diameter, which protrude 28 mm (c₀) beyond the edge of the PVC mounting and the probe spacing r is 6 mm. The PVC box they used had a size of 10.2 by 10.2 by 17.6 cm, i.e., a = b = 10.2 cm and c = 17.6 cm, and the value of c₀/c = 0.159. For this condition, we find that the corresponding model error in C is from 0.02 to 0.69% (Fig. 8a) and from 0.20 to 3.02% in k (Fig. 8b) for the ZST boundary condition. Here we assume that the soil sample of Bristow et al. (1993) had the same thermal properties (C and k) and heating parameters (q' and t₀) as we used for the simulation, and we ignore the difference between the lengths and widths used by Bristow et al. (1993) and those of our calculation (10.2 vs. 5 cm). The influence of lengths and widths will be discussed below. Bristow et al. (1993) used a heating duration of 8 s, while our simulation used 15 s. Below (Fig. 15), we report that this heating time difference (8 vs. 15 s) as well as the use of different q' does not alter the results considerably. The relatively small errors in k and C as shown in Fig. 8 indicate that the ILS theory (Eq. [5–6]) used by Bristow et al. (1994) was a good approximation for the particular probe and PVC geometry that they used under ZST boundary conditions. Note, however, that the corresponding model error is about 10.5% (Fig. 9a) in C and from 4.1 to 4.6% in k (Fig. 9b) under ABC. The ILS method can lead to overestimation of both k and C compared with ABC (Fig. 9). By assuming that the difference in temperature–time curves between soil columns with rectangular cross-section and those with circular cross-section is negligible, we can evaluate model error results for the soil column (5.2-cm i.d. and 6.0 cm in height) that was used by Ren et al. (2003). The probe they used consisted of three parallel stainless steel needles, 1.3 mm in diameter and 40 mm in length and spaced 6 mm apart. For soil with the same thermal properties as used to produce Fig. 8 and 9, the c₀/c of 0.667 corresponds to 0.01% error in C and 0.1% error in k for ZST boundary conditions and 0.02% error in C and 0.013% error in k for ABC.

Figure 8 shows that c₀/c = 0.1 results in model errors of 7.6% (c = 20 cm) to 100% (c = 4 cm) in k and model errors of 2.6% (c = 20 cm) to 424% (c = 4 cm) in C. The corresponding model error in Fig. 9 is 5.2% (c = 20 cm) to 28.4% (c = 4 cm) in k and 21.3% (c = 20 cm) to 44.8% (c = 4 cm) in C. These larger errors show that the ILS theory (Eq. [5–6]) breaks down for small c₀/c. The relatively large errors in Fig. 8 (especially for small c₀/c values) compared with those in Fig. 9 indicate that when c₀/c is small, the ZST boundary condition will have more influence on the evolution of observed temperature than does the ABC. The reason is that the heaters with smaller c₀/c will lose heat more easily through the ZST surface, while there is no heat flux across the ABC boundaries. The humpbacked part of Fig. 9b may result from the difference between heat propagation in radial and z axis directions.

Just as the temperatures of ABC and ZST boundary conditions give upper and lower bounds, respectively, for estimating actual temperature in experiments, the error of ABC and ZST give upper and lower bounds, respectively, for estimating the error when the model of Bristow et al. (1994) is used to compute k and C.

Error Due to Different Soil Column Geometry
In our model, the soil column size was controlled by three parameters (a, b, and c); if we suppose that a = b, then the effect of different sample sizes can be examined by varying one geometry parameter (sample height c, or the length and width of the soil samples) and keeping the others fixed. We first examine the effect of finite length and width by varying a (or b) of the soil samples and choose c = 5 cm and c₀ = 40 mm. In this case, evaluating Eq. [19–22] yields the curves shown in Fig. 10 (ZST) and 11 (ABC). Results are shown for samples with lengths and widths of a = b = 25, 28, 31, 35, 40, and 60 mm and heating durations of 15 s. Because of the characteristics of ZST and ABC, these curves do not represent actual errors in most experiments but estimated error bounds. Thus, Fig. 10 and 11 show bounds on the relative errors in k and C that result from using Eq. [5–6] to compute k and C rather than the model for pulsed heating from a line heat source of finite length in a parallelepiped. For a heat pulse experiment with more general boundary conditions (Eq. [3]), the corresponding model error will be between the curves of Fig. 10 and 11 that share the same probe and geometry parameters. As before (Fig. 8 and 9), for ZST conditions, the relative errors in k and C are always positive (Fig. 10) and indicate overestimation of k and C. For the case of ABC, however, the relative errors in k and C (Fig. 11) are negative for soil samples with small cross-section (a = b = 25, 28, and 31 mm) and for relatively large probe spacing (r 7.5 mm). Therefore, treating the probes in smaller soil samples of ABC as ILS might lead to underestimation of k and C. This can be understood by the fact that heat will accumulate near the surfaces of the soil samples, which leads to higher temperature there, causing the negative parts in Fig. 11. In contrast, the heat that reaches the ZST surfaces is removed to keep the temperature constant. Therefore, the values for ZST are always positive.

To examine the effects of sample heights on model errors of k and C, we fixed the sample length and width (a = b = 5 cm) and kept c as a changeable variable. In this case, evaluating Eq. [19–22] yields the curves shown in Fig. 12 (ZST) and 13 (ABC). Results are shown for sample heights of c = 30, 32, 34, 120, and 210 mm and a heating duration of 15 s. The relative errors in k and C are always positive, which indicates parameter overestimation. As sample heights increase, errors decrease rapidly at first, then they seem to become stable for ZST but they continue to decrease for ABC. This distinct deviation between the two kinds of boundary conditions stems from the fact that the ZST boundary will cause heat loss on the surfaces. This can be further understood from Fig. 14.

Like Fig. 7, Fig. 14 shows the variation with z of temperature distribution T(r, z, ) in the parallelepiped. Figure 14 differs from Fig. 7, however, because both r and c₀ are fixed, while c is changeable. As can be seen in Fig. 14a, the temperature distribution along the z axis changes continuously with varying c values (i.e., different c₀/c) for ABC, and will approach a limit when c₀/c 0.7 for ZST (Fig. 14b). For a given c₀, the influence of ABC on the temperature distribution along the z axis changes with c values, and there is no tendency to become stable. In contrast to ABC, the influence of ZST boundaries on the temperature distribution along the z axis becomes stable with increasing c. The arrow in the figure indicates the position of the midpoint of a heater with c₀ = 3 cm. Figure 14 illustrates that, for ABC with a fixed c₀, a larger c (a taller sample) leads to significant deviations from ILS model predictions. Increasing the height of a ZST sample does not increase deviation from the ILS model. This is consistent with the results shown in Fig. 12 and 13 . For an ABC, the boundary influence can be decreased or eliminated by increasing the values of a and b, as shown in Fig. 14a when a = b = 10 cm.

Error Due to Different Probe Spacings
Figures 10 through 13 indicate the magnitude of relative errors in k and C variations with probe spacing r. In these four figures, the error decreases rapidly and becomes <<1% as r approaches 3 mm for both ZST and ABC; this is because when c₀/r , Eq. [14–15] and [17–18] approximate the model for an ILS. The tendency is not the same when r is increasing. In Fig. 10, 12, and 13a, the model error increases monotonically with r. For ABC, the errors in k and C first increase as r increases, but for smaller sample sizes the errors reverse their sign because of heat accumulation (Fig. 11). The accumulation of heat and competition between radial and z axis heat fluxes contribute to the non-monotonic curves shown in Fig. 13b. For ZST systems with relatively short heater probes, no matter how tall the samples, there are noticeable errors in k and C (Fig. 12). This is probably due to the heat loss at the surfaces, especially the surface adjacent to the heater. Some effect can be seen on temperature in the sensing needle (Fig. 14b).

Error Due to Different Heating Period and Heating Power Strength
So far our analyses have been with t₀ = 15 s and q' = 16.823 J m^–1 s^–1. Now we examine the effects of t₀ and q' on model errors of k and C. To do this, we fix the sample geometry size (a = b = c = 5 cm) as well as k and C and vary t₀ or q'. Two heating needle lengths, c₀ = 2.8 cm (Bristow et al., 1994) and c₀ = 4 cm (Ren et al., 2000), are considered. The results in Fig. 15a indicate that solely changing the heating strength q' from 1 to 50 J m^–1 s^–1 does not cause any noticeable model error variation for either ABC or ZST. Although these results were obtained for specific parameters (C, k, a, b, c, and t₀), it is reasonable to deduce that model error for C and k is constant if only q' changes. This is associated with the principle of superposition (Carslaw and Jaeger, 1959) for linear differential equations. Once again we find that the shorter heater needle (c₀ = 2.8 cm) has a relatively large error compared with the longer needle (c₀ = 4 cm). When varying only the parameter t₀, i.e., changing the heating period from 1 to 50 s, the model error in both C and k with the shorter heater needle (c₀ = 2.8 cm) increases monotonously and nonlinearly with t_0. With longer heating periods, more heat energy reaches the boundaries and thus the influence of the boundaries increases.

Compared with the extent (50 times) that t₀ (Fig. 15b) varies, however, the change in model error for C and k is small. The error in C for the 2.8-cm heater needle with t₀ = 8 s is 0.89 (ZST) and 0.30% (ABC); it becomes 0.93% (ZST) and 0.33% (ABC) when t₀ is changed to 15 s. Based on these relationships of error in C and k as functions of q' and t₀, our previous analyses from Fig. 2 to 14 for specific q' and t₀ can be generalized for a reasonable range of values (for example, 1 q' 50 J m^–1 s^–1 and 8 t₀ 15 s).

	CONCLUSIONS

TOP NOTES ABSTRACT INTRODUCTION MATHEMATICAL MODEL RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

 
Analytical solutions were derived by the method of Green's function for a finite cylindrical source in a parallelepiped sample of size a by b by c cm³ with either ABC or ZST boundary conditions. These solutions are useful for evaluating the potential error caused by using the ILS model to describe the DPHP system with specific parameters (C, k, q', and t₀) and sample size (a, b, and c).

Application of these solutions are presented in examples to investigate the influence of boundary conditions, various parameters (C, k, a, b, c, c₀, r, q', and t₀), and the location of the point at which the temperature is simulated in a temperature-sensing needle for an assumed air-dried soil sample. In addition, for specific parameters (C, k, q', and t₀) and sample size (a, b, and c), the errors in both C and k that were introduced by approximating the finite linear source with a pulsed ILS were investigated. The temperature of ABC and ZST boundary conditions give upper and lower bounds, respectively, for estimating the real temperature in experiments. For a given parallelepiped (or soil column) with all other parameters (C, k, a, b, c, q', and t₀) fixed, the longer the heater probe, the smaller the boundary influence on the temperature rise at the mid-needle temperature sensing location, and the smaller the errors introduced by approximating the finite linear source with an ILS. The accumulation of heat near the surfaces for the ABC case and the loss of heat at the surfaces for the ZST case caused the magnitudes of the temperature increases to differ. The errors in ABC and ZST boundary conditions gave upper and lower bounds, respectively, for estimating the real error in experiments that use the ILS model as approximation. As the heating strength changed, the error in both k and C was relatively constant for a fixed set of parameters. The error in both C and k slowly increased monotonically when t₀ was increased.

	APPENDIX A

TOP NOTES ABSTRACT INTRODUCTION MATHEMATICAL MODEL RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

 
For the heat conduction problem, Green's function G(x, y, z; x', y', z'; t – ) is defined as the temperature at (x, y, z) at time t due to an instantaneous point source of strength unity generated at the point P(x', y', z') at time . Green's function for a rectangular parallelepiped, 0 < x < a, 0 < y < b, 0 < z < c, with a ZST boundary condition is (Carslaw and Jaeger, 1959, p. 362)

[A1]

where k is the thermal diffusivity (m² s^–1) of the medium between the needles.

If heat is produced in the solid at point (x, y, z) for t > 0 at the rate A(x, y, z, t) per unit time, the temperature distribution in the parallelepiped is

[A2]

where

[A3]

For the case of dual thermal probes with length c₀, if we use a cylindrical source to approximate the heater probe with the following short-duration heat pulse input:

[A4]

where H() is similar to the Heaviside step function. Substituting Eq. [A1] and [A3] into Eq. [A2], we have

[A5]

After evaluating the integral analytically, the temperature distribution for t < t₀ becomes

[A6]

where

[A7]

[A8]

and V is the volume of the parallelepiped. In the limit of r₀ 0, this integral can be simplified as

[A9]

As to t > t₀, using the formula for temperature distribution at (x', y', z') at time t due to the initial distribution f(x, y, z) and the surface temperature (x, y, z, t),

[A10]

where /n_i denotes differentiation along the outward-drawn normal, and [T_P]_t stands for the value of temperature at the point P(x', y', z') at time t (Carslaw and Jaeger, 1959, p. 354). Substituting Eq. [A1] into Eq. [A10], we have the temperature distribution for t > t₀:

[A11]

When we use Eq. [A6] for the expression of the initial distribution f(x, y, z) and finish the integral, finally we obtain the exact temperature distribution:

[A12]

so the solution for temperature distribution will consist of two parts:

[A13]

where

[A14]

[A15]

	APPENDIX B

TOP NOTES ABSTRACT INTRODUCTION MATHEMATICAL MODEL RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

 
For rectangular parallelepiped 0 < x < a, 0 < y < b, 0 < z < c with no flow of heat over the surface, i.e., the ABC, Green's function is

[B1]

where

[B2]

If we define , ß, and as

[B3]

[B4]

[B5]

Green's function can be rewritten as

[B6]

where

[B7]

[B8]

[B9]

[B10]

If heat is produced by a cylindrical source for a duration of t₀ for t > 0 in the rectangular parallelepiped along the z axis at the rate q' at point (x, y, z) per unit time, by substituting Green's function (Eq. [B6]) into Eq. [A2], we have the temperature for t t₀:

[B11]

where

[B12]

[B13]

[B14]

[B15]

[B16]

[B17]

[B18]

[B19]

and

[B20]

[B21]

[B22]

In the limit of r₀ 0, these three integrals will reduce to

[B23]

[B24]

[B25]

Substituting Eq. [B11] into Eq. [A10], we obtain the temperature for t t_0, which is given by the following expression:

[B26]

The final result consists of eight terms:

[B27]

where

[B28]

[B29]

[B30]

[B31]

[B32]

[B33]

[B34]

[B35]

so the solution will consist of the following two parts

[B36]

where

[B37]

[B38]

	ACKNOWLEDGMENTS

 
This work was supported by the program for Changjiang Scholars and Innovative Research Team in University (Project IRT0412) of P.R. China. The financial support of China National Natural Science Foundation (Grant no. G40671085 [G. Liu]) and the U.S. National Science Foundation (Grant no. 0337553 [R. Horton]) are gratefully acknowledged.

	NOTES

TOP NOTES ABSTRACT INTRODUCTION MATHEMATICAL MODEL RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

 
All rights reserved. No part of this periodical may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permission for printing and for reprinting the material contained herein has been obtained by the publisher.

Received for publication November 13, 2006.

	REFERENCES

TOP NOTES ABSTRACT INTRODUCTION MATHEMATICAL MODEL RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

 

• Bennacer, R., H. Beji, F. Oueslati, and A. Belghith. 2001. Multiple natural convection solution in porous media under cross temperature and concentration gradients. Numer. Heat Transfer A 39:553–567.
• Bristow, K.L., G.S. Campbell, and K. Calissendorff. 1993. Test of a heat-pulse probe for measuring changes in soil water content. Soil Sci. Soc. Am. J. 57:930–934.[Web of Science]
• Bristow, K.L., G.J. Kluitenberg, and R. Horton. 1994. Measurement of soil thermal properties with a dual-probe heat-pulse technique. Soil Sci. Soc. Am. J. 58:1288–1294.[Abstract/Free Full Text]
• Campbell, G.S., K. Calissendorff, and J.H. Williams. 1991. Probe for measuring soil specific heat using a heat-pulse method. Soil Sci. Soc. Am. J. 55:291–293.[Abstract/Free Full Text]
• Carslaw, H.S., and J.C. Jaeger. 1959. Conduction of heat in solids. 2nd ed. Clarendon Press, Oxford, UK.
• Dane, J., and C. Topp. 2002. Methods of soil analysis. Part 4. SSSA Book Ser. 5. ASA and SSSA, Madison, WI.
• Ham, J.M. 2001. On the measurement of soil heat flux to improve estimates of energy balance closure. Abstr. B51A-0183. In Fall Meet. 2001 Suppl. Am. Geophys. Union, Washington, DC.
• Ham, J.M., and E.J. Benson. 2004. On the construction and calibration of dual-probe heat capacity sensors. Soil Sci. Soc. Am. J. 68:1185–1190.[Abstract/Free Full Text]
• Heitman, J.L., J.M. Basinger, G.J. Kluitenberg, J.M. Ham, J.M. Frank, and P.L. Barnes. 2003. Field evaluation of the dual-probe heat-pulse method for measuring soil water content. Vadose Zone J. 2:552–560.[Abstract/Free Full Text]
• Hewitt, G.F., G.L. Shires, and Y.V. Polezhaev. 1997. International encyclopedia of heat and mass transfer. CRC Press, Boca Raton, FL.
• Holman, J.P. 1981. Heat transfer. 5th ed. McGraw-Hill, New York.
• King, J.E., I. Preston, and L. Paterson. 2005. Onset of convection in anisotropic porous media subject to a rapid change in boundary conditions. Phys. Fluids 17:084107–084115.[CrossRef]
• Kluitenberg, G.J., K.L. Bristow, and B.S. Das. 1995. Error analysis of the heat pulse method for measuring soil heat capacity, diffusivity, and conductivity. Soil Sci. Soc. Am. J. 59:719–726.[Abstract/Free Full Text]
• Kluitenberg, G.J., J.M. Ham, and K.L. Bristow. 1993. Error analysis of the heat pulse method for measuring soil volumetric heat capacity. Soil Sci. Soc. Am. J. 57:1444–1451.[Abstract/Free Full Text]
• Kluitenberg, G.J., and J.R. Philip. 1999. Dual thermal probes near plane interfaces. Soil Sci. Soc. Am. J. 63:1585–1591.[Abstract/Free Full Text]
• Knight, J.H., and G.J. Kluitenberg. 2004. Simplified computational approach for dual-probe heat-pulse method. Soil Sci. Soc. Am. J. 68:447–449.[Abstract/Free Full Text]
• Mori, Y., J.W. Hopmans, A.P. Mortensen, and G.J. Kluitenberg. 2005. Estimation of vadose zone water flux from multi-functional heat pulse probe measurements. Soil Sci. Soc. Am. J. 69:599–606.[Abstract/Free Full Text]
• Ochsner, T.E., R. Horton, G.J. Kluitenberg, and Q. Wang. 2005. Evaluation of the heat pulse ratio method for measuring soil water flux. Soil Sci. Soc. Am. J. 69:757–765.[Abstract/Free Full Text]
• Philip, J.R., and G.J. Kluitenberg. 1999. Errors of dual thermal probes due to soil heterogeneity across a plane interface. Soil Sci. Soc. Am. J. 63:1579–1585.[Abstract/Free Full Text]
• Press, W.H., B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling. 1986. Numerical recipes. Cambridge Univ. Press, Cambridge, UK.
• Ren, T., G.J. Kluitenberg, and R. Horton. 2000. Determining soil water flux and pore water velocity by a heat pulse technique. Soil Sci. Soc. Am. J. 64:552–560.[Abstract/Free Full Text]
• Ren, T., K. Noborio, and R. Horton. 1999. Measuring soil water content, electrical conductivity, and thermal properties with a thermo-time domain reflectometry probe. Soil Sci. Soc. Am. J. 63:450–457.[Abstract/Free Full Text]
• Ren, T., T.E. Ochsner, R. Horton, and Z. Ju. 2003. Heat-pulse method for soil water content measurement: Influence of the specific heat of the soil solids. Soil Sci. Soc. Am. J. 67:1631–1634.[Abstract/Free Full Text]
• Song, Y., J.M. Ham, M.B. Kirkham, and G.J. Kluitenberg. 1998. Measuring soil water content under turfgrass using the dual-probe heat-pulse technique. J. Am. Soc. Hortic. Sci. 123:937–941.
• Tarara, J.M., and J.M. Ham. 1997. Measuring soil water content in the laboratory and field with dual-probe heat-capacity sensors. Agron. J. 89:535–542.[Abstract/Free Full Text]

This article has been cited by other articles:

				G. Liu, B. Li, T. Ren, R. Horton, and B. C. Si Analytical Solution of Heat Pulse Method in a Parallelepiped Sample Space with Inclined Needles Soil Sci. Soc. Am. J., September 1, 2008; 72(5): 1208 - 1216. [Abstract] [Full Text] [PDF]

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 Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via HighWire Citing Articles via Google Scholar Google Scholar Articles by Liu, G. Articles by Horton, R. Search for Related Content PubMed Articles by Liu, G. Articles by Horton, R. Agricola Articles by Liu, G. Articles by Horton, R. Related Collections Heat Transport Analytical Solutions