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Published online 29 March 2006
Published in Soil Sci Soc Am J 70:711-717 (2006)
DOI: 10.2136/sssaj2005.0174
© 2006 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
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Soil Physics

Correcting Wall Flow Effect Improves the Heat-Pulse Technique for Determining Water Flux in Saturated Soils

Jianying Gao, Tusheng Ren and Yuanshi Gong*

College of Resource and Environmental Sciences, China Agricultural Univ., Beijing, China 100094

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


   ABSTRACT
TOP
 NOTES
 ABSTRACT
INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Heat-pulse sensors are promising tools that can provide rapid measurements of soil water fluxes. The main objective of this study was to determine whether wall flow could be a main reason for discrepancies between measured and heat-pulse water flux densities. Heat-pulse measurements were obtained with a range of water flux densities imposed on packed soil columns of three saturated media: glass beads, a sandy loam soil, and a sandy clay loam soil. Water flux density was calculated from the thermal properties of the media and the ratio of temperature changes at downstream and upstream positions. A novel finding from this study was that in packed columns, wall flow was responsible for the deviations between water flux estimates from heat-pulse data and water flux measured from outflow, and the magnitude of wall flow was largely determined by soil texture. An amplification factor, 1.12 for the sandy loam and 1.24 for the sandy clay loam, was introduced to correct the influence of wall flow, which reduced the errors of the heat-pulse measurements to within 5%. We demonstrated that wall flow was able to explain quantitatively the "reduced convection" theory proposed by earlier researchers. Under the experimental conditions, heat transfer by dispersion was of minor importance.


   INTRODUCTION
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
SOIL WATER FLUX DENSITY is required for quantifying infiltration, runoff, solute transport, and subsurface hydraulic processes. Conduction and convection heat transfer theory, after step application of power to a line heater, has received considerable attention for measuring soil and ground water flow (Byrne et al., 1967, 1968; Melville et al., 1985; Ballard, 1996). Recently, Ren et al. (2000) used a three-cylinder, heat-pulse probe to determine water flux density on saturated soils. The cylinders were parallel, aligned in a common plane, and embedded in a waterproof epoxy body. A heat pulse was generated by passing electrical current through the central cylinder that contained a resistance heater. Temperature increases upstream and downstream of the heater position were measured by thermocouples in the two outer cylinders. Due to convective heat transfer by the flowing water, the temperature increase of the downstream sensor was larger than that of the upstream sensor. Ren et al. (2000) derived an analytical solution that related soil water flux density to soil thermal properties, the temperature difference between the downstream and upstream positions, and the thermocouple-to-heater spacing. Kluitenberg and Warrick (2001) improved the Ren et al. (2000) solution by converting the equations to well function. A further simplified form was provided by Wang et al. (2002), who established an exponential function that related the ratio of downstream and upstream temperatures to soil water flux density, thermal diffusivity, and the thermocouple-to-heater spacing. Alternatively, Hopmans et al. (2002) applied an inverse technique to determine soil thermal properties and water flux density simultaneously.

A few experimental studies have demonstrated the effectiveness of the heat-pulse technique in determining soil water flux (Ren et al., 2000; Mori et al., 2003, 2005; Ochsner et al., 2005). On fine texture soils, however, substantial errors exist in the measured data, and there is discrepancy in the explanations for the difference between measured and predicted temperature responses to soil water flux. Results from Ren et al. (2000) and Ochsner et al. (2005) showed that, in general, there was good agreement between measured and predicted thermal response on sand soils, but the heat-pulse method underestimated water flux density on fine texture soils. Mori et al. (2003, 2005) showed that the heat-pulse technique was accurate on a Tottori Dune sand in the water flux range between 0.056 and 27.0 m d–1. Theoretical analysis by Hopmans et al. (2002) showed that the underestimation of water flux density in Ren et al. (2000) was a result of ignoring thermal dispersion and the physical size of the heater cylinder. On the other hand, Ochsner et al. (2005) pointed out that inclusion of thermal dispersion could not explain the differences between heat-pulse data and water flux results from outflow measurements. They illustrated that the errors could be explained by reducing the magnitude of the convective heat transfer. Ochsner et al. (2005) also demonstrated that the temperature ratio method introduced by Wang et al. (2002) was superior to the single temperature difference procedure of Ren et al. (2000) in terms of parameter requirement, calculation process, and precision.

In view of the small number of actual measurements, additional information is required to evaluate the heat-pulse method for measuring water flux density on saturated soils. The major objectives of this study were to further investigate the performance of the conduction–convection heat transfer theory in describing the temperature dynamics in determining the water flux density and to explore wall flow as the possible mechanism to the discrepancy between heat-pulse estimated water flux densities and actual measurements.


   MATERIALS AND METHODS
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Thermo-TDR Probe
The thermo-TDR probe developed by Ren et al. (1999) was used in this study. The sensor consists of three parallel stainless-steel cylinders 0.04 m in length and 0.0013 m in diameter, with the two outer cylinders spaced 0.006 m from the central cylinder. The central cylinder contains a resistance line heater (40 AWG, Nichrome 80 with enamel; Pelican Wire Company Inc., Naples, FL), and each outer cylinder encloses a chromel-constantan thermocouple at the midpoint. The resistance of each completed heater is 887.6 {Omega} m–1. The space inside the cylinders is filled with high-thermal-conductivity epoxy glue (Omegabond 101; Omega Engineering, Stanford, CT), which keeps the heater and thermocouples in position and provides a water-resistant and electrically insulated environment.

To determine the apparent distances from the heater to the downstream temperature sensor (xd) and the upstream temperature sensor (xu), heat-pulse measurement was conducted in 5 g L–1 agar-immobilized water. The heat pulse was generated by applying a DC current to the central heater for 15 s with a direct current supply (Model HY1791-3s; Huaiying Electronics Equip. Corp., Huaiying, China). A data logger (Model CR23X; Campbell Scientific, Logan, UT) was used to control the heat input through a relay, and the current was obtained by monitoring voltage drop through a precision resistor (1 {Omega}). The data logger also recorded the upstream and downstream temperatures at 1-s intervals for 300 s. The parameter estimation method of Welch et al. (1996) was applied to estimate the apparent spacing, assuming the heat capacity of the agar gel was 4.1819 MJ m–3 K–1 (Campbell et al., 1991). The xd and xu values are 6.073 mm and 6.140 mm for the probe used on the glass beads, 5.673 mm and 5.842 mm for the probe used on the sandy loam soil, and 6.268 mm and 6.287 mm for the probe used on the sandy clay loam soil, respectively.

Experimental Procedure
Water flow experiments were conducted on packed columns of glass beads, a sandy loam soil, and a sandy clay loam soil in a temperature-regulated room (20 ± 1°C). Table 1 presents the particle size distribution, organic matter content (OM), and bulk density of the test materials. Soil samples were air dried, ground, and sieved to pass a 2-mm screen. Air-dry samples were wetted to a water content of approximately 0.15 kg kg–1 and mixed. The wet soils were packed into Plexiglas pipes 35-cm long and 8.2 cm in diameter. Both ends of the pipes were sealed with Plexiglas lids that were connected to a sleeve attached to the cylinder. To ensure uniform flow across the entire cross-section, a layer (1.5-cm thick) of glass beads and four layers of nylon cloth were placed at both sides of the soil sample. Finally the thermo-TDR probe was installed horizontally at 17.5 cm depth of the column through a pre-drilled hole on the pipe. To avoid the influence of probe body on water flow, only the three cylinders were pushed into the soil. The cylinders formed a vertical plane that was parallel to the flow direction but perpendicular to the top column surface. The empty space between the probe and the pipe was filled with wax.


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  		Table 1. Particle-size distribution, bulk density, and organic matter content of the experimental materials.

 
Soil columns were saturated by introducing water at the column bottoms for 72 h. Soil thermal properties were determined from heat-pulse measurements under no-flow condition (Bristow et al., 1994). A parameter estimation method (Welch et al., 1996) was used to estimate soil thermal diffusivity ({alpha}, m2 s–1), volumetric heat capacity (C, J m–3 K–1), and thermal conductivity ({lambda}, W m–1 K–1) from the temperature-by-time data. Table 2 lists the means and standard deviations of soil thermal properties from seven repeated measurements.


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  		Table 2. Thermal diffusivity ({alpha}), thermal conductivity ({lambda}), and volumetric heat capacity (C) of the saturated soil columns.

 
For the water flow experiment, a range of water flux density was provided by a syringe peristaltic pump (Model DHL-A; Shanghai Electronics Equip. Corp., Shanghai, China). The water flux ranged from 7.89 x 10–8 to 4.40 x 10–5 m s–1 for the glass beads, from 3.58 x 10–7 to 3.31 x 10–5 m s–1 for the sandy loam soil, and from 2.84 x 10–7 to 2.68 x 10–5 m s–1 for the sandy clay loam soil. Water flow direction was from the column bottom to the top. Steady state flow was verified by monitoring effluent volumes with respect to time. When the flux from outflow measurement became constant, a 15-s heat pulse was applied, and temperature-by-time data at upstream and downstream sensors was recorded at 1-s interval for 300 s. The heating power was in the range of 53.7 to 54.9 W m–1.

Data Processing
We used the Wang et al. (2002) model to estimate soil water flux from heat-pulse measurements. In the model, the convective heat-pulse velocity V is related to Td/Tu, the ratio of downstream temperature change to upstream temperature change. Wang et al. (2002) showed that theoretically Td/Tu was time dependent but approached a constant value as t -> {infty}. For large times, V is expressed as a function of Td/Tu:

Formula 1[1]
In this study, we selected the data points between 80 s and 90 s to calculate V. We picked up 80 to 90 s because at this time period, the Td/Tu ratio seemed relatively constant and there was no apparent noise in the data.

Finally, V was converted to soil water flux density based on the following relationship (Marshall, 1958; Melville et al., 1985; Ren et al., 2000):

Formula 2[2]
where Cw is volumetric heat capacity of water (J m–3 K–1), Vw is pore water velocity (m s–1), and J is water flux density (m s–1). The water flux density values reported in the Results and Discussion section represent the means of three replicated measurements.


   RESULTS AND DISCUSSION
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
CONCLUSIONS
 REFERENCES
 
Response of Temperature Change to Water Flux
Figure 1 shows the responses of measured and predicted temperature changes to three water fluxes at upstream and downstream locations. Temperature change at downstream positions rises with increasing water flux, and temperature change at upstream positions declines with increasing water flux. The response of temperature changes to water flux is the largest at the maximum point and becomes less significant at larger times. With smaller C and {alpha} values (Table 2), the glass beads show a relatively large magnitude of temperature increase compared with the sandy loam and sandy clay loam soils.


Figure 1
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  		Fig. 1. Transient temperature changes at the downstream and upstream positions as related to the water flux density. The symbols represent measured data, and the lines indicate predicted results using the conduction–convection model.

 
We estimated temperature change as a function of time using water flux results from outflow measurements and thermal properties determined under no-flow conditions. The results are presented in Fig. 1. In general, the measured and predicted temperature changes agree well at smaller water fluxes. At larger fluxes, the theory overestimated temperature change at the downstream and underestimated temperature change at the upstream. At downstream positions, the maximum temperature change was overestimated by 4.51, 4.76, and 10.02% for glass beads, sandy loam, and sandy clay loam with water flux densities of 3.09 x 10–5, 3.00 x 10–5, and 2.68 x 10–5 m s–1, respectively. At upstream positions, the maximum temperature change was underestimated by 6.85, 8.11, and 8.85% for glass beads, sandy loam, and sandy clay loam, respectively, at the corresponding water flux densities. The discrepancy between model prediction and measurement is relatively small for coarse-textured soils and relatively large for fine-textured soils. These results are in agreement with Ren et al. (2000), who showed that the performance of the conduction–convection heat transfer model was soil texture and water flux dependent.

Estimated Water Flux
A comparison of calculated water flux from Td/Tu versus water flux measurements from outflow (designated as actual water flux density hereafter) is presented in Fig. 2 . The results demonstrate the following characteristics first, a linear relationship exists between estimated water flux density and actual water flux density on all three media. Second, the method performs better on coarse-texture materials than on fine-texture materials, as the slope of the regression line is in the order of glass bead > sand loam > sandy clay loam. The root mean square errors were 9.01 x 10–7, 1.35 x 10–6, and 2.04 x 10–6 m s–1 on the glass beads, sand loam, and sandy clay loam. Third, the model is biased toward underestimating water fluxes, as indicated by the less-than-unity slopes from linear regression analysis (Table 3). Similar observations were recorded by Ren et al. (2000) and Ochsner et al. (2005). Nevertheless, the calculated water flux density is in better agreement with outflow data than the results of Ren et al. (2000) and Ochsner et al. (2005). For example, the slopes of the regression lines are 0.925, 0.871, and 0.787 for the glass beads, sandy loam, and sandy clay loam, respectively. These are greater than the slopes of 0.739, 0.224, and 0.342 reported by Ochsner et al. (2005) for a sand, a sandy loam soil, and a silt loam soil, respectively.


Figure 2
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  		Fig. 2. Water flux density (J) estimated from the Td/Tu versus actual values measured at column outlet.

 

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  		Table 3. Slopes and intercepts from linear regression of water flux estimates using the Td/Tu ratio method versus the actual water flux densities. The numbers are means and standard deviations of three repeated measurements.

 
Explanation of the Discrepancy between Measured and Actual Water Flux
Several theoretical assumptions or experimental constraints may have contributed to the underestimation of water fluxes by the heat-pulse method on the sandy loam and sandy clay loam soils. It is difficult to verify whether local thermal equilibrium between the liquid and solid phases is always maintained or not. It seems that the assumption of an isothermal and homogeneous test medium is reasonable because the experiment was conducted in a temperature-regulated room (20 ± 1°C) and efforts were made to pack the soils uniformly. We consider that ignoring the finite physical size of the thermo-TDR probe by the conduction-convection model is acceptable because Hopmans et al. (2002) have showed that error in water flux estimates from excluding the size of the heater cylinder was within 3%.

Inconsistency exists in the literature regarding the role of heat transfer by thermal dispersion in determining water flux density using the heat-pulse method. Sisodia and Helweg (1998) tested a conduction-convection-dispersion heat transfer model that simulated the temperature field of a "heat sense flowmeter" on a sand soil. At relatively higher flux densities (3.94 x 10–4 to 1.84 x 10–3 m s–1), the predicted temperature response agreed well with the experimental result. Theoretical analysis by Hopmans et al. (2002) also indicated that by including thermal dispersion in the conduction-convection equation, water flow velocities could be determined accurately for water flux densities ranging from 1.16 x 10–5 to 1.16 x 10–4 m s–1 (1.0–10 m d–1). On the other hand, Ochsner et al. (2005) demonstrated that an increasing thermal conduction term (i.e., including thermal dispersion) could not explain the difference between experimental results and water flux estimates from the heat-pulse technique. In this study, we applied the theory of Hopmans et al. (2002) to test whether the contribution of thermal dispersion to heat transfer was significant or not. Instead of using an anisotropic thermal dispersivity (Hopmans et al., 2002), we assumed that thermal dispersion occurred in the flow direction only. The effective thermal conductivity, including the stationary thermal conductivity and a thermal dispersion coefficient, was applied in Eq. [11], [12], [14], and [15] of Ren et al. (2000) to simulate the temperature dynamics. For a given water flux density, an arbitrary value of thermal dispersion coefficient was picked up to force the theoretical maximum temperature changes at downstream and upstream positions to match the experimental data. Figure 3 compares the measured and simulated Td and Tu results for the three media. When thermal dispersion is considered, the model does produce maximum Td and Tu values that match the experimental data and creates earlier arrival of the peak temperature. These results support the conclusion of Ochsner et al. (2005) that the thermal dispersion theory is insufficient to explain the discrepancies between estimated and actual water flux data.


Figure 3
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  		Fig. 3. Transient temperature changes at the downstream and upstream positions at the largest flux densities. The symbols represent measured data, and the lines indicate predicted results of the conduction–convection–dispersion model as suggested by Hopmans et al. (2002). A thermal dispersion coefficient ({lambda}d) is arbitrarily selected to force the theoretical maximum temperature changes agree with the measured data.

 
Wall Flow Effect on Water Flux Measurement
For the packed soil columns, the air gaps at the soil–Plexiglas interface may be larger than the average pore size of the bulk soil. Thus, there is a possibility that a higher permeability annular region exists near the Plexiglas wall, which allows water flow at higher rate (Tokunaga, 1987). Because it measures temperatures 2 cm away from the Plexiglas wall, the thermo-TDR probe may not be able to detect this wall flow and therefore may tend to underestimate water flow rate. A simple tracer test was conducted to investigate if wall flow modified water flow distribution in the packed soil column. We added some red ink in the inflow water and monitored the color change at the top of the soil column. The sandy loam was used, and the applied water flux density was 1.57 x 10–5 m s–1. Figure 4 shows three pictures taken at different times after the color became noticeable at the top surface of the column. The red color first appeared at the soil–Plexiglas interface and then moved toward the inner area (Fig. 4a). After about 6 min, the red color reached the column center, but considerable difference in color intensity existed between the central part and the wall area (Fig. 4b). The color change lasted for about 9 min, when the darkness of the column surface tended to become uniform (Fig. 4c). The dye pattern indicated that wall flow occurred at the soil wall interface.


Figure 4
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  		Fig. 4. Pictures from the ink-dying test, showing the color change (a) 1.5 min, (b) 6 min, and (c) 9 min from the moment when the color is noticeable at the top surface of a packed column.

 
The influence of wall flow on the estimated water flux was confirmed by an additional experiment. Two groups of Plexiglas cylinders, one with epoxy (Bakelite Gooey Type; Haizhou Chemical Plant, Beijing, China) coating inside the cylinder and the other without coating, were used for packing the sandy loam and sandy clay loam soils. For the epoxy-coating treatment, a layer (about 0.5 mm thick) of epoxy was smeared uniformly over the inner surface of the Plexiglas wall with a knife. Then wet soil packing was performed following the procedure as previously. Soil saturation and heat-pulse measurement was made after the epoxy coating hardened. We expected that the epoxy coating would not only serve as an adhesive agent that bound the soil particles but would also create a larger surface roughness for the Plexiglas wall. These factors would reduce the formation of the high-permeability annular region around the wall. Table 4 compares the water flux densities estimated from the heat-pulse technique against the outflow values. With epoxy coating, the errors in water flux estimates are reduced from 12.5 to 16.6% to 2.8 to 5.2% on the sandy loam soil and from 20.1 to 24.2% to 2.6 to 5.1% on the sandy clay loam soil. The reduced discrepancy between estimated and actual water flux implies that wall flow does account for the underestimation of water flux density by the heat-pulse technique.


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  		Table 4. Comparison of water flux density estimated from the heat pulse technique, with and without epoxy coating, against actual water flux density from outflow data.

 
To quantify the extent to which the wall flow might modify heat-pulse water flux measurements, we introduce an amplification factor (Fa), which is defined as the ratio of water flux estimates between epoxy coating treatment and no epoxy coating treatment. The larger the Fa, the more influence wall flow has on water flux estimates from the heat-pulse method. A notable feature of the experimental results is that Fa of the sandy clay loam soil is considerably larger than that of the sandy loam soil, whereas little Fa difference is observed between different water flux densities (Table 4). This indicates that soil texture is the major factor determining the magnitude of the amplification factor. Soil bulk density and the diameter of the soil column may also influence the magnitude of Fa. For the current study, the mean Fa value is 1.12 for the sandy loam and 1.24 for the sandy clay loam.

The mean Fa value is then used to correct wall flow effect on water flux data of Fig. 2b in which heat-pulse results are obtained without epoxy coating treatment. The results are reported in Fig. 5 . There is good agreement between water flux estimates from heat pulse method and actual water flux densities. For both soils, the data points distribute randomly along the 1:1 line. A least-squares fit of a straight line has a slope of 0.973, an intercept of 2.12 x 10–7 m s–1, and an r2 of 0.996 for the sandy loam soil. For the sandy clay loam soil, the slope, intercept, and r2 of the fitted line are 0.963, 2.14 x 10–7 m s–1, and 0.998, respectively. The root mean square error, 6.53 x 10–7 m s–1 for the sandy loam and 5.08 x 10–7 m s–1 for the sandy clay loam, is reduced considerably from the previous values of 1.35 x 10–6 m s–1 and 2.04 x 10–6 m s–1. We therefore conclude that at the water flux ranges considered in this study, the heat-pulse technique provides accurate water flux results when wall flow is taken into consideration.


Figure 5
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  		Fig. 5. Water flux density (J) estimated from the Td/Tu, with correction of wall flow effect, versus actual values at column outlet on the sandy loam and sandy clay loam soils.

 
Wall Effect and the "Reduced Convection" Assumption
Ochsner et al. (2005) attributed the difference between heat-pulse water flux density and the actual water flux density to overestimation of the convective heat transfer around the temperature sensor by the standard conduction–convection model. They therefore introduced an empirical "reduced convection" factor, b = S (where S is the slope from linear regression of heat-pulse water flux density versus the actual water flux density), to correct this type of error. However, Ochsner et al. (2005) were not able to explain the mechanism that created this discrepancy. Experimental results from this study indicate that the "reduced convection" at the temperature sensor is caused by wall flow that creates increased water flux at the soil–Plexiglas interface. As a result, the reciprocal of b (i.e., 1/b) should be at the same magnitude as Fa. For example, using the slopes listed in Table 3, the calculated 1/b value is 1.15 for the sandy loam and 1.27 for the sandy clay loam, which are similar to the Fa values of 1.12 (sandy loam) and 1.24 (sandy clay loam) obtained here.


   CONCLUSIONS
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
REFERENCES
 
The heat-pulse technique for measuring water flux density in saturated soils is evaluated experimentally. Our results indicate that in laboratory column studies, wall flow contributes most of the differences between estimated and actual water flux densities on fine soils. The major factors that determine the magnitude of wall flow are soil texture and water flux density. Wall flow is more pronounced on the sandy clay loam soil than on the sandy loam soil. For a given soil, wall flow becomes greater with increasing water flux density, but the amplification factor remains constant.

When wall flow is not considered, the measured temperature tends to be lower than the predictions from the conduction–convection heat transfer model at the downstream position and higher than the predictions at the upstream position on fine-textured soils. Consequently, measured water flux density from the heat-pulse method is smaller than the actual values. When an amplification factor, Fa, is applied as a multiplier to correct the influence of wall flow, the errors of water flux estimates from the heat-pulse method are reduced to within 5%. The value of Fa, computed as the ratio of heat-pulse flux measurement on epoxy-coated columns to that of regular column, was 1.12 for the sandy loam and 1.24 for the sandy clay loam. For the current study, the Fa could also be obtained from the comparison of outflow measurement against water flux estimates from heat-pulse method.

We demonstrate that the "reduced convection" assumption of Ochsner et al. (2005) can be explained by the influence of wall flow on water flux density. On the other hand, the water flux range considered in this study, including a thermal dispersion term in the conduction–convection heat transfer equation, does not improve the model significantly.


   NOTES
TOP
 NOTES
ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
This article is based on work supported by the National Natural Science Foundation of China under Grant No. 50479010 and the Program for Changjiang Scholars and Innovative Research Team in University (IRT0412).

Received for publication June 5, 2005.


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




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