Published in Soil Sci. Soc. Am. J. 68:447-449 (2004).
© 2004 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
DIVISION S-1—NOTES
SIMPLIFIED COMPUTATIONAL APPROACH FOR DUAL-PROBE HEAT-PULSE METHOD
J. H. Knighta and
G. J. Kluitenberg*,b
a Mathematical Sciences Institute, Australian National Univ., Canberra, ACT 0200, and CSIRO Land and Water, GPO Box 1666, Canberra, ACT 2601, Australia
b Dep. of Agronomy, Kansas State Univ., Manhattan, KS 66506
* Corresponding author (gjk{at}ksu.edu).
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ABSTRACT
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Two equations are currently available for estimating soil volumetric heat capacity (
c) with the dual-probe heat-pulse (DPHP) method. One is simple but gives only approximate results because it assumes that the DPHP sensor releases an impulse of heat instantaneously. The other explicitly accounts for the finite duration of heating and gives exact results. Unfortunately, the equation that gives exact results involves the exponential integral function, which is not available in most computer spreadsheet software packages or data logger function libraries. In this note we introduce an approximation of the exact equation that contains only simple algebraic functions. The approximation consists of the first five terms of a Taylor series, which are written as a telescoped polynomial for computational purposes. For most applications of the DPHP method, the polynomial approximation gives estimates of c that are at least an order of magnitude more accurate than estimates obtained from the simple equation based on instantaneous heating.
Abbreviations: DPHP, dual-probe heat-pulse
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INTRODUCTION
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THE VOLUMETRIC HEAT capacity of soil can be measured with the DPHP method (Kluitenberg, 2002). This method, first suggested by Campbell et al. (1991), involves measuring the maximum temperature rise at a fixed distance from a cylindrical heater probe that contains enamel-coated resistance wire. An impulse of heat is released from the probe by passing electrical current through the wire. Temperature rise is then monitored with a thermocouple or thermistor in a second cylindrical probe oriented parallel to the heater probe. Campbell et al. (1991) developed an inverse relationship between the maximum temperature rise and the volumetric heat capacity by assuming, among other things, that heat is released from the heater probe instantaneously. The relationship is
 | [1] |
where c is the soil volumetric heat capacity (J m–3 °C–1), q is the quantity of heat released per unit length of heater (J m–1), r is the spacing between the heater and temperature probes (m), and Tm is the maximum temperature rise (°C). In practice, heating takes place over a time interval of several seconds. That is, heat is released at the constant rate q' (W m–1) during the time interval 0 < t 
t0, and the heat input per unit length is calculated from q = q't0. Measurements of r, q', t0, and Tm are therefore needed to evaluate Eq. [1].
A shortcoming of Eq. [1] is that it was derived from theory for instantaneous heating. Kluitenberg et al. (1993) developed an alternative expression that explicitly accounts for the finite heating duration. This expression is
 | [2] |
where 
is the soil thermal diffusivity (m2 s–1), tm is the time (s) from the initiation of heating to the occurrence of the maximum temperature rise, and –Ei(–x) is the exponential integral function with argument x (e.g., Gautschi and Cahill, 1972). Kluitenberg et al. (1993) used Eq. [2] to quantify the error introduced by calculating c from Eq. [1]. They found that Eq. [1] always overestimates c and that the relative error is approximately 0.01 for intermediate values of thermal diffusivity and common values of probe spacing (r = 0.006 m) and heating duration (t0 = 8 s). Their results also showed that relative error increases for larger thermal diffusivities, approaching 0.03 for = 1.0 x 10–6 m2 s–1 and the values of r and t0 given above. Errors are larger for smaller probe spacings or longer heating durations.
Bristow et al. (1994) showed that Eq. [2] can be used to estimate c if tm is measured in addition to r, q', t0, and Tm. Although a value of is required in Eq. [2], they derived an exact expression for the thermal diffusivity
 | [3] |
that depends on r, t0, and tm. Thus, c can be calculated from Eq. [2] by using Eq. [3] and measurements of r, q', t0, tm, and Tm. Equation [2] gives more accurate c estimates than Eq. [1] because it explicitly accounts for the finite duration of heating. But some investigators prefer to use Eq. [1] because of its simpler mathematical form. The exponential integral function is not available in most computer spreadsheet software packages or data logger function libraries. The objectives of this note are (i) to present an approximation of Eq. [2] that contains only simple algebraic functions and (ii) to report the accuracy that can be achieved with this approximation.
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Theory
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We approximate Eq. [2] with a Taylor series in the small quantity 
= t0/tm (see Appendix). Retaining the first five terms of the Taylor series gives the approximation
 | [4] |
Higher-order terms can be included in Eq. [4], but the coefficients associated with these terms are unwieldy. Notice that Eq. [4] amounts to Eq. [1] minus correction terms of order two and higher. More accurate approximations of Eq. [2] can be derived, but, in all cases, greater accuracy comes at the expense of simplicity of form. The recommended computational form of Eq. [4] is the telescoped polynomial
 | [5] |
which minimizes the number of required arithmetic operations (Burden and Faires, 1988, p. 18).
The accuracy of Eq. [5] can be evaluated by comparing c estimates obtained from Eq. [5] with c estimates obtained from Eq. [2]. Subtracting Eq. [2] from Eq. [5] and dividing the result by Eq. [2] yields
 | [6] |
This expression gives the relative error in c resulting from the use of Eq. [5]. An analogous expression can be used to evaluate the accuracy of Eq. [1]. Subtracting Eq. [2] from Eq. [1] and dividing the result by Eq. [2] yields
 | [7] |
Equation [7], which is identical to Eq. [19] of Kluitenberg et al. (1993), gives the relative error in c resulting from the use of Eq. [1]. Neither of the two relative error expressions contains q' or Tm. Thus, the relative error resulting from the use of Eq. [1] or [5] is independent of the magnitude of the heat input or maximum temperature rise.
A natural quantity useful for calculating 
and ß is r2/4. By selecting values of t0 and r2/4, the value of tm can be obtained from Eq. [3] and then used to compute . The relative errors and ß therefore can be expressed solely in terms of the quantities r2/4 and t0. We used a Van Wijngaarden–Dekker–Brent iterative technique (Press et al., 1989) to solve Eq. [3] for tm. Formulas 5.1.53 and 5.1.56 of Gautschi and Cahill (1972) were used to evaluate –Ei(–x) for 0 x 1 and x > 1, respectively.
Results and Discussion
Figure 1
shows the relative error in volumetric heat capacity when Eq. [1] is used to estimate c. The results are identical to those presented by Kluitenberg et al. (1993), but are included here for comparison. Figure 2
shows the relative error in heat capacity when Eq. [5] is used to estimate c. The errors in Fig. 1 and 2 are positive, indicating that Eq. [1] and [5] both overestimate c. Relative error resulting from the use of Eq. [1] or [5] also decreases as heating duration decreases. This result is expected because heating over a finite time interval more closely approximates instantaneous heating as t0 decreases. Figures 1 and 2 also show that errors decrease logarithmically as the ratio r/
increases. Thus, approximation errors decrease markedly as probe spacing increases or as thermal diffusivity decreases.
Comparing the results in Fig. 1 and 2 reveals that the relative error resulting from the use of Eq. [5] is consistently smaller than that resulting from Eq. [1]. With Eq. [5], relative error also decreases more rapidly as r/ increases. Select numerical results are provided in Table 1. Results are given for 3 < r/ < 5 because values of r/ < 3 are unlikely in practice and because the error resulting from the use of Eq. [1] becomes insignificant for r/ > 5. For t0 8 s and 3 < r/ < 5, c estimates resulting from Eq. [5] are at least one order of magnitude more accurate than those obtained from Eq. [1]. The same is true for t0 10 s, but in the slightly smaller range 3.2 < r/ < 5. The range for 10x or greater accuracy is 3.5 < r/ < 5 for t0 12 s.
We conclude that Eq. [5] offers a useful alternative for estimating c with the DPHP method. It eliminates evaluation of the exponential integral function required if using Eq. [2] yet provides better accuracy than can be achieved with Eq. [1]. We note, however, that the use of Eq. [5] requires a measurement of tm whereas this measurement is not required when using Eq. [1].
Inasmuch as the DPHP method is often used to estimate volumetric water content (
) from measurements of c, it is useful to consider the error in estimates of caused by error in c. Estimates of are usually obtained from an expression of the form
 | [8] |
where (c)w is the volumetric heat capacity of water, b is the soil bulk density, and cs is the specific heat of the soil solid constituents. From Eq. [8] we can derive the expression
 | [9] |
which is useful for calculating the absolute error in water content, 
, corresponding to a known relative error in heat capacity, (c)/c. For example, a relative error of 0.02 in heat capacity gives a water content error of 0.01 m3 m–3 for a soil with c/(c)w = 2.
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Appendix
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An approximation of Eq. [2] for small values of t0 can be obtained by expanding it in the form of a Taylor series. First, we introduce the change-of-variables t' = tm – (t0/2) to give
 | [A1] |
Next, we expand the difference of the exponential integral terms in Eq. [A1] in the form of a Taylor series centered at time t'. This is accomplished by expanding the first term in powers of –t0/2 and the second term in powers of t0/2. Subtracting the two series leads to the approximation
 | [A2] |
where
 | [A3] |
Simplification of Eq. [A2] is accomplished by expressing , 
, and exp(–) in the form of Taylor series expansions in the small quantity = t0/tm. The series approximation of from Eq. [A3] is
 | [A4] |
The series approximation of is obtained by using Eq. [3] in Eq. [A3] to give the alternate form
 | [A5] |
Expanding the right hand side of Eq. [A5] gives the result
 | [A6] |
Substituting Eq. [A6] into the standard Taylor series for the exponential function gives
 | [A7] |
Substituting Eq. [A4], [A6], and [A7] into Eq. [A2] and simplifying gives Eq. [4]. The Taylor command in MATLAB's Symbolic Math Toolbox (ver. 2.1.3; The MathWorks, Inc.; Natick, MA) was used to confirm that Eq. [A4], [A6], and [A7] are correct, and that Eq. [4] is equivalent to Eq. [A2].
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ACKNOWLEDGMENTS
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The authors gratefully acknowledge funding from the Kansas State University President's Faculty Development Awards Program and NASA Grant NAG 9-1399.
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NOTES
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Contribution no. 03-235-J from the Kansas Agric. Exp. Stn., Manhattan, KS.
Received for publication January 2, 2003.
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REFERENCES
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- 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]
- Burden, R.L., and J.D. Faires. 1988. Numerical analysis. 4th ed. PWS-KENT Publishing Co., Boston.
- Campbell, G.S., C. 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]
- Gautschi, W., and W.F. Cahill. 1972. Exponential integral and related functions. p. 227–254. In M. Abramowitz and I.A. Stegun (ed.) Handbook of mathematical functions. Dover, New York.
- Kluitenberg, G.J. 2002. Heat capacity and specific heat. p. 1201–1208. In J.H. Dane and G.C. Topp (ed.) Methods of soil analysis. Part 4. SSSA Book Ser. 5. SSSA, Madison, WI.
- 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]
- Press, W.H., B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling. 1989. Numerical recipes in Pascal. The art of scientific computing. Cambridge Univ. Press, New York.
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TOP
ABSTRACT
INTRODUCTION
for heating durations of t0 = 2, 4, 6, 8, 10, and 12 s. Results are from 