A model of heat transfer in sapwood and implications for sap flux

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Tree Physiology 31, 669–679doi:10.1093/treephys/tpr051

A model of heat transfer in sapwood and implications for sap flux density measurements using thermal dissipation probes†

Stan D. Wullschleger1,3, Kenneth W. Childs2, Anthony W. King1 and Paul J. Hanson1

1Environmental Sciences Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6301, USA; 2Computational Sciences and Engineering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6085, USA; 3Corresponding author ([email protected])

Received January 28, 2011; accepted May 12, 2011; published online July 8, 2011; handling Editor Nathan Phillips

A variety of thermal approaches are used to estimate sap flux density in stems of woody plants. Models have proved valuable tools for interpreting the behavior of heat pulse, heat balance and heat field deformation techniques, but have seldom been used to describe heat transfer dynamics for the heat dissipation method. Therefore, to better understand the behavior of heat dissipation probes, a model was developed that takes into account the thermal properties of wood, the physical dimen-sions and thermal characteristics of the probes, and the conductive and convective heat transfer that occurs due to water flow in the sapwood. Probes were simulated as aluminum tubes 20 mm in length and 2 mm in diameter, whereas sapwood, heartwood and bark each had a density and water fraction that determined their thermal properties. Base simulations assumed a constant sap flux density with sapwood depth and no wounding or physical disruption of xylem beyond the 2 mm diameter hole drilled for probe installation. Simulations across a range of sap flux densities showed that the dimensionless quantity k [defined as (ΔTm − ΔT )/ΔT, where ΔTm is the temperature differential (ΔT ) between the heated and unheated probe under zero-flow conditions] was dependent on the thermal conductivity of the sapwood. The relationship between sap flux density and k was also sensitive to radial gradients in sap flux density and to xylem disruption near the probe. Monte Carlo analysis in which 1000 simulations were conducted while simultaneously varying thermal conductivity and wound diameter revealed that sap flux density and k showed considerable departure from the original calibration equation used with this technique. The departure was greatest for variation in sap flux density typical of ring-porous species. Depending on the spe-cific combination of thermal conductivity and wound diameter, use of the original calibration equation resulted in an 81% under- to 48% overestimation of sap flux density at modest flux rates. Future studies should verify these simulations and assess their utility in estimating sap flux density for this widely used technique.

Keywords: heat transfer, sap flow, sap velocity, thermal properties, water transport.

Introduction

A number of thermal-based techniques have been developed to estimate the movement of water through herbaceous and woody plant stems, branches and roots (Wullschleger et al. 1998, Burgess et al. 2001, Green et al. 2003). The heat pulse

(Cohen et al. 1981), heat balance (Baker and van Bavel 1987, Steinberg et al. 1990), heat field deformation (Nadezhdina et al. 2002) and heat dissipation (Granier 1987) methods are among the most widely used in forestry, agriculture and horti-culture. Although considerable differences exist in how each of

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Tree Physiology Volume 31, 2011

these techniques is deployed, they all use heating elements and thermocouples or thermistors to estimate sap flow from measured temperature profiles.

Executing measurements of sap flux density with the various techniques is a relatively simple process, but the theory and mathematical treatment of the heat transfer characteristics that underlie each of the methods can be complex (Swanson 1994, Smith and Allen 1996, Cermák et al. 2004). Sophisticated approaches that account for the heat transfer properties of sap-wood when estimating sap flux density have been developed for the heat pulse and heat field deformation techniques (Marshall 1958, Dye et al. 1996, Köstner et al. 1996, Tributsch et al. 2006). Interestingly, and no doubt contributing to its pop-ularity, information on the thermal properties of wood is not required for the heat dissipation method. Instead, this technique measures the temperature differential (ΔT) between heated and unheated probes and a dimensionless quantity (k) is related to sap flux density through a universal calibration equation:

k T T T= ( – ) /∆ ∆ ∆m (1)

F kd = 0 0119 1 231. . (2)

where ΔTm is the ΔT under zero-flow conditions and Fd is sap flux density in units of g cm−2 s−1 (Steppe et al. 2010). The coefficient a (0.0119) and scaling exponent b (1.231) in Eq. (2) were determined by fitting a nonlinear regression to the relationship between Fd and k measured experimentally on sec-tions of wood excised from three tree species (Granier 1985).

While the strengths of the heat dissipation technique are numerous, a potential weakness is that compared with other approaches, calculations of Fd are based on an empirical rela-tionship rather than on the explicit heat transfer properties of sapwood. For this reason, Smith and Allen (1996) recommended that the calibration equation be checked when the technique is used on new plant species. Investigators have heeded this rec-ommendation and, as a result, it has been shown that estimates of Fd derived using the heat dissipation method are comparable to those from other sap flow techniques (Köstner et al. 1996) and to independent estimates from gravimetric and micrometeo-rological measurements (Diawara et al. 1991, Goulden and Field 1994, David et al. 1997, Lu and Chacko 1998). However, an increasing number of reports have identified instances where substantial deviations from the empirical calibration can occur (Clearwater et al. 1999, Do and Rocheteau 2002, McCulloh et al. 2007, Taneda and Sperry 2008, Bush et al. 2010, Steppe et al. 2010). Foremost among these are errors and uncertainties associated with variation in sap flux density along the length of the probe, improper placement of probes into heartwood, physi-cal disruption of xylem during probe installation, inadvertent impacts on ΔT and ΔTm caused by placement of heated and unheated probes too close to one another, and the presence of temperature gradients within sapwood.

The heat dissipation technique has been used to estimate plant transpiration and stand water use in hundreds of labora-tory and field studies, yet there have been few attempts to model the underlying physics of these probes. Several investi-gators have used numerical models to evaluate the transient behavior of heat dissipation probes where the emphasis was on heat storage within the vicinity of the heated probe and temporal sensitivity of the technique given rapid changes in environmental conditions (Burgess and Dawson 2008, Sevanto et al. 2008, Chu et al. 2009, Phillips et al. 2009, Sevanto et al. 2009). Tatarinov et al. (2005) conducted a detailed numerical analysis of the heat dissipation technique. These authors con-structed a steady-state model that considered wood as a two-phase medium (i.e., solid and liquid fractions) and then used that model to assess performance of the heat dissipation tech-nique compared with the tissue heat balance approach (Cermák et al. 1973, Kucera et al. 1977). Tatarinov et al. (2005) con-cluded that due to the small heat field generated by the heat dissipation method, it was more sensitive than the tissue heat balance technique to a range of uncertainty factors, including thermal conductivity of sapwood, natural thermal gradients and variation in sap flux density with sapwood depth. Unfortunately, while this analysis was both informative and thorough, the authors analyzed factors one at a time and did not simulate the consequences that might arise due to varying all uncertainty factors simultaneously. This, of course, would more accurately reflect expected errors and uncertainties associated with the use of this technique in an experimental setting. Therefore, our objectives were to (i) develop a model of heat transfer that takes into account the thermal properties of wood, the physical dimensions and thermal characteristics of the probes, and the heat transfer that occurs due to water moving through the sap-wood, (ii) use this model to assess the sensitivity of Fd and k to singular and simultaneous changes in thermal conductivity of the sapwood, radial gradients in sap flux density with sapwood depth, and physical disruption to xylem due to wounding that occurs during probe installation, and (iii) use Monte Carlo anal-ysis to describe the relationship between Fd and k for species with different radial patterns of sap flux density (i.e., ring- and diffuse-porous trees) while simultaneously varying thermal conductivity and wound diameter. Our intent was to simulate the behavior of the heat dissipation technique across a range of scenarios, reveal sensitivities of the technique that would otherwise be difficult to assess experimentally and make rec-ommendations as to how this information could be used to improve measurements of water use in woody perennials.

Material and methods

Physical and thermal parameters used in the simulations

Heat dissipation probes were modeled as aluminum tubes that were 20 mm in length and 2 mm in diameter. Sidewall

670 Wullschleger et al.


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thickness was 0.2 mm. The heated probe was located entirely within the sapwood. The distance between the heated and unheated probe is assumed to be great enough so that the temperature of the unheated probe is not influenced by the heated probe. In practice, this distance is 10–12 cm. A con-stant 200 mW energy was delivered to the heated probe. The stem was composed of a 10-mm layer of bark, a 20-mm layer of sapwood and a 20-mm layer of heartwood. The probe was inserted into the sapwood through an oversized hole drilled into the bark. Heat transfer due to this hole was not explicitly modeled. In developing the model it was assumed that the outer radius of the stem was large enough so that curvature effects did not significantly influence the heat transfer and con-sequently the temperature distribution observed around the probe. Base simulations assumed that sap flux density was uniform with sapwood depth and that there was no physical disruption of xylem or wounding beyond the diameter of the hole drilled for probe installation (i.e., 2 mm). All other physical and thermal parameter values used in the simulations are shown in Table 1.

Inputs required by the model were the thermal conductivi-ties of the sapwood, heartwood and bark. The thermal conduc-tivity of sapwood (Ksw) was calculated according to Scholz et al. (2002):

K K m K msw s dw s dw dw s( )= +( / ) ( – / )ρ ρ ρ ρ1 (3)

where Ks and Kdw are thermal conductivities of xylem sap and dry wood at 20 °C, respectively, m is water fraction of wood, ρdw is density of dry wood and ρs is density of sap. Since sap contains only a limited concentration of dissolved minerals, it is treated here as pure water. The thermal conductivities of water and dry wood are shown in Table 1 and were obtained from Jones et al. (2004). Sapwood and heartwood density was 600 kg m−3 and water fraction [i.e., (wet wt. − dry wt.)/dry wt.] was set to 1.0. Thermal conductivity of sapwood was calcu-lated with these values and Eq. (3) was 0.383 W m−1 °C−1 (Table 1). Heartwood had the same physical and thermal prop-erties as sapwood. We acknowledge that the density and water fraction of heartwood does often differ from that of sapwood (Fromm et al. 2001). Simulations show that making a distinc-tion between water fraction, density or thermal conductivity of sapwood and heartwood does little to change the results of our model (data not shown). Axial heat transfer was assumed to be twice that of radial thermal conductivity following the observations of Steinhagen (1977). Bark was considerably drier and less dense than sapwood or heartwood and there-fore had a lower thermal conductivity (Table 1).

Numeric simulation of heat transfer

Steady-state energy balance calculations for the heat dissipa-tion technique were modeled with the general-purpose heat

transfer code HEATING (Childs 1993). The name HEATING is an acronym for Heat Engineering and Transfer in Nine Geometries. The version of the code used in this study was HEATING 7.3 and, like earlier versions, it solves steady-state and transient heat transfer problems in one-, two- or three-di-mensional Cartesian, cylindrical or spherical coordinates. The model can include multiple materials, and the thermal conduc-tivity, density and specific heat of each may be both time and temperature dependent. It is written in FORTRAN 90. The mesh spacing can be variable along each axis. In the case of our simulations, spatial resolution of the model varied according to probe orientation. Along the length of the probe, resolution was 1.0 mm. In the horizontal direction perpendicular to the probe axis and in the vertical direction the spacing is 0.1 mm. For the steady-state simulations conducted here, all calculations were solved using a point-successive-overrelaxation iterative method with extrapolation (Childs 1993). In order to obtain a steady-state solution it was necessary to apply a boundary condition

Dependence of sap flux density measurements on thermal properties of sapwood 671

Table 1. Physical and thermal characteristics used to simulate heat transfer from thermal dissipation probes installed in sapwood. Parameters are given for base simulations and for three scenarios in which model sensitivity to changes in thermal conductivity, radial gradients and wound diameter were evaluated.

Thermal dissipation probes Aluminum tubes with 0.2-mm wall thickness 20 mm in length and 2 mm in diameter Thermal conductivity of aluminum 190 W m−1 °C−1

Constant energy supply 200 mW

Wood Thickness of sapwood (2 cm), heartwood (2 cm) and bark (1 cm) Density of sapwood and heartwood 600 kg m−3

Density of bark 440 kg m−3

Water fraction [(wet wt. − dry wt.)/dry wt.] of sapwood and heartwood 1.0

Boundary conditions Ambient air temperature 20 °C Convective heat transfer coefficient 5 W m−2 °C−1

Base simulations Thermal conductivity of water 0.5984 W m−1 °C−1

Thermal conductivity of dry wood 0.1276 W m−1 °C−1

Calculated thermal conductivity of sapwood and heartwood 0.383 W m−1 °C−1

Thermal conductivity of bark 0.0683 W m−1 °C−1

Constant radial gradient in Fd was assumed Wound diameter set to diameter of drilled hole (2 mm)

Scenarios 1—Sensitivity to changes in thermal conductivity Thermal conductivity varied between 0.283 and 0.493 W m−1 °C−1

2—Sensitivity to changes in radial gradients of sap flux density Radial gradients in Fd simulated as gradual or abrupt

(Clearwater et al. 1999) 3—Sensitivity to changes in wound diameter Wound diameter varied from 2.2 to 4.4 mm 4—Monte Carlo analysis Thermal conductivity and wound diameter varied simultaneously



on the model so that there is a mechanism for heat removal from the stem. This condition was specified on the surface of the stem as an ambient air temperature of 20 °C and a convec-tive heat transfer coefficient of 5 W m−2 °C−1 (Table 1). These boundary conditions reflect expectations for natural convection from a heated vertical surface.

Given the physical and thermal properties specified for the thermal dissipation probes, bark, sapwood and heartwood, it was possible to simulate steady-state temperature profiles under zero-flow conditions due to heat conduction from the heated probe surface into the surrounding wood matrix. In the case of the base simulations, when Fd is zero and conductive heat transfer dominates, ΔTm is typically in the range 12.5–13.0 °C. In order to assess the influence Fd on steady-state temperature profiles and the relationship between Fd and k sim-ulated by the model, it was necessary to account for convective heat transfer that occurs due to water moving through the sap-wood. This was done by simulating the movement of water through the sapwood at sap flux density that ranged from 0.002 to 0.018 g cm−2 s−1 in increments of 0.002 g cm−2 s−1. Sap flux density in the base simulations was assumed to be constant across sapwood depth.

Sensitivity of the model to uncertainty factors

Scenario 1: Thermal conductivity of wood is a function of ρdw and m for the species under consideration. Simpson and TenWolde (1999) indicate that ρdw and m are in fact interde-pendent and that across many hardwood and softwood spe-cies, m varies from 2.67 at a ρdw equal to 300 kg m−3 to 0.44 at a ρdw equal to 900 kg m−3 (see their Eq. (3-1)). Thermal con-ductivities calculated using these values and our Eq. (3) are 0.493 and 0.283 W m−1 °C−1, respectively (Table 1). We con-sidered these estimates to represent the upper and lower bounds on expected thermal conductivity for sapwood and heartwood. These values were used in Scenario 1 to bracket simulations that were designed to examine the sensitivity of ΔT and the Fd and k relationship to changes in the thermal proper-ties of wood. All other parameters were identical to those used in the base simulations.

Scenario 2: Spatial variation in Fd with sapwood depth is a common observation for many hardwood and softwood species (Phillips et al. 1996). In general, Fd is low in the immature xylem beneath the cambium, it increases to a maximum with depth into the sapwood, and then declines to zero as it nears the sapwood–heartwood transition. The extent and complexity of these patterns can, as shown by Clearwater et al. (1999), have a significant effect on ΔT and on the relationship between Fd and k. In order to explore the consequences of sapwood radial gradients, gradual and abrupt gradients in sap flux density were simulated to approximate typical radial patterns observed for diffuse- and ring-porous species, respectively (Clearwater et al. 1999, their Figure 7). As was done in the base simulations, the

movement of water through the sapwood was varied from 0.002 to 0.018 g cm−2 s−1 in increments of 0.002 g cm−2 s−1, but unlike the base simulations, Fd varied spatially according to the radial pattern specified. All other parameters were identical to those used in the base simulations.

Scenario 3: Initial simulations assumed that there was no physical disruption or wounding of xylem beyond the diameter of the hole drilled for installation of the probe (i.e., 2 mm). Several experiments using either staining or microscopy have shown, however, that wounding does occur beyond the original diameter of the hole and that the extent of physical disruption can be a significant factor in determining the performance of heat pulse probes (Barrett et al. 1995). In Scenario 3, the model was run for cases where wound diameters resulting from the installation process varied from 2.2 to 4.4 mm. These wound diameters are broadly consistent with previous reports (Olbrich 1991, Dunn and Connor 1993, Barrett et al. 1995, Kalma et al. 1998, Zeppel et al. 2008, Zhao et al. 2009). Wounding was physically characterized in the model as an adjacent interruption of sap flux density near the heated probe. No change in thermal conductivity or heat transfer at the sap-wood–probe interface due to wood shrinkage (i.e., loose probe) was implemented in these simulations. All other para-meters were identical to those used in the base simulations.

Scenario 4: The sensitivity of ΔT and the Fd and k relationship to simultaneous variations in thermal conductivity (Scenario 1) and wound diameter (Scenario 3) were evaluated for each of the radial profiles in sap flux density (Scenario 2). Latin hyper-cube sampling (McKay et al. 1979), a constrained form of Monte Carlo sampling (Helton and Davis 2003), was used to generate 1000 values for thermal conductivity and wound diameter. The sampling distributions were defined by the fre-quency distributions and statistics of the thermal conductivity and wound diameter data compiled from the published litera-ture (Figure 1). Estimates of thermal conductivity were derived using Eq. (3) and values of water fraction and wood density reported for 45 deciduous and coniferous tree species (Simpson and TenWolde 1999). Wound diameters compiled were from investigators who reported this parameter for the heat pulse technique in which probe diameter ranged from 1.7 to 2.2 mm and across all studies averaged 1.95 mm (Green and Clothier 1988, Olbrich 1991, Dunn and Connor 1993, Dye and Olbrich 1993, Barrett et al. 1995, Hatton et al. 1995, Becker 1996, Dye 1996, Dye et al. 1996, Cook et al. 1998, Kalma et al. 1998, O’Grady et al. 1999, Pausch et al. 2000, Wullschleger and King 2000, Wullschleger and Norby 2001, Medhurst et al. 2002, Green et al. 2003, O’Grady et al. 2005, Fernández et al. 2006, O’Grady et al. 2006, McJannet et al. 2007, González-Altozano et al. 2008, Zeppel et al. 2008, Zhao et al. 2009). We assumed that measurements were sta-tistically independent and defined the two parameters as inde-pendent continuous random variables. The approach of Iman




and Conover (1982) was used to impose independence (non-significant, approximately zero correlation between the two variables) on the Monte Carlo sampling.

On examination of the measurement distributions (Figure 1a), it was determined that the probability distribution of thermal conductivity was normal or Gaussian and assumed a univariate normal probability distribution function. In contrast, the frequency distribution of wound diameter measurements (Figure 1b) did not approximate any known probability distribution. The mea-surements appeared bi-modal with one asymmetric cluster of values near the minimum drill size of 2 mm (with values ≥2 mm but none, obviously, <2 mm) and another larger symmetric cluster near a value of 3.3 mm, but with a long tail of values approaching 4 mm. We accordingly chose to define the sam-pling distribution for wound diameter as the observed fre-quency distribution, converting that distribution into its cumulative probability density function for Monte Carlo sam-pling as a continuous variable. The frequency distributions and statistics of the resulting Monte Carlo samples (n = 1000) very closely approximated those of the measurements for both thermal conductivity (n = 45) and wound diameter (n = 65). The code used in generating the Monte Carlo samples was an implementation of the Iman and Shortencarier (1984) program written by Robert H. Gardner and others at Oak Ridge National Laboratory (Gardner et al. 1983).

Results

Steady-state simulations under zero-flow conditions showed realistic temperature profiles for the probe and the adjacent

bark, sapwood and heartwood (Figure 2). Temperatures were highest along the surface of the heated probe where the abso-lute temperature at the middle of the probe was 32.9 °C. Simulated temperatures along the length of the heated probe were fairly uniform; deviations were <0.1 °C (data not shown). The reference probe, which was separated by 12 cm from the heated probe, was unaffected by the heated probe and main-tained a temperature of 20 °C. As a result, ΔTm between the heated and unheated probe was 12.9 °C. Simulated tempera-tures decreased with increasing distance from the heated probe surface, with fairly uniform and gradual temperature pro-files in sapwood and heartwood, and across the sapwood–heartwood interface (Figure 2). In comparison, temperature profiles at the sapwood–bark transition were less gradual, reflecting the marked difference in thermal conductivity between bark and sapwood. The elongated nature of the tem-perature profiles along the vertical axis of the sapwood reflects the higher thermal conductivity of sapwood in the axial versus the radial direction.

The relationship between ΔT and Fd modeled in the base simulations was consistent with expectations (Figure 3). There was a decline in ΔT as Fd increased from 0 to 0.018 g cm−2 s−1 and the predicted pattern followed that of the original empirical relationship determined by Granier (1985). There was a slight offset in ΔT across the range of sap flux densities, with simu-lated temperature differentials being 0.5–0.8 °C higher than those calculated from the equation of Granier (1985). However, despite this offset, the simulated relationship between Fd and k


Figure 1. Distribution of observations for thermal conductivity (a) and wound diameter (b) as either estimated from or reported in the pub-lished literature. Monte Carlo analyses in which sap flux density was simulated while simultaneously varying thermal conductivity and wound diameter were conducted using probability functions derived from these distributions.

Figure 2. Simulated tangential (left) and radial (right) temperature contours where the length of the heat dissipation probe is 20 mm and sap flux density is zero. Temperature contours under zero-flow condi-tions are the sole function of conductive heat transfer. Bark, sapwood and heartwood thickness is 10, 20 and 20 mm, respectively.



closely followed that of the original empirical relationship (Figure 3 inset).

Simulated temperature profiles in the base case showed the spatial patterns of heat transfer from the probe as Fd varied from 0.002 to 0.008 g cm−2 s−1 (Figure 4). Absolute tempera-ture at the surface of the heated probe was 29.6 °C where Fd = 0.002 g cm−2 s−1 and declined sharply as Fd increased to 0.004, 0.006 and 0.008 g cm−2 s−1 (Figure 4). Heat dissipa-tion was, not surprisingly, almost exclusively in the upward direction and largely restricted to the sapwood, reflecting the

movement of energy away from the heated probe via sap flow. There was only minimal heat transfer in the downward direc-tion when Fd > 0. However, under zero-flow conditions heat transfer in the downward direction was apparent with higher temperatures observed within 5–6 cm of the heated probe.

Compared with the base simulations where Ksw = 0.383 W m−1 °C−1, the relationship between Fd and k sim-ulated by our model was sensitive to variation in Ksw (Figure 5). The relationship deviated from the base case as Ksw varied from 0.283 to 0.493 W m−1 °C−1. Deviations were also observed in ΔT and ΔTm across a range of sap flux densities as Ksw changed (Figure 5 inset).

Base simulations considered that Fd was uniform across the sapwood into which the heat dissipation probes were inserted. Simulations that accounted for depth-dependent variation in Fd typical of ring-porous and diffuse-porous species (Figure 6 inset) demonstrated that radial gradients in Fd could, depending on severity, impact the relationship between Fd and k (Figure 6). Deviations from the base simulations were modest for the grad-ual radial gradient typical of diffuse-porous tree species, but deviated more dramatically for simulations that considered abrupt radial gradients often observed for ring-porous tree spe-cies. The magnitude of this effect was less evident at k < 0.5, but increased exponentially at k > 0.5 (Figure 6).

The relationship between Fd and k was also sensitive to assumptions about the diameter of any xylem disruption or wounding that might occur during the probe installation pro-cess (Figure 7). Compared with the base simulation where no physical disruption was modeled, deviations became more


Figure 3. Modeled temperature differentials between the heated and unheated probes (ΔT) for known sap flux densities contrasting base simulations with the relationship reported originally by Granier (1985). The inset shows the modeled versus measured relationship between sap flux density and the dimensionless coefficient k. ΔT under zero-flow conditions for the Granier equation was assumed equal to 12 °C (see Clearwater et al. 1999).

Figure 4. Simulated radial temperature contours for heat dissipation probes under sap flux density conditions of 0.002, 0.004, 0.006 and 0.008 g cm−2 s−1 (left to right), respectively. In these simulations, temperature contours are a function of both conductive and convective heat transfer. The heated needle length is 20 mm and is located entirely in the hydro-active sapwood. Bark, sapwood and heartwood thickness is 10, 20 and 20 mm, respectively.



apparent and the relationship between Fd and k became steeper as the wound diameter increased from 2.2 to 4.4 mm.

Monte Carlo analysis, in which thermal conductivity and wound diameter were varied simultaneously, showed marked differences between the lower and upper boundaries of the Fd and k relationship (Figure 8). These differences were similar for uniform and gradual radial variation in sap flux density, but were more exaggerated for abrupt gradients typical of ring-porous species (Figure 8c). Depending on the specific combi-nation of thermal conductivity and wound diameter, use of the original calibration equation resulted in an 81% under- to 48% overestimation of sap flux density at modest flux rates. Coefficients for Eq. (2) are provided in Table 2.

Discussion

Numeric models have been used to develop and evaluate ini-tial designs for the heat pulse, heat balance and heat field deformation methods (Marshall 1958, Swanson 1972, Baker

and van Bavel 1987, Cermák et al. 2004) and helped lay the foundation for many of the techniques used today in forestry, agriculture and horticulture. These studies have greatly advanced our ability to deploy thermal-based techniques in pursuit of better understanding the biotic, edaphic and envi-ronmental controls on whole-plant and stand water use. We are aware, however, of only one study in which a detailed numerical analysis has been conducted for the heat dissipa-tion technique (Tatarinov et al. 2005). Similar to the results of


Figure 5. Simulated relationships between sap flux density and the dimensionless coefficient k for a range of sapwood thermal conductivi-ties (Ksw). The inset shows the dependence of ΔT on sap flux density for the same three Ksw values.

Figure 6. Simulated relationships between sap flux density and the dimensionless coefficient k for hypothetical ring- and diffuse-porous species having abrupt or gradual flow gradients from the cambium to the sapwood–heartwood transition (inset). Radial gradients in sap flux density used in these simulations were as reported by Clearwater et al. (1999, their Figure 7).

Figure 7. Simulated relationships between sap flux density and k for a range of reported wound diameters. No convective heat transfer occurred in xylem within the area of xylem disruption.

Figure 8. Simulated relationships between sap flux density and k obtained from simultaneously varying thermal conductivity and wound diameter in the Monte Carlo analysis. The lower and upper bounds on the distribution are plotted for three patterns of radial variation in sap flux density: i.e., uniform, gradual and abrupt. The dashed line is the original calibration equation of Granier (1985).



these authors, our simulations indicate that ΔT and ΔTm and subsequent calculations of the dimensionless quantity k are sensitive to (i) changes in thermal conductivity of the sap-wood, (ii) variation in Fd with sapwood depth and (iii) physical disruption or wounding of xylem at the probe–sapwood inter-face. Individually these can introduce significant error and uncertainty into estimates of sap flux density. When varied at the same time, as would occur under actual practice, sap flux densities could be greatly under- or overestimated depending on wood thermal properties.

Given that thermal, anatomical and physical characteristics are known to vary widely across tree species (Phillips et al. 1996, Simpson and TenWolde 1999, Ford et al. 2004) and that disruption of xylem routinely occurs during the installation process (Barrett et al. 1995), our results raise a number of concerns about the heat dissipation technique. Foremost among these is the degree to which uncertainties are uninten-tionally introduced into estimates of Fd knowing that the heat dissipation technique does not explicitly account for variation in many important thermal and physical properties of wood. Our analysis shows that the negative relationship between ΔT and Ksw is strong, particularly when Fd = 0 (i.e., ΔTm), and that the relationship persists, albeit to a lesser degree, at higher sap flux densities. One of several consequences of this depen-dence is that the relationship between Fd and k cannot be described by a single equation and that, in reality, a family of curves exists for the method driven by species-specific varia-tion in moisture fraction, wood density and sapwood thermal conductivity. Although estimates of Ksw are not widely avail-able, typical values of wood heat conductivity have been reported to range between 0.15 and 0.40 W m−1 °C−1 (Steinhagen 1977). This variation coupled with the modeled dependence of ΔT on Ksw would introduce considerable uncer-tainty into estimates of whole-tree transpiration and stand

water use derived using the heat dissipation technique. This would be a problem when using a single calibration equation to estimate water use for trees grown in monocultures (i.e., plantations), especially if species-dependent Ksw deviates sig-nificantly from the test species used to calibrate the original calibration equation (Granier 1985). Scaling estimates of water use are likely to be even less accurate when multiple species are measured and calculations of Fd are made using only a single equation (Wilson et al. 2001, Wullschleger et al. 2001, Meinzer et al. 2004, Bovard et al. 2005, Herbst et al. 2008). Under such conditions, not only are quantitative estimates of stand-level water use biased, but the proportional contribution of different species would be uncertain.

As was shown by Clearwater et al. (1999), heat dissipation probes underestimate k when part of the probe is in contact with inactive xylem. Our model could easily reproduce this observation (data not shown). These authors also speculated that depth-dependent variation in Fd might similarly contribute to an underestimation of k and sap flux density. It has repeat-edly been shown that radial gradients in sap flux density exist across a variety of non-porous, diffuse-porous and ring-porous tree species (Phillips et al. 1996, Cermák et al. 2008, Cohen et al. 2008). Clearwater et al. (1999) noted that deviations from the original calibration of Granier (1985) were small, about an 8% underestimate in k at a mean Fd = 0.015 g cm−2 s−1, for the diffuse-porous pattern, but that they were more signifi-cant for the ring-porous pattern. They showed that given a ring-porous pattern, k at a mean Fd = 0.015 g cm−2 s−1 was underestimated by 45%. Our results and those of Clearwater et al. (1999) suggest that radial variation in Fd and the resul-tant impact on the relationship between k and Fd can introduce considerable uncertainty into estimates of transpiration and water use. Taneda and Sperry (2008) recently acknowledged that the relationship between Fd and k for five co-occurring ring- and diffuse-porous tree species differed considerably between species and between individuals within a species. They attributed these differences to probe placement into heartwood and subsequent effects on coefficients of the Granier (1985) equation. Although this is certainly plausible, it could also be that differences arise due to variation in Fd with sapwood depth. Either of these findings would be consistent with the results of our model-based experiments.

Drilling holes for the purpose of installing thermal-based probes into xylem causes obvious mechanical damage (Barrett et al. 1995). Such damage is characterized by the direct removal of wood by the drill bit and the disruption of vessels at the edge of the drilled hole. In addition, intact vessels adjacent to the hole may become occluded over time by the formation of tylo-ses, as the plant responds to wounding. The resulting region of non-conducting sapwood around the site of probe insertion can be expected to affect the heat transfer properties of sapwood and to influence the relationship between Fd and k. Indeed, of all


Table 2. Coefficients of sap flux density (g cm−2 s−1) versus k relation-ship as determined by Monte Carlo analysis. Coefficients for the lower and upper boundaries are shown for uniform, gradual and abrupt gra-dients in sap flux density.

Radial gradient Boundary Coefficient (Eq. (2))

a b

UniformLower 0.00545 1.7162Upper 0.01778 1.7556Mean 0.01049 1.7691

GradualLower 0.00544 1.7536Upper 0.01798 1.7971Mean 0.01055 1.8124

AbruptLower 0.00720 1.9218Upper 0.02642 2.1287Mean 0.01482 2.0709



the factors examined in our study, physical disruption of xylem had the greatest impact. In our simulations and when compared with base conditions, k at a mean Fd = 0.015 g cm−2 s−1 was underestimated by 20–25% for a wound diameter of 3.0 mm. Unlike the heat pulse technique, which requires wound diameter as an input for estimating sap flux density, we are aware of no studies in which wound diameters have been characterized when using the heat dissipation technique. It is reasonable to expect that wound diameters for a given drill bit size could vary considerably across non-, diffuse- and ring-porous tree species. We know from experience that mechanical disruption encoun-tered when drilling holes in ring-porous oaks can be greater than in diffuse-porous maples and yellow poplar (unpublished). A separate but related effect could also arise due to poor ther-mal contact between the heated probe and sapwood due to improper hole size (Tatarinov et al. 2005) or localized effects of wounding and tyloses formation over long periods of time which may impair probe performance (Moore et al. 2010).

Monte Carlo analysis indicated that the Fd and k relationship could, depending on the combination of thermal conductivity and wound diameter, be higher or lower than the original equa-tion proposed by Granier (1985). The coefficient a was more sensitive to the thermal and physical properties of sapwood than was the scaling coefficient b (see Table 2). Across the three radial patterns of sap flux density, the coefficient a varied from 0.00545 to 0.02642 whereas the coefficient b varied from 1.7162 to 2.1287. These values, especially for the upper range of the a coefficient, are in agreement with previous reports where estimates of 0.0230 (Steppe et al. 2010) and 0.0240 (Hultine et al. 2010) were reported for American beech and tamarisk, respectively. In these studies, the authors indi-cate that use of the Granier equation resulted in a substantial underestimate of sap flux density. Results from our simulation studies support this conclusion, but also show that use of the Granier equation can just as likely lead to an overestimation of sap flux density; the direction and magnitude of the error or uncertainty depends on the exact thermal properties of the wood at the time of measurement. It is presently not possible to tell whether the heat dissipation technique is more prone to over- or underestimate sap flux density in ring- or diffuse-porous species without further study. Our modeling efforts put highly variable responses observed in the literature into the context of factors that influence the thermal properties of sap-wood and, in turn, that could influence estimates of sap flux density. Similarly, the recent study by Bush et al. (2010) begins to provide insights into the role of xylem anatomy in determin-ing the shape and magnitude of the Fd and k relationship. That study raised a series of timely concerns about how estimates of sap flux density could potentially be influenced by the pres-ence of compression or tension wood, vessel diameter and dis-tribution, early- and late-season vessels, growth rings and tyloses. Although Bush et al. (2010) could do little to resolve

the contribution of these variables to observed patterns of sap flux density, a modeling framework that allowed integration of thermal, physical and anatomical components that together drive estimates of sap flux density could be a productive way of advancing our understanding of sap flux density in trees, especially as it applies to methodological challenges likely to be faced when using the thermal dissipation technique.

Finally, model-facilitated analyses presented here indicate that the Granier-style approach to estimating Fd is, like all ther-mal-based techniques, sensitive to a suite of anatomical and physical characteristics that, in turn, impact the conductive and convective heat transfer properties of sapwood. Based on the results from previous empirical studies and those from our numerical analysis, we are coming to understand the factors that contribute to this sensitivity. Furthermore, depending on the combination of factors involved, we know that the magni-tude of uncertainty in ΔT, ΔTm and the relationship between Fd and k can be considerable. Our simulations clearly show that the single universal calibration equation proposed by Granier (1985) could be replaced by one or more alternatives derived more directly from first principles of heat transfer. These equa-tions would depend on known thermal characteristics of the sapwood, radial gradients in sap flux density, wound diameter, and a host of other physical and thermal properties that arise from the unique combination of bark, sapwood and heartwood in a complex biological matrix. Perhaps this level of sophistica-tion would detract from the original intent of the heat dissipa-tion technique and discourage its continued widespread use. This would be unfortunate and we would certainly not advo-cate such an outcome. Nonetheless, the common use of a single calibration equation for calculating Fd with the heat dis-sipation technique should be reexamined, especially knowing that the technique is sensitive to species-specific differences in thermal wood properties. Just as they have for other thermal-based methods, empirical studies coupled with numerical anal-yses should be encouraged and therein characterize the direction and magnitude of these uncertainties.

Acknowledgments

Editorial assistance was kindly provided by Tara A. Hall. More information on the model and a spreadsheet containing results from the Monte Carlo simulations (Scenario 4) can be obtained by contacting the senior author.

Funding

Research was sponsored by the US Department of Energy, Office of Science, Biological and Environmental Research Program. Oak Ridge National Laboratory is managed by UT-Battelle, LLC, for the US Department of Energy under con-tract DE-AC05-00OR22725.




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A model of heat transfer in sapwood and implications for sap flux