Foliage influences forced convection heat transfer in conifer branches and buds

Authors

  • S. T. Michaletz,

    1. Department of Biological Sciences and Kananaskis Field Stations, University of Calgary, 2500 University Drive NW, Calgary, Alberta T2N 1N4, Canada
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  • E. A. Johnson

    1. Department of Biological Sciences and Kananaskis Field Stations, University of Calgary, 2500 University Drive NW, Calgary, Alberta T2N 1N4, Canada
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Author for correspondence: Sean T. Michaletz Tel: +1 403 220 7635 Fax: +1 403 289 9311 Email: sean.michaletz@ucalgary.ca

Summary

  • • Conifer foliage structures affect branch and bud temperature by altering the development and convective resistance of the thermal boundary layer. This paper examines foliage effects on forced convection in branches and buds of Picea glauca (Moench) Voss and Pinus contorta Dougl. Ex. Loud., two species that represent the range of variation in foliage structure among conifers.
  • • Forced convection is characterized by a power law relating Nusselt (heat transfer) and Reynolds (boundary layer development) numbers. Data were collected in a laminar flow wind tunnel for free stream velocities of 0.16–6.95 m s−1. Scaling parameters were compared against literature values for silver cast branch replicas, a bed of real foliage, cylinders, and tube banks.
  • • Foliage structures reduced Nusselt numbers (heat transfer) relative to cylinders, which are typically used to approximate leafless branches and buds. Significantly different scaling relationships were observed for all foliage structures considered.
  • • Forced convection scaling relationships varied with foliage structure. The scaling relationships reported here account for variation within populations of branches and buds and can be used to characterize forced convection in a forest canopy.

Introduction

The temperature of a branch or bud controls many physiologically important processes. For example, branch temperature controls rates of woody tissue respiration (Ryan et al., 1994, 1996; Meir & Grace, 2002) and radial wood formation (Wodzicki, 1971; Antonova & Stasova, 1993, 1997), affects the hydraulic conductivity of xylem (Cochard et al., 2000), determines the extent of xylem embolism during freeze–thaw events (Sperry & Sullivan, 1992; Wang et al., 1992; Lemoine et al., 1999), and drives the mechanism of vascular cambium necrosis during forest fires (Dickinson & Johnson, 2004). Bud temperature controls the rate of ontogenetic bud development (Sarvas, 1972, 1974), determines flower bloom and leaf flush phenology (Kozlowski, 1971; Hänninen, 1994; Saxe et al., 2001), and governs patterns of tree crown mortality during forest fires (S. T. Michaletz & E. A. Johnson, unpubl. obs.).

Foliage influences branch and bud temperatures by affecting the convection heat transfer processes which contribute to the overall heat budget (Nobel, 1974; Landsberg et al., 1974; Phillips et al., 1983; Hamer, 1985; S. T. Michaletz & E. A. Johnson, unpubl. obs.). Radiation heat transfer is also affected by foliage (Tibbals et al., 1964; Gates et al., 1965) but is not considered here. Convection is a two-part process: conduction between the surface and air molecules within the thermal boundary layer, and mass transport of air molecules between the thermal boundary layer and the free stream (cf. Incropera & DeWitt, 2002). Thus, convection rates are governed by boundary layer development around the branch or bud. For a given set of free stream conditions, boundary layer development is determined by the size, geometry, and orientation of the branch or bud (cf. Nobel, 1999). Foliage also affects boundary layer development because it creates aerodynamic interference (Landsberg & Thom, 1971; Grant, 1983, 1984) which causes a pressure drop, reduces stream velocity through the foliage, increases branch and bud boundary layer thickness, and increases the convective resistance of branches and buds. These effects are dependent upon the structural arrangement of foliage (e.g. length, density and porosity; Landsberg & Thom, 1971; Grant, 1984) and are especially pronounced in coniferous species, which have a less porous foliage structure (volume of air per unit volume of foliage; cf. Kaviany, 1995) than deciduous species.

Conifer foliage has been shown to reduce convection heat transfer for individual silver cast branch replicas (Tibbals et al., 1964; Gates et al., 1965) and to decouple in situ branches from free stream conditions (Martin et al., 1999). However, the structural arrangement of foliage varies among individual branches and buds within a forest canopy, so boundary layer development and the associated convection heat transfer effects should also vary among individual branches and buds. Silver and wood (cellulose) also have different thermophysical properties which should affect measured rates of convection heat transfer. Foliage effects on convection heat transfer have not yet been investigated for populations of real conifer branches and buds.

This paper reports forced convection regression results for branches and buds of white spruce [Picea glauca (Moench) Voss] and lodgepole pine [Pinus contorta Dougl. Ex. Loud.], two species that represent the range of variation in foliage structure among conifers. Data were collected in a laminar flow wind tunnel for a Reynolds number range of approx. 100–2000 (characteristic length = branch or bud diameter; free stream velocities of 0.16–6.95 m s−1). Results are presented in dimensionless form and are compared against existing reports for silver cast branch replicas (Abies concolor, Picea pungens and Pinus ponderosa), a bed of real foliage (Pinus pinaster), and geometries that are commonly used to approximate conifer branches and buds (cylinder and tube bank). This dimensionless form permits comparison of regression results regardless of differences in dimensional variables such as geometry, diameter, and free stream velocity. The engineering approach used here identifies the functional relationships between governing variables and overcomes many of the problems associated with traditional statistical approaches.

Theory

The complexity of fluid dynamics within conifer foliage makes analytical convection calculations impractical. However, dimensional analysis suggests that convection heat transfer can be characterized by the set of dimensionless numbers

image(Eqn 1)

(Nomenclature is defined in Table 1.)

Table 1.  Nomenclature
SymbolDefinitionUnits
Roman letters
AWetted surface aream2
aWood–water bond adjustment factorkJ kg−1 °C−1
CNormalization constantDimensionless
cSpecific heat capacitykJ kg−1 °C−1
DDiameterm
gGravitational accelerationm s−2
hForced convection heat transfer coefficientW m−2 °C−2
kThermal conductivityW m−1 °C−1
LLengthm
MWater content (per cent dry mass)Dimensionless
mMasskg
NPure numberDimensionless
nScaling exponentDimensionless
TTemperature°C, K
tTimes
UVelocitym s−1
VVolumem3
Greek letters
βVolumetric thermal expansion coefficientK−1
θExcess temperature ratioDimensionless
µDynamic viscositykg m−1 s−1
νKinematic viscositym2 s−1
ρMass densitykg m−3
Dimensionless groups
GrGrashof number [Gr = gβ(T − Ts)D3/inline image]Dimensionless
NuNusselt number (Nu = hD/ka)Dimensionless
PrPrandtl number (Pr = cpµa/ka)Dimensionless
ReReynolds number (Re = UD/νa)Dimensionless
Subscripts
Free stream 
aAir 
bBranch or bud 
cbCured branch or bud 
fFilm 
fbFresh branch or bud 
iIteration 
nNeedle or fascicle 
pConstant pressure 
sSurface 
totTotal 
wWater 

The average Nusselt number Nu is a measure of thermal similarity given by the ratio of conductive (D/ka) to convective (1/h) resistance in the thermal boundary layer

image(Eqn 2)

(h, the average heat transfer coefficient; D, the branch or bud diameter; ka, the thermal conductivity of air.)

The Reynolds number Re is a measure of dynamic similarity given by the ratio of inertia (inline image/D) to viscous (µaU/D2) forces in the boundary layer

image(Eqn 3)

a, the air mass density; U, the free stream velocity; D, the branch or bud diameter; µa, the dynamic viscosity of air; νa, the kinematic viscosity of air.)

The Grashof number Gr is a measure of dynamic similarity given by the ratio of buoyancy to viscous forces in the boundary layer

image(Eqn 4)

(g, the gravitational acceleration; β, the volumetric thermal expansion coefficient; T, the free stream temperature; Ts, the branch or bud surface temperature; D, the branch or bud diameter; νa, the kinematic viscosity of air.)

The Prandtl number Pr is a relative measure of the hydrodynamic and thermal boundary layer thicknesses given by the ratio of momentum and thermal diffusivities

image(Eqn 5)

(cp, the specific heat of air at constant pressure; µa, the dynamic viscosity of air; ka, the thermal conductivity of air.)

The total amount of heat transferred by convection is the sum of free and forced convection components. In free convection, air flow is driven by buoyancy forces arising from density (temperature) differences near the surface of a branch or bud, while in forced convection air flow is driven by external forces such as wind. The relative contributions of free (Gr) and forced (Re) convection processes are expressed by

image(Eqn 6)

For cylinders perpendicular to the free stream flow, forced convection processes dominate for Gr/Re2 < 0.5 and free convection processes dominate for Gr/Re2 > 40 (Fand & Keswani, 1973). Here we are concerned with forced convection heat transfer, and, assuming that Gr/Re2 < 0.5, we can neglect the influence of the Grashof number on the Nusselt number. The influence of the Prandtl number on the Nusselt number can also be neglected because the Prandtl number is constant for air (Pr = 0.69; cf. Incropera & DeWitt, 2002). Thus, forced convection heat transfer in conifer branches and buds can be characterized by

image(Eqn 7)

where the normalization constant C and scaling exponent n are obtained empirically.

Materials and Methods

Data collection

Forced convection data were collected in a laminar flow wind tunnel at the University of Calgary. The wind tunnel has a cylindrical working section 0.5 m in diameter and 2.0 m in length. Free stream temperature and velocity are constant across the working section. Free stream velocity ranges from approx. 2.75 to 20 m s−1 and is controlled by louvers on the intake fans. Because we were interested in free stream velocities representative of light to moderate winds (0.16–6.95 m s−1), velocities were reduced using six #200 mesh 0.0021″ (0.05 mm) diameter stainless steel wire cloth screens (Continental Wire Cloth, Calgary, AB, Canada) that were secured onto the air intake to create a pressure drop. Screens were spaced 1 cm apart to allow for decay of fine-scale turbulence between screens.

Picea glauca (Moench) Voss (Ntot = 184) and Pinus contorta Dougl. Ex. Loud. (Ntot = 180) branches were collected during April 2004 at the Kananaskis Field Stations of the University of Calgary in the southern Canadian Rocky Mountains of Alberta, Canada. Tree crowns were accessed using a Swedish cone picking ladder, and branches were excised using a tree pruner and 3.6-m extension pole. To account for variation in foliage structure among branches in a forest canopy, each branch was collected from a random location within the crown of a different tree. Each branch contained a terminal bud so that it could be used as either a branch (NP. glauca = 93; NP. contorta = 92) or bud (NP. glauca = 91; NP. contorta = 88) specimen in heating experiments. Branches were trimmed to a length of approx. 0.75 m, wrapped in plastic, and transported under ice to the laboratory. Branches were stored at −18°C in the laboratory and wind tunnel measurements were completed within 1 wk of collection date.

In preparation for wind tunnel data collection, the terminal end of each branch was trimmed to a length of 0.30 m (i.e. single shoots). A 0.203″ (0.52 cm) shank diameter wood screw was screwed into the base of each branch. This screw was used to mount branches in the wind tunnel and was custom-manufactured to be compatible with existing wind tunnel mounts. Thus, tunnel mounts did not interfere with boundary layer or wake stream development around the branch. An unsheathed 30-gauge type-K chromel-alumel thermocouple was inserted into the center of each branch or bud (i.e. branch and bud experiments varied only by thermocouple placement). Thermocouples were inserted into a 4-cm-deep hole that was drilled at ∼10° along the longitudinal axis of each branch using a #52 × 6″ (15.24 cm) OAL drill bit (CenturyVallen, Calgary, AB, Canada). This hole helped to secure the thermocouple in place and also to further insulate the thermocouple leads from free stream convection. Glass braid sheathing insulated thermocouple leads from free stream convection. To minimize contact resistance between the branch and thermocouple, holes were filled with Omegatherm® 201 silicone heat sink paste (Omega Engineering Inc., Stamford, CT, USA) before thermocouple insertion. Branches were then cooled to −45°C in a cooler under dry ice. During this time the branch or bud thermocouple temperature was monitored using a HH2002AL digital thermometer (Omega Engineering Inc.).

Equation 6 was used to verify that free convection was negligible relative to forced convection. To calculate Reynolds numbers (Eqn 3), free stream velocity U was varied across the range observed during heating experiments (P. glauca= 0.22–6.95 m s−1; P. contorta = 0.16–6.95 m s−1; see Heating experiment below), the diameter D was taken as the mean of all branch (D̄P. glauca ± SEM = 2.48 × 10−3 ± 4.08 × 10−5 m; D̄P. contorta ± SEM = 3.15 × 10−3 ± 5.99 × 10−5 m) and bud (D̄P. glauca ± SEM = 3.33 × 10−3 ± 5.79 × 10−5 m; D̄P. contorta ± SEM = 4.29 × 10−3± 1.16 × 10−4 m) samples used for heating experiments, and the kinematic viscosity of air was taken as νa = 1.5 × 10−5 m2 s−1 (Vargaftik, 1975). To calculate Grashof numbers (Eqn 4), the gravitational acceleration was taken as g = 9.8 m s−2, the volumetric thermal expansion coefficient as β = 1/T, and the free stream temperature as T = 23°C (from heating experiments). The diameter D and kinematic viscosity of air νa were calculated as described for the Reynolds number. We used a surface temperature of Ts = 0°C in order to account for the maximum temperature gradient that drove free convection during the heating experiments (i.e. we calculated the largest potential Grashof numbers for the experiments; see Mean T, Re, and Nu calculations, below for the temperature data truncation procedure).

To measure convection heat transfer rates, frozen branches were mounted in the center of the wind tunnel working section and oriented so that the bottom (abaxial) side of the branch was facing perpendicular to free stream flow. Branches were mounted quickly so that branch temperature was still very low (typically ∼−30°C) when data collection began. Frozen branches were heated to free stream temperature (T = 23°C) using ambient air at velocities 0.22 ≤U ≤ 7.95 m s−1 for P. glauca and 0.16 ≤ U ≤ 7.95 m s−1 for P. contorta. Condensation on branches was not observed during heating experiments. Free stream temperatures were measured using an unsheathed 30-gauge type-K chromel-alumel thermocouple mounted approx. 10 cm above the branch (i.e. outside of the foliage boundary layer and wake stream). Branch and free stream temperatures were sampled 10 times per second and logged as a time series using a DAQBOOK-216® data acquisition system (Omega Engineering Inc.) and a desktop PC.

Mean temperatures, Reynolds numbers, and Nusselt numbers were then calculated for each 1-s time-step (taking the leafless branch or bud as the control volume). To avoid errors from heat loss to the latent heat of fusion and also differences in the thermophysical properties of ice and water, time-series data were truncated to branch or bud temperatures between 2 and 22°C. The truncated data were then used to calculate the mean Reynolds number and mean Nusselt number for each branch or bud during the heating experiment. Thus, each branch or bud yielded a single mean (Re, Nu) data-point.

To calculate the Nusselt number (Eqn 2) for each 1-s time-step, branch and bud diameters D were measured to the nearest 0.01 mm using a digital caliper. The temperature-dependent thermal conductivity of air ka was calculated for each 1-s time-step using a quadratic fit to handbook data (R2 = 1.0; Vargaftik, 1975):

image(Eqn 8)

Here, Tf is the film temperature (the arithmetic mean between the branch or bud surface temperature and the free stream temperature) given by

image(Eqn 9)

(Tb, the branch or bud temperature; T, the free stream temperature.)

Average heat transfer coefficients h were measured using a lumped capacitance analysis of branch and bud heat transfer (cf. Incropera & DeWitt, 2002). Lumped capacitance analyses assume a uniform internal temperature distribution, which was validated for branches and buds by S. T. Michaletz & E. A. Johnson (unpubl. obs.). The analysis equates the rate of convection heat transfer to the branch or bud surface with the rate of internal heat accumulation according to

image(Eqn 10)

(h, the average heat transfer coefficient; A, the branch or bud wetted (convective) surface area; Tb, the branch or bud temperature; T, the free stream temperature; ρ, the branch or bud mass density; c, the branch or bud specific heat capacity; V, is the branch or bud volume; dTb/dt, the rate of internal temperature change.) Condensation and evaporation processes were neglected because condensation was not observed during heating trials.

To calculate h for each 1-s time-step, the temperature derivate was replaced with the finite difference approximation

image(Eqn 11)

(Ti, the branch or bud temperature at time-step i; Δt, the length of each time-step (1 s); Οt)4, the fourth and higher order terms which are neglected.)Substituting Eqn 11 into Eqn 10 and rearranging gives

image(Eqn 12)

Following wind tunnel data collection, foliage and buds were dissected from each branch. The mass density ρ of each dissected branch or bud was calculated using

image(Eqn 13)

(m, the branch or bud mass measured to the nearest 0.0001 g using a digital balance.) Volume V was measured by water immersion (ASTM, 1993).

Specific heat capacity c was calculated following Simpson & TenWolde (1999):

image(Eqn 14)

Here, cw is the specific heat of water (4.18 kJ kg−1 °C) and ccb is the temperature-dependent specific heat capacity of cured wood given by

image(Eqn 15)

(Tb, the branch or bud temperature measured in Kelvin.) M is the water content (per cent dry mass) given by

image(Eqn 16)

[mfb, the fresh mass; mcb, the cured mass (oven-dried at 90°C to a constant mass).]

The adjustment factor a (for Eqn 14) accounts for the additional energy in the wood–water bond according to

image(Eqn 17)

(b1 = −0.06191; b2 = 2.36 × 10−4; b3 = −1.33 × 10−4; Tb, branch or bud temperature measured in Kelvin.)

The wetted surface area A of branches was calculated using

image(Eqn 18)

[D̄, the mean branch diameter calculated from a total of four diameter measurements (two measurements at right angles on both ends of the branch); L, the dissected branch length (not including the terminal bud); Nn, the number of P. glauca needles or P. contorta fascicles on the branch; Ān, the mean cross-sectional basal area of the needles or fascicles (calculated for each branch using five needles or fascicles).] The cross-sectional basal area of a single needle or fascicle was calculated using

image(Eqn 19)

(Dn, the needle or fascicle diameter.)

Branch, needle, and fascicle diameters D were all measured to the nearest 0.01 mm using a digital caliper. The wetted surface area A of buds was measured using 80- to 100-mesh Porapak™ (VWR International, Edmonton, AB, Canada), which forms a monolayer on the bud surface that can be massed and converted to area using a predetermined surface area to mass ratio.

To calculate Reynolds numbers (Eqn 3) for each 1-s time-step, the free stream velocity U of each experiment was measured using a Florite 700 hot wire anemometer (Bacharach Inc., Pittsburgh, PA, USA). Branch and bud diameters D were measured to the nearest 0.01 mm using a digital caliper. The temperature-dependent kinematic viscosity of air νa was calculated for each 1-s time-step using a quadratic fit to handbook data (R2 = 1.0; Vargaftik, 1975):

image(Eqn 20)

[Tf, film temperature (Eqn 9).]

Data analysis

To estimate the normalization constants C and scaling exponents n in Eqn 7, the equation was log10-transformed and rearranged to give the linear form

image(Eqn 21)

where the slope n and y-intercept log(C) were obtained by Model II reduced major axis (RMA) regression (Sokal & Rohlf, 1995). Model II (RMA) regression minimizes residual variation in both the x- and y-dimensions and is less biased than Model I ordinary least-squares (OLS) regression in estimates of the functional relationship between two variables subject to error. Model II (RMA) analyses were performed using (S)MATR (Falster et al., 2003).

We compared regression results by testing for differences in the slope and elevation (y′- or y-intercepts) of Eqn 21. Slopes and intercepts were not reported for P. pungens, A. concolor, and P. ponderosa (Tibbals et al., 1964; Gates et al., 1965), so points were estimated from the published plots and fit using Model II (RMA) regression.

To compare P. glauca and P. contorta regressions, a common RMA slope was estimated using a likelihood ratio method (Warton & Weber, 2002). The significance of this slope was then determined using permutation to test for significant differences among group slopes (Manly, 1997). When slopes were not significantly different, differences in elevation were tested using one-sample analysis of variance (ANOVA) on the group mean y′, where y′ was transformed for each group as y − bx and b was the common slope (i.e. slopes were transformed so n = 0 and group means were compared; Wright et al., 2002). This test is analogous to analysis of covariance (ANCOVA) in Model I (OLS) regression.

Experimental data-points were not available for literature regressions, so one-sample t-tests were used to compare RMA slopes for P. glauca and P. contorta against literature slopes. When slopes were not significantly different, differences in elevation (y-intercept) were tested using one-sample t-tests on RMA intercepts and reported literature intercepts.

All statistical tests were significance tested at α = 0.05.

Results

Forced convection dominates in P. glauca and P. contorta over the range of free stream velocities used in the heating experiments (Fig. 1). As expected, mixed convection occurs at low free stream velocities (UP. glauca branches < 0.22 m s−1, UP. glauca buds < 0.25 m s−1, UP. contorta branches < 0.24 m s−1, and UP. contorta buds < 0.29 m s−1), but this range of mixed convection is negligible relative to the range of forced convection for all experiments (per cent of total velocity range where mixed convection occurs: < 0.00% for P. glauca branches, 0.45% for P. glauca buds, 1.18% for P. contorta branches, and 1.92% for P. contorta buds). Thus, the influence of the Grashof number on the Nusselt number (Eqn 1) can be neglected and convection can be characterized by Eqn 7.

Figure 1.

Grashof number (Gr)/Reynolds number (Re)2 as a function of free stream velocity for Picea glauca and Pinus contorta branches and buds. The horizontal line distinguishes mixed and forced convection regimes (Gr/Re2 = 0.5; Fand & Keswani, 1973). Forced convection dominates for P. glauca and P. contorta across the range of free stream velocities observed in heating experiments (per cent of total velocity range where mixed convection occurs: < 0.00% for P. glauca branches, 0.45% for P. glauca buds, 1.18% for P. contorta branches, and 1.92% for P. contorta buds) so that convection can be characterized by Eqn 7.

Table 2 gives the normalization constants C and scaling exponents n that relate Reynolds and Nusselt numbers (Eqn 7) for P. glauca and P. contorta, three silver cast branch replicas (A. concolor, P. pungens and P. ponderosa), a bed of foliage (P. ponderosa), and two geometries commonly used to approximate conifer branches and buds (cylinder and tube bank). For P. glauca and P. contorta, RMA results are reported for branches, buds, and branches and buds combined. Different normalization constants and scaling exponents are required for branches and buds of both species because RMA slopes (scaling exponents n) are significantly different for P. glauca branches and buds (Table 3; Fig. 2) and there is a significant shift in elevation (y′) between branches and buds of P. contorta (Table 4; Fig. 2).

Table 2.  Normalization constants C and scaling exponents n for Eqn 7 relating the Reynolds number (Re) and the Nusselt number (Nu)
Structure or geometryU (m s−1)ReCn (95% CI)R2
  1. Regression results are reported for Picea glauca and Pinus contorta branches and buds, four conifer structures from the literature (Abies concolor, Picea pungens, Pinus pinaster and Pinus ponderosa), and two geometries commonly used to approximate conifers (cylinder and tube bank). For P. glauca and P. contorta regressions, scaling exponents (reduced major axis slopes) are reported with upper and lower 95% confidence intervals (CIs); confidence intervals were not reported for literature regressions. Nomenclature is defined in Table 1.

  2. NR, not reported.

  3. Literature sources: aHilpert (1933); bGlaser (1938); cKnudsen & Katz (1958); dTibbals et al. (1964); eGates et al. (1965); fMendes-Lopes et al. (2002).

Picea glauca branches, perpendicular flow0.22–6.95133–12420.0700.677(0.588–0.779)0.538
Picea glaucabuds, perpendicular flow0.22–6.95152–14360.0170.912(0.795–1.045)0.582
Picea glauca branches and buds combined, perpendicular flow0.22–6.95133–14360.0270.831(0.756–0.913)0.585
Pinus contorta branches, perpendicular flow0.16–6.9595–20070.0440.714(0.632–0.806)0.661
Pinus contorta buds, perpendicular flow0.16–6.95139–14080.0500.664(0.563–0.783)0.406
Pinus contorta branches and buds combined, perpendicular flow0.16–6.9595–20070.0510.674(0.607–0.749)0.498
Abies concolor silver cast, crossflowd0.4–1.430–1050.3280.605NR
Abies concolor silver cast, longitudinal flowd0.4–1.430–1050.2010.610NR
Abies concolor silver cast, combined flowd0.4–1.430–1050.2570.607NR
Picea pungens silver cast, crossflowd0.4–1.425–860.0400.999NR
Picea pungens silver cast, longitudinal flowd0.4–1.425–770.0410.978NR
Picea pungens silver cast, combined flowd0.4–1.425–860.0410.988NR
Pinus pinaster bed of foliagef0.95–7.5050–4000.1420.6050.760
Pinus ponderosa0–1.230–1250.1620.712NR
silver cast, combined flowe cylinder,NR4–400.9110.385NR
perpendicular flowcsensuaNR40–40000.6830.466NR
tube bank, perpendicular flowbNRNR0.3000.578NR
Table 3.  Results of tests for common slope (scaling exponents n) of the linear form of Eqn 7[log(Nu) = log(C) +n log(Re)]
 P. glauca, branchesP. glauca, budsP. glauca, combinedP. contorta, branchesP. contorta, budsP. contorta, combinedA. concolor, silver cast, combined flowdP. pungens, silver cast, combined flowdP. pinaster, bed of foliagefP. ponderosa, silver cast, combined floweCylinder, perpendicular flowcsensuaTube bank, perpendicular flowb
  1. Bold numbers are results of likelihood ratio tests for common reduced major axis (RMA) slope (Warton & Weber, 2002). Nonbold numbers are results of one-sample t-test for RMA slopes against literature slopes.

  2. , not tested; NS, not significant at α= 0.05.

  3. Literature sources: aHilpert (1933); bGlaser (1938); cKnudsen & Katz (1958); dTibbals et al. (1964); eGates et al. (1965); fMendes-Lopes et al. (2002).

P. glauca, branchesP = 0.004P = 0.573(NS)P = 0.862(NS)P = 0.954(NS)F = 2.364F = 29.813F = 2.510F = 0.500F = 29.088F = 5.057
d.f. = 92 d.f. = 92 d.f. = 92 d.f. = 92 d.f. = 92 d.f. = 92
P = 0.128 (NS)P < 0.001P = 0.117 (NS)P = 0.481 (NS)P < 0.001P = 0.027
P. glauca, budsP = 0.004P = 0.018P = 0.006P = 0.002F = 36.692F = 1.368F = 37.323F = 13.144F = 109.891F = 46.932
d.f. = 88 d.f. = 88 d.f. = 88 d.f. = 88 d.f. = 88 d.f. = 88
P < 0.001P = 0.245 (NS)P < 0.001P = 0.001P < 0.001P < 0.001
P. glauca, combinedP = 0.052(NS)P = 0.022P = 0.003F = 44.741F = 13.279F = 45.571F = 10.566F = 163.978F = 60.434
d.f. = 182 d.f. = 182 d.f. = 182 d.f. = 182 d.f. = 182 d.f. = 182
P < 0.001P < 0.001P < 0.001P = 0.001P < 0.001P < 0.001
P. contorta, branchesP = 0.573(NS)P = 0.018P = 0.052(NS)P = 0.484(NS)F = 7.070F = 29.407F = 7.364F = 0.002F = 51.833F = 12.193
d.f. = 91 d.f. = 91 d.f. = 91 d.f. = 91 d.f. = 91 d.f. = 91
P = 0.009P < 0.001P = 0.008P = 0.965(NS)P < 0.001P = 0.001
P. contorta, budsP = 0.862(NS)P = 0.006P = 0.022P = 0.484(NS)F = 1.137F = 23.882F = 1.223F = 0.701F = 18.682F = 2.782
d.f. = 85 d.f. = 85 d.f. = 85 d.f. = 85 d.f. = 85 d.f. = 85
P = 0.289 (NS)P < 0.001P = 0.272 (NS)P = 0.405 (NS)P < 0.001P = 0.099 (NS)
P. contorta, combinedP = 0.954(NS)P = 0.002P = 0.003F = 3.930F = 54.379F = 4.144F = 1.055F = 50.662F = 8.483
d.f. = 178 d.f. = 178 d.f. = 178 d.f. = 178 d.f. = 178 d.f. = 178
P = 0.049 (∼NS)P < 0.001P = 0.043 (∼NS)P = 0.306 (NS)P < 0.001P = 0.004
Figure 2.

Forced convection data and Type II reduced major axis (RMA) regression lines for Picea glauca and Pinus contorta branches and buds. Each point represents an individual specimen. A different regression equation is required for branches and buds of both species (Tables 3 and 4). (a) P. glauca branches and buds; (b) P. contorta branches and buds; (c) combined branches and buds for P. glauca and P. contorta. For P. glauca (a), RMA slopes are significantly different (Table 3) and for P. contorta (b), y-intercepts are significantly different (Table 4).

Table 4.  Results of tests for shift in elevation of the linear form of Eqn 7[log(Nu) = log(C) +n log(Re)]
 P. glauca, branchesP. glauca, budsP. glauca, combinedP. contorta, branchesP. contorta, budsP. contorta, combinedA. concolor, silver cast, combined flowdP. pungens, silver cast, combined flowdP. pinaster, bed of foliagefP. ponderosa, silver cast, combined floweCylinder, perpendicular flowcsensuaTube bank, perpendicular flowb
  1. Bold numbers are results of one-sample analysis of variance (ANOVA) tests for differences in group means y′(Wright et al., 2001). Nonbold numbers are results of one-sample t-test for reduced major axis (RMA) intercepts [log(C)] against literature intercepts [log(C)].

  2. , not tested.

  3. Literature sources: aHilpert (1933); bGlaser (1938); cKnudsen & Katz (1958); dTibbals et al. (1964); eGates et al. (1965); fMendes-Lopes et al. (2002).

P. glauca, branchesF = 22.252F = 58.395F = 45.971t = 7.354t = 2.719t = 3.197
   d.f. = 186d.f. = 180d.f. = 273d.f. = 92 d.f. = 92d.f. = 92  
   P < 0.001P < 0.001P < 0.001P < 0.001 P = 0.004P = 0.001  
P. glauca, budst = 2.179
       d.f. = 88    
       P = 0.016    
P. glauca, combinedF = 36.915
   d.f. = 276        
   P < 0.001        
P. contorta, branchesF = 22.252F = 36.915F = 15.192t = 5.002
d.f. = 186 d.f. = 276 d.f. = 179    d.f. = 91  
P < 0.001 P < 0.001 P < 0.001    P < 0.001  
P. contorta, budsF = 58.395F = 15.192t = 6.798t = 3.023t = 3.412t = 5.180
d.f. = 180  d.f. = 179  d.f. = 85 d.f. = 85d.f. = 85 d.f. = 85
P < 0.001  P < 0.001  P < 0.001 P = 0.002P < 0.001 P < 0.001
P. contorta, combinedF = 45.971t = 10.658t = 5.276
d.f. = 273     d.f. = 178  d.f. = 178  
P < 0.001     P < 0.001  P < 0.001  

Figure 3 compares P. glauca and P. contorta regressions against literature reports for three silver cast branch replicas (A. concolor, P. pungens and P. ponderosa), a bed of foliage (P. ponderosa), and two geometries commonly used to approximate conifer branches and buds (cylinder and tube bank). For a given Reynolds number, conifer foliage (real or replica) reduces Nusselt numbers when compared against cylinders or tube banks. RMA slopes (scaling exponents n) for P. glauca and P. contorta are also significantly different from slopes for cylinders and tube banks. In many cases, RMA slopes (scaling exponents n) for P. glauca and P. contorta branches and buds are not significantly different from published conifer values (Table 3), indicating that Reynolds and Nusselt numbers scale the same in these conifer structures. For those cases where RMA slopes (scaling exponents n) were not significantly different, significant differences in elevation [y′- or y-intercepts log(C)] were observed (Table 4).

Figure 3.

Comparison of forced convection regressions for conifers and their common geometric approximations. Literature sources: 1Hilpert (1933); 2Glaser (1938); 3Knudsen & Katz (1958); 4Tibbals et al. (1964); 5Gates et al. (1965); 6Mendes-Lopes et al. (2002).

Discussion

Foliage was shown to increase the convective resistance (1/h; Eqn 2) of conifer branches and buds (Fig. 3). Convection in leafless branches and buds is typically characterized using regression coefficients for cylinders (Landsberg et al., 1974; Nobel, 1974; Hamer, 1985), and, when compared against cylinder regressions, conifer structures have lower Nusselt numbers (larger convective resistance) for a given Reynolds number (Fig. 3). Conifer foliage increases the convective resistance of branches and buds by creating aerodynamic interference (Grant, 1984), which been shown to also reduce momentum and mass transfer coefficients relative to cylinders (Landsberg & Thom, 1971). Foliage should cause qualitatively similar reductions in heat, momentum, and mass transfer, as described by the Reynolds analogy relating heat, momentum, and mass boundary layer parameters (Thom, 1968; Incropera & DeWitt, 2002). Tube banks have been used to account for the effects of foliage on forced convection (Glaser, 1938; Tibbals et al., 1964; Gates et al., 1965), but, when compared against tube bank regressions, conifer structures again have lower Nusselt numbers, indicating that conifer structures offer more resistance to convection than tube bank geometries (Fig. 3).

Conifer structures had consistently lower Nusselt numbers for a given Reynolds number, regardless of differences in material composition (Fig. 3). Silver cast branch replicas, real branches and buds, and the bed of real foliage all increased the convective resistance when compared against cylinders and tube banks. This indicates that the increased convective resistance is attributable to foliage effects on boundary layer development (Grant, 1984) and not to differences in thermophysical properties between metals (cylinders, tube banks and cast branch replicas) and cellulose (real branches and a bed of real foliage). Silver cast branches have a larger Nusselt number than real branches and the bed of real foliage, because silver has a much higher thermal conductivity than cellulose, which results in lower internal temperature gradients, higher overall heat transfer rates, and an apparently higher Nusselt number (Incropera & DeWitt, 2002).

Conifer foliage increases the convective resistance of branches and buds by changing the fluid mechanics of boundary layer development (Re) and the associated convection heat transfer effects (Nu). This relationship between Re and Nu is characterized by a power function in which the scaling exponent n and normalization constant C (Eqn 7) vary with foliage structure and flow characteristics. Variation in the structural arrangement of foliage among species or the branches and buds within a species causes significant differences in either slopes (scaling exponents n; Table 3) or elevation [log(C); y′- or y-intercept; Table 4] of Eqn 21, so different scaling parameters are required for each of the conifer structures considered here.

The scaling exponents n describe how a change in boundary layer development (Re) causes a change in heat transfer characteristics (Nu). For conifers, regression estimates of scaling exponents n were all between approx. 0.6 and 1.0 (Table 2). Thus, for a unit change in Re, some branches and buds had approximately a square-root change in Nu, while others has an almost proportional relationship between Re and Nu. Significantly larger scaling exponents were observed for conifer foliage structures than for cylinders and tube banks, which approximate cube and square-root relationships, respectively (with the exception of P. contorta buds and tube banks; Table 3). Thus, for a unit change in Re, conifers have a larger change in forced convection characteristics. Conifer scaling exponents n are larger than those of cylinders and tube banks because there is an inverse relationship between free stream velocity and foliage porosity (Grant, 1983, 1984), which is attributable in part to drag-induced streamlining of flexible foliage (Rudnicki et al., 2004). Foliage structure varies among species and also among the branches and buds within a species, so the significantly different scaling exponents that were observed among some conifer structures were expected (Tables 2 and 3). However, many conifers exhibit the same scaling exponents regardless of differences in foliage structure, which indicates that the processes linking boundary layer development and heat transfer effects work in the same way in these cases.

Unlike the scaling exponent n which relates the scale of Reynolds and Nusselt numbers, the normalization constant C has a direct relationship with the Nusselt number and affects the elevation of the relationship between Reynolds and Nusselt numbers. Returning to Eqn 21, log(C) is the y-intercept of this relationship on log10-transformed axes. As the free stream velocity approaches zero (Re approaches an undefined value), log(C) should theoretically begin to reflect the influence of the Grashof number (free convection) on the Nusselt number (Eqn 1). However, the y-intercepts of the regressions presented here do not account for free convection because they were obtained for a convection regime where the Grashof influence is negligible (Fig. 1). In fact, the y-intercepts of forced convection regressions result in negative normalization constants C which would suggest a negative Nusselt number (heat transfer from the branch to the free stream). This is a good example of why regression results should not be applied outside their range of validity. The forced convection regressions reported here should only be used for 0.22 ≤ U ≤ 7.95 m s−1 for P. glauca and 0.16 ≤ U ≤ 7.95 m s−1 for P. contorta (Fig. 1); for free stream velocities approaching zero, free convection regressions for silver cast branch replicas should be useful (Tibbals et al., 1964; Gates et al., 1965).

The regression results presented here were obtained under laminar free stream conditions which allow for reproducibility and comparison with published results. However, free stream flow in forest canopies is turbulent, comprising coherent structures (eddies) from the scale of leaf boundary layers to the scale of the canopy height (Finnigan, 2000). At the smallest scale, eddies increase boundary layer conductance via vortex amplification (Sutera, 1965), while at the largest scale eddies appear to the branch as variations in the mean velocity (Finnigan & Raupach, 1987). Together, various turbulence scales contribute to turbulence intensity (the ratio of standard deviation of free stream fluctuations to the mean free stream velocity) which is related directly to leaf boundary layer turbulence (Grace & Wilson, 1976) and inversely to branch boundary layer thickness (Nobel, 1974). Turbulent flows thus enhance boundary layer transfer rates over laminar flows (Parlange et al., 1971; Pearman et al., 1972; Schuepp, 1972; Chamberlain, 1974; Wigley & Clark, 1974; Grace & Wilson, 1976; Murphy & Knoerr, 1977; Chen et al., 1988a,b), so in a forest canopy we expect that forced convection in conifer branches and buds will be greater than reported here. The effects of turbulent free stream flows should be explored in future heat transfer studies of conifer branches and buds.

The forced convection regressions reported here for P. glauca and P. contorta account for the variation in foliage structure among individual branches and buds. The effect of this variation is reflected by the residual variation in the data (Table 2; Fig. 2), which is greater than typically observed in convection regressions where the same specimen is used in each experiment and only the free stream velocity is varied (e.g. Hilpert, 1933; Tibbals et al., 1964; Gates et al., 1965). The use of a single specimen is appropriate for geometries such as cylinders and tube banks where the absolute size can be varied while preserving geometric similarity, but is less appropriate for geometries such as conifer branches and buds where each specimen is structurally unique. Although less residual variation is observed in conifers when a single specimen is used (Tibbals et al., 1964; Gates et al., 1965), the resulting relationship is valid only for the original specimen and caution must be exercised when extending the results to other branches. The regressions presented here for P. glauca and P. contorta characterize forced convection in populations of branches and buds from a forest canopy.

Acknowledgements

The authors thank R. J. Hugo for technical advice and wind tunnel access. Comments from three anonymous referees improved the manuscript. This work was supported by an NSERC operating grant (EAJ), an Alberta Conservation Association challenge grant (STM), and the G8 Legacy Chair in Wildlife Ecology (EAJ).

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