## 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

(Nomenclature is defined in Table 1.)

Symbol | Definition | Units |
---|---|---|

Roman letters | ||

A | Wetted surface area | m^{2} |

a | Wood–water bond adjustment factor | kJ kg^{−1} °C^{−1} |

C | Normalization constant | Dimensionless |

c | Specific heat capacity | kJ kg^{−1} °C^{−1} |

D | Diameter | m |

g | Gravitational acceleration | m s^{−2} |

h | Forced convection heat transfer coefficient | W m^{−2} °C^{−2} |

k | Thermal conductivity | W m^{−1} °C^{−1} |

L | Length | m |

M | Water content (per cent dry mass) | Dimensionless |

m | Mass | kg |

N | Pure number | Dimensionless |

n | Scaling exponent | Dimensionless |

T | Temperature | °C, K |

t | Time | s |

U | Velocity | m s^{−1} |

V | Volume | m^{3} |

Greek letters | ||

β | Volumetric thermal expansion coefficient | K^{−1} |

θ | Excess temperature ratio | Dimensionless |

µ | Dynamic viscosity | kg m^{−1} s^{−1} |

ν | Kinematic viscosity | m^{2} s^{−1} |

ρ | Mass density | kg m^{−3} |

Dimensionless groups | ||

Gr | Grashof number [Gr = gβ(T_{∞} − T_{s})D^{3}/] | Dimensionless |

Nu | Nusselt number (Nu = hD/k_{a}) | Dimensionless |

Pr | Prandtl number (Pr = c_{p}µ_{a}/k_{a}) | Dimensionless |

Re | Reynolds number (Re = U_{∞}D/ν_{a}) | Dimensionless |

Subscripts | ||

∞ | Free stream | |

a | Air | |

b | Branch or bud | |

cb | Cured branch or bud | |

f | Film | |

fb | Fresh branch or bud | |

i | Iteration | |

n | Needle or fascicle | |

p | Constant pressure | |

s | Surface | |

tot | Total | |

w | Water |

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

(*h*, the average heat transfer coefficient; *D*, the branch or bud diameter; *k*_{a}, the thermal conductivity of air.)

The Reynolds number Re is a measure of dynamic similarity given by the ratio of inertia (/*D*) to viscous (µ_{a}*U*_{∞}/*D*^{2}) forces in the boundary layer

(ρ_{a}, the air mass density; *U*_{∞}, the free stream velocity; *D*, the branch or bud diameter; µ_{a}, the dynamic viscosity of air; ν_{a}, 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

(*g*, the gravitational acceleration; β, the volumetric thermal expansion coefficient; *T*_{∞}, the free stream temperature; *T*_{s}, the branch or bud surface temperature; *D*, the branch or bud diameter; ν_{a}, 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

(*c*_{p}, the specific heat of air at constant pressure; µ_{a}, the dynamic viscosity of air; *k*_{a}, 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

For cylinders perpendicular to the free stream flow, forced convection processes dominate for Gr/Re^{2} < 0.5 and free convection processes dominate for Gr/Re^{2} > 40 (Fand & Keswani, 1973). Here we are concerned with forced convection heat transfer, and, assuming that Gr/Re^{2} < 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

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