Tranzflo NZ Ltd 
Specialists in Sap Flow Sensors 
Heat pulse theory—idealized 
Tranzflo NZ Ltd, 15 Parata St, Palmerston North 4410, NEW ZEALAND Tel: +6463574600 Fax: +6463574601 Email: stevegreen@inspire.co.nz 
Heatpulse methods date back some 70 years to the work of Huber (1932) who first conceived the idea of using heat as a tracer of sap flow. Some fifty years later Marshall (1958) developed a theoretical framework for heatpulse, based on a set of analytical solutions to the following heat flow equation, Eq. [1] that ideally represents the two dimensional pattern of temperature surrounding a line heater of zero dimension that is inserted into a section of sapwood of uniform physical and thermal properties. Here, T is the temperature departure from ambient [K], t is time [s], λ is the thermal conductivity [W m^{1} K^{1}] in the axial (x) and tangential (y) directions, a is the fraction of xylem crosssectional area occupied by sap streams moving with a velocity u in the xdirection, and Q is the amount of internal heat that is released from the heater [W m^{3}].
There is a simple relationship between the heatpulse velocity, V [m s^{1}], and the sap flux density, J_{S} = au [m s^{1}], that is given by Eq. [2] where ρ is the density [kg m^{3}], c is the specific heat capacity [J s^{1} m^{1} K^{1}] and the subscripts s and w refer to the sap and the fresh wood, respectively. Following the application of a heatpulse, the temperature rise, T, at a distance of from the line heater is given by (Marshall, 1958) Eq. [3] where is the thermal diffusivity [m^{2} s^{1}] of the sapwood. Swanson (1962) was one of the first to utilize Marshall’s analytical solutions, in his analysis of the ‘compensation’ heatpulse method where two temperature sensors are placed asymmetrically either side of a line heat source. Swanson showed that if the temperature rise is measured at distances x_{U} [m] upstream and x_{D} [m] downstream from the heater, then the heatpulse velocity can be calculated from Eq. [4] where t_{Z} [s] is the time delay for the temperatures at points x_{D} and x_{U} to become equal. Equation [4] implies that the centre of the heatpulse is convected downstream from the heater to reach a point midway between the two temperature sensors after a time t_{Z}. Marshall’s (1958) analytical theory was also used by Cohen et al. (1981) to develop an alternative ‘improved heatpulse method’ that relies on measuring the time, t_{M}, for a maximum temperature rise to be recorded by a single sensor located a distance x_{D} downstream from a line heater. We shall refer to this as the ‘Tmax’ method. The heatpulse velocity, V_{M} [m s^{1}] is calculated from Eq. [5] The only other factor required to determine V_{M} is the thermal diffusivity, κ, which is determined from the following equation Eq. [6] that is calculated at times when zero sapflow occurs. The condition V_{M} = J_{S} = 0 normally occurs at night, when vapour pressure deficits are low, leaf stomata have closed and transpiration losses are close to zero. We refer to the estimate of V [m s^{1}] from Eqs 4 and 5 as the ‘raw’ heatpulse velocity.

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