A new formulation of the combined macro and microvascular model for heat transfer in a human arm developed in Song et al.  is proposed using a recently developed approximate theory for the heat exchange between countercurrent vessels embedded in a tissue cylinder with surface convection . The latter theory is generalized herein to treat an arm with an arbitrary variation in cross-sectional area and continuous bleed off from the axial vessels to the muscle and cutaneous tissue. The local microvascular temperature field is described by a “hybrid” model which applies the Weinbaum-Jiji  and Pennes  equations in the peripheral and deeper tissue layers, respectively. To obtain reliable end conditions at the wrist and other model input parameters, a plethysmograph-calorimeter has been used to measure the blood flow distribution between the arm and hand circulations, and hand heat loss. The predictions of the model show good agreement with measurements for the axial surface temperature distribution in the arm and confirm the minimum in the axial temperature variation first observed by Pennes  for an arm in a warm environment.
A Three-Dimensional Variable Geometry Countercurrent Model for Whole Limb Heat Transfer
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Zhu, M., Weinbaum, S., and Lemons, D. E. (August 1, 1992). "A Three-Dimensional Variable Geometry Countercurrent Model for Whole Limb Heat Transfer." ASME. J Biomech Eng. August 1992; 114(3): 366–376. https://doi.org/10.1115/1.2891397
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