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Research Papers

One-Dimensional Transient Heat Conduction in Composite Living Perfuse Tissue

[+] Author and Article Information
S. M. Becker

Department of Mechanical Engineering,
University of Canterbury,
Private Bag 4800,
Christchurch 8401, NZ
e-mail: sid.becker@canterbury.ac.nz

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received December 1, 2011; final manuscript received October 27, 2012; published online June 21, 2013. Assoc. Editor: Franz-Josef Kahlen.

J. Heat Transfer 135(7), 071002 (Jun 21, 2013) (11 pages) Paper No: HT-11-1542; doi: 10.1115/1.4024063 History: Received December 01, 2011; Revised October 27, 2012

Modeling the conduction of heat in living tissue requires the consideration of sudden spatial discontinuities in property values as well as the presence of the body's circulatory system. This paper presents a description of the separation of variables method that results in a remarkably simple solution of transient heat conduction in a perfuse composite slab for which at least one of the layers experiences a zero perfusion rate. The method uses the natural analytic approach and formats the description so that the constants of integration of each composite layer are expressed in terms of those of the previous layer's eigenfunctions. This allows the solution to be “built” in a very systematic and sequential manner. The method is presented in the context of the Pennes bioheat equation for which the solution is developed for a system composed of any number of N layers with arbitrary initial conditions.

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Figures

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Fig. 1

Steady state two-layer system composite representation in which transport in layer i − 1 is governed by pure conduction and transport within layer i has both perfusion and conduction

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Fig. 2

Composite system representation of the perfuse tissue slab

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Fig. 3

Schematic of the three layered slab in which the outer layers are exposed to convection at the periphery, the first layer experiences no perfusion, and the third layer experiences volumetric energy generation

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Fig. 4

Maximum error at small dimensionless times defined by Eq. (52): comparison between the test case solution and the semi-infinite Green's function solution of Eq. (51)

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Fig. 5

Transient solution of the case study at different dimensionless times corresponding to test case parameter values: δ1 = 9/200, δ2 = 1/50, δ3 = 1/1000, HO = 1, HL = 10, m1 = 0, m2 = 1, m3 = 4, K2/K1 = 5/9, K3/K1 = 5/18, Δθ2 = 0, Δθ3 = −1, ΔθO = 1, ΔθL = 1, and ϕ3 = 5

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