Date of Award

Winter 2011

Document Type


Degree Name

Doctor of Philosophy (PhD)


Computational Analysis and Modeling

First Advisor

Weizhong Dai


Heat transfer in living tissue has become more and more attention for researchers, because high thermal radiation produced by intense fire, such as wild fires, chemical fires, accidents, warfare, terrorism, etc, is often encountered in human's daily life. Living tissue is a heterogeneous organ consisting of cellular tissue and blood vessels, and heat transfer in cellular tissue and blood vessel is quite different, because the blood vessels provide channels for fast heat transfer. The metabolic heat generation, heat conduction and blood perfusion in soft tissue, convection and perfusion of the arterial-venous blood through the capillary, and interaction with the environment should be also considered in heat transfer in living tissue. To understand the effect of high thermal radiation on biological tissues, specifically, thermo-mechanical damage to the tissue, a mathematical model for skin injury induced by radiation heating has been developed by W. Dai et al. in 2008 [19], where the skin was considered to be a 3D triple-layered structure with an embedded three-level dendritic countercurrent vascular network.

Since there are up to seven layers of blood vessels in the skin tissues [25], the motivation of this dissertation research is to extend the mathematical model developed by W. Dai et al. in 2008 [19] to the case that considers a seven-level dendritic countercurrent vascular network, where the dimensions and blood flow of the blood vessels are determined based on the constructal theory of multi-scale tree-shaped heat exchangers. As such, the number of the blood vessels is increased from eight to one hundred and twenty eight. This makes the computation much more complicated. To this end, blood flow oriented coordinates system was first designed, so that a simple energy equation in blood vessels can be obtained and solved using the fourth-order Runge-Kutta method. Coupled with the mathematical model and numerical schemes developed in W. Dai's paper [19], the temperature distribution in a living skin tissue embedded with a seven layered dendritic countercurrent vascular network is able to be predicted, and hence the skin burn injury induced by radiation heating can also be predicted. Furthermore, the numerical scheme is proved to be unconditional stable and the Preconditioned Richardson iteration developed for the computation is convergent. Unconditional stable scheme (no restriction on mesh ratio) is particularly important in this research since the thickness of the first layer of the skin structure is small and hence the grid size in the thickness direction can be very small. The developed Precondition Richardson iteration allows us to transform a complicated solution system to a tridiagonal linear system, so that the conventional Thomas Algorithm can be easily used, and hence the computation cost can be reduced.

Numerical results show that there is no difference between the current study and the previous study regarding the area of the high degree burn injury. However, the areas of the first and second degree burn injury are different from those obtained in W. Dai's paper [19], because of the more complex countercurrent vascular network that is used in the present model. The obtained model and numerical method in this dissertation could be used in future studies: e.g., by considering a larger area of skin structure with complicated dendritic countercurrent multi-level blood vessels, as well as modeling such well documented effects of thermal damage as skin wrinkles and tissue shrinkage.