Mathematical Models for Heat Transfer in Human Body

Abstract

The work is focused on human thermal comfort, which depends not only on physical, biological, and environmental heat transfer mechanisms but also on clothing parameters. In excessively hot or cold climatic conditions, the physiological imbalance and overall human thermal comfort can be maintained through the proper management of well-designed protective clothing. We have developed both linear and nonlinear mathematical models for heat transfer in the human body. For the linear case, Pennes’ model is extended incorporating clothing thermal conductivity, the thickness of cloth, and clothing area factors. The extended model attempted to use the interface condition between the human body and cloth. The usual Robin’s boundary condition is modified by incorporating effective clothing area factor and convective heat transfer coefficient as two important factors. Where the effective clothing area factor includes clothing insulation, air insulation, and clothing area factor. The convective heat transfer coefficient includes air velocity and the walking speed of a person. The interface between two non-homogeneous materials and the radiation-induced nonlinearity in Robin’s boundary condition makes the problem difficult and intricate to solve analytically even in a simple geometry except for the steady state case. As such, we investigated the thermal responses of clothing insulation, air velocity and walking speed, metabolic and sweating effect, using numerical methods to the extended transient model and in varying spatial dimensions in cylindrical coordinates: one-dimensional radial, two-dimensional axisymmetric, and three-dimensional. First, we solved the one-dimensional model in radial direction numerically employing an implicit finite difference method. Solvability, consistency, stability, and convergence of the numerical scheme are established. Numerical simulation results exhibited that the light garment system would be comfortable and easy for sweat drainage. To address more realistic problems of finding both radial and longitudinal variations of the temperature profile in human limbs, we extended our model to two-dimensional axisymmetric case with time-dependent metabolism, temperature-dependent sweating, and clothing effects during physical exercise. In the radial direction, the numerical simulation results agree with the one-dimensional model whereas in the longitudinal direction there seem no remarkable variations observed which were also expected as the ratio of radial and axial length scales are significantly different. Finally, we further extended the model to three-dimension with temperature-dependent thermophysical parameters to address non-symmetric temperature variations in the abnormal tissue. The simulated results using Finite Volume (FV) energy conservation techniques showed the nonlinear behavior of temperature in an abnormal tissue with an initial high temperature. Numerical tests with different lateral boundary conditions exhibit some non-symmetric variations in temperature in the cross-section. Various simulations for the axisymmetric model in (r, z)-direction were performed with zero flux at the inner, bottom, and top of the limb and Robin’s boundary condition at the lateral surface to reveal an axially symmetric temperature profile. While implementing Robin’s boundary condition at the top and the lateral surface of the limb, non-symmetric temperature variation in the skin surface and the top of the longitudinal cross-sectional slice is obtained. Additionally, numerical experiments have shown that various coefficients of temperature-dependent parameters, thermal conductivity $K(T), metabolic heat generation qm(T) and blood perfusion wb(T), are directly proportional to the temperature of abnormal tissue.