# Improved Smart Pillow for Remote Health Care System
## 2 Materials and Methods
### Bioheat Transfer Principles  
Heat transfer [19] within the human body is called bioheat transfer [20] that is more complex than in conventional environments because of the metabolism and blood flow, both of which are related to heat exchange. Heat transfer occurs as a difference of temperature, with non-invasive MCBT being difficult to determine as ambient temperature is volatile. Conduction, convection, and radiation are three types of heat transfer processes. [PERSON]'s Law of Heat Conduction describes the amount of energy being transferred per unit time in the following equation:  
\[\frac{dq}{dt}=-k\frac{dT}{dx} \tag{4}\]  
where the heat flux, \(dq/dt\), is the flow of heat through a unit area per unit of time, \(dT\) is the temperature difference along a distance \(dx\) in the direction of conduction and \(k\) is the thermal conductivity of the material. Likewise, [PERSON]'s Law of Cooling is the expression for convection:  
\[\frac{dq}{dt}=h(T_{s}-T_{\infty}) \tag{5}\]  
where \(T_{s}\) is the temperature of surface, \(T_{\infty}\) is the temperature of the environment, and the parameter \(h\) is the convection heat transfer coefficient. Radiation is the surface emissive power of matter; \(E\) is prescribed by the Stefan-Boltzmann Law:  
\[E=\varepsilon\sigma T_{s}^{4} \tag{6}\]  
where \(T_{s}\) is the absolute temperature (\(K\)) of the surface and \(\sigma\) is the Stefan Boltzmann constant, \(\varepsilon\) is a radiative emissivity of the surface, whose values are in the range of \(0\leq\varepsilon\leq 1\). Density (\(\rho\)) and specific heat (\(c_{p}\)) are two properties used extensively in thermodynamic analysis. The product \(\rho c_{p}\), commonly called the volumetric heat capacity, measures the ability of a material to store thermal energy. Thermal diffusivity \(\alpha\) is the ratio of the thermal conductivity \(k\) to the heat capacity \(\rho c_{p}\):  
\[\alpha=\frac{k}{\rho c_{p}} \tag{7}\]  
Large \(\alpha\) means a quick change in thermal environment, whereas small \(\alpha\) means a slow response. The other concept of thermal transfer is thermal resistance:  
\[R=\frac{\Delta T}{q} \tag{8}\]\(\Delta T\) is the difference of temperature, similar to \(U\) in electricity, and \(q\) is the heat flux which is similar to electric current \(I\). The balance of thermal energy for a control volume at an instant can be expressed as:  
\[q_{st}=q_{in}-q_{out}+q_{g} \tag{9}\]  
where \(q\) stands for the sum of thermal and mechanical energy, \(q_{st}\) as stored energy, \(q_{in}\) and \(q_{out}\) as energy input and output and \(q_{g}\) for energy generation. The expansion of this equation results to the Penn equation:  
\[\rho c\frac{\partial T}{\partial t}=\
abla(k\
abla T)+\omega_{b}\rho_{b}c_{b} (T_{a}-T)+q_{m} \tag{10}\]  
With the value of \(q_{st}\), \(q_{in}\) includes the first two terms in the right, the first term is from thermal conduction and the gradient in temperature within the tissue, while the second term is from blood perfusion, where \(\omega\) is the blood perfusion rate, \(\rho_{b}\) and \(c_{b}\) are the density and specific heat of the blood, and \(T_{a}\) is the temperature of the arterial blood. \(q_{m}\) represents \(q_{g}\), which is the metabolic heat generation in the skin tissue. Heat transfer at the outer surface of the human body can consist of a combination of all kinds of boundary conditions (\(BC\)) that are convection, radiation and sweat. It can be written as:  
\[-k\frac{\partial T}{\partial x}=h(T_{\infty}-T)+\varepsilon\sigma\Big{(}T_{ \infty}^{4}-T^{4}\Big{)}+q_{e} \tag{11}\]  
The parts of convection and radiation are explained before, with \(q_{e}\) being the evaporative heat losses due to sweat. According to the analysis from Pompei [21], \(q\) heat lost from the skin to the environment can be modeled as:  
\[q=HA(T_{skin}-T_{env}) \tag{12}\]  
\(H\) is a combined heat transfer coefficient, \(A\) is the area of surface, and \(T_{skin}\) and \(T_{env}\), is the temperature of skin and environment, respectively.