Mathemat ical model

17.4 Mathemat ical model

It is assumed, that the heat exchange between the sectors is negligible at every cross section of the human body. Therefore both the vertical and cir- cumferential heat transfer processes are not taken into account. The heat transfer along the radius x within each sector is described by the one di- mensional heat conduction equation derived from the energy conservation consideration. The change of the internal energy in the sector element with coordinates x and x C dx due to heat transport within the time dt is

@T dQ 1 D k.x C dx/S.x C dx/

@T

@T

k.x/S.x/

k.x/S.x/ dt

where T .x; t/ is the temperature, k is the thermal conductivity, S.x/ D ˛xh is the sector arc length multiplied with its height h and ˛ is the central angle

175

F igure 17.2: Horizontal cross section of the human body represented as an ellipse with five layers: 1- inner core, 2- outer core, 3- muscles, 4- fat, 5- skin.

Figure 17.3: Ice protection construction.

1- polyurethane foam, 2- ice briquette, 3- human body, 4-special overheating protection clothes, 5- polyurethane net (air layer).

Table 17.2: Radii of the elliptical cross sections used in simulations radial

upper elliptical lower elliptical

line

cylinder, m

cylinder, m

Table 17.3: Radii of layers used in simulations in fraction of the skin thickness

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Elliptical Element Area

Layer

Radius

cylinder length, m

Skin Core

of the sector. Within the material we have additionally the heat release with the heat density F (heat per unit volume):

(17.2) Due to the heat transfer and heat release the internal energy of the material

dQ 2 D SF dxdt

is changed by dQ 3

where c p By conservation of energy

C S.x/f .x/

is the thermal diffusivity and f .x/ D F .x/=.c p . The equation (17.4) is solved at the initial condition

(17.5) and the boundary conditions formulated at the sector center

T .x; 0/ D '.x/

(17.6) and at the outer boundary of the clothes

T .0; t / D T core

T .x out er ;t/DT med

The temperature of the surrounding medium was T med D 50 0 . It was as- sumed that the core center (at x D 0) temperature was constant in time T core D 36:7 0 .

The heat transfer process is subdivided into three steps. Within the first step the ice is warmed up to the melting temperature. The melting process is followed by the absorption of the heat coming from both the surrounding medium Q med

and the human body Q hum caused by metabolism

med is the thermal diffusivity of the clothing layer adjacent to the ice layer, A med is the surface between the ice layer and adjacent clothing ski n is the thermal diffusivity of the human skin and A ski n is the skin surface in the sector. The second step is the ice layer melting. The phase transition in matter is described by the Stefan problem [40]. In this paper we used a simplified approach based on the heat balance conditions. It is assumed that the water and ice mixture is uniform with the constant temperature of zero degree of Celsius. The dynamics of the boundary between the water and ice is not considered. The heat fluxes (17.8) and (17.9) are constant in time. The second step is finished in time t once the ice is melted:

(17.10) Here Q mel t is the heat necessary to melt the whole ice. After that, within

Q mel t D.P Q med CP Q ski n /t

the third step the water from the melted ice is warming up due to heat fluxes (17.8) and (17.9) continuing in time. The calculations within the third step are running as long as the comfort conditions are fulfilled.

The equation (17.4) is solved using the finite differential method on the numerical implementation utilizes the central differential scheme for space

derivatives and Crank Nicolson implicit representation of the unsteady term. An inhouse code was developed for this purposes.