External Heat Transfer

This section describes the external heat transfer model.

The human body exchanges heat with the environment through the following heat exchange mechanisms:

Heat transfer due to convection and radiation
Heat losses due to diffusion, transpiration, and respiration.

Each heat loss is consolidated to a single value for the global heat balance.

Convection

The heat flux of an undressed skin element of a segment j is computed as:

{\dot{Q}}_{{C O N V}_{j}} = α_{{C O N V}_{j}} \cdot (T_{4, j} - T_{{A I R}_{j}})

(32)

where:


${\dot{Q}}_{{C O N V}_{j}}$	Convective heat transfer coefficient of segment j
$T_{4, j}$	Skin temperature of segment j
$T_{{A I R}_{j}}$	Temperature of air

Radiation

Heat transfer due to radiation takes place between the skin surface or the clothes surface of the human body and the surrounding surfaces. By use of the Stefan-Boltzmann law, a radiation heat balance for each skin segment can be formulated as follows:

{\dot{Q}}_{{R A D}_{j}} = ϵ_{{S K I N}_{j}} \cdot σ \cdot A_{j} \cdot (T_{4, j}^{4} - T_{{E N V}_{j}}^{4})

(33)

where:


$ϵ_{{S K I N}_{j}}$	Emissivity of skin of segment j
$σ$	Stefan-Boltzmann constant
$A_{j}$	Skin surface of segment j
$T_{{E N V}_{j}}$	Environment temperature of segment j

For temperature differences between the skin and the environment that are smaller than 20 K, the following linear approximation can be used:

{\dot{Q}}_{{R A D}_{j}} = α_{{R A D}_{j}} \cdot A_{j} \cdot (T_{4, j} - T_{{E N V}_{j}})

(34)

where $α_{{R A D}_{j}}$ is a linearized heat transfer coefficient for radiation.

Using the equations for convection and radiation above, the two exterior heat fluxes are combined to an overall value — the so-called dry heat flux ${\dot{Q}}_{D R Y}$ . The heat transfer coefficients for convection and radiation are combined to a joint coefficient $α_{{D R Y}_{j}}$ :

α_{{D R Y}_{j}} = α_{{R A D}_{j}} + α_{{C O N V}_{j}}

(35)

With this dry heat transfer coefficient, an effective temperature $T_{{E F F}_{j}}$ is calculated for each skin element. The effective temperature is formed by using the temperature of the ambient air $T_{A I R}$ and the temperature of the surrounding surfaces $T_{E N V}$ :

T_{{E F F}_{j}} = \frac{α_{{C O N V}_{j}} \cdot T_{{A I R}_{j}} + α_{{R A D}_{j}} T_{{E N V}_{j}}}{α_{{D R Y}_{j}}}

(36)

The equation for the heat loss due to convection and radiation can now be written as:

{\dot{Q}}_{{D R Y}_{j}} = α_{{D R Y}_{j}} \cdot A_{j} \cdot (T_{4, j} - T_{{E F F}_{j}})

(37)

With the above equation, all dry heat losses are described for the total heat balance.

Respiration

Heat loss due to respiration occurs because of the difference in humidity and temperature between incoming and outgoing air.

The heat loss due to the humidity difference is proportional to the water vapor pressure gradient between the lung and the ambient air. It also depends on the inhaled air volume and, as such, from the metabolic rate. The heat loss due to respiration is given by:

{\dot{Q}}_{R E S P, H U M} = 1.752 \cdot 10^{- 5} \cdot {\dot{Q}}_{M E T} \cdot (p_{S A T, L U N G} - p_{A I R})

(38)

The heat loss due to the temperature difference of the inhaled air is given by:

{\dot{Q}}_{R E S P, T E M P} = 0.0014 \cdot (34 - T_{A I R})

(39)

The total heat loss due to respiration is therefore:

{\dot{Q}}_{R E S P} = {\dot{Q}}_{R E S P, H U M} + {\dot{Q}}_{R E S P, T E M P}

(40)

The total heat loss due to respiration is extracted one half each from the head core and the torso core.

{\dot{Q}}_{{R E S P}_{1, 1}} = \frac{1}{2} {\dot{Q}}_{R E S P, H E A D}

(41)

{\dot{Q}}_{{R E S P}_{1, 2}} = \frac{1}{2} {\dot{Q}}_{R E S P, T O R S O}

(42)

Diffusion

Because of perspiratio insesibilis, the diffusion of water through the human skin, there is a permanent heat loss ${\dot{q}}_{D I F F}$ to the environment. According to [10], this heat loss can be specified as 5.6 W/m² for the entire body. As the heat loss is distributed to all skin elements, an equal rate of 5.6 W/m² can be assumed for all skin elements. The heat flux due to diffusion is thus given by:

{\dot{Q}}_{{D I F F}_{j}} = {\dot{q}}_{D I F F} \cdot A_{j}

(43)

where $A_{j}$ is the area of a segment.

Transpiration

A further form of heat release is the evaporation of sweat on the surface of the skin during transpiration. During transpiration, water vapor transport takes place through the boundary layer. The driving force is the difference between the vapor pressure at the body surface and the vapor pressure of the ambient air. The heat flux due to transpiration is thus given by:

{\dot{q}}_{T R A N S} = β \cdot ρ \cdot r \cdot (X_{S K I N} - X_{A I R})

(44)

where:


$X_{S K I N}$	Moisture content of air at the skin surface
$X_{A I R}$	Moisture content of ambient air
$β$	Mass transport coefficient
$r$	Heat of evaporation of water

Taking into account the analogy between heat transfer and mass transport:

β = \frac{α_{C O N V}}{ρ \cdot c_{p} \cdot {L e}^{0.67}}

(45)

The Lewis number Le does not deviate much from a value of one for the considered range of application. Therefore, the following relation can be assumed for the surface heat flux due to transpiration evaporation:

{\dot{q}}_{T R A N S} = \frac{α_{C O N V}}{c_{p}} \cdot r \cdot (X_{S K I N} - X_{A I R})

(46)

Environmental conditions lead to a maximum evaporation rate. When the environment is saturated, sweat can no longer evaporate. Drops of sweat form on the skin that do not contribute to the heat loss. The maximum evaporation rate is given by:

Q_{{E V A P, \max}_{j}} = k \cdot α_{C O N V} \cdot A_{j} \cdot (p_{S A T, S K I N} - p_{A I R})

(47)

where $k = 0.0165 K / P a$ is the evaporation coefficient [4].

The heat losses due to diffusion and transpiration are summed as:

{\dot{Q}}_{{E V A P}_{j}} = {\dot{Q}}_{{D I F F}_{j}} + {\dot{Q}}_{{T R A N S}_{j}}

(48)