Thermal Transpiration

This section describes the Thermal Transpiration model.

The global controlled variable, transpiration, is calculated as:

S_{T R A N S P} = G_{T R A N S P, K K} \cdot θ_{1, 1} + G_{T R A N S P, H A} (θ_{W A, T O T} - θ_{C O, T O T})

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The basis for this global controlled variable and the used coefficients are experiments described in [11].

The following assumptions are made:

Transpiration is proportional to the head core temperature at a constant average skin temperature.
Transpiration is proportional to the average skin temperature at a constant head core temperature.

The global constants for calculating the transpiration signal have the following values:

G_{T R A N S P, K K} = 372.2 \frac{W}{K}

G_{T R A N S P, H A} = 33.7 \frac{W}{K}

The above equation means that the influence of the core head temperature is approximately 11 times higher than of the skin temperature.

The local effect of transpiration is accounted for by a factor that depends on the control deviation and consequently on the temperature of each skin element. The factor behaves in such a way that when a local control deviation of the skin temperature occurs, an n-fold amplification of transpiration takes place. The value of this amplification factor can vary in literature. For this model, the value from [10] is used, which is 2.

{\dot{Q}}_{{T R A N S P}_{j}} = (L_{T R A N S P} S_{T R A N S P}) \cdot 2^{\frac{θ_{4, j}}{10}}

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The distribution of sweat formation across the skin surface is assumed to correspond to the distribution of perspiratory glands on the skin surface . The values of this local controlled variable are listed in the following table:


Segment	$L_{T R A N S P}$
Head	0.081
Torso	0.481
Upper arm	0.039
Forearm	0.039
Hand	0.016
Thigh	0.064
Lower leg	0.043
Foot	0.018

The local transpiration rate is limited by the maximum possible evaporation. The evaporation also includes diffusion. This maximum evaporation rate tells how large the total evaporation heat loss of the body to the environment can be at maximum. This limitation results from the vapor pressure difference between the skin surface and the environment and additionally depends on the convective heat transfer coefficient.

If the computed value for the evaporation heat loss exceeds the sum of the constant diffusion heat flux ${\dot{Q}}_{D I F F}$ and the evaporation through transpiration ${\dot{Q}}_{T R A N S P}$ , the maximum possible value is taken. Therefore, the heat loss due to transpiration is reduced:

{\dot{Q}}_{{T R A N S P}_{j}} = {\dot{Q}}_{{E V A P, M A X}_{j}} - {\dot{Q}}_{{D I F F}_{j}}

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The ratio of effective transpiration to maximum possible transpiration yields the degree of moisture $w_{j}$ , that is, the fraction of moistened skin to total skin surface.

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

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