# Numerical analysis of thermal processes in the system protective clothing - biological tissue subjected to an external heat flux

### Bohdan Mochnacki

,### Mateusz Duda

Journal of Applied Mathematics and Computational Mechanics |
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NUMERICAL ANALYSIS OF THERMAL PROCESSES IN THE SYSTEM PROTECTIVE CLOTHING - BIOLOGICAL TISSUE SUBJECTED TO AN EXTERNAL HEAT FLUX

Bohdan Mochnacki, Mateusz Duda

University of Occupational Safety
Management in Katowice, Katowice, Poland

bmochnacki@wszop.edu.pl, mduda@wszop.edu.pl

**Abstract.** The non-homogeneous fragment of biological tissue is considered. Its
shape roughly corresponds to the fragment of cross-section
of the upper or lower limb. The tissue domain is protected by a layer of protective clothing. The
purpose of numerical computations is to examine the effectiveness of the
clothing insulation layer on the action of
the external heat fluxes of differing intensity. Thermal
processes in the tissue domain
are described by the system of the Pennes equations. This system is
supplemented by the appropriate boundary-initial conditions and the energy
equations determining the transient temperature field in the fabric and air gap
sub-domains (the air gap is treated as a solid body). At the stage of numerical
computations, the program MSC.Marc has been used.
In the final part of the paper, the examples of numerical simulations and also
the conclusions are presented.

*Keywords: **bio-heat transfer,
protective clothing, tissue heating, numerical simulations*

1. Introduction

Heat transfer processes proceeding in the domain of biological tissue can be described by the Fourier-type equation called the Pennes one [1-4]. The Pennes equation contains two internal source functions connected with the blood perfusion and the metabolism. The mathematical form of perfusion heat source results from the assumption that the tissue is supplied with a large number of capillary blood vessels uniformly distributed throughout its volume (a soft tissue model). The metabolic heat source can be considered in the form of a temperature-dependent function but, as a rule, is treated as the constant value.

The transient temperature field in the domain of protective clothing is determined by the Fourier equation (or in the case of multi-layered fabric by the system of these equations). In this paper the air gap (a trapped air) between the clothing and skin tissue is treated as a solid body, while the thermal conductivity of this domain is defined in a special way.

The external surface of protective clothing
is subjected to the external heat flux *q _{b}*

_{ }[W/m

^{2}] and the different values of

*q*

_{b}_{ }have been considered. The aim of successive numerical simulations was to determine the permissible residence times near the heat sources ensuring safe working conditions for the people at risk of burns. The acceptable times result from the temporary temperatures on the surface of the skin tissue and the internal surface of fabric.

At the stage of numerical computations, the commercial code MSC. Marc has been used.

2. Governing equations

The transient temperature field in the 2D non-homogeneous tissue domain oriented in the Carthesian co-ordinate system is described by the partial differential equations (the Pennes equations) in the form

(1) |

where *e* = 1,…,4 distinguishes
the tissue sub-domains, this means skin, fat, muscle and bone, respectively, *c _{e}*
is the volumetric specific heat,

*λ*is the thermal conductivity,

_{e}*Q*and

_{p,e}*Q*are the capacities of volumetric internal heat sources resulting from the blood perfusion and metabolism, and

_{m,e}*T*,

*x*,

*y,*

*t*denote temperature, spatial co-ordinates and time, respectively. The perfusion heat source is given by the formula:

(2) |

where *G** _{b,e}* is the blood
perfusion rate [m

^{3}blood/(s m

^{3}tissue)],

*c*is the blood volumetric specific heat and

_{b }*T*is the arterial blood temperature. Metabolic heat source

_{B }*Q*can be treated both as the constant value or the temperature-dependent function.

_{m,e }On the contact surface between the tissue sub-domains, the ideal thermal contact is assumed, namely

(3) |

where ¶*T*/ ¶*n* denotes a temperature derivative in a
normal direction.

The temperature field in the domain of
homogeneous fabric (*T _{f}*) and air gap domain (

*T*) is described by the well-known Fourier equations

_{a}(4) |

and

(5) |

where* c _{f}* ,

*c*are the volumetric specific heats,

_{a}*λ*,

_{f}*λ*are the thermal conductivities.

_{a}The problem of the air thermal conductivity requires additional explanation. The trapped air between the fabric and tissue is not moving (the convection does not occur). The heat transfer occurs by the radiation and conduction, in particular

(6) |

where is
the radial heat transfer coefficient, is
the air thermal conductivity, while_{ }_{ }is
the thickness of the air gap. The total heat flux is equal to

(7) |

Now, the substitute air thermal conductivity_{
}_{ }is introduced:

(8) |

or

(9) |

wherein

(10) |

and

(11) |

For example: the
mean temperatures of fabric and skin surfaces are equal to
52 and 32°C (325 and 305 K). Mean air temperature equals 42°C. For this
tem-perature thermal conductivity *λ** _{a}* = 0.027 W/mK.
Additionally,

_{ }= 0.95,

_{ }=

**0.9 and = 0.005 m. For the above data the substitute thermal conductivity**

*λ*

*= 0.057 (see Fig. 2.1). The values of temperature-dependent air specific heat and mass density can be found in tables (e.g. [5]).*

_{z}Fig. 2.1. The substitute thermal
conductivity *λ** _{z}* in temperature relation

Between the fabric, air gap and skin sub-domains, the boundary condition (3) is assumed, while between blood vessels and tissue the Robin condition is taken into account

(12) |

where *α _{B}* is the heat
transfer coefficient,

*T*

*is the blood temperature. The value of*

_{B}*α*has been found under the assumption that the Nusselt number for blood vessels is equal to 4 [6], the values of arterial and vein temperatures values have been accepted intuitively. The problem of thermal interactions between blood vessels and soft tissue is considered in the case of the so-called vessel models applications (e.g. [7]) and will not be discussed here.

_{B}The initial conditions are also given:

(13) |

At the stage of numerical computations, the initial temperatures in domains of fabric and air gap have been assumed as the constant values. It is also possible to introduce the initial temperature distribution resulting from the solution corresponding to the steady state problem solved for the assumed set of external parameters [4].

3. Numerical solution of the problem

The considered domain is shown in Figure 3.1. It presents the repeatable segment of a circular cross-section containing the successive tissue layers in a sequence typical for limbs anatomy [8, 9]. Assumed dimensions of sub-regions approximately correspond to the forearm section. Taking into account the geometrical and thermal symmetry of the segment considered, on the left and right boundaries of the model the no-flux conditions can be accepted. One may notice that the layers corresponding to protective clothing and trapped air are also taken into account.

Fig. 3.1. Left: The domains of human limb secured by protective clothing. Right: The temperature response [°C] of human forearm exposed to the non-hazardous environmental conditions. The domains of fabric and air gap have been disabled from the results visualisation

Physical parameters of
human tissue were taken from the paper [9]. The parameters of air and protective clothing (Cotton/Spandex) have been generated based on the online calculator [10]
using the information collected in [5, 11-13]. The courses of the fabric
thermal conductivity *l _{f}* [W/mK] and the specific heat

*C*[J/kgK] in relation to temperature [°C] are shown in Figures 3.2 and 3.3 (the mass density of fabric is equal to

_{f}*r*= 365 kg/m

^{3}).

At the first stage of computations, the
initial condition for tissue domain has been determined for the non-hazardous
ambient temperature *T _{a}*

_{0 }= 20

^{o}C (heat transfer coefficient

*a*= 3 W/m

^{2}K). The solution corresponding to the steady state conditions is shown in Figure 3.1 and it is the starting point for further computations.

Fig. 3.2. The conductivity *l* of
Cotton/Spandex protective clothing in relation
to temperature

Fig. 3.3. The
heat capacity *c _{p} *of Cotton/Spandex protective clothing in
relation
to temperature

4. Results of computations

The external heat fluxes_{ }_{ }from the range [900, 2100 W/m^{2}] warm the
external face of the fabric at the second stage of
computations. The results below presented correspond to
points A and B marked in Figure 3.1. The heat is transferred by a layer of
protective clothing and trapped air to the biological tissue. The border
temperature when the burn of tissue could occur is assumed on the different
level depending on the authors, i.e. 42^{o}C [14], 44^{o}C [15]
(temperature between epidermis and dermis). To estimate the burn degree the
residence time at this temperature
is essential (Henriques integral [3]). The permanent damage of biological
tissue (destruction of human protein) occurs when the temperature reaches 55°C.
In this study, the border temperature was established as 40^{o}C due to
the fact that not only
the temperature of biological tissue should be taken into account but also the
tem-perature of the internal surface of the fabric layer - the heated
protective clothing could indirectly cause the burns, too.

Fig. 4.1. The heat response of human skin exposed on different heat flux in time relation. Point B from Figure 3.1

Fig. 4.2. The heat response of fabric (internal surface - point A) exposed on different heat fluxes in time relation

Fig. 4.3. Example of results: the temperature
response [°C] of human forearm exposed
to heat flux (_{ }= 1500 W/m^{2}; *t*
= 282 s and *t *= 324 s). Corresponding domains
to the fabric and air gap have been turned off from
results visualisation

5. Conclusions

The first or second
degree burn occurs when the heat penetration reaches
the dermis region. The time of hazardous
temperature appearance (e.g. 40°C)
in the above region is the different one, of course. In the case of the highest
value of heat flux = 2100 W/m^{2} the critical
time is equal to 240 s, while the lowest value of heat flux = 900 W/m^{2} this time is equal
to 360 s. The difference of
2 minutes shows the influence of the heat flux level on the heat penetration
process. The high temperatures of tissue could lead not only to burns, but also
to the heat stroke (excessive accumulation of heat in body caused by intense
influx from the environment with a difficult outflow to the environment at the
same time).
The temperature when the heat stroke occurs is equal to 43^{o}C [16, 17]
but it should be noticed that the high temperature should appear in the
considerable parts
of tissue subdomains. This can cause an increase of
the blood temperature
which leads directly to overheating of the whole body. The protective clothing
has, as a rule, the layered structure and the
introduction of the fabric heterogeneity to the computer program is
quite easy and uncomplicated.

Acknowledgements

*This work is supported by the project No.
PB3/2013 sponsored by WSZOP
Katowice.*

References

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