Thermal transfer in a multilayer plate with source of metabolic heat

In this article heat transfer in a multi-layered plate (human skin and thermal insulation) with various boundary conditions is considered. This allows you to justify the choice of air temperature in the production premises.


Introduction
The air temperature (T A ) in the production room is the most significant factor during design heating and ventilation systems, the TA value is selected in such a way as to ensure the requirements of the "thermal comfort" of the worker are met [1].But by now the choice of T A is based on empirical data on human sensations and this is "subjective" (it is assumed that TA should be in the range 18 °C to 24 °C [2][3][4]), and the mathematical justification is "not enough" described [5].In this connection, the mathematical modelling of heat transfer within the framework of the heat conduction model (which allows us to describe the process of heat exchange of a person with the environment quite accurately) is an actual research direction.
The purpose of this work is numerically research of the heat transfer within a multilayer plate (layers of human skin and external thermal insulation), with different heat exchange options at the boundaries of the layers.

Formulation of the problem
In this paper is considered two options of formulation the problem: the temperature of the main part of the human body T 1 varies in a small range of 35.5-35.8°C;from the surface of the human body the heat flows and equal to q=M/A [5] (M is the part of metabolic heat leaving the surface of the skin, and A is the area of the human body).
It was assumed that the temperature of the air in the production premises is constant and does not change with time (T A = const), and heat exchange on the outer surface of clothing occurs due to natural convection and thermal radiation.
As a parameter that serves as a "comfort characteristic" is used the surface temperature of the T S kin.On the basis of the theory [1] it is assumed that the worker feels "comfort" at T S = 306 ÷ 307 K.
Mathematical statement of the first problem: Here ρ i, λ i , c ithe density, thermal conductivity and specific heat of the material system of i-layer, Ticurrent temperature value (i = 1 ... 5).
The boundary condition of the first kind (T 1 = 309 K) is given on the left boundary x=0, heat transfer due to natural convection and radiation is taken into account at x=L, boundary conditions of the fourth kind (equality of temperatures and heat fluxes) are given on the interfaces of the layers.
The initial and boundary conditions are:  , ; The main difference between these two statements is that in the first one only layers of external isolation are considered, and a boundary condition of the second kind (q=M/A) is given at the boundary x=0.
The heat transfer coefficient was calculated by the criterion eqution for the conditions of natural convection near a plane wall [7]: Pr ; Ra Gr (2) 3 ( ) ; Here Gr -the number of Grazgof, l -a characteristic size (height of a man), ν -kinematic viscosity coefficient (air), β -the temperature coefficient of volume expansion of the air.
From the determination of the Nusselt number [8]: At this stage of the research it was assumed that the person is at rest (the heat given off by evaporation is small [8], because M/A -for a resting state of 50 W/m 2 ): 0.49 ( / (1 ) 50); The system of differential equations with appropriate initial and boundary conditions is solved by finite difference method, using an algorithm [8,9] developed for solving heat transfer problems in the areas of multi-layer discontinuous thermal conductivity.
At the solution of the problem the temperature dependence of thermal characteristics of materials and layer materials are not taken into consideration, since the latter was only a change from 1 K to 10 K.

Results of numerical modelling
When modelling TA changed in the really possible range (from 277 K to 304 K), the parameters of clothing varied.3

HMTTSC 2017 -
The thermo physical characteristics of the layers of the system are accepted in accordance with the data [11,12,13 The thicknesses of the layers of skin accepted in accordance with [7]: Typical initial temperatures: The duration of the modeled process chosen within the limits: t = 7000 s.The degree of emissivity of human skin [13]: 0.7.

H
Dependences of skin surface temperature on time at different ambient temperatures for the adopted formulations of the problem are shown in Fig. 1.It is established that the T S decreases significantly faster and depends more on T A in a system with a boundary condition of the second kind.Dependences of skin surface temperature on time at different ambient temperatures for the adopted formulations of the problem are shown in Fig. 1.It is established that the T S decreases significantly faster and depends more on T A in a system with a boundary condition of the second kind.
"Comfortable" air temperatures for easily dressed people are given in [1], obtained in experimental studies.It was found that T A is in the range of 17-24 °C.Dependences of the surface temperature of the heat supply object on T A are shown in Fig. 2 It can be concluded that a mathematical model with a boundary condition of the first kind (Fig. The thickness of clothes plays a big role in choosing the air temperatures of production premises.The connection between the "comfort" T A and the thickness of the layer of external thermal insulation (woollen clothing) for the two tasks discussed above is shown in Fig. 3.It is established that for a system with a boundary condition of the second kind at x=0, the influence of the clothing thickness on the "comfort" T A is greater than for conditions of the first kind at this boundary.

Fig. 1 .
Fig. 1.Dependence of T S on time at: 1 -T A =291 K; 2 -T A =294 K; 3-T A =302 K: a) a first-kind boundary condition for x=0; b) a boundary condition of the second kind for x=0.

Fig. 2 .
Fig. 2. Dependence of T S on ambient temperature (single-layer wool clothing): 1 -T A =291 K; 2 -T A =294 K; 3-T A =302 K: a) a first-kind boundary condition for x=0 (1 -results obtained in this paper, 2 -obtained in [15]); b) a boundary condition of the second kind for x=0.

Fig. 3 .
Fig. 3. Dependence of "comfortable" air temperature on the thickness of wool clothing: a) a first-kind boundary condition for x=0; b) a boundary condition of the second kind for x=0. ]: