2.1. Thermodynamic model
The human body thermal model developed for this article is based on the model used by [
24], described in [
26]. The human body is modelled as two control volumes, as shown in
Figure 1. CV1 represents the macroscopic part of the body, while CV2 represents the cellular metabolism. Focusing on CV2, the oxidation of nutrients during the cellular respiration generates two therms: the heat,
, transferred to the body (CV1) to keep its temperature in a narrow range, and the work,
, used in activities. CV1 interacts with the environment, thus it is considered the convection,
, and the radiation,
, heat transfers from the skin. There is also an enthalpy flow rate to environment,
, due the vaporization of water at the skin surface. The respiratory system operation generates an enthalpy flow rate entering the body,
, and an enthalpy flow rate leaving the body,
.
The energy balance of a control volume involving CV1 and CV2 results in Equation (
1). In this equation the metabolism,
, is considered an enthalpy variation over time of the reactions of oxidation inside the control volume. The enthalpy flow rate caused by food and water ingestion and by wastes and urine are disregarded, since it is considered that the body does not have any of these mass transfers during the period of study. The exergy balance of the same control volume is shown in Equation (
2). The terms inside the parentheses in Equation (
1) are usually agglutinated in one term called energy transfer to environment,
, and in Equation (
2) , in one term named exergy transfer to environment,
.
2.2. Energy and Exergy Metabolisms
Mady and Oliveira Junior [
26] evaluated the enthalpy and exergy changes of the nutrients’ reaction of oxidation during the cellular respiration, based in the article of [
27]. The nutrients chosen by the authors were glucose, as a representative of carbohydrates, palmitic acid, representing the lipids and an amino-acid with average composition (
) representing the proteins. It was considered that the glucose and the palmitic acid suffer complete oxidations in the body, while the amino-acid oxidizes only until the formation of urea.
The energy metabolism can be calculated using Equation (
3), while the exergy metabolism is obtained by applying Equation (
4). In this article the energy metabolism is given by the body composition, and its calculation will be explored forward (
Section 2.5). Equation (
3), is used herein to calculate the nutrients consumption rate, necessary to obtain the exergy metabolism and the respiratory enthalpy exchange.
Applying the hypothesis made by [
28], that there is a daily excretion of 12 g of nitrogen in urea produced by amino-acids oxidation, and using the equations of the reactions of oxidation, it is possible to obtain the mass of amino-acids consumed in one day and its rate, considered constant. To obtain the carbohydrate and lipid consumption rates it is necessary to solve the remaining system, composed by Equation (
3), the respiratory coefficient,
, and the oxidation reaction equations. The
represents the ratio between the carbon dioxide generation and the oxygen consumption during the respiration at volumetric basis (or molar basis using the ideal gas model). As discussed in [
29] a typical value of
is 0.83 for a person in daily activities.
2.4. Exergy Transfer to Environment
The environmental conditions are adopted as reference during the calculation of the exergy transfer to environment. Therefore, for a given environment, the reference of the relative humidity (
), atmospheric pressure (
) and operative temperature (
) are determined. The exergy transfer rates associated with the convection and radiation heat transfers are evaluated as Equations (
5) and (
6), respectively. The evaporative exergy flow rate to environment is given by Equation (
7). The exergy of the expired air is calculated using Equation (
8). Note that
due the use of the environmental conditions as reference (
).
where, in these equations
is the reference environmental temperature,
is the skin temperature,
is the enthalpy of vaporization of water at the skin temperature,
is the entropy of vaporization of water at the skin temperature,
is the gas constant of the water,
is the rate of sweat eliminated through skin,
of the partial pressure of water vapor in the skin,
is the partial pressure of water vapor in the environment; these equations are analysed in [
23,
24]. For Equation (
8) the index
i refers to the gases of the respiration, which is oxygen, carbon dioxide, water vapor and nitrogen. In this equation
is the partial pressure of the expired gas
i and
is the partial pressure of the gas
i in the envinroment.
2.5. Human Thermal Model
To obtain the temperature profile of the body, essential for the application of the exergy analysis, it is necessary to develop a human body heat transfer model. The model that follows in this section is a simplification of [
30]. The human body in this article is considered as a four-layer cylinder, which layers consist, from inner to outer, in core, muscle, fat and skin.
Figure 2 indicates a representation of the human body as a cylinder where each layer is represented, accordingly.
The initial geometry of the body was calculated using the data for the standard man and was based on the previous work of [
31]. Werner and Buse [
32] present some physical and thermal properties of many organs and tissues of the standard man, and define him as a man with 1.76 m height, 67 kg weight, 1.8 m
area and 67 dm
volume. Since the body is modelled as a cylinder, it is impossible to keep all the parameters of the standard man. Tests made with the model showed that the temperature profile is closer to reality if the height of the standard man is maintained instead of its area. Therefore, the height of the model is fixed at
m, and its area and volume should be calculated for each geometry using geometrical relations.
Most of the human body blood is distributed along tissues and organs in the data provided by [
32]. Considering that the volume of this element is the same than provided by the authors, their mass must be corrected. The correction is made by estimating the total blood volume using the equation proposed by Nadler [
33], and considering that this blood is equally distributed along all organs and tissues. The volume of the layers is calculated by adding the volume of each element that composes it. Note that, after the correction, the blood volume is disregarded for the body volume calculation, but its mass should be taken into account.
Table 1 demonstrates these weightings and the results of each tissue thermophysical properties.
It is considered that there is not a variation of the mass of the organs (or volume) as a function of the increase/decrease of lean/fat body masses. Therefore, the core layer has a constant radius,
cm. Besides that, the skin layer is considered always with the same thickness, 0.28 cm, independent of the body composition. These values were calculated using the data for the standard man. The muscle and fat layers are free to vary. The body metabolism is then obtained by summing the metabolism of each layer, as shown in Equation (
9). The blood volume,
(in m
), should be evaluated for every body composition using Equation (
10), obtained from [
33].
The human body mechanisms in order to adapt to the environment where it is submitted is called thermoregulation or control system of the body. These mechanisms exist in order to control the heat exchanges with the environment, preserving the internal temperatures as close as possible of the normothermia conditions [
4]. The human body control system is activated when the body departs from the thermal neutrality condition (for the model, thermal neutrality can be interpreted as the same of thermal comfort conditions). The thermal neutrality condition is obtained by submitting the model set for the standard man (naked) to an environment at 30
C and 50% relative humidity, as indicated by [
30]. The model developed in this article considers that the temperature profile generated by this condition represents the thermal neutrality for every composition studied. In other words, it means that the temperature profile of the model for the thermal neutrality condition does not change from person to person. This profile consists in
C,
C,
C,
C and
C.
The thermoregulatory system is composed by a sweat modeling, adapted from [
35], a vasoconstriction and vasodilatation model, adapted from [
36], and a shivering model, obtained from [
37].
The energy balance of the core layer is given by Equation (
11). Note that the enthalpy variation due the respiration is considered uniformly distributed all along the core volume. In this equation,
is the specific mass of the core,
is the volume of the core,
is the heat transfer between core and muscle and
is the heat transfer between the blood and tissue in the small vessels according to the model proposed by [
38]
The energy balances of the muscle and fat layers have the same form, and are given by Equations (
12) and (
13), respectively.
The skin layer energy balance is given by Equation (
14).
Eventually the First Law of Thermodynamics is also applied to a central reservoir of blood [
30]. The heat exchange between blood and tissue (
) is given by Pennes’ model [
38]. The heat exchange between tissues,
, is calculated using the thermal conductivities weighted by the volume of the layers in contact, and the temperatures of the layers are considered constant. To solve the differential equations a C++ program was developed for the explicit Euler method.
The variation of the environmental conditions causes the variation of the body internal temperature profile over time [
23], but herein the focus is given only in the steady-state points, obtained when the body attains an equilibrium state with the new environment. In other words, for the body perspective, steady state may be defined when there is no variation of the temperature of each tissue over time. The transient conditions may be used in future analysis where modifications in the actual environment may be evaluated. It is important to highlight that in the period considered, minutes of simulations, the water mass lost in sweat is negligible (larger periods of time, such as one day this is not true), nevertheless its energy must be considered.
2.6. Thermal Comfort Indexes
Although it is difficult to obtain a rational definition of thermal comfort conditions [
29], for the model perspective it is possible to define the thermal comfort as the same of thermal neutrality (since the environment analysed will not have any local discomfort).
Fanger [
39] proposed the Predicted Mean Vote (PMV) and the Predicted Percent Dissatisfied (PPD) as alternatives to evaluate the environment aiming to predict a “satisfactions” or “dissatisfaction” with the thermal environment. The PMV index uses the Ashrae scale [
29], a scale that is defined on integers from −3 to 3. The negative values of PMV represent the so called “cold" sensation while the positive values represent the “hot” sensation. The further PMV is from zero, the bigger the thermal discomfort.
In this article, the predicted mean vote (PMV) was calculated according to Fanger method as indicated in Equation (
15). Where, according to [
29] “L is the thermal load on the body, defined as the difference between internal heat production and heat loss to the actual environment for a person hypothetically kept at comfort values”. Therefore, the predicted mean vote evaluate the influence of control system in the energy balance.
In order to quantify the number of people who are not satisfied with the environment Equation (
16) indicates the predicted percent dissatisfied (PPD) [
29].
These physical quantities will be compared with the exergy analysis results (destroyed exergy and exergy transfer to the environment).