Thermal management topics like air conditioning and thermal comfort become more and more important during vehicle development. As a consequence the need for suitable tools and methods to predict the human comfort behaviour in cabins increases rapidly. THESEUS-FE is a finite element based software tool with a fully coupled implementation of Fiala’s thermal sensation manikin presented in [1]. This paper will demonstrate the finite element based implementation of the new manikin FIALA-FE in an existing solver environment, various thermal comfort topics will be discussed in a second step. Additionally this paper explains the application of the finite element theory on the complete framework of formulas that represents the thermoregulatory human system. Because the major challenge was not the numerical solution of the bio heat equation based on the finite difference method, as shown in [1], but on a method that allows to create a special finite element type, representing a single human material layer. Finally this paper will discuss different aspects of global and local thermal comfort prediction, based on mathematical models from literature. The usage of certain human state variables (like the mean skin temperature) will be discussed in terms of comfort prediction. At the end the idea of the ‘equivalent temperature’ by using simulated skin and cloth temperatures leads to a quite simple-to-use method of assessing local thermal comfort at given boundary conditions, typical for a vehicle simulation. Considering not only surface-to-surface radiation and convection but also sun radiation, seat contact and evaporation the equivalent temperature can then be derived at each body part of a thermal manikin.
1 General
The manikin FIALA-FE (as a new feature of the software tool THESEUS-FE) provides all the thermo-physiological effects of the human body model published in the frequently cited PhD thesis [1] of 1998. Dr. Dusan Fiala acts as our advisor during the implementation of his theories. After the model’s realization an extensive validation period of the manikin’s results took place , in which a good fit with the huge number of diagrams shown in literature was reached. From the mathematical point of view, the human organism can be separated into two interacting systems of thermoregulation: the controlling active system and the controlled passive system. The active system is simulated by means of cybernetic models predicting regulatory responses, i.e., shivering, vasomotion, and sweating, as discussed in [1]. The passive system is modelled by simulating the physical human body and the heat transfer phenomena occurring inside and at its surface. The physical body consists of body elements that are approximated as cylinders or sphere (head), they are separated into sectors and subdivided into material layers like skin, fat, muscle, bone. Fiala`s advanced manikin combines both, the passive and active system, in a very complex model that reaches a good fit towards field measurements of human thermal responses in a wide range of environmental conditions. Several hundreds of physical parameters1 that build up the standard humanoid represent an averaged human. This is the manikin THESEUS-FE works with by default, but it is also possible for the user to build his own manikin with an arbitrary number of body elements, sectors and layers: e.g., it is thinkable to create fingers, nose and ears as new body elements. Hence reasonable physical parameters have to be extracted from literature or measurements.
In a transient coupled simulation, the manikin FIALA-FE permanently interacts with its environment via convection, evaporation, radiation and contact. Convective heat exchange typically takes place between the manikin’s outer surface and the surrounding airzones2. Evaporative heat loss on the skin results from a certain steam mass flow that increases the absolute humidity of the airzones, with distinctive mass flow rates due to sweating in a hot environment. The pre-defined contact regions might lead to additional heat conduction, e.g. between the upper legs and the seat. Sun radiation can be considered as well as thermal radiation between manikin surfaces and the surrounding structures via view factors, pre-calculated with the shell model shown in Fig. 1. The total number of manikins of the new type “FIALA-FE” used in one transient fully coupled analysis is unlimited and the comparable small number of additional degrees of freedom (about 650 per manikin) will not lead to a serious increase in CPU time.
2 Fiala’s Passive System
2.1 Discretization
As shown in Fig. 1 THESEUS-FE uses two different models for the manikin FIALA-FE. A shell model used during pre- and post-processing in the GUI to apply boundary conditions, define clothing, assign body parts or to visualize field results, e.g. skin or cloth temperatures. A solver-internal manikin model is based on the ideas of Fiala, who presented such a model in table 2 of [2]. This model uses a half sphere for the head and cylindrical solid bodies for the rest of the humanoid, as shown in Fig. 2 . Both models are linked via heat fluxes that are first calculated on the shell model and then given to the solver-internal manikin. To simulate the human heat conduction that transports metabolic heat from the inner regions to the outer skin or garment surfaces FIALA-FE uses special finite elements (FE) that build up the layers, sectors and body elements as shown in Fig. 2. These elements provide not only temperature approximation functions for radial heat conduction but also arterial blood heating and special links to the blood pool temperature.
2.2 Bio Heat Equation
State-of-the-art modelling of heat transport mechanisms in living tissues uses the bio heat equation. This partial differential equation states that internal energy changes at each material point inside the manikin result from conduction, metabolism and arterial blood heating effects. The heat conduction is only considered in a radial direction, taking into account only temperature derivation along r. The equation describes the law of energy conservation at each material point of a layer that might represent skin, fat,muscle, bone, brain etc. The magnitude of the physiological variables qm and wbl are affected by responses of the active system, as shown in [1] and [3]. The metabolic heat source qm collects influences from different human phenomena: the basal metabolism, working, shivering and Q10-effect. The metabolic heat typically reaches high basal values at the brain and the abdomen core. Additional metabolism from working and shivering appears at muscle layers. The arterial blood temperature Tbl,a arises from the actual overall thermal state of the body and results from simulating the human blood circulatory system, as shown in [1]. Arterial and venous blood temperatures can be derived for each body element of the humanoid by counter flow heat transfer. The blood pool temperature appears as an additional degree of freedom of each manikin, representing the temperature of the blood leaving the human heart. For a numeric solution of the partial differential equation (1) THESEUS-FE uses finite elements specially developed for this type of problem.
2.3 Human Heat Exchange with the Environment
Most heat of a human is lost through the body surfaces3: radiation, convection, skin evaporation and contact heat fluxes will be considered in THESEUS-FE. All these different kinds of boundary conditions are visualized in Fig. 3, acting on a clothed body element sector, e.g. upper leg anterior.
2.3.1 Contact
If the user defines thermal contact for a body element sector heat conduction with the contact partner (e.g. the car seat) takes place at the outer surfaces, and all other types of boundary conditions will be switched off automatically.
2.3.2 Convection
Convective heat exchange between a body surface area Asf of temperature Tsf and the ambient air of temperature Ta considers both, free (natural) and forced convection using combined convection coefficients hc,mix
For this coefficient D. Fiala uses a well validated analytical function from literature:
That depends on the location on the body, the surface and air temperature, and the effective air speed vair,eff (m/sec). The coefficients anat,j , afrc,j and amix,j are listed in table 2 in [2]. They have been derived from experiments and provide different values for each human body element.
2.3.3 Radiation
In a coupled simulation THESEUS-FE derives radiation heat fluxes for the shell model (Fig. 1, left) and applies them on body element sectors of the internal manikin model (FIALA-FE). The following radiation phenomena might be considered: i) radiation heat from sun, ii)transmission (e.g. through car glass), iii)view factor radiation (black- or grey-body). In an uncoupled simulation the user defines a fixed radiation wall temperature Tw. The black-body approach leads then to. Boltzmann coefficient σ and a wall emissivity εw. The body surface emission coefficients εsf and the view factors ψsf-w are listed in table 2 in [2]. Fiala uses different values depending on body location and position (sedentary or standing).
2.3.4 Evaporation
The wet heat loss4, as defined in [1] and [2], through the clothes is driven by the evaporative potential between skin and ambient air (eqn). Where psk is the water vapor pressure at the skin, pa is the vapor pressure of the ambient air, and U*e,cl is the resultant evaporative heat transfer coefficient. The mathematical relations that define those three functions are not shown here, but can be derived from [1], [2] and [5]. Finally the following quantities effect the evaporative heat loss; skin temperature: Tsk, ambient air temp., velocity, humidity: Ta , ϕa ,va, local evaporative resistance of the clothes: R*e,cl, local sweating rate : dmsw/dt. Evaporative heat loss in a hot environment strongly depends on the local sweating rate that can be derived from the global sweating rate SW that is a part of the thermoregulatory response of the active system.
2.4 Finite Element Implementation
To solve the bio heat equation (1), considering the actual set of boundary conditions, a special finite element type has to be established that allows only radial heat conduction. For the radial temperature field we assume a linear function separating nodal temperatures Tei from the spatial approximation functions. This simple approach will then be used to model the temperature distribution in a material layer, as shown in Fig. 4 (red line). Such finite elements might represent a material layer like fat, skin or bone with a linear temperature approach or a set of such finite elements might build up one single material layer, as shown for the muscle layer in Fig. 2. Using more than one finite element per layer deals with the fact that in most cases transient loading and inner heat sources result in a non-linear temperature distribution, as shown in Fig. 5. A larger number of sub-layers (FE) are chosen to build up the muscle temperature distribution in such a smooth way as shown in the figure above. Replacing the temperature T in (1) by the simple linear approach (6) and integrating over the finite element’s solid region while multiplying with φi one can show that after some mathematical operations5 a time differential equation system can be derived, that holds for one single finite element ‘e’: The underlined quantities in (8) are vectors or matrices. T stands for tT∂∂. Components result from. The sum over all boundary heat flux densities Σqbc from convection, radiation, evaporation or contact will be integrated on the bounding surfaces:
3 Fiala’s Active System
The active system controls regulatory responses of shivering, sweating and peripheral vasomotion in terms of global manikin functions depending on three state variables: the mean skin temperature: Tsk,m, the hypothalamus (head core) temperature: Thy and derivations of the mean skin temperatureversus time: dTsk,m/dt. The non-linear active system function in [1] and [3] had been derived from regression analysis, taking into account a huge number of tests. A detailed descriptions on the manikin’s active system (with its influences on the passive system) can be found in literature and will not be shown here.
4 Validation
In his PhD thesis [1] Fiala collected a large number of experiments from literature where humans had been exposed to different (and changing) environmental conditions. An extensive validation program showed the applicability of his thermal manikin. In Fig. 6 one single experiment is presented that shows the global evaporative response of manikins (n=3) on hot environmental conditions. The effect of rising evaporative heat loss results from sweating, that is one of the active system response functions Finally a good correlation between the THESEUS-FE simulation results and those presented in [1] are reached. A special manual [6] shows simulation results derived from the new manikin in THESEUS-FE that had been compared with the complete experimental data base in [1]
5 Thermal Comfort
The final purpose of the HVAC (Heating Ventilation Air Conditioning) –system is to provide comfortable thermal conditions, irrespective of the environmental climate outside the vehicles cabin. Although most models for thermal comfort prediction only take into account the global thermal state, the thermal boundary conditions of the passengers are always time dependent and asymmetric, dominated by convection and radiation. Human discomfort often results from the local appearance of thermal loads hitting body parts with extreme heat fluxes from sun, ventilation or contact. Of that reason the prediction of thermal comfort is one of the major aims of thermal simulation today, and especially for vehicle passengers. In THESEUS-FE such manikin simulations typically start from thermal neutrality.
5.1 Thermal Neutrality
Fiala defines the thermal neutrality as a state of optimal thermal comfort at steady state boundary conditions. Responses from the active system like shivering, sweating and vasomotion do not occur at thermal neutrality. Manikin’s mean skin and hypothalamus temperature are listed in Tab. 2. The temperatures at neutrality are typically are used as set-point temperatures for the active system and the thermal comfort prediction. That means that deviations from the set-points lead to active system responses (like sweating and shivering) on the one side, and states of discomfort on the other hand. Local derivations of the skin temperatures Tsk versus their set-point temperatures Tsk,0 can give the user a first indication of discomfort. In such a context a ΔTsk value of zero represents the optimal state of thermal neutrality for a body element sector. Modern models for thermal comfort prediction will be discussed in the following two sub-chapters: Both models use set-points deviations and some dynamic effects, too. That means a proper simulation of the temperatures at thermal neutrality is essentially needed for further thermal comfort prediction.
5.2 Global Thermal Comfort Prediction (by Fiala)
Fiala’s global dynamic thermal sensation index DTS uses set-point deviations of the mean skin temperature and the hypothalamus temperature together with the dynamic variable dTsk,m/dt. The three state variables used for the DTS index calculation are the same as for the active system (see chapter 3). This index directly correlates to the standardised ASHRAE thermal sensation scale from -3 (cold) to +3 (hot) and is for many load cases comparable with the well known PMV index [8]. In standard simulations the influence of ΔThy is quite low and the mean skin temperature deviations affects the DTS index mainly. Relations between ΔTsk,m and the DTS index are shown in the figure below. The manikin’s skin temperatures are initialized at thermal neutrality (Fig. 7) and the boundary conditions start with high cabin air and radiation wall temperatures. A sudden rise of skin temperatures at the beginning makes the DTS increase to its maximum value of +3. This dynamic effect results from the third state variable in (15).
After about 5 minutes simulation ventilations (blowing cold air inside the cabin) effect a decrease of the mean skin temperatures that reach the state of neutrality at 34.4°C after 40 minutes. Finally the DTS indices shows well-being in a range of -1..+1. Nevertheless a skin temperature distribution as shown in Fig. 7 (right hand side) makes it clear that a certain value for the mean skin temperature often hides information about local asymmetries that might be responsible for local discomfort: See Fig. 7, a cooled right arm and a heated left arm neglect each other in the mean temperature calculation. Finally there is a strong need for a local thermal comfort model today, as each car driver can imagine. Such a model had been presented by Zhang in [7]…
5.3 Local Thermal Comfort Prediction (by Zhang)
As the result of her PhD thesis H. Zhang presented in 2003 a very complex mathematical framework for local thermal comfort prediction in [7]. The principle idea of her model is visualized in Fig. 9. At a first step the human hypothalamus and skin temperatures must be derived from measurement or a thermal manikin simulation. In a second step local thermal sensation indices can be calculated for certain body parts (e.g. head, face, upper arm). The state variables used therefore are temperature deviations versus neutrality and derivatives with respect to time, as shown in Fig. 9. An overall thermal sensation index can then be found with pre calculated Sl values. Finally the local thermal comfort index is a function of the overall thermal sensation and the related local thermal sensation index of the body part. All these mathematical functions consider a huge number of local weighting factors derived from experiments and regression analysis. In [10] a method had been presented that shows how one could derive seat temperatures (at the contact area) that would realize a maximum local comfort index at the manikins back. Therefore the workflow shown in Fig. 9 must be reversed, starting with a local comfort index +4 with the final aim to derive contact temperatures that realize this maximum comfort at given boundary conditions. The simple question behind such an analysis is: “what is the optimal contact temperature in a cold vehicle”.
5.4 Local Equivalent Temperatures
The principle idea is to take simulated heat fluxes as an input and derive more meaningful temperatures… In [9] the definition of an equivalent temperature is given: “The equivalent temperature Teq is the temperature of an imaginary enclosure with the same mean radiant temperature and air temperature and very low air velocity in which a person exchanges the same heat loss by convection and radiation as in the actual condition”. This method can be used in a global way, deriving one Teq value from known over all human heat fluxes. A local approach would derive Teq values at different body parts from known heat fluxes and Tsf: The left hand side of (16) builds the sum over all boundary heat fluxes from the manikin simulation, shown in Fig. 3. On the right hand side the equations (2) – (4) build together the heat loss of a certain body part in an imaginary enclosure with Teq=Tw=Ta. The effective air speed va,eff can then be replaced by zero or a low value like 0.05m/s. Equation (16) holds for an equivalent temperature prediction at the clothed body part, replacing Tsf by the skin temperature Tsk would lead to an equivalent temperature related to the nude body part. For given manikin temperatures and heat fluxes the only unknown remaining in (16) is Teq, that can easily be derived from an iterative scheme like the Newton Method. The original concept of the equivalent temperature does not consider evaporative heat loss in the imaginary enclosure. Based on a thermal manikin simulation, an extension of the standard method considering (5) on the right hand side of (16) would be easily possible, because all quantities listed in chapter 2.3.4 are given. The relative humidity of the enclosure should then be a pre defined value, e.g. 40%. Finally the idea of the ‘equivalent temperature’ leads to a quite simple-to-use method of assessing local thermal comfort at very complex boundary conditions. For example Teq = 30°C related to a nude body part, tells us everything is O.K., concerning the thermal neutrality definition, see Tab. 1.
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