A dynamic model predicting
human thermal responses in cold, cool, neutral, warm, and hot
environments is presented in a two-part study. This, the
first paper, is concerned with aspects of the passive system:
1) modeling the human body, 2) modeling heat-transport mechanisms within the body and at its periphery, and 3) the
numerical procedure. A paper in preparation will describe the
active system and compare the model predictions with experimental data
and the predictions by other models. Here, emphasis is given to a
detailed modeling of the heat exchange with the environment:
local variations of surface convection, directional radiation exchange,
evaporation and moisture collection at the skin, and the
nonuniformity of clothing ensembles. Other thermal effects are also
modeled: the impact of activity level on work efficacy and the change
of the effective radiant body area with posture. A stable and
accurate hybrid numerical scheme was used to solve the set of
differential equations. Predictions of the passive system model are
compared with available analytic solutions for cylinders and spheres
and show good agreement and stable numerical behavior even for large time steps.
dynamic simulation; human heat transfer; asymmetric thermal
environments; exercise; numerical modeling
 |
INTRODUCTION |
IN THE PAST THREE DECADES, several
multisegmental mathematical models of human thermoregulation have been
developed (e.g. Refs. 17, 34, 41) and have become valuable tools in the hands of their authors in contributing to a deeper understanding of
regulatory processes. These models, however, have not yet gained widespread use. Reasons for this may include lack of confidence in
their predictive abilities, limited range of applicability, and poor
modeling of the heat exchange with the environment. This paper and the
paper in preparation make a contribution to research efforts to
formulate a more precise, flexible, and universal model of the human
thermoregulatory system.
The main stimulus for the model is the need to predict the thermal
response of human beings to the conditions that they experience in and
around buildings. Other well-established models can predict ambient air
and surface temperatures, air speeds, and humidity; therefore, these
parameters are the boundary conditions that will act on the
thermoregulatory model.
The research sought to draw together the most appropriate models
describing individual parts of the whole thermoregulatory system. These were then combined within a mathematical framework to
produce a robust, computationally fast, and accurate model that will be
of practical value in the design of buildings and urban
environments. The research thus seeks to deliver the products of
fundamental physiological research to scientists working in other disciplines.
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 elsewhere (16). The passive system, the subject of this
paper, is modeled by simulating the physical human body and the
heat-transfer phenomena occurring in it and at its surface. Comparisons
between the model's predictions and known analytic solutions are made.
Only the complete model, i.e., the passive system plus active system,
can be compared with field measurements of human responses to
environmental conditions and with the predictions of whole body models
produced by others; such comparisons are, therefore, a subject of the
upcoming paper.
The metabolic heat is produced within the body. This heat is
distributed over body regions by blood circulation and is carried by
conduction to the body surface, where, insulated by clothing, the heat
is lost to the surroundings by convection, radiation, and evaporation
(see Fig. 1). Additionally, some heat is
lost to the surroundings by respiration. To simulate adequately all these heat-transport phenomena, the present model of the passive system
accounts for the geometric and anatomic characteristics of the human
body and considers the thermophysical and the basal physiological
properties of tissue materials.
Whereas the model was inspired by the desire to evaluate
environmental conditions in buildings, it also has applications in other areas, e.g., in the car industry, in textile industries, in the
aerospace industry, in meteorology, in medicine, and in military
applications. In these disciplines, the model can serve for research
into human performance, thermal acceptability and temperature
sensation, clinical and therapeutic treatments, safety limits, etc.
| |
BODY CONSTRUCTION |
The passive system of the model was developed by using thermophysical
and basal physiological properties of tissue materials, based on
literature (10, 17, 40). The humanoid was formulated to represent an
average human (see Table 1 for overall body
data). The body was idealized as 15 spherical or cylindrical body
elements: head, face, neck, shoulders, arms, hands, thorax, abdomen,
legs, and feet. The use of lumped data (e.g., if body elements are
divided into a core and a shell) is prone to errors during temperature transients and was thus avoided. In accordance with former studies (17), a division was therefore made whenever a significant change of
body tissue properties occurred. As a result, the present multilayer model consists of annular concentric tissue layers and uses seven different tissue materials: brain, lung, bone, muscle, viscera, fat,
and skin. Each of these is subdivided into one or more tissue nodes
(see Table 2).
In accordance with the work of Weinbaum et al. (39), the skin was
modeled as two layers with distinctly different physiological properties: the inner skin and the outer skin. The inner skin is ~1
mm thick and simulates the cutaneous plexus, a region in which
metabolic heat is generated and blood is perfused (see Table 2). The
outer layer of skin is of similar thickness; however, it contains
neither any heat source nor any thermally significant blood vessels.
This superficial cutaneous layer plays a role when evaporative heat
losses through the skin are modeled. The layer contains sweat glands
and simulates the vapor barrier for moisture diffusion through the skin.
Asymmetric removal of bodily heat is often experienced in practice. It
may be due to inhomogenous ambient conditions, e.g., solar irradiation,
cold surfaces, draught effects, nonuniform clothing, or to
physiological effects on the skin, such as local differences in
evaporative heat losses, etc. Except for the face and shoulders, each
body element was, therefore, subdivided spatially into sectors. Most of
the body elements were divided into three sectors: anterior, posterior,
and inferior (see Table 2). Anterior and posterior segments permit the
treatment of lateral environmental asymmetries. Inferior segments
account for body sides, which are "hidden" by other body parts
(e.g., section A-A' in Fig. 1) and thus have reduced radiant heat
exchange with the environment.
The sectors were coupled thermally by introducing a core element around
the cylinder axis (see Fig. 1, section A-A') or the midpoint of the
sphere. In the model, except for the head, the radius of the core was
defined to be identical to the radius of the innermost tissue material
layer in each body element. In extremities, for instance, the radius of
this domain is identical to the radius of the bone. However, the brain,
which has an outer radius of r = 8.6 cm (Table 2), was given a
core of 4-cm radius.
Glossary
Symbols
, 
, 
 |
Short-wave absorptivity
|
 |
Factor of the blood perfusion rate
(W · K 1 · m3)
|
 |
Difference
|
 |
Stefan-Boltzmann constant
(W · m2 · K4)
(=5.67051 × 108
W · m2 · K4)
|
 |
Long-wave emissivity
|
| , , |
Time-independent coefficients arising from the numerical formulation of
the bioheat equation
|
 H2O |
Heat of vaporization of water (= 2,256 × 103
J/kg3)
|
| µbl |
Proportionality factor (K1)
|
 |
Meachanical efficiency of the human body
|
 |
Density (kg/m3)
|
 |
View factor
|
 |
Linear operator
|
 |
Angle of a body element sector (°)
|
 |
Sum
|
 |
Geometry factor of an isothermal core domain of a body element
|
 |
Geometry factor of the bioheat equation
|
 |
Partial derivative
|
| A |
Area (m2)
|
| a |
Distribution or weighting coefficient
|
| act |
Activity level (met)
|
| b |
Regression coefficient
|
| Bx |
Arterial blood temperature parameter arising from hx
|
| C |
Heat loss by convection (W)
|
| c |
Heat capacitance
(J · kg1 · K1)
|
| Cc |
Coefficient of the blood pool temperature column of the major
coefficient matrix
|
| CCX |
Countercurrent heat exchange
|
| CO |
Cardiac output (l/m)
|
| CplC |
Coupling coefficient of the major coefficient matrix
|
| E |
Heat loss by evaporation (W)
|
| fcl |
Overall clothing surface area factor
|
| f*cl |
Local value of fcl
|
| feff |
Effective radiant area of the body
|
| h |
Surface heat transfer coefficient
(W · m2 · K1)
|
| hx |
Countercurrent heat exchange coefficient (W/K)
|
| H |
Internal whole body workload (W)
|
| icl |
Overall moisture permeability index from the skin to the clothing
surface
|
| i*cl |
Local value of icl
|
| Icl |
Overall intrinsic insulation from the skin to the clothing surface
(m2 · K · W1)
|
| I*cl |
Local value of Icl
(m2 · K · W1)
|
| k |
Conductivity
(W · m1 · K1)
|
| L |
Lewis constant (La = 0.0165) (K/Pa)
|
| Lc |
Coefficient of the blood pool temperature line of the major coefficient
matrix
|
| m |
Mass (kg)
|
| M |
Whole body metabolism (W)
|
| MRT |
Mean radiant temperature (°C)
|
| N |
Number of nodes
|
| P |
Water vapor pressure (Pa)
|
| q |
Heat flux density (W/m2)
|
| qm |
Metabolic rate (W/m3)
|
| Q |
Heat flux (W)
|
| r |
Radius (m)
|
| Re |
Moisture permeability resistance
(m2 · Pa · W1)
|
| rh |
Relative humidity (%)
|
| s |
Short-wave irradiation (W/m2)
|
| t |
Time (s)
|
| T |
Temperature (°C)
|
| T* |
absolute temperature (K)
|
| U* |
Local clothing insulation coefficient from the skin to the ambient air
(W · m2 · K1)
(or)
(W · m2 · Pa1)
|
| v |
Airspeed (m/s)
|
| V |
Volume (m3)
|
| w |
Perfusion rate (s1)
|
| wt |
Wetted area ratio
|
Subscripts and superscripts
impl
 |
Last node next to an interface between two inhomogenous tissue shells
|
| + |
First node next to an interface between two inhomogenous tissue shells
|
| 0 |
Thermal neutrality
|
| air |
Air
|
| acc |
Accumulation
|
| bas |
Basal value
|
| bd |
Body
|
| bl |
Blood
|
| bl,a |
Arterial blood
|
| bl,p |
Central blood pool
|
| bl,v |
Venous blood
|
| c |
Convection
|
| cl |
Clothing
|
| CN |
Crank-Nicolson
|
| Du |
Dubois
|
| e |
Evaporation
|
| eff |
Effective
|
| expl |
Explicit
|
| f |
Fabric
|
| frc |
Forced
|
| ifc |
Interface
|
| j |
Running number variable
|
| hy |
Hypothalamus
|
| impl |
Implicit
|
| m |
Mean value
|
| mix |
Mixed
|
| mus |
Muscle
|
| n |
Total number
|
| nat |
Natural
|
| o |
Operative
|
| osk |
Outer skin layer
|
| r |
Node number
|
| R |
Long wave radiation
|
| re |
Rectal
|
| rsp |
Respiration
|
| sat |
Saturated
|
| sf |
Surface of body sector
|
| sh |
Shivering
|
| sk |
Skin
|
| sR |
Short-wave radiation
|
| sr |
Surrounding
|
| sw |
Sweating
|
| t |
Time step number
|
| ty |
Tympanic
|
| w |
Work
|
| x |
Countercurrent heat exchange
|
| |
HEAT TRANSFER WITHIN THE TISSUE |
The heat transport mechanisms occurring in the living tissue
have been formulated by Pennes (28) in the so-called "bioheat equation." This differential equation describes the heat dissipation in a homogenous, infinite tissue volume
|
(1)
|
According to Eq. 1, from left to right, the radial heat flow
from warmer to colder tissue regions [heat-conduction term, where k is the tissue conductivity
(W · m1 · K1),
T is tissue temperature (°C), r is radius (m), is a
geometry factor (dimensionless); = 1 for polar coordinates and
= 2 for spherical coordinates (head)] is overlayed by metabolism
qm(W/m3) and blood perfusion
[heat-convection term, where bl is density of blood
(kg/m3), wbl is blood perfusion rate
(s1), cbl is heat capacitance of blood
(J · kg1 · K1),
and Tbl,a is arterial blood temperature (°C)]. This
combined effect is balanced by the storage of heat within the tissue
mass [right-hand side of Eq. 1, where is tissue density
(kg/m3), c is tissue heat capacitance
(J · kg1 · K1)
and t is time (s)].
The bioheat equation was applied to all tissue nodes by using the
appropriate material constants k, , and c, the basal
heat-generation term qm, and basal blood perfusion
rate wbl for each tissue layer (Table 2). It should
be noted that the magnitudes of the physiological variables
qm and wbl are affected by
responses of the active system, as described elsewhere (16). The
arterial blood temperature Tbl,a in Eq. 1 arises
from the actual overall thermal state of the body and is obtained by
simulating the human blood circulatory system.
Heat conduction.
The heat-conduction term in Eq. 1 considers temperature
variations in the radial direction. Angular heat flows were also
neglected in the present model, even though the body elements were
divided into sectors, and environmental asymmetries were considered.
There were two reasons for this simplification. First, at the periphery (i.e., predominantly in the skin), although significant angular temperature gradients may occur due to asymmetric boundary conditions, the total amount of the heat conducted angularly from one
skin sector to another is negligible, compared with the total amount of
heat transferred radially through a sector band. This is because peripheral sectors have large shell areas, compared with the small interface areas between adjacent skin sectors. Second, ambient influences are insignificant in internal tissue layers where heat dissipation is dominated by metabolism and by blood circulation (28,
33). Therefore, inhomogeneities of qm and
wbl between individual organs of the abdominal
viscera (40), for instance, which can hardly be considered by
symmetrical geometry models, govern local temperature profiles within
the body and swamp the effect of environmental asymmetries.
Metabolism.
In the model, the metabolic heat production of a finite tissue volume
is treated as a sum of the basal value qm,bas,0 and the additional heat qm, which may be produced by
local autonomic thermoregulation, while the subjects are exercising
and/or shivering
|
(2)
|
The basal metabolic rates of individual tissue materials
qm,bas,0 were obtained from literature (10, 17, 40)
and are listed in Table 2. In thermal neutrality
(Tair = 30°C; for a detailed specification see Table
3), where no regulation occurs, the overall
basal metabolism Mbas,0 = 
qm,bas,0 dV of the passive system results in a
value of 87 W, which agrees with the (standardized) whole body
metabolism of a reclining average human, according to ASHRAE Standards
55 (1).
In muscles, the extra heat qm may contain three
components; i.e., changes in the basal metabolism
(qm,bas) and in additional metabolism by
shivering and working (qm,sh and
qm,w, respectively)
|
(3)
|
The change in the basal metabolism qm,bas is the
difference between the actual basal rate and the basal rate
corresponding to neutral thermal conditions. It appears also in
nonmuscular tissues where qm,bas,0 is present and
where the tissue temperature differs from its setpoint T0.
This local regulation arises from the van't Hoff Q10
effect with a sensitivity coefficient of 2 (40), which reflects the
dependence of biochemical reactions on local tissue temperature T
![&Dgr;<IT>q</IT><SUB>m,bas</SUB> = <IT>q</IT><SUB>m,bas,0</SUB> · [2<SUP>(T−T<SUB>0</SUB>)/10</SUP> − 1]](/content/vol87/issue5/fulltext/1957/img004.gif)
|
(4)
|
The shivering term qm,sh in Eq. 3 is a portion of the overall regulatory response elicited by the
active system (16). The qm,w term can be written as
|
(5)
|
where am,w (see Table 2) is a
coefficient distributing to body elements the overall extra metabolism
due to exercise, Vmus is the corresponding body-element
muscle volume, and H is the internal whole body workload. The
values of am,w for standing activities were
obtained from Stolwijk (34). The present model uses other estimates of
am,w for sedentary activities, because the extra
metabolism shifts from the muscles of the legs to the musculature of
the trunk and arms when the subject is seated.
The workload H in Eq. 5 represents that part of the
overall extra energy which is produced by muscular activity but which does not appear as external work. It is obtained as the difference between the actual overall heat-generation rate remaining in the body
and the basal metabolism Mbas,0
|
(6)
|
The calculation of the actual metabolic heat production utilizes the
activity level units act (measured in met) as the rate between the body
heat produced at that activity and the average metabolism when the
subject is seated, defined as 58.2 W/m2 (2), i.e., 1 met.
In Eq. 6, the whole body metabolism is calculated from the
basal metabolic rate Mbas,0 for a reclining,
resting individual, corresponding to actbas = 0.8 met. The
act units, used as (time-dependent) input variables to the model, are
commonly known and can easily be obtained by the users from standard
literature (e.g., Refs. 1, 2).
The term in parentheses (1 ) in Eq. 6 represents that
fraction of the overall metabolic heat actually produced which remains for warming the body, i.e., which is not converted into external mechanical power. The human mechanical efficiency is not constant in reality but rises with increasing activity levels. For the present
model, we developed the following formula for , by means of
regression analysis from measurements of Wyndham et al. (42)
|
(7)
|
The regression coefficients were b1 = 0.39 ± 0.13 met1 and b0 = 0.60 ± 0.28, with
a correlation coefficient of r = 0.86. This equation reflects
the behavior of the human work efficacy during normal activities
[e.g., stepping or cycling (42)]. At low activity levels
(act < 1.6 met), where humans work most inefficiently, is
zero. At higher activity levels (~1.6 < act < 5.0 met), rises almost linearly with increasing activity level. At high activity
levels, begins to approach a theoretical maximum of ~0.2; i.e.,
~20% of the whole body metabolism is transferred into external work.
Blood circulation.
Blood circulation is of vital importance for the dissipation of heat
within the human body. The dissipation effect was examined by
simulations using the present model without blood circulation: the core
temperature of the brain, which has a high metabolic rate (Table 2),
exceeded 73°C when the passive system was exposed to steady neutral
ambient temperatures of 30°C.
In the present model, the human blood circulatory system was simulated
as three main components: 1) the central blood pool, 2)
countercurrent heat exchanges CCX, and 3) pathways to
individual tissue nodes. Body elements are supplied with warm blood
from the central pool by the major arteries. However, before the tissue of the extremities and shoulders are perfused, the blood is cooled by
heat lost to the countercurrent bloodstreams in the adjacent veins. The
arterial blood exchanges heat by convection in the capillary beds,
according to the bioheat Eq. 1. The venous blood then collects
in the major veins and is rewarmed by heat from the adjacent arteries
as it flows back to the central blood pool. It mixes with blood from
other elements to produce the new central blood pool temperature.
The blood perfusion term of the bioheat Eq. 1 is based on
Fick's first principle, assuming that heat is exchanged with the tissue in the capillary bed and that no heat storage occurs in the
bloodstream. The blood perfusion term has been critically discussed in
the literature (see, e.g., Ref. 11) and has been subjected to various
improvements (13, 33). Also, alternative models of microvascular
structures have been developed (see, e.g., Ref. 39). However, the
traditional blood perfusion term seems preferable in whole body models.
It is simple and produces an accuracy that is comparable (9) to more
detailed models that require comprehensive data on the vascular
architecture of tissues.
The simulation of the countercurrent heat exchange CCX was introduced
into the model in an attempt to obtain a more realistic distribution of
the arterial blood temperature Tbl,a, instead of assuming a
constant arterial blood temperature for all body elements equal to the
temperature of the central blood pool Tbl,p. A number of
models of the CCX between pairs of adjacent vessels exist in the
literature [for a review see, e.g., Eberhart (13)], but these are
seldom incorporated into whole body models.
Assuming mass continuity in blood vessels, the decrease in the
temperature of the blood in the arteries of an element
Tbl,p Tbl,a corresponds to the increase in
temperature Tbl,v,x Tbl,v of the blood in
the veins of the same body element after CCX
|
(8)
|
Here, wbldV represents the blood
flow in the veins as the volume-integral of capillary blood flows
wbl over the body element. The blood temperatures
after undergoing CCX, Tbl,a and Tbl,v,x, are
the two unknowns in Eq. 8. To obtain an element's arterial blood temperature Tbl,a in Eq. 8, the net heat
exchange between adjacent arteries and veins Qx was
defined by using the equation of Gordon (17)
|
(9)
|
where hx is the corresponding countercurrent
heat exchange coefficient and Tbl,v is the venous blood
temperature before CCX. The arterial blood temperature
Tbl,a of a body element is then obtained by substituting
the right-hand side of Eq. 8 by the right-hand side of Eq. 9 and rearranging
|
(10)
|
The computation of the central blood pool temperature Tbl,p
utilized a special calculation process (see NUMERICAL
METHODS). Because the bioheat equation assumes that capillary
blood reaches an equilibrium with the surrounding tissue,
Tbl,v of a body element in Eq. 10 is obtained as
follows
|
(11)
|
The heat-exchange coefficients hx in Eq. 10 are
zero for central body elements (17) (i.e., no countercurrent heat
exchange). Few reliable data on hx of extremities exist in
literature. Therefore, hx values of extremities and
shoulders were estimated by a trial-and-error procedure, in which the
coefficients were adjusted to obtain agreement between local skin
temperatures as predicted and measured. For legs, feet, arms, and
hands, hx were obtained from measurements by Mayer and
Schwab (21) and Nielsen and Pedersen (25). An experiment of Budd and
Warhaft (6) provided data for the shoulders. The individual
coefficients hx are listed in Table 2.
In thermal neutrality, tissues are supplied at basal perfusion rates
wbl,0 (see Table 2). In nonneutral conditions or
during work, the blood flows, wbl in Eq. 1,
vary with changes in regional metabolic rates qm.
Experimental estimations (30) indicate a linear relationship between
wbl and qm. Instead of dealing
with blood perfusion rates wbl, the present model
considers directly their energy equivalent = blcblwbl
(W · m3 · K1).
The increased demand for oxygen is therefore accounted for by
calculating the change in the gain factor = blcblwbl as a
function of the change in metabolism qm using a
proportionality constant µbl = 0.932 (K1),
which was obtained from Stolwijk (34)
|
(12)
|
Utilizing , the blood perfusion term in Eq. 1
becomes
|
(13)
|
where
|
(14)
|
The very variable blood perfusion rates in the cutaneous plexus arise
from human temperature-regulation mechanisms, as discussed elsewhere
(16).
| |
HEAT EXCHANGE WITH THE ENVIRONMENT (BOUNDARY CONDITIONS) |
At the body surface, heat is exchanged by convection with the ambient
air qc, by radiation with surrounding surfaces
qR, by irradiation from high-temperature sources
qsR, and by evaporation of moisture from the skin
qe. The rate of heat exchange varies over
the body and is affected by the clothing ensemble worn. Hence, in the
model, heat balances were established for each skin sector of each body
element as boundary conditions to Eq. 1. In general, the net
heat flux qsk (W/m2) passing the
surface of a peripheral sector is equivalent to the sum of these
individual heat exchanges
|
(15)
|
The elements of this equation are considered in detail in following sections.
Convection.
The convective heat exchange qc between a skin
sector of surface temperature Tsf and ambient air of
temperature Tair was modeled by considering both natural
and forced convection using combined convection coefficients
hc,mix
|
(16)
|
In the present model, the convection coefficients hc,mix
(W · m2 · K1)
are a function of the location on the body, the temperature difference
between the surface and the air, and the effective airspeed
vair,eff (m/s)
|
(17)
|
The basis for hc,mix are the experiments of Wang
(36-38), who measured local convective heat losses on a heated
full-scale manikin with a realistic skin temperature distribution for
both natural and forced convection. Coefficients
anat, afrc, and
amix (see Table 2) were obtained by means of
regression analysis in which Wang's results were modified and
transformed into expressions more appropriate for computer simulation,
as shown by Eq. 17. To check the reliability of the convective
heat-exchange model, the overall, i.e., the mean, convection
coefficient of the human body, hc,m
(W · m2 · K1),
was obtained by integrating the local convective heat losses qc = hc,mix
(Tsk Tair) over the (naked) body
surface ADu
|
(18)
|
Here, Tsk,m is the area-weighted mean skin
temperature, i.e., Tsk,m = TskdAsk/ADu,
where Tsk is the model's local skin temperature, dAsk is the corresponding local surface area at a
body element of the humanoid, and ADu = dAsk is the Dubois area of the passive system,
which is 1.86 m2.
The overall coefficient was computed both for conditions of the natural
convection (vair,eff < 0.05 m/s) and for forced
convection. Figure 2 compares the
simulation results for the free convection with the data of Fanger (14)
and Bach (3) and the data quoted by McIntyre (24), i.e., free
convection coefficients for a 2-m-high cylinder obtained from Birkebak
(5) and coefficients obtained by Rapp (29), which are recommended for
sedentary people by McIntyre (24). Simulated forced convection
coefficients were compared with formulas used by Fanger (14), Bach (3),
Colin and Houdas (12), and Nishi (26) and are illustrated in Fig. 3. As apparent from Fig. 3, the model's
overall coefficient for forced convection correlates better with
results of more recent literature (3) indicating higher values (see
also Refs. 8 and 21), compared with earlier investigations.
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|
Fig. 2.
Overall convective heat transfer coefficient (hc,m) for
natural convection. Tsk,m, mean skin temperature;
Tair, air temperature; Fanger, Ref. 14; Bach, Ref. 3;
Birkebak, Ref. 5; Rapp, Ref. 29.
|
|
The effective air velocity vair,eff in Eq. 17 is a composite of the local room airspeed and the relative
airspeed due to body motion. Earlier investigations of walking subjects
(26) suggested an increase in hc,m with increasing body
movement. More recent experiments (8), however, have illustrated
constant or even decreasing hc coefficients during walking.
Similarly, simulations using the present model did not show evidence of
functional dependency of the convective heat loss on the activity
level. Obviously, hc is affected by the type of activity
rather then by its level. Hence, instead of introducing a fixed
relationship between hc and the activity level, which could
produce systematic errors in predictions, users of the model may modify
the input variable vair,eff appropriately, if
detailed information on hc for a specific task is available.
Radiation.
The exchange of heat by long-wave radiation qR is
usually of similar importance in the heat balance of the human body as
the heat exchange by convection.
In asymmetric environments, qR of a body element
sector represents the sum of the partial heat exchanges between this
sector and the surrounding structures (walls, windows, etc.), which may have different surface temperatures. Thus the exact simulation of
qR requires the computation of view factors (which
describe the influence of geometry on the radiative heat exchange)
between the surface sectors of the body and each surface segment of the surroundings. Unfortunately, calculation of the view factors between the curved surfaces of the body and surrounding structures is very
computer intensive. Hence, the concept of direction-dependent average
temperatures for the surrounding structures was adopted. In the model,
either Tsr,m, the mean temperature of the surrounding surfaces, or mean radiant temperature MRT can be used. Each of these
will differ from one body sector to another, and they may vary with
time. The value of Tsr,m is defined as the temperature of a
fictitious uniform envelope "seen" by a sector, which causes the
same radiative heat exchange with the sector as the actual asymmetric
enclosure. MRT is defined similarly, but it refers to a fictitious
uniform black envelope. Employing Tsr,m, and rearranging the Stefan-Bolzmann law by introducing the (local) radiative
heat-exchange coefficient hR
(W · m2 · K1),
the energy exchange is calculated by
|
(19)
|
where
|
(20)
|
and = 5.67 × 108
W · m2 · K4 is
the Stefan-Bolzmann constant; sf and sr,m
are the emission coefficients of the body surface sector considered and
of the surrounding surfaces, respectively; sf-sr is the
corresponding view factor; T*sf and
T*sr,m are the absolute temperatures (K) of the
body surface sector and of the surrounding surfaces seen by the body sector.
The long-wave emissivity of the body segments sf depends
on the covering material. The values for skin and hair in Table 2 were
obtained from Refs. 2 and 17. Clothing usually has a value close to
0.95 (14). The emissivity of the surroundings, which is an input
variable into the model, is typically sr = 0.93 for
indoor spaces. Note that if MRT is used in place of Tsr,m, sr has to be set to unity (black envelope).
The concept of using mean surrounding temperatures Tsr,m
makes it possible to introduce the view factors sf-sr
between sectors and the surroundings seen by them. The view factors
sf-sr of the cylindrical and spherical body elements
(Table 2) were calculated
for each sector separatelyby a specially
developed finite-element program (15). They vary from 0.1 to unity,
depending on the degree to which individual skin sectors are
"hidden" by other body parts. View factors are given for postures
with stretched limbs (reclining, standing activities) and for the
seated position that is characterized by a decreased body-radiation
area. As a check on the reliability of the view factors, the effective
radiant area ratio of the (nude) body feff = sf-srdAsk/ADu
was computed. The resultant whole body values of
feff = 0.80 for the standing position and
feff = 0.74 for the seated position are somewhat higher than provided, e.g., by Fanger (14), but they are in excellent
agreement with results of recent, more detailed investigations (19).
The mean surrounding temperatures are provided together with other
environmental variables in a so-called climate-input file. For
inhomogeneous environments, Tsr,m should be given
separately for the half of the room in front of the person (to be
applied to the anterior sectors of the body elements) and the half
behind the person (for posterior sectors). However, sometimes, it might be useful to determine Tsr,m for additional directions. For
example, Tsr,m would be calculated separately for the upper
part of the room when investigating the radiative effects of
heating/cooling ceilings, which may cause occupants to perceive local
discomfort to the head and shoulders.
While the MRT of a half room can be measured directly, the
corresponding Tsr,m must be calculated from measurements of
individual surface temperatures. A simple approach to calculate
Tsr,m is to weight the surface temperatures
Tsr, j by the corresponding surface areas
Asr, j
|
(21)
|
A more detailed method is to consider the effect of the body's
location in the room by using the view factors
bd-sr, j
|
(22)
|
The overall view factors bd-sr,j between the
body and wall segments have been measured experimentally by means of photographic projection techniques for both standing and sitting postures (14) and can either be taken from graphs or calculated by
using appropriate algorithms (7) and computer models (35).
The irradiation of the body by high-temperature sources (sun,
fireplaces, etc.) was also considered in the formulation of the passive
system. The short-wave radiation intensity and its direction is a
characteristic of the environment and is provided, together with other
environmental variables, in the climate-input file. The amount of heat
qsR (W/m2) absorbed at a sector
surface is taken into account in the heat balance of superficial body
element sectors by the term
|
(23)
|
where sf is the surface absorption
coefficient and depends on the color of the covering material, s
(W/m2) is the radiant intensity, and sf-sr
is the view factor between the sector and the surrounding envelope (see
Table 2).
Clothing insulation.
Because clothing plays an important role in the behavioral
thermoregulation of human beings, efforts were made to model garments in some detail. Unfortunately, standard literature provides only overall clothing parameters, i.e., the intrinsic clothing insulation Icl, the clothing area factor fcl, and
the moisture permeability index icl (for exact
definitions see, e.g., Ref. 22). These do not reflect the real
insulating properties of the garments but provide characteristics of a
uniform (imaginary) clothing layer covering the entire body. Therefore,
a simple computer model was developed that uses the present passive
system and resimulates the experimental procedures for measuring the
overall insulation characteristics of garments undertaken by McCullough
et al. (22, 23). The program computes the required local insulation
values I*cl,
f *cl, and
i*cl of trousers, shirts, underwear, socks,
etc., from measured uniform layer data Icl,
fcl, and icl for these garments and from the evaporative resistance of the fabric
Re,f. These local values include the insulation
effect provided by any layer of air trapped between the skin and the
fabric. A database on clothing items was created, which was used to
compose clothing ensembles in the present multisegmental model.
The local effective heat-transfer coefficient
U*cl
(W · m2 · K1)
of a j-layered clothing ensemble worn on a body-element sector is computed in the model as
|
(24)
|
where (I*cl)j
(W · m2 · K1)
is the "local" heat resistance of j-th clothing layer,
f *cl (dimensionless) is the local clothing area factor of the outer clothing layer, and
hc,mix and hR
(W · m2 · K1)
are the actual local coefficients for convection and radiation, respectively.
The corresponding evaporative coefficient
U*e,cl
(W · m2 · Pa1)
is obtained by using the local values i*cl
and I*cl of individual clothing layers applied to
the body sector, f *cl of the outer clothing layer, the local convection coefficient hc,mix,
and the Lewis constant for air Lair = 0.0165 K/Pa (24)
|
(25)
|
As a check on the validity of the new clothing model, the overall
insulation Icl of clothing ensembles and of individual
garments was computed by using a similar procedure to that
employed for obtaining the overall convection coefficient
hc,m. However, qcdAsk,
ADu, and Tair in Eq. 18 were
replaced by the integral of the local heat fluxes through the clothing
qcldAcl, the overall surface area Acl of the clothed body, and the
corresponding mean surface temperature Tcl,m, respectively.
The comparisons with results of experiments on clothed subjects showed
good agreement with published data. For instance, simulating the
thermal comfort experiment of Olesen and Fanger (27), in which the
subjects wore the so-called "KSU-uniform," Icl = 0.6 clo, icl = 0.34, the overall
insulation was predicted to be Icl = 0.59 clo and
icl = 0.34.
Evaporation.
The present skin evaporation model ensures heat and mass transfer
balances at each skin sector of each body element. The latent energy
transport from a skin sector of area Ask is given
by
|
(26)
|
The left-hand side of Eq. 26 represents the net energy transfer
as driven by the evaporative potential between skin and air, where
Psk is the water vapor pressure at the skin surface,
Pair is the vapor pressure of the ambient air, and
U*e,cl is the resultant evaporation
coefficient of garments covering the sector.
The first right-hand term of Eq. 26 considers the evaporation
of sweat from the skin surface, whereby
H2O = 2,256 kJ/kg is the heat of
vaporization of water and dmsw/dt is the rate of
sweat production over Ask, as elicited by the
active system (16).
The last term of Eq. 26 describes the heat transport by
moisture diffusion through the skin where Posk,sat is the
saturated vapor pressure within the outer skin layer of the body sector considered. In this superficial cutaneous layer, where sweat glands are
located, moisture is always present. Hence, the vapor pressure at this
location is set to its saturated value Posk,sat depending on the tissue temperature of the outer skin, Tosk (24)
|
(27)
|
The model uses a skin moisture permeability equal to
1/Re,sk = 0.003 W · m2 · Pa1,
which is also the published value (14). Simulations using the present
model showed that this value produces the model's basal skin
wettedness of wtsk = 0.06, which is a basic
property of the human skin (2).
The net evaporative heat loss of a skin sector arises from the actual
vapor pressure Psk (Pa) found at the skin surface. This quantity is calculated by using a rearranged Eq. 26
|
(28)
|
The present evaporation model accounts for the storage of sweat liquid
at the skin surface, as described by Jones and Ogawa (20). Moisture is
accumulated when Psk, as computed from Eq. 28,
exceeds the saturated vapor pressure Psk,sat, which is
calculated from Tsk by using an equation similar to Eq. 27. The rate of moisture storage dmacc/dt
(kg/s) is then equivalent to the rate of moisture production
dmsw/dt minus the rate of moisture evaporation
|
(29)
|
According to the experimental results quoted by Jones and Ogawa (20),
the maximum amount of sweat liquid on the skin is limited to about
macc/Ask = 35 g/m2.
Quantities exceeding this threshold run off and are not considered in
the model.
Respiratory heat losses.
Most heat is lost through the body surface; however, heat exchange with
the environment also occurs by respiration. The model for respiratory
heat loss was obtained from the work of Fanger (14), taking into
account the loss by evaporation and convection. The latent heat
exchange is calculated according to Ref. 14 from the pulmonary
ventilation as a function of the whole body metabolism [i.e., qmdV (W)]; latent heat of vaporization of water;
and the difference between humidity ratio of the expired and inspired
air, which depends on the ambient air temperature Tair
(°C) and the partial vapor pressure of the ambient air,
Pair (Pa)
|
(30)
|
Because the enthalpy of the expired air still depends to a certain
degree on the condition of the inspired air, Ersp
appears as a function of Tair and Pair,
allowing Ersp to be calculated for a wide range of
ambient conditions.
The dry heat loss of respiration due to the temperature difference
between expired and inspired air can be expressed (14) as a function of
the pulmonary ventilation rate and the temperature and vapor pressure
of the ambient air, Tair and Pair,
respectively
|
(31)
|
The total respiratory heat loss Ersp + Crsp was distributed over body elements of the pulmonary
tract: the lung and the muscle layers of the face and of the neck. The
distribution coefficients arsp were estimated from
the work of Scherer and Hanna (31). This approach recognizes that
inspired air is mainly conditioned in the nasal cavity where it reaches
~60% of its final enthalpy deep in the lung. By considering also the
rewarming of the nasal region by expiration, the following distribution
percentages were obtained: 25% in the outer face muscles, 20% in the
inner face muscles, 25% in the muscle band of the neck, and 30% in
the lung. These heat losses appear in the nodal heat balances for these tissues as the volume derivative in the heat-generation term
qm in Eq. 1
![<IT>q</IT><SUB>m</SUB> = <IT>q</IT><SUB>m,bas,0</SUB> + &Dgr;<IT>q</IT><SUB>m</SUB> − <FR><NU>∂[<IT>a</IT><SUB>rsp</SUB>(<IT>E</IT><SUB>rsp</SUB> + C<SUB>rsp</SUB>)]</NU><DE>∂V</DE></FR>](/content/vol87/issue5/fulltext/1957/img035.gif)
|
(32)
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NUMERICAL METHODS |
TOP
ABSTRACT
INTRODUCTION
