SCOPE AND CONSTRAINTS

TOP ABSTRACT SCOPE AND CONSTRAINTS THE STANDARD AT MODEL:... TOWARD AN AT SCALE... RESULTS SUMMARY AND CONCLUSIONS REFERENCES

Glossary

First Part of Term

A	Apparent
C	Convective
	Difference between surface and environment
E	Evaporative
F	Clothing
G	Absorbed/emitted extra radiation
I	Internal, core, or rectal (body)
K	Conductive
L	Forced convective
M	Metabolic
N	Natural convective
O	Outer
P	Proportion
R	Radiative
S	Skin surface
T	Skin tissue
U	Outer surface (skin or clothing)
V	By breathing
Y	Mean of boundary layer next to body's surface, whether clothed or bare
Z	Ambient
	Sum of T, F and Y; F component is zero on bare parts
	Second Part of Term (With Order of Magnitude in Still Conditions and Units)

B	Bare (proportion 0.05-0.7)
C	Clothed (proportion 0.3-0.95)
D	Diameter of body component (0.06-0.33 m)
E	Efficiency (proportion, 0.3-0.87)
g	Acceleration due to gravity (not necessarily 9.81 m/s2)
H	Heat transfer coefficient (0-8 W · m2 · K1)
J	Relative humidity (proportion, 0-1)
K	Conductivity (0.025-0.05 W · m1 · K1); K is also a standard abbreviation for degrees Kelvin
L	Change in enthalpy of moisture evaporating at skin (2.4 MJ/kg)
m	Ordinal number of cylinder representing one hundredth of skin surface
P	Vapor pressure (0-6,500 Pa)
Q	Heat flux density (40 to +150 W/m2)
R	Resistance to "dry" heat flow or "R" factor (0-0.3 m2 · K · W1)
S	Wind speed at person (0, normally, to 0.4 m/s when fan used)
S10	Wind speed at anemometer 10 m above ground
T	Temperature (20-55°C; or 293-328°K)
U	Radial thickness of clothing (0-0.01 m)
V	Virtual temperature (21-56°C)
W	Concentration of water vapor in air (0-29 g/m3)
Z	Resistance to moisture flow, in thermal units (0-100 m2 · Pa · W1)
	Thermal diffusivity of air (21-24 × 106 m2/s)
	Coefficient of expansion (1/aYT K1 for air)
	Kinematic viscosity of air (15-17 × 106 m2/s)
D	Diffusion constant of water vapor in air (22-26 × 106 m2/s)
Ptot	Total atmospheric pressure (101,350 Pa at sea level)
R	Gas constant for water vapor (467 m3 · Pa · kg1 · K1)
	Prefixes

a	Absolute (degrees Kelvin)
b	Bare part of body
c	Clothed part of body
d	While dressing, R increasing
i	Initial value
m	Mean
n	Neutral value of
s	Saturation
u	While undressing, R diminishing
	Dimensionless Numbers

Gr =	× OD3 × V × g/Y2 (106-108)
Nu =	CH × OD/YK (1-30)
Re =	OD × ZS/Y (not applicable in still conditions, but equivalent to 100-600)
Pr =	/ (0.72)
Sc =	/D (0.58-0.61)
St =	R ×  × OD/D/L/YZ (1-28)
	Designed to serve as a heat-stress index, a comfort scale and a measure of windchill, apparent temperature (AT) derives its validity from a wide range of measurements in many countries over the period from 1940 to 1995. Moderate extrapolation is indulged in, although others have extended it further at the hot end of the heat index. Because even the fittest human subjects cannot be made to endure the more severe conditions, the paucity of data at the extremes limits accuracy there. "Data" include such components as the ability of persons to perspire copiously, effects of extreme wind penetration into apparel as worn, effects of gustiness of strong winds, effect of certain reactions to extreme cold, and the effect of the inherent nonuniformity of sunshine on the person. As better data become available, they will be incorporated into the scale. The AT scales, and all conclusions reached in this paper, are applicable only within the following limits: 1) 70  AT  55°C, or 35.1  IT  39.6°C; 2) 40  ZT  50°C; 3) ZP < 4 kPa (dew point < 29°C) and ZP < sZP; 4) ZS < 11 m/s, i.e., ZS10 < 20 m/s; and 5) 40  GQ  150 W/m2 of body surface.
Furthermore, results are regarded as approximate if IT > 39°C, if at any place SJ > 0.90, or if, in the triple-solution region, one solution of the heat-balance equation gives AT > 55°C. These last regions are included on the charts, but with dotted lines.	A set of comparative tables encompassing the above ranges, although showing the effect of extra radiation when GQ = 100 W/m2, is available free of charge (steadman{at}sub.net.au). The set includes outdoor AT, equivalent temperature, temperature-humidity index, corrected effective temperature, wind-chill equivalent temperature, and wet-bulb globe temperature, each covering the range for which it was claimed to be designed; only AT covers all conditions likely to be encountered outdoors. Parsons (8) reviews heat-stress indexes more broadly, to include those that use measures other than temperature.

	THE STANDARD AT MODEL: A SUMMARY

TOP ABSTRACT SCOPE AND CONSTRAINTS THE STANDARD AT MODEL:... TOWARD AN AT SCALE... RESULTS SUMMARY AND CONCLUSIONS REFERENCES

The model represents an "average" person walking outdoors, at 1.4 m/s, generating heat at 165 W/m² in a set of environmental conditions ZT, ZP, ZS, and GQ. Average refers particularly to body parameters derived from Fanger's results (3) in 256 adults, representing equal numbers of men and women of various ages in the US and Denmark. The range of activity levels and clothing enables derivation of the variation in human thermal parameters used in this work and elsewhere, subject to the reasonable assumption that exogenous and endogenous heat stresses have the same effects. The model has been applied since 1971 with minor revisions as needed.

The AT model is elaborated from the work of Höschele (4). It integrates physiological factors in the body's interior and skin tissues, the physics of the clothing and internal air layers covering part of the body, and the meteorological factors in the environment. The last three factors, ZP, ZS, and GQ, have a combined effect that is added to the dry-bulb temperature ZT and expressed as AT3 ("in sunlight"). The level of AT, i.e., the number of the three secondary parameters used in determining AT, is the numeral after AT.

The scale is established by setting GQ = 0, ZS at a level due only to the person's movement (1.4 m/s) and zero external wind speed, and ZP at a neutral level nZP (12), given by as in most of the present paper. This corresponds to the same vapor concentration as at ZJ = 0.65, ZT = 20, the conditions specified for temperate-zone textile-testing laboratories.

In an evaluaton of AT, steady-state conditions are used, with the body losing heat at the same rate as it is produced metabolically. This steady state does not imply thermal comfort, except when IT  37°C. Contrary to a view that appears occasionally in the literature, AT, unlike effective temperature, with the latter's base of ZJ = 0.5, is based on absolute, not relative, humidity. Tables show ZJ as a concession to convention and to simplify application.

In an application, if the clothing requirement for 30°C, for example, is obtained at these neutral or base levels, any other set of conditions calling for the same insulation has an AT of 30°C. Another paper (14) quantifies four reflex and seven conscious ways in which the person regulates heat loss. Moreover, breathing, a mechanism for regulating heat loss for some other mammals, is a part of convective and evaporative heat exchange and is proportional to MQ in humans. The lungs function as a recuperative heat and moisture exchanger having 94% efficiency, giving

(1)

Of the four ambient variables, breathing heat loss depends only on ZT and ZP, and the evaporative component is the larger when ZT > 40°C. Total heat loss by breathing, VQ, accounts for as little as 3% of heat loss in hot or humid, and as much as 12% in cold, conditions at all activity levels.

In an unbiased scale, such as AT, effects of humidity, wind, and extra radiation can be positive, zero, or negative, although warming effects of wind are rare and slight. AT is obtained in two ways: by fitting values of FU (a measure of the amount of clothing needed) and of IT (a measure of physiological response) to the "neutral" ZT values. The AT values are quintic curve fits, having SDs in both calibration and validation <0.09 K. If the two values, which always differ by <1 K, are denoted by AT' and AT'', then AT = PC × AT' + PB × AT''. Conventional AT has an approximately one-to-one correspondence with IT, IP, TR, FR, FU, TZ, FZ, PC, and PB. Curve fitting is the smallest of eight sources of error inherent in the measurement and evaluation of the three levels of AT and the components. Along with two others that are derived from the effects of rounding off measured relative humidity and cloud cover in Australian official data, nine sources of error are quantified and summed in a table available free of charge from the author. Under less favorable conditions, for example, a value of AT3 is expressed ±1.1 K, AT1 is ±0.7 K, and the effect of wind is ±0.6 K, provided that ZT is measured to ±0.1 and wet-bulb to ±0.2 K.

Of the four independent variables, ZT is the most important, although there are conditions where it over- or underestimates the effect of weather, even climatic averages, on a person by >10 K. Although AT3 takes account of all four, some users find it useful to have simpler measures. If only ZT and ZP are evaluated, as in the US heat index, the effect of humidity, for which a universal 95% confidence interval is approximately 4 to +5 K, is obtained by subtracting the dry-bulb temperature ZT from the "indoor" AT1. Introducing wind speed at the person gives a second ("outdoor") level, AT2, hence the effect of wind, AT2  AT1, in an analogous range 8 to 0 K. This difference, the windchill, is more moderate than values presented by the media when ZS₁₀ < 19 m/s or ZS < 10 m/s. The popular scale, however, dangerously underestimates windchill at higher wind speeds, and efforts are being made to replace it (e.g., Ref. 9).

Complexities of estimating extra radiation GQ are explained elsewhere (12). When applied to solar radiation, it has four components, taken as acting evenly on the person: direct (typically 0-100 W/m² of total body surface); diffuse (0-40); terrestrial (0-30); and long-wave outgoing (30 to 0). In total, values are 30  GQ  120 for the walking human and 40  GQ  200 for quadrupeds or a prone, sunbathing person. When GQ is introduced, AT3 is obtained, giving an effect of extra radiation, AT3  AT2, usually in the range 2 K on a calm clear night to +6 K on the walking person exposed to sunshine at altitude angles near 50°. A 95% universal population-weighted interval for the combined effect, i.e., the amount by which AT3 as sensed by an active person exceeds ZT, is 10 to +9 K.

In contrast to some older systems, which either assume or ignore values for YR, ZJ, YZ, IT, IP, ST, and their derivatives, all the key physiological parameters are evaluated as part of the process of deriving AT. Other indexes, if clothing is considered at all, usually assume PC = EE = GE = 1. In the popular windchill scale, VQ = EQ = PC = 0. The AT printout can be set up to provide as many as 20 physiological parameters and 6 clothing variables. As AT changes, the corresponding change in IT is only 5 (cool) -10% (warm conditions) as great and is ignored by most workers in human biometeorology. IT is both worthy of inclusion and indirectly important because it is the key stimulus to the other 10 regulatory mechanisms that come into play.

Quantitative Aspects

In many publications, even contemporary ones, EQ is calculated (with perfect evaporative efficiency assumed) to express the discrepancy between two other heat flows and sometimes comes up negative, and then is taken as zero, as if the person were impermeable to "active" sweat. If it exceeds "E_max," a fresh set of assumptions is engaged. The AT system does not introduce such immeasurable terms as wetted area of skin or wetness factor of clothing, although SJ and UJ can be evaluated. Because accuracy of the AT scale cannot be guaranteed if SJ > 0.90, it is commonly checked. Only 80% of the surface is exposed to full convection and 72% to full radiation, because heat transfer of much of the surface, especially of the limbs, is limited by other parts.

Estimation of physiological parameters is based partly on the work of Fanger (3). Clothing parameters are based on physical properties of average fabric ensembles. To provide a seamless analysis, i.e., with no gap or overlap between winter and summer scales, these ensembles are qualitatively the same for hot and cold conditions, varying in thickness and coverage, but the effects of temperature (including the effect of extra radiation on clothing temperature) and wind penetration on conductivity are allowed for (Eqs. 12 and 12a, below).

Figure 1 illustrates moisture flow from the body's core to the environment. The potential is vapor pressure, and the three variable resistances are in series or additive. The flow follows Ohm's law, even though the flow is in liquid form through TZ and vapor through FZ and YZ. More important than the flow is the evaporation at the skin surface; for each gram that evaporates, a latent heat of 2,400 J is abstracted from both sides of the skin. This enthalpy change refers to the latent heat of evaporation at ST and the change of sensible heat as moisture temperature changes from IT to ST. It amounts to 2,400 ± 12 kJ/kg in all the conditions considered. Because the system is thermodynamically closed, isothermal heat lost in vapor expansion can be ignored (6).

View larger version (6K): [in this window] [in a new window]   Fig. 1.   Diagram of moisture flow from body core to environment. See the GLOSSARY for definitions.

This reasoning is now applied to heat transfer in Fig. 2. Sweat evaporates at the skin, and extra radiation is absorbed at the surface, which has 0.7 absorptivity and 0.97 emissivity. Ohm's law still applies but is combined with Kirchhoff's law at the junctions. TQ and EQ are always in the direction shown, but the other arrows in Fig. 2 may reverse, especially in the hot conditions that are the focus of this paper. To obtain steady state and to clarify TQ

(2)

The following equations are expressions of Kirchhoff's law and apply to the whole body, or to each of the body elements into which it is divided: at the skin

(3)

at the outer clothing surface

(4)

at bare parts and

(5)

Combining Eqs. 3 and 4 shows Eq. 5 to be valid also for clothed parts.

View larger version (8K): [in this window] [in a new window]   Fig. 2.   Diagram of heat flow in presence of evaporation and extra radiation.

The three temperature differences can be expressed as and, where applicable It follows by addition that the evaporative efficiency

(6)

and the efficiency with which absorbed extra radiation, positive or negative, adds to the person's heat load (to engineers, the "inward-flowing fraction") is

(7)

When a weighted average over the whole body surface in steady state is obtained, representative values are 0.7  EE  0.85 and 0.3  GE  0.7, averaged over the walking person. Both efficiencies, especially GE, tend to diminish as wind speed increases, causing a reduction in YR. Re (the ratio of inertial to viscous forces) > 10,000 almost always, in sharp contrast to the free-convection scenario (see the GLOSSARY).

Many results are conveniently portrayed on psychrometric charts, such as Fig. 3, which shows isotherms of AT1. In two dimensions, only two independent variables can be displayed. The use of three-dimensional graphs has been tried, but with such a loss of clarity and ease of measurement by the reader that contour charts of the type used here, for which there is no satisfactory available software, continue to be drawn by hand, for the same reason as isobars on weather maps are drawn by hand. Figure 3 is more commonly presented as a table, often in Fahrenheit temperatures, as the heat index. Wet-bulb globe thermometer (WBGT) readings provide one of the few other four-factor measures of heat stress. Although WGBT is insensitive to wind, oversensitive to humidity, inappropriate in freezing temperatures, and gives results some 20% below the perceived temperature, it correlates well with AT (SD about linear regression line = 2.0 K in the range defined in SCOPE AND CONSTRAINTS, with the further restraint ZT > 0) but has to be adjusted upward to give a realistic impression of temperature. The conversion AT = 1.25 WBGT is a useful one in above-freezing conditions.

View larger version (24K): [in this window] [in a new window]   Fig. 3.   Heat index for walking person. ZS = 1.4; GQ = 0; MQ = 165. Solid lines, effect of humidity (K); dashed lines, apparent temperature (AT).

The human model has a mass of 72 kg, a volume of 73 liters, and a surface area of 1.9 m². These values are not critical in the relative assessment of the four independent variables ZT, ZP, ZS, and GQ. Applying standard engineering reasoning gives a mean effective diameter (mSD) for the sides of a long cylinder, of 4 × 0.073/1.9 = 0.17 m. The separate variable diameters for the bare and clothed parts are obtained, with the significant diameter for the clothed parts being OD = SD + 2FU. The component bare and clothed diameters are not detailed here, because a more refined analysis of body dimensions was used in the free-convective scenarios. The AT scale is not applicable to infants, who have lower significant diameter SD and, especially if premature, lower TZ.

As the body's core temperature changes, its vapor pressure changes appreciably. Saturation vapor pressure is given by As IJ = 0.9 (1)

(8)

In view of the limit that Eq. 8 imposes on skin humidity of both clothed and bare parts, any result in which SJ > 0.90 at the wettest part of the skin is suspect and is possibly associated with free liquid on, or even dripping off, wetter parts of the skin. Such results are denoted by dotted lines (figures) and suffixes in the author's tables. A necessary but insufficient condition for this outcome at the metabolic levels considered here is ZP > 2,400 Pa. The internal temperature, which is used as a measure of AT, is related to the body's mean thermal resistance by

(9)

[In cold conditions, the erythemal effect limits bTR to a maximum of 0.053 to mitigate frostbite (7)]. Such conditions are beyond the scope of this paper, which will address comfort and heat stress, where TR < 0.048, but are included in the comparative values in SCOPE AND CONSTRAINTS.

Under all conditions, regardless of whether moisture transfer by perspiration is called "sensible," "insensible," or a combination

(10)

Observation of persons' thermally appropriate clothing indicates

(11)

Conductivity of the clothing ensemble is given by

(12)

This conductivity is well above the corresponding value for still air, 0.0240 + ZT/13,000, because of the higher conductivity of fibers but also because of internal radiation and air motion due to the person's movement. An earlier paper (11) shows the conversion of any anemometer wind speed ZS₁₀ to ZS for a moving person when all directions of movement relative to wind direction are equally likely.

The corresponding clothing resistance to moisture, being more sensitive to wind penetration, is given by

(13)

The clothing thickness, which is used as a second measure of AT, is given by and the proportion of the body's surface left unclothed by

(14)

The heat balance formula equates the heat production, in the absence of net useful work, to the "dry" and evaporative heat losses by breathing, and through the bare and clothed parts of the body

(15)

The surface resistance YR, referring to the parallel flows of convective and radiative heat from the surface to the environment, adjusted for clothing thickness but referred always to the skin surface area, is given by The convective dry heat-transfer coefficient for 80% of full convection is derived by fitting linear relationships to YK and Y for air at 60% saturation as functions of YT, substituting, because air is a perfect gas, 1/aYT for , and applying Nu = 0.22 Pr^0.31 Re^0.58 for the range of wind speeds encountered outdoors

(16)

The radiative coefficient for 72% of full radiation is obtained by factorizing the equation describing radiant exchange between UT and ZT for skin and clothing emissivities of 0.97 

(17)

If the radiant temperature of the surroundings differs from that of the ambient air, the difference is expressed as extra radiation. Another publication (13) shows how extra radiation is related to operant temperature and mean radiant temperature. Equation 15 is solved iteratively by entering an initial value of TR to find a trial value of the right-hand side (RHS) and repeatedly adjusting TR by (RHS  MQ) (TR/1.6)^1.6 until the two sides of Eq. 15 differ by <0.02 W/m². To avoid a fluke solution before the parameters have stabilized, this condition is prescribed for two successive iterations. More than 30 iterations are seldom needed. Corresponding values of FU and IT are substituted into curve fits to get AT.

Trials in which Reynolds' analogy (10) was applied to the dimensionless numbers (see the GLOSSARY) verified that the psychrometric constant is close to 61 Pa/K over the range of conditions pertinent to the free-convective anomaly, although as low as 59 in saturated cool conditions. The analogous resistance to moisture transfer is

(18)

In the outdoor conditions in which the heat index is determined, FR is commonly the highest dry resistance and TZ is the highest "moist" resistance. The effect is to keep the skin warm but dry. The rest of this paper will examine stagnant or still conditions where YR and especially YZ are on the same order of magnitude as the body and clothing resistances. In still conditions, YZ sometimes accounts for more than one-half of Z and becomes the limiting factor in moisture transfer.

	TOWARD AN AT SCALE FOR INDOOR CONDITIONS

TOP ABSTRACT SCOPE AND CONSTRAINTS THE STANDARD AT MODEL:... TOWARD AN AT SCALE... RESULTS SUMMARY AND CONCLUSIONS REFERENCES

There is much interest, especially from industrial engineers and medical specialists, in heat stress associated with indoor work. In the absence of much air movement, a person is more sensitive to extra radiation; when the person is resting or less active than a walking person, lower perspiration renders the person generally less sensitive to humidity than in the outdoor AT model. Partly, too, for forensic purposes and in sports medicine, an AT scale encompassing low air movement was needed. Work began in 1985 on this apparently routine task, with the intention of publishing scales at various levels of GQ within a year. Unexpected and puzzling complications soon arose. They led to refinements, such as the division of the body into 100 equal areas having different "hydraulic" or effective diameters.

The analysis performed here refers to the most stagnant conditions that a person is likely to encounter, and the reader's professional judgment is needed in interpolating between the outdoor AT scale and the "still" scale (see Evaluation). Potential applications of "phone-booth conditions," in which purely free convection is approached, are to such scenarios as the following: 1) a person working at a furnace or oven, or in a confined space, e.g., an attic; 2) a heat-stressed collapsed athlete; 3) children and pets in parked cars; 4) animals closely confined, such as battery hens and live-animal cargoes; 5) a person sunbathing in the absence of wind; and 6) a person engaged in underground mining.

Because the anomaly occurs at AT levels at which the sustained use of human subjects would be forbidden, there is scant experimental evidence to support quantitatively the conclusions reached theoretically in this paper. In view of the heat fatalities that occur indoors, especially in heat waves and occupationally, the prompt publication of an unconfirmed and partly unconfirmable set of results is better than a large gap in our quantitative knowledge. Refinements to the theory should obviate the need for any human experiments in such distressing conditions. However, a few pieces of anecdotal evidence pointing to "superheating" in still conditions have come to the author's attention and have been edited to remove some colorful adjectives and nonstandard units: 1) "I felt cool enough until I collapsed." (Runner in "fun run" at AT  35°C); 2) "It must be 60°C in that attic." [Electrician. Measurement showed 41°C with ~40 W/m² of extra radiation (AT3  47)]. Later it will become clear that he was in the middle of the free-convective anomaly. An advertisement for attic fans claims an attic temperature of 70°C; 3) "It got hot each time I went down the straight," i.e., when the tailwind had the same velocity as the athlete. (Walker in 10,000-m track event with ZT  33°C, conventional AT  38°C); and 4) "Once it goes above ~40°C, you don't notice much difference." (Expatriate manager working in Dubai; the author would argue that the claim is true only in the range 41  ZT  45°C).

Alternative Models

Clothing Requirements

(11a)

(12a)

(19)

View larger version (23K): [in this window] [in a new window]   Fig. 4.   Key body and clothing parameters when ZT = AT = 40. ZT= 40, neutral conditions.

Although FZ is usually a small part of Z, the effects of increasing FR are the following: 1) generally to reduce the outward dry flow of heat when ZT  36; 2) to reduce inward dry flow when ZT  38; 3) to increase EE slightly, especially important in conditions when EQ > MQ; 4) to reduce GE, thereby reducing inward-flowing solar and other radiation if GQ > 0; 5) to increase PC, increasing EE further; 6) to increase surface area, hence slightly reducing YR and YZ, which are referred to skin area; 7) to impede moisture vapor flow only slightly; and 8) to bring UT closer to ZT, hence reducing CH and raising YZ, often the chief effect.

These sometimes conflicting effects help explain why increasing FR sometimes tends to increase heat loss, but only when UV ~ ZV and when YR and YZ are dominant resistances. These conditions were found to appear only when MQ < 100 and low CH < 2.2.

Peculiarities of Free Convection

(20)

All values of Gr in this work are below 10⁸ and correspond to laminar flow. Gr is physically the ratio of buoyancy to the square of viscous forces. At this level, convective heat transfer from cylinders having vertical axes with 80% of full exchange is described by Nu = 0.44 (Gr × Pr)^0.25. From Reynolds' analogy, St = 0.44 (Gr × Sc)^0.25. Because Sc  Pr, St for free convection is ~5% less than Nu in air. Nu, in physical terms, is the ratio of the significant length (OD, here) to the thickness of the equivalent boundary layer of still air.

To improve the precision of the analysis, especially at the very low values of Nu where the anomaly is most apparent, the conventional formulas for free convection were split into a conductive part (with Nu = 1) and a remaining convective part. The small conduction has as its potential UT  ZT and its equivalent coefficient is

(21)

Substituting the relevant physical properties for the free-convective part gives

(22)

In practice this parameter is highly variable in an iterative solution if |UV  ZV| is near zero. In the program it is damped by a factor of nine to avoid instability, at the cost of generally slower convergence to the final value.

Pure natural convection from the living person is impossible even if there is complete protection from ambient air movement, including that due to heating, ventilating, and air conditioning systems. Involuntary movements such as breathing correspond to a minimum relative movement of ZS = 3 mm/s, averaged over the surface. At this speed, Re < 100, and forced convection, similarly over 80% of the area, is described by Nu = 0.32 Pr^0.31 × Re^0.5. This small component translates to

(23)

KQ and the much larger RQ are heat flows in parallel or antiparallel with CQ, but the composition of CQ is more complex. LQ, especially in the "fan" scenario, is essentially horizontal, whereas NQ is vertical, upward or downward, allowing the heat flows to be composed as vectors. Figure 5, averaged over all 100 cylinders, gives a clearer picture of the convective flows as ZT changes, positive values of Q corresponding to the outward flow of heat.

View larger version (23K): [in this window] [in a new window]   Fig. 5.   Components of convection in neutral conditions. A-D, zones of convection. See text for details.

In the conventional literature, only the larger of NQ or LQ is considered, leading to substantial errors in some of the conditions considered here and rendering continuous slopes in the scales impossible. Because the coefficients are at right angles, CH, on which YZ depends, is always given by

(24)

The determination of CQ is less simple. Only when NQ and LQ > 0, CQ =  (Fig. 5, zone A, left of ZT = 35.5, at which point LQ = 0 and CQ = NQ). If both components are negative, CQ =  (zone B, right of ZT = 40.6, at which point NQ = 0 and CQ = LQ). The scenario LQ > 0, NQ < 0 is a null set in human biometeorology, but if LQ < 0, NQ > 0, then, if NQ > |LQ|, CQ =  (zone C); but if NQ < |LQ|, CQ =  (zone D). At the boundary of these last regions, there is no net convection, CH is minimal and corresponds to the highest YZ = 1/(CH + KH) for that cylinder (as high as 90 for 1 cylinder), with mYZ never >52 in the anomalous region, but <30 elsewhere.

Evaluation

This work uses a resting level of MQ = 60 W/m² to provide a continuous scale, a near-minimal level for a standing person. MQ would be greater if the person were well enough to resist metabolically either extreme cold or extreme heat, and in an active person who had just collapsed. Another modification is a slight one, in recognition of the notion that the lower activity level affects vasoconstriction more than it affects body temperature

(9a)

In the original AT model, the person's surface is divided into only clothed and bare parts. In the present work, erratic results conduced refinement of the model, to have five different thicknesses of clothing on the clothed parts. This was, in turn, superseded by a model illustrated in Fig. 4. As before, cylinder theory opens access to a body of heat-transfer literature.

The iterative procedure is more complicated. The procedure described in Quantitative Aspects is merely an outer framework within which an inner iteration first occurs. Beginning with initial values of the 100 surface temperatures (m has values from 0 to 99) given by each value of UT is changed iteratively until Kirchhoff's law is obeyed at the surface (Fig. 2); i.e., both sides of Eq. 3 for the clothed parts and Eq. 5 for the bare parts agree within 0.002 W/m².

After this pair of equalities is achieved for all 100 rings, an adjustment is made to TR; this more sensitive adjustment is TR = (RHS  MQ) × (TR/2.6)^2.8. Typically, several hundred iterations are needed to satisfy Eq. 15 to within 0.002 W/m². The program, in C++ and run on a personal computer, is written to stop if it reaches 8,000 iterations, and results of the last few iterations are then inspected. If steady state is being approached, i.e., the differences are consistently <0.01 W/m² and falling, an approximate result is recorded. If there is oscillation, the program is modified until convergence occurs. In over 99% of cases, a result is obtained with <8,000 iterations, the remainder being all in the anomalous region. A commercial user of the program has reported no stopped evaluations in meteorological work. In the absence of wind, and its penetration into clothing, substitution of either IT or FU would yield the same value of AT, so only FU is used.

Offsetting this simplification is the finding, to be explained in more detail in Resistance in the Anolamous Range, that there are sometimes three solutions to Eq. 15. The central one, alone, is less stable and calls for either more or less thermal resistance until another steady state is reached. To find these two values, one begins with a value of iTR that must be above the higher result, and another iTR that must be below the lower result, and makes the iterative adjustment of the previous paragraph small, to avoid jumping from the domain of one solution to that of the other. The two extreme solutions for TR (and for all the incidental physiological and clothing parameters) are then averaged and substituted into the curve fit of TR against ZT at the neutral levels of ZP and GQ as before, except that ZS is input only as 0.003 in the free-convection scenario. The two initial values, developed by trial and error to ensure that they were always higher and lower, respectively, than the final values, are and This curve fit is achieved by omitting results in the anomalous region, which is not automatically or objectively defined. Depending secondarily on ZP and GQ, the anomaly peaks at ~41°C and is evident in the range 36  ZP  45°C when ZP and GP have neutral levels, i.e., levels used in preparing the curve of best fit. Accordingly, input values for preparing this fit were in the ranges 20  ZT  34 and 46  ZT  57°C. The curve fit was extended beyond 55°C because an earlier fit ending at 55°C had a regression, long undetected, in the relationship between X (defined below) and AT; it was removed by a minor change to Eq. 17. This yielded the general equation

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where X = ln (140 FD₉₉) and FD₉₉ is the thickness of clothing on the thickest and most clothed ring; the multiplier 140 is chosen to minimize the coefficients in Eq. 25. SE of the curve fit is 0.09 K. This smooth curve is then applied to the whole range 22  AT  55°C, although it gives results 0.20 and 0.09 K high when validated at 34 and 46°C, respectively, showing evidence of the anomaly even at the fringes.

Resistance in the Anomalous Range

View larger version (28K): [in this window] [in a new window]   Fig. 6.   Effect of varying resistances when heat loss varies.

When ZT  40 and ZT  45°C, the results are "normal", i.e., increasing R causes diminishing Q. But the curves of intermediate temperatures show inflexions and regions where increasing R also increases Q. These are places where UV  ZV, at least at many values of OD_m, and some of the usually minor effects in Alternative Models predominate. A further requirement for a triple solution is that the line MQ should intersect the curve in three places. That this can happen only at fairly low MQ explains why only stationary people are sensitive to the superheating. At least as important is the movement associated with higher levels of activity, which provides forced convection at a much higher rate than that due to breathing. The curves also show that appreciable displacement of resistance from the middle solution is likely to lead to a new steady state corresponding to the highest or lowest resistance, hence the use only of those two, uFD₉₉ and dFD₉₉, which are first averaged to determine AT from Eq. 25.

	RESULTS

TOP ABSTRACT SCOPE AND CONSTRAINTS THE STANDARD AT MODEL:... TOWARD AN AT SCALE... RESULTS SUMMARY AND CONCLUSIONS REFERENCES

Advantages of the theoretical approach are that harsh and dangerous conditions can be safely explored; hundreds of subjects can be averaged without employing any; no time is wasted in bringing subjects to steady state; complete consistency is ensured in comparisons; and an almost unlimited number of conditions can be examined quickly, e.g., for preparing charts. Although the chief emphasis is on AT, the output file can include any other variables of interest, such as IT, bST, cST, UT, SP, SJ, UV, TR, FR, FU, CH, RH, YR, TZ, YZ, PC, or PB, VQ/MQ, OD/SD (the "clothing factor"), TQ, EQ, EE, and GE, usually averaged over the whole person. It is worth repeating that many publications shed little light on these subsidiary parameters; indeed, they are treated in one of two ways: assumed or ignored. As an example, the popular US windchill scale sets ST = UT = 33°C, then declares in places, "Exposed flesh freezes." The logical process makes it easy to obtain by subtraction in the following order: 1) effect of humidity (by using conventional outdoor AT1  ZT; solid lines in Fig. 3); 2) effect of stagnation and inactivity (Fig. 7); and 3) effect of extra radiation.

View larger version (36K): [in this window] [in a new window]   Fig. 7.   Effect of stagnation and inactivity, illustrating free-convective anomaly. Dashed lines, AT in still conditions; solid lines, effect of stagnation and inactivity. GQ = 0.

The following observations are made on the above three points. The still AT scale confounds the effects of humidity, stagnation, and inactivity. The first is already available, at least for the active person, relative to the base humidity ZW = 12. The other two are not readily separable, because any level of steady activity having MQ 60 is likely to be accompanied by movement. The chief caution is that, owing to the lower rate of activity and especially perspiration, the resting person is less sensitive to humidity changes, especially when ZT < IT. Hence, at the lower temperatures considered here, near ZT = 28, the person has a relative sensation of AT as much as 3 K warmer as ZP  4,000. When ZT  38 and sweating becomes copious, the differences can be fairly interpreted as due mostly to stagnation.

Inner and Outer Solutions in Anomalous Region

View larger version (27K): [in this window] [in a new window]   Fig. 8.   Differences between AT in triple region. ZS = GQ = 0.

The most extreme example of this discontinuity to be found when GQ = 0 was at ZT = 43, ZP = 2,663, when uTR = 0.02538, giving uAT = 51.61 and umYZ = 39.9. The slightest increase in AT, whether due to ZT, ZP, or GQ, upsets this unstable equilibrium. In this instance ZP was changed incrementally to 2,664, with umYZ now 42.8 and uTR now 0.02430, corresponding to uAT = 53.97. There was little change in dAT = 55.59 at both vapor pressures, so that AT, on the basis of the mean of uTR and dTR, showed a step change from 53.46 to 54.72. Such steps are all <1 K when AT < 50, and have little effect on the contour charts, except that in the AT model, there is no exact value of ZP corresponding to the contour AT = 55 when ZT = 44.

Within the loosely defined anomalous region, the region where dAT  uAT is clear (Fig. 8), although sensitive to the scenario's specifications. Conditions where (uAT  dAT) > 1 K are relatively rare, but there is some uncertainty about the vortex of the anomaly, because of the three solutions, the high skin humidity, and AT (Fig. 7, which includes the same conditions). The numbers shown in Fig. 8 refer to the differences between the two extreme solutions.

Figure 7 illustrates the regions where relief from stagnation causes AT to fall despite an increase in ZT. There are a few places, with ZT ~ 41, where an increase in ZP causes a slight reduction in AT, but no instances were found where increasing extra radiation lowered AT, although microscale reductions could conceivably occur in the anomalous region for a small change in GQ. Only steps of 20 W/m² were investigated.

An interesting corollary observation was that, although most triple solutions are associated with slight absolute values of GQ, they occurred when GQ = 60, ZT > 47, but in reverse, i.e., apparently because of the protection that the body's resistances give to extra radiation, uAT > dAT in this very limited and oppressive range, where an experimental check of the result would be hazardous.

Fundamental Findings

The physical conditions are examined more closely in a series of parameters graphs, which refer only to "neutral conditions," corresponding to GQ = 0 and the neutral vapor pressures shown by the zero line in Fig. 3, and described by ZW = 12. Figure 9 examines the changes in temperatures, averaged over the body when necessary, at the various nodes of Fig. 2. The anomaly, the vertical difference between the AT curve and the dashed line showing ZT, peaks at ZT = 41. There is a slight cooling effect as ZT increases further, until increasing inward heat flow adds to AT. With a range in ZT of 22 K, the sensitivities of the other temperatures to change in ambient temperature are 0.09 for ZT, 0.13 for mST, and 0.35 for mUT. The last becomes so great at high ZT that it exceeds ST, even though the two are identical on the predominant bare parts of the body, i.e., heat is transferred inward through clothes. mST < IT always, as most of the body's heat output must be conducted through the skin. [Observations of poikilotherms (cold-blooded animals) and, occasionally, of panting quadrupeds have shown measurements of ST > IT in extreme conditions.]

View larger version (21K): [in this window] [in a new window]   Fig. 9.   Dependent temperatures. ZW = 12, GQ = 0.

There is no crossing in the lines of Fig. 10, which shows corresponding vapor pressures. These curves illustrate the heat stress of the anomalous region, where intense perspiration occurs. As more of the surface is bared, the lines of UP and UT approach those of SP and ST, respectively. Perspiration to compensate for stagnation is such that, at least when ZP = nZP, UP reaches a maximum at ZT = 42. Figures 8 and 9 can be used to derive approximately the corresponding mean values of V and J (not illustrated).

View larger version (22K): [in this window] [in a new window]   Fig. 10.   Nodal vapor pressures. ZW = 12, GQ = 0.

Figure 10 illustrates thermal resistance cumulatively, the total resistance to heat flow being a major controlling factor. As ZT increases, so does RH, offsetting the reduction in CH as UV approaches ZV. After the anomalous region, both RH and CH increase, helping to make the consequent "fall" in most parameters steeper than the rise in the approximate range 35  ZT  41. These resistances are arithmetic, not harmonic, averages of the 100 rings, and in general differ from values obtained by dividing the mean heat flow (TQ, FQ, or YQ) into the mean temperature difference.

Figure 11 is the corresponding depiction of resistances to moisture transfer. Because of its superficial similarity to Fig. 12, its peculiarities are described. Although TZ is the controlling resistance in cool and moderate conditions, and FZ of normal clothing is always small, the rise of mYZ to >40 when UV  ZV limits sweat dissipation. In the absence of radiation, Z, unlike R, reaches a maximum in the anomalous region.

View larger version (23K): [in this window] [in a new window]   Fig. 11.   Resistances to moisture flow. ZW = 12, GQ = 0.

View larger version (23K): [in this window] [in a new window]   Fig. 12.   Resistances to heat flow. ZW = 12, GQ = 0.

To clarify the influence of mYZ at all vapor pressures, it is plotted in Fig. 13. When ZT  45 with GQ = 0, the rapid restoration of free convection as all parts of the surface develop a temperature difference from the surroundings becomes apparent. Because any extra radiation warms the surface, this transition occurs at slightly higher values of ZT. The lowest values of YZ occur when GQ  20. The next section examines effects of extra radiation more fully.

View larger version (30K): [in this window] [in a new window]   Fig. 13.   Surface resistances to moisture flow (YZ). ZW = 12.

Effects of Extra Radiation

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View larger version (29K): [in this window] [in a new window]   Fig. 14.   Effect of moderate extra radiation. Dashed lines, AT when GQ = 60. , Change.

View larger version (22K): [in this window] [in a new window]   Fig. 15.   Effect of strong extra radiation. Dashed lines, AT when GQ = 120.

View larger version (32K): [in this window] [in a new window]   Fig. 16.   Effect of outgoing extra radiation. Dashed lines, AT under clear night sky. GQ = 40.

At a mild level (GQ = 60, Fig. 14), warming of a still person covers a wide range of effects, possibly reaching 14 K in the anomalous region. At this level, there are no triple solutions except in the marginal range ZT > 47. The synergism of both temperature and humidity with extra radiation is apparent in the narrowing gaps between the AT isotherms as ZT and ZP increase.

The strong extra radiation in Fig. 15 (GQ = 120) refers especially to full sunshine, a level likely to be exceeded only when the sun's altitude is between ~50 and 75°, and more likely near the perihelion. The epicenter of the anomaly has now moved off-scale, but AT > 55 at all levels to the right of ZT = 43.

Even higher levels of GQ, near 200, are absorbed by a prone white person sunbathing when the sun's altitude is high, but the present model is not suited to the sunbathing scenario, where there is much contact with the ground, of vaguely specified temperature. Modifying rigid traditional formulas to recognize the variability in skin temperature, de Freitas (2) has examined heat transfer in sunbathing.

Exposure to a clear night sky corresponds to GQ  30, depending on temperature and especially humidity. The extreme value GQ = 40 is evaluated in Fig. 16. Some of the greatest cooling effects are where the greatest warming effects of stagnation occurred at GQ = 0. The range of that chart limits the comparisons in Fig. 16. When GQ = 40, and when GQ = 120, there is only one solution of Eq. 15 throughout the figures. In indoor industrial practice, GQ is seldom below 10, because the insulation around refrigerated spaces brings the inner temperature of walls close to ZT.

Figure 17 summarizes generally the effect of extra radiation when ZP is neutral. An approximate locus describing the displacement of the anomaly as GQ increases is sketched. Examination of the isolines, especially in the anomalous region, shows that, in general, unlike outdoor AT, the effect of GQ is not exactly proportional to GQ.

View larger version (31K): [in this window] [in a new window]   Fig. 17.   General effect of extra radiation at neutral humidity. ZW = 12. Dashed lines show AT.

The information developed so far enables compilation of Table 1, which describes the "stagnant" heat index as a function of ZT and relative humidity, as well as estimating the effects of 100 W/m² of extra radiation when ZT = 40 and 30, subject to the restrictions in SCOPE AND CONSTRAINTS.

Relieving Heat Stress by Air Movement

View larger version (21K): [in this window] [in a new window]   Fig. 18.   Effect of slight air movement. Dashed lines, AT when ZS = 0.4, GQ = 0.

In view of an opinion, apparently deriving from an extension of the earlier effective temperature scale, that fans may aggravate heat stress, the key result in Fig. 18 is that air movement reduces AT in all conditions. There is much variation in the cooling effect, which is greatest where the anomaly occurred in still conditions. Even at this low wind speed, there are no triple solutions but a one-to-one relationship between resistance and AT. Higher speeds provide more cooling, although not in proportion to fan speed.

Alternative Measures of Heat Stress

Accordingly, the perspiration rate and average skin humidity were plotted as contours in Fig. 19. Perspiration through the skin (not the lungs) is appreciable at all temperatures, and gross perspiration EQ exceeds MQ (60) when ZT  38. As ZT increases, perspiration intensifies; i.e., the solid isolines become closer. At low temperatures, when perspiration is more passive, it is greater at low humidities, i.e., when P is high, but at higher temperatures the dependence on perspiration to relieve the stress of both temperature and humidity is so great that a reverse tendency becomes apparent. Clearly, the isolines of EQ have slopes so similar to those of ZT that perspiration rate is little more than a measure of ZT alone.

View larger version (21K): [in this window] [in a new window]   Fig. 19.   Alternative measures of heat stress. Solid lines, EQ; dashed lines, SJ. ZW = 12, GQ = 0.

The isolines of skin humidity, although distorted by the anomaly, are essentially horizontal and provide merely a measure of absolute humidity, ZP. If one needs a measure of total heat stress, where the slope of the isolines is between those of the wet-bulb or adiabatic cooling lines and the dry-bulb lines, measures of perspiration and skin wetness are both unsatisfactory, a fact that is even more obvious in outdoor AT, where there is no free-convective anomaly.

Analysis of Heat Flows

View larger version (22K): [in this window] [in a new window]   Fig. 20.   Heat flows to and from surface. ZW = 12, GQ = 0.

Evaporation is described in Fig. 21, which shows it to greatly exceed breathing loss due to both heat and moisture exchange. When evaporation occurs at the skin under still conditions, where ZR, hence also EE, is high, ~85% of the cooling is abstracted from the body, only ~15% from the surroundings. The slightly lower net EQ from the body is also shown, because only it can be directly compared with MQ and VQ. Only if CQ, RQ, and KQ are multiplied by (FR + YR)/R, the various heat losses can be added to confirm that the sum equals MQ.

View larger version (22K): [in this window] [in a new window]   Fig. 21.   Heat flows from interior and skin. ZW = 12, GQ = 0.

	SUMMARY AND CONCLUSIONS

TOP ABSTRACT SCOPE AND CONSTRAINTS THE STANDARD AT MODEL:... TOWARD AN AT SCALE... RESULTS SUMMARY AND CONCLUSIONS REFERENCES

The one-to-one relationship between total resistance and AT that is the basis of the outdoor AT scale fails when applied to still conditions. When ZT just exceeds AT, especially when ZV  UV, there is little air movement due to free convection, hence limited means of removing perspiration vapor. Furthermore, the complexity of heat and moisture flows leads, in an anomalous zone, to three solutions of the human heat-balance equation, where an increase in the resistance ensemble sometimes leads to an increase in heat loss. An immediate and serious effect is for the AT in the range 38-44°C to be several degrees hotter than expected when there is no external air movement around a person. Extra radiation has a greater and more variable effect on the still person than the walking person and shows synergy with both temperature and humidity. Use of a fan, or any means of providing air movement, always provides a stationary person with relief from heat.