On the Problems Related to Natural Wet Bulb Temperature ...

Ann. Occup. Hyg., Vol. 56, No. 9, pp. 1063–1079, 2012© The Author 2012. Published by Oxford University Press

on behalf of the British Occupational Hygiene Societydoi:10.1093/anhyg/mes036

1063

*Author to whom correspondence should be addressed.Tel: +39-0-89-96-41-07; fax: +39-0-89-96-40-37;e-mail: [email protected]

On the Problems Related to Natural Wet Bulb Temperature Indirect Evaluation for the Assessment of Hot Thermal Environments by Means of WBGTFRANCESCA ROMANA D’AMBROSIO ALFANO1,*, BORIS IGOR PALELLA2 and GIUSEPPE RICCIO2

1DIIN-Dipartimento di Ingegneria Industriale, Università di Salerno, Via Ponte Don Melillo, 84084 Fisciano (Salerno), Italy; 2DETEC-Dipartimento di Energetica, Termofluidodinamica Applicata e Condizionamenti Ambientali, Università degli Studi di Napoli Federico II, Piazzale Vincenzo Tecchio 80, 80125 Napoli, Italy

Received 28 April 2012; in final form 17 January 2012; published online 17 July 2012

This paper deals with the indirect evaluation of the natural wet bulb temperature, tnw, one of the two quantities forming the basis of the well-known wet bulb globe temperature (WBGT) index, considered worldwide to be a suitable and user-friendly tool for the preliminary assess-ment of hot thermal environments. This quantity can be measured by a wet bulb thermometer (a temperature sensor covered with a wetted wick naturally ventilated) or, if this is not available, calculated from other microclimatic parameters (i.e. the air temperature, the globe temperature, the air velocity, and the humidity) using a quite trivial energy balance equation. Because of the strong non-linear structure of such an equation, the risk of a multiplicity of steady state solutions could result in the failure to obtain a reliable index evaluation. To dispel all doubts, this work car-ries out an in-depth analysis of the heat balance equation to be solved for the indirect evaluation of the natural wet bulb temperature. A preliminary investigation of each heat flow term involved in the heat balance on the sensor has been carried out; in a second phase a special continuation method has been implemented, highlighting the effect of microclimatic parameters on the multi-plicity of solutions. Results show that under free convection the evaluation produces a single solu-tion only under uniform conditions, whereas in the presence of even slight differences between the air temperature and the mean radiant temperature, there can be as many as three solutions. This phenomenon, if confirmed by a further experimental investigation, could become a difficult matter since a sensor, in principle, has to read a unique value of the quantity measured. In any case, from a numerical point of view, the presence of many values of tnw greatly reduces the pos-sibility of an indirect WBGT calculation from the other involved physical quantities; as a conse-quence, the indirect evaluation of WBGT should be clearly avoided based on ISO 7243 Standard.

Keywords: heat stress; natural wet bulb temperature; WBGT

INTRODUCTION

The need for a quick and reliable index able to assess the environmental heat stress is a topic still fiercely debated by both occupational hygiene and biometeor-

ology specialists. This has been further encouraged by the COST Action 730, completely committed to the formulation and the validation of the Universal Ther-mal Climate Index (Bröde et al., 2009, 2011). As a consequence, the different approaches have led to the formulation of several indices in the effort to reach a balanced compromise between a reliable assessment, an easy calculation, and a reduced number of meas-urements (d’Ambrosio Alfano et al., 2011).

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1064 F. R. d’Ambrosio Alfano, B. I. Palella and G. Riccio

In 1957, Yaglou and Minard introduced the wet bulb globe temperature (WBGT) index proposed by CEN and ISO as a preliminary tool for the assess-ment of hot thermal environments (ISO 7243, 1989). Its formulation, despite continuous investigation by the scientific community over the last 50 years, has not changed (Epstein and Moran, 2006; Parsons, 2006; Mochida et al., 2007; Budd, 2008; Gaspar and Quintela, 2009). This index combines the measure-ment of two derived quantities, the natural wet bulb temperature, defined as the temperature indicated by a sensor covered with a wetted wick naturally venti-lated, and the globe temperature or the air tempera-ture; in this way, all of the heat transfer phenomena affecting the thermal sensation (evaporation, con-vection, and radiation) are summarized in only three measurements. The index is calculated as follows:

(indoor) WBGT nw g0 7 0 3. . .t t (1)

(outdoor) WBGT nw g a0 7 0 2 0 1. . . .t t t (2)

To assess the thermal environment with the value of WBGT obtained with equation (1) or (2), a com-parison is required with limit values related to the metabolic rate, the acclimatization status, and the air movement (see Table 1).

Although WBGT seems not to be related to other physical quantities, which usually appear in the heat balance of the human body (McIntyre, 1980; Parsons, 2003), it is noteworthy to highlight that tg depends on the air temperature, the mean radiant temperature, and the air velocity (ISO 7726, 2002), whereas tnw is also related to the humidity (Sullivan and Gorton, 1976; Azer and Hsu, 1977); therefore, its indirect evaluation requires the measurement of four quantities.

According to a review by Budd (2008), an impres-sive number of papers have been devoted to the WBGT index: some have concerns about its reliabil-ity and seek to introduce special corrections related to its inability to take into account the metabolic rate of the subject as well as both the thermal insu-

lation and the vapour permeability of clothing (Ber-nard, 1999; d’Ambrosio et al., 2004; Bernard et al., 2009), whereas other authors attempt to introduce other indices as more reliable indicators of hot stress induced by both the special working situation and climatic impact (Moran and Epstein, 2006) through combined experimental and theoretical investiga-tions. Finally, due to its worldwide dissemination, other papers address its indirect evaluation from the above-quoted microclimatic quantities (Buonanno et al., 1998, 2001; Gaspar and Quintela, 2009). Previous papers devoted to this topic proposed easy numeri-cal procedures based on the heat balance equation on the sensor and special correlations based on the analogy between heat and mass transfer phenom-ena (Malchaire, 1976; Sullivan and Gorton, 1976; Azer and Hsu, 1977). Malchaire, in an earlier study, used a combined experimental and regression analy-sis to find a correlation between the main physical quantities and both the globe and the natural wet bulb temperatures. This possibility was discussed several years later by Brake (2001) and Buonanno et al. (2001) who highlighted a significant uncertainty in the calculation of the tnw by means of correlations (up to 1.5°C) under forced controlled conditions even greater than the accuracy specifications required by the ISO 7243 Standard (see Table 2). In a paper by Gaspar and Quintela (2009), aimed at the formula-tion of a special procedure for the assessment of the WBGT from meteorological data, an enhanced model of radiative heat flows combined with new heat and mass transfer correlations was introduced. Notwith-standing this enhanced modelling, the authors dem-onstrate a systematic overestimation of both tg and tnw of about 1.0–1.4 and 0.6–1.1°C, respectively, which they try to eliminate by introducing a special correc-tion for the wind profile under outdoor conditions.

This structural uncertainty seems to be quite a tricky matter not only for the evaluation in situ of the working situation in the absence of a globe or a natural wet bulb sensor but also for a preliminary

Table 1. WBGT limit values reported in ISO 7243 Standard (ISO 7243, 1989).

Metabolic rate class

Metabolic rate (M; W m−2)

Reference WBGT value (°C)

Person acclimatized to heat Person not acclimatized to heat

0 (resting) M < 65 33 32

1 65 < M < 130 30 29

2 130 < M < 200 28 26

No sensible air movement

Sensible air movement

No sensible air movement

Sensible air movement

3 200 < M < 260 25 26 22 23

4 M > 260 23 25 18 20

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Problems in indirect valuation of natural wet bulb temperature by means of WBGT 1065

assessment of the workplace and the prediction of such situations recognized as dangerous for peo-ple (d’Ambrosio Alfano et al., 2007). Moreover, although the ISO 7726 Standard makes explicit reference to which correlation must be used for the calculation of the heat transfer coefficient of the globe, hc,g, no equations have been formulated for the wet wick sensor, both under still and forced convection. This omission is even more problematic because the ISO 7726 Standard does not specify the minimum air velocity value that is consistent with forced convection. In any case, regarding the indirect calculation of tg, ISO 7726 suggests using the greater value of hcg from among the natural and the forced ones, and, according to Table 3, the measurement ranges of the air velocity in Class S would seem to reflect situations in which the air conditions are quite still, such as va < 0.20 m s−1 (ISO 7726, 2002).

There is another crucial factor that must be high-lighted: the systematic uncertainty of tg and tnw reported in the literature could be related to the highly non-linear mathematical structure of heat bal-ance equations on which both quantities are based rather than the unreliability of heat and mass trans-fer correlations or measurements. In other words, are we really sure that a microclimatic situation characterized by one value of the air temperature, the mean radiant temperature, the air velocity, and humidity has only one value of tg and tnw? Concern-ing the globe temperature the answer is almost trivial because the temperature read by the sensor installed inside the black globe (ISO 7726, 2002) is the result of the equilibrium between radiative and convective

flows. On the other hand, the steady state value read by the wet bulb sensor is the result of the equilibrium between the energy flow required for the evapora-tion of the water and the overall heat flow by con-vection and radiation. Therefore, in principle, more values of convective and radiative heat flows could return the same heat flow required for water evap-oration on the wick. As a consequence, the sensor placed inside the wet wick could measure different tnw values under the same microclimatic condition in terms of wet heat loss; conversely, the same tnw value could be related to more microclimatic conditions. This phenomenon is well known in several chemi-cal and mechanical engineering applications (Aris and Amundson, 1958; Levenspiel, 1972; Hale and Koçak, 1991; Pirone et al., 2000).

The above-cited information focuses the analysis on the equations on which the indirect evaluation of the natural wet bulb temperature is based. This study is carried out in three steps: in the preliminary phase, we deal with the behaviour of each heat flow term involved in the definition of both tg and tnw in order to highlight the result of a multiplicity of solutions for the heat balance equation on both sensors. In the second phase, in order to discover the range of micro-climatic conditions leading to multiple values of tnw for the same thermal environment, numerical results of a bifurcation analysis carried out through a spe-cial continuation algorithm are thoroughly discussed. Finally, the results of the indirect evaluation of the tnw have been used to highlight the ambiguity of the heat stress assessment by means of the WBGT index.

METHODS

In principle, the evaluation of the WBGT index requires neither a globe nor a wet bulb thermometer; in fact, the globe temperature can be evaluated from the mean radiant temperature, the air temperature, and the air velocity (ISO 7726, 2002), whereas the wet bulb temperature can be obtained from the air temperature, the humidity, and the mean radiant tem-perature (ISO 7243, 1989). In any case the indirect

Table 2. Main requirements for the instruments devoted to the WBGT index evaluation (ISO 7243, 1989).

Globe Natural wet bulb thermometer

Shape Sphere Cylinder

Diameter 150 mm 6 mm ± 1 mm

Thickness As thin as possible —

Measurement range 20–120°C 5–40°C

Accuracy 20–50°C:±0.5°C50–120°C:±1°C ±0.5°C

Mean emissivity 0.95 1.0

Length of the sensor — 30 mm ± 5 mm

Table 3. Characteristics of measuring instruments devoted to the air velocity according to ISO 7726 Standard (ISO 7726, 2002).

Class Measuring range Accuracy

Class C 0.05÷1.0 m s−1 Required: ±(0.05 + 0.05va) m s−1

Desirable: ±(0.02 + 0.07va) m s−1

Class S 0.2÷20 m s−1 Required: ±(0.1 + 0.05va) m s−1

Desirable: ±(0.05 + 0.05va) m s−1

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evaluation of WBGT index by measuring the dry bulb temperature, the mean radiant temperature, the humidity ratio of the moist air and, finally, the air velocity requires an in-depth analysis of mass and energy balance equations related to the calculation of the globe and the natural wet bulb temperatures. In the first elementary phase, each heat flow term has been analysed as a function of the four microclimatic parameters by which each is affected (ta, tr, va, RH).

Energy balance on the globe thermometer

Concerning the globe temperature, defined as the temperature indicated by a sensor placed in the cen-tre of a globe having standard characteristics (see Table 2), the energy balance equation under steady state conditions applied to the external surface of the sensor can be written as follows:

q qr c+ = 0, (3)

where

q h t tc c,g g a= −( ) (4)

and

q t tr g r g= + − +ε σ[( ) ( ) ]273 2734 4 (5)

The calculation of the convective heat transfer coefficient of the globe has been carried out by mak-ing use of special correlations suggested by ISO

7726 Standard (ISO 7726, 2002) under both natural and forced convection (see Table 4 for details).

Energy balance on the natural wet bulb thermometer

Regarding the natural wet bulb temperature, the heat balance equation structure appears slightly differ-ent due to the presence of the evaporative heat flow and the different correlations related to the different geometry of the sensor (see Table 2):

q q Er c+ = (6)

with

q h t tc c,n a nw( ),− (7)

q t tr n r nwε σ[( ) ( ) ],273 2734 4− (8)

and

E h p t p t= − ⋅e as nw as a[ ( ) ( )].RH (9)

Special care must be given to the equations (7) and (8) because neither ISO 7243 nor ISO 7726 specify equations/correlations for the calculation of the heat transfer coefficient; this omission is not an insignifi-cant matter because under natural convection con-ditions the geometry of the wick (a vertical slender cylinder) requires the use of non-standard correla-tions. In particular, the reduced diameter–height ratio of the wick invalidates the use of correlations, which

Table 4. List of correlations used for the evaluation of the coefficients in heat and mass transfer equations at the base of globe and natural wet bulb temperature.

Parameter Symbol Equation References

Convective coefficient of the globe under free convection

hc,g1 4

0 25

.

,t t

Dg a

g

− ISO 7726 (2002)

Convective coefficient of the globe under forced convection

6 30 25

0 6,,

,

Dv

ga ISO 7726 (2002)

Nusselt number for the natural wet bulb sensor under free convection Nu =

h d

kcn

f

0 360 0 392 0 25

2 3

2

. . ,+

=−R

R

a

agf f a nw

f

ρ βµt t d

Churchill and Chu (1975)

Nusselt number for the natural wet bulb sensor globe under forced convection

0 676 0 466 0 31. Re Pr

Re Pr

. .

= =ρµ

µv D ck

a P

Dernedde and Gilbert (1991)

Saturated water vapour pressure pas 0 610517 27

237 3. exp

.

.

t

t+

ISO 7726 (2002)

Lewis number L Lh

h= =e

c,n

16 5. ISO 7726 (2002) and ASHRAE (2009)

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usually refer to vertical walls where the characteristic dimension for the heat transfer is the height (Church-ill and Chu, 1975; Dernedde and Gilbert, 1991; Bejan, 1993; Corcione, 2005). Due to these peculi-arities, Churchill and Chu’s correlation has been used in this paper (see Table 3 for details), with air prop-erties evaluated at the film temperature (ASHRAE, 2009). Finally, the evaluation of the evaporative mass transfer coefficient has been effected using the Lewis equation (ISO 7726, 2002; ASHRAE, 2009), based on the well-known Chilton–Colburn analogy. The investigation of equations (3) and (6) will be primar-ily discussed under both uniform (ta = tr) and non-uniform conditions (ta tr) by varying ta, tr, RH, and va (only under forced conditions).

Bifurcation analysis

In order to highlight the presence of more steady state values of tnw under a specific microclimatic situation, a special continuation method (Hale and Koçak, 1991) has been implemented (see Appendix for details).

RESULTS AND DISCUSSION

In Figs 1–4 the behaviours of thermal flows involved in equations (3) and (6) were depicted as a function of the main microclimatic parameters and the tem-perature of the sensors.

Heat flow analysis around the globe

According to profiles depicted in Fig. 1, for a fixed value of ta and tr under natural and forced convection conditions, both heat flows involved in equation (3) exhibit a monotonic profile as a function of the globe temperature. As a trivial consequence, equation (3) shows only one steady state solution consistent with the attainment of the equilibrium between the heat flows by convection and by radiation.

Heat flow analysis around the wet bulb

Based on the curves shown in Fig. 2, the indirect evaluation of the natural wet bulb temperature is a

Fig. 1. Radiative heat flow, qr (continuous lines), and convective heat flow, qc (dash-dotted lines), involved in the heat balance equation on the black globe as a function of

the sensor temperature under natural and forced convention conditions. ta = 25°C; RH = 50%.

Wet bulb temperature, °C15 20 25 30

0

40

80

120

160

Hea

t flo

ws a

roun

d th

e w

et b

ulb,

W/m

2

0

40

80

120

160 0

40

80

120

160

15 20 25 30

0

40

80

120

160

d)

c)

b)

a)

tnw,1

tnw,1= tnw,2 tnw,3

tnw,3

tnw,1

tnw,2 tnw,3

Eqc+qr

2D Graph 1

24.8 25.0 25.240

45

50

55

60

tnw,2 tnw,3

Fig. 2. Heat flows involved in the heat balance equation on the natural wet bulb thermometer as a function of the sensor

temperature under natural convention conditions. ta = 25.0°C; RH = 50%, tr = ta (a), tr − ta = 7.0°C (b), tr − ta = 17.0°C

(c), tr − ta = 20.0°C (d). tnw,1, tnw,2, and tnw,3 are the tnw values consistent with the equation (6).

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delicate matter because of the cusp shape of the evap-orative term under natural convection. In fact, only under uniform conditions (Fig. 2a) and under low or very high values of the tr−ta difference (Fig. 2d) does the heat balance equation on the sensor result in a single equilibrium value, whereas in a range of tr – ta values from 7.0° (Fig. 2b) to 17°C (Fig. 2c), as many as three equilibrium values (tnw,1, tnw,2, and tnw,3) consistent with the energy balance expressed by equation (6) have been found at 25°C. Moreover, the difference between the three obtained solutions appears very significant (from 1.2 to 5.0°C in Fig. 2d and c, respectively), with the consequent ambiguous assessment of the WBGT value. In fact, under indoor conditions for tnw = tnw,1 the WBGT value correspond-ing to the microclimatic situation as given in Fig. 2b is 23.4°C instead of 26.5°C for tnw = tnw,2.

At this point it is a legitimate question of which of them can be considered acceptable from a physi-cal point of view. Following the same approach of continuous stirred tank reactors (Levenspiel, 1972), a survey of the local stability of each value can pro-vide some useful indications.

Low- and high-temperature solutions: tnw,1 and tnw,3

The stability of the minimum and the maximum tnw values can be illustrated by altering each steady state solution by a small amount. Particularly if tnw,1 and tnw,3 are altered by a small positive amount, one observes that the evaporative heat flow E exceeds the overall convective and radiative flow (qr + qc). Thus, the wick will tend to become cold as a result of the lack of energy necessary to sustain the evaporation of water. Similarly, a small decrease of tnw,1 and tnw,3 will result in the thermal flow coming from the envi-ronment being exceeded by the heat flow required for the evaporation. As a consequence the wick will adapt to the new equilibrium through its heating. In brief, both high- and low-temperature solutions are stable and physically meaningful.

Middle-temperature solution: tnw,2

Due to the shape of the curve modelling the evapora-tive flow, it is easy to observe (see the box in Fig. 2b) that in the presence of small positive shifts of tnw,2, the wick will receive an overall heat flow by radiation and evaporation higher than the evaporative (qr + qc > E). As a consequence the wick will reach a new equilibrium by warming up to the high-temperature solution tnw,3. Similarly, a shift to the left of tnw,2 will cause the wick to reach an equilibrium up to tnw,1. As a consequence the middle-temperature value must be considered unstable. A simple analytical expression

demonstrating the local stability of the solution is as follows:

¶

¶

¶

¶tE

tq q

nw nwc r( ) ( ).> + (10)

The analysis of thermal flows in the tnw previously reported evaluation highlights the effect of the non-uniformity of the environment on the phenomenon of multiple tnw values, which is consistent with the energy balance around the sensor under steady state conditions. In any event, the attainment of the equilibrium between the evaporative and the over-all convective–radiative flows around the sensor is also affected by humidity; therefore, the analy-sis of thermal flows has been extended to different humidity ratio values as shown in Fig. 3. From the curves in Fig. 3, three different solutions consistent with equation (6) in a fixed microclimatic condition

04080120160200240

Hea

t flo

ws a

roun

d th

e w

et b

ulb,

W/m

2

04080

120160200240

15 20 25 30

04080

120160200240

04080120160200240

Wet bulb temperature, °C15 20 25 30

04080

120160200240

tnw,1

tnw,1 tnw,2 tnw,3

tnw,1= tnw,2 tnw,3

tnw,3

a)

b)

c)

d)

e)

Eqc+qr

tnw,1 tnw,2 = tnw,3

Fig. 3. Heat flows involved in the heat balance equation on the natural wet bulb thermometer as a function of the sensor temperature under natural convention conditions. ta = 25°C,

tr = 32°C; RH = 30% (a), RH = 45% (b), RH = 60% (c), RH = 77% (d), RH = 90% (f). tnw,1, tnw,2, and tnw,3 are

the tnw values consistent with the equation (6).

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characterized by a [ta, tr] pair can be revealed within two critical values of humidity (45 and 77% shown in Fig. 3b and d, respectively). Therefore, the phe-nomenon of manifold values for the natural wet bulb temperature is strongly activated by the differ-ence [tr − ta] and humidity under natural convection; whereas, from the curves in Fig. 4, when the analy-sis under forced convection is carried out, only one steady state solution has been found. This occurrence not only suggests avoiding the indirect evaluation of tnw at negligible air velocities (below 0.20 m s−1, according to the ranges reported in Table 3) but also reduces the reliability of the measurement, because in principle the sensor could measure the high- or low-temperature value, depending on the initial con-ditions of the measurement (Aris, 1958; Levenspiel, 1972). Therefore, the systematic difference between measured and predicted tnw values reported in litera-ture (Buonanno et al., 2001, Gaspar and Quintela, 2009) could be traced not to a lack of modelling heat and mass transfer phenomena but to the physi-cal nature of the system that structurally exhibits more than one equilibrium condition (Aris, 1958; Levenspiel, 1972).

In principle, we cannot exclude that thermoregula-tion models based on the heat balance equation on an irradiated surface in the presence of both convection

and evaporation flows lead to different steady state solutions under natural convection conditions on naked skin or a clothed surface. In fact, on a naked surface the mathematical structure of the heat trans-fer equation is similar to that exhibited by a wet wick (Stolwijk, 1970), but in the presence of clothing (Fiala et al., 2001; Tanabe et al., 2002; d’Ambrosio Alfano et al., 2008), the evaporation rate is obtained, taking into account the vapour resistance of cloth-ing, the air boundary layer, and the pumping effect (Parsons et al., 1999). As a consequence, the math-ematical structure of equation (9) changes slightly as the driving force of the evaporation should be divided by the total water vapour resistance, Re,T, defined as follows (Alfano et al., 1989):

R RR

fe,T e,cle,a

cl

= + . (11)

On the basis of equation (11), the appearance of more steady state values for tnw is strictly related to the Re,cl/Re,a ratio. In other words, in the presence of low values for the clothing vapour resistance, the evaporation rate is mainly due to the vapour resist-ance of the air boundary layer, and then Re,T = 1/he. As a consequence, natural convection conditions will produce the same behaviour as a wet wick. On the contrary, at high Re,cl values the vapour resistance of the air becomes negligible, and the evaporation rate is calculated by dividing the driving force in equa-tion (9) by a constant value with a numerical behav-iour similar to that observed for the evaporation rate under forced convection.

From this perspective the PHS model (Malchaire et al., 2002) does not seem to be affected by this problem, since both the required and the maxi-mum evaporation rates are calculated by means of the preliminary calculation of the total evapo-rative resistance of the clothing and the boundary air layer (Holmér et al., 1999; Parsons et al., 1999; Malchaire, et al., 2002).

Bifurcation analysis of heat balance equations on the wick

In order to highlight the effect of the mean radiant temperature and the humidity ratio on the tnw evalua-tion, a bifurcation analysis based on the continuation method, reported in the appendix, has been imple-mented. The results have been summarized in Fig. 5 by plotting tnw versus tr for different values of the air temperature.

The values plotted in Fig. 5 clearly prove that at each investigated air temperature value the presence of a saddle-node bifurcation (Hale and Koçak, 1991;

Fig. 4. Heat flows involved in the heat balance equation at the base of natural wet bulb temperature as a function of

the sensor temperature under forced convection conditions. ta = 25°C; RH = 50%, va = 0.10 m s−1 (a), va = 0.20 m s−1

(b), va = 0.50 m s−1 (c). tnw is the value obtained by solving equation (6).

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Continillo et al., 1995) can be revealed. In particular, at each air temperature value, a range of mean radi-ant temperature values [tr,crit,1, tr,crit,2] has been found, which, in equation (6), exhibits three solutions. It is noteworthy that the range of values of the mean radi-ant temperature consistent with such phenomenon appears to be promoted by the air temperature. In fact, the difference between the two critical values increases from 7.8 to 13.2°C by increasing the air temperature from 20 to 30°C. The minimum differ-ence between the critical lower tr value and the air temperature seems to be promoted by the air tem-perature as well. In any case, such effect appears to be milder, because by increasing the air temperature value from 20 to 30°C, the difference, tr,crit,1− ta, only increases from 5.2 to 8.2°C.

Since the humidity obviously affects the heat bal-ance equation on the sensor, in order to highlight its effect on the above-discussed behaviour, the bifur-cation analysis has also been carried out by varying

the hygrometric ratio in the range from 10 to 90% as shown in Fig. 6. The analysis of the profiles shown clearly demonstrates how the saddle-node bifurca-tion can take place, in principle, at every humidity value. In particluar, increasing the relative humid-ity results in a significant reduction of the differ-ence trcrit,1− ta, which varies from 11.5 to 1°C when RH is increased from 10 to 90%. Moreover, as an obvious consequence of the decreased efficiency of evaporative heat exchange from the wet wick, the saddle-node bandwidth is significantly reduced by the RH rise (from 20°C at RH = 10% to just 1.9°C at RH = 90%). Finally, the analysis of profiles shown in Fig. 6 clearly shows that in the saddle-node bifurcation zone, the difference between the high and the low tnw values is markedly reduced by the increasing humidity; in fact at RH = 10% and tr = tr,crit,1 the difference [tnw,3 − tnw,1] was about 9.0°C, whereas at RH = 90% the difference [tnw,3 − tnw,1] was just 1.0°C.

Fig. 5. Bifurcational analysis of the wet bulb temperature as a function of the mean radiant temperature under natural convection conditions for different air temperature values at RH = 50%.

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Fig. 6. Bifurcational analysis of the wet bulb temperature as a function of the mean radiant temperature under natural convection conditions for different humidity ratio values (air temperature, ta = 25°C).

The effect of humidity has been finally investigated in a wider range of conditions also by changing the air temperature (from 15 to 30°C) as shown in Fig. 7. Results seem to further confirm the above-cited phe-nomenon with an additional promoting effect due to the air temperature. In fact, at RH = 50% the hyster-esis bandwidth changes from 5.7 to 12.8°C by chang-ing the air temperature from 15 to 30°C.

The relationship between the natural wet bulb temperature and the psychrometric wet bulb temperature

Although under forced convection conditions the indirect evaluation of tnw is not affected by the cir-cumstance of a variety of steady state solutions, fur-ther discussion of this matter is necessary since the natural wet bulb temperature is misused as an indica-tor of the humidity of the environment in place of the psychrometric wet bulb temperature, tw (ISO 7726, 2002; ASHRAE, 2009).

This aspect has not been investigated much in the past, and the only two papers devoted to the matter tried to find an empirical correlation (Malchaire, 1976) between tnw and the main microclimatic param-eters (particularly tg, ta, RH, and va) or a numerical method to obtain tnw from standard psychrometric measurements (Brake, 2001).

To check the reliability of above-cited correla-tions under forced convection conditions (where only a tnw value can be found) and find a certain relationship between tnw and tw, we have compared the difference between tnw and tw with those pre-dicted by a correlation proposed by Malchaire (1976) as reported here:

t tt t

tnw wg a

aRH−−

− − −=+0 16 0 8

200560 2 5 0 8

. ( ) .( ) . . (12)

This analysis (see Table 5) has been carried out in the range of va from 0.1 to 1.0 m s−1, both under uniform and non-uniform conditions, with (tr− ta) up to 20°C.

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Fig. 7. Minimum (open symbols) and maximum (closed symbols) critical mean radiant temperature values for the onset of the saddle-node bifurcation under natural convection conditions as a function of the relative humidity.

Table 5. Comparison between the natural wet bulb temperature and the psychrometric wet bulb temperature predicted according to equation (12) and the correlations quoted in Table 4.

ta (°C) tr (°C) tg (°C) tnw − tw (°C)

Malchaire (1976) In this work

RH = 10% RH = 90% RH = 10% RH = 90%

va = 0.1 m s−1

20.0 20.0 20.0 1.0 0.3 2.0 0.1

20.0 30.0 26.2 3.2 1.7 3.5 1.2

20.0 40.0 32.1 5.2 3.0 5.2 2.4

30.0 30.0 30.0 0.8 0.1 1.6 0.1

30.0 40.0 36.4 2.8 1.3 3.0 1.0

30.0 50.0 41.5 4.7 2.4 4.6 1.9

va = 1.0 m s−1

20.0 20.0 20.0 1.0 0.3 0.8 0.0

20.0 30.0 23.0 2.0 1.0 1.4 0.4

20.0 40.0 26.3 3.2 1.7 2.1 0.9

30.0 30.0 30.0 0.8 0.1 0.2 0.0

30.0 40.0 33.2 1.8 0.7 0.8 0.4

30.0 50.0 36.7 2.9 1.4 1.4 0.7

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On the basis of the obtained results we can state the following:

• Due to the presence of the radiative flow on the wick, the equilibrium temperature reached by the sensor is always greater than the psychro-metric wet bulb temperature (Malchaire, 1976; Brake, 2001). As a consequence of the increase of the radiative heat flow, the difference [tnw − tw] appears to be promoted by the mean radiant temperature (Brake, 2001). In fact, increasing the mean radiant temperature from 20 to 40°C resulted in an increase of the difference [tnw − tw] from 2.0 to 5.2°C for ta = 20°C, RH = 10%, and va = 0.10 m s−1. Such a phenomenon is obviously reduced by the increase of the relative humidity as a consequence of the reduction of the evapora-tive heat flow.

• From a quantitative point of view, the equa-tion proposed by Malchaire (1976) appeared to strongly agree with our predictions. A mean-ingful difference has been revealed only under uniform conditions (tg = tr = ta) and at low values of the hygrometric ratio, probably due to the mathematical structure of equation (12) that cannot take into account the direct effect of the radiation when tg − ta= 0. In particular, under the conditions posited by equation (12), the difference between the psychrometric wet bulb and the natural wet bulb temperature is only affected by the air temperature and the hygrometric ratio. Conversely, the natural wet bulb temperature is affected by the air velocity and, especially, by the mean radiant tempera-ture [see equations (7) and (8)]. At higher air velocity (va = 1.0 m s−1) due to the increase of the heat transfer by convection (and evapora-tion), the predicted difference between tnw and tw is often negligible under very humid condi-tions and appeared only slightly different to that obtained by Malchaire.

Effect of the tnw multiplicity on the assessment of the working situation using the WBGT index

In the previous sections it was unambiguously proved that under natural convection conditions and in non-uniform environments in a fixed measured microcli-mate (e.g. at a fixed triplet of ta, tr, RH values) the indirect evaluation of the natural wet bulb tempera-ture does not result in a single solution. The conse-quent effect on the thermal environment assessment by the WBGT index has been preliminarily studied by evaluating the WBGT index on the basis of the set of ta, tr, RH triplets reported in Table 6. According to these data, the existence of two different WBGT

values precludes a reliable use of the method, espe-cially if the WBGT limit value of the working situ-ation (which depends on the acclimatization status and the metabolic rate of the person given in Table 2) is in the range of the two WBGT evaluated values, as several cases reported in Table 6 seem to confirm. Particularly, concerning the analysis of an 8-h con-tinuous work situation for non-acclimatized persons we can state as follows:

• At low metabolic rate (WBGTlim = 29°C), although the presence of two tnw stable solutions results in two different WBGT values, they are both less than WBGTlim. Only for ta = 30°C and high tr − ta differences does the limit value fall into the range of the two evaluated values. For instance at ta = 30°C, tr = 45°C, and RH = 30%, the lower WBGT value (28.1°C) is consistent with 8 h continuous work, whereas the second (32.9°C) has to be considered dangerous.

• At a moderate metabolic rate (WBGTlim = 26°C) the ambiguity induced by the saddle-node bifur-cation starts to be more frequent with a dramatic increase of ambiguous assessments of working situations. In fact, for ta = 25°C, only a slight tr − ta difference at RH = 70% results in two conflict-ing WBGT values (24.5 and 26.1°C).

In order to highlight the ambiguous assessment of WBGT graphically, Fig. 8, on a psychrometric chart, the air temperature–humidity ratio curves corresponding to a fixed WBGTlim value under still air have been displayed for different tr − ta val-ues. Usually these curves split the psychrometric chart as safe and unsafe areas (Sullivan and Gor-ton, 1976; d’Ambrosio Alfano et al., 2007, 2011); unfortunately the presence of the cusp related to the mathematical structure of equations used for the cal-culation of the heat transfer coefficient makes them quite useless and further emphasizes the impossibil-ity of using the WBGT index for an indirect assess-ment of the heat stress. In fact, according to the limit curves depicted in Fig. 8, for RH = 50% and tr − ta = 15°C, it is impossible to calculate a limit air temperature consistent with 8 h of continuous work because the same WBGTlim value can be obtained for the same humidity value (50% in the figure) at three different air temperature values (26.1, 26.3, and 28.3°C for WBGTlim = 29°C and 23.1, 23.5, and 24.4°C for WBGTlim = 26°C). Moreover, according to above-quoted findings, the region of the psychro-metric chart characterized by this ambiguous assess-ment appears to be made greater by the increase of the difference between the mean radiant tempera-ture and the air temperature and the decrease of the humidity ratio.

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The sensitivity of tnw at lowest air velocity and its effect on the WBGT index

The ambiguity resulting from cusp-shaped relation-ship of the heat transfer coefficient on the wet wick under natural convection conditions led us to inves-tigate the possibility of smoothing the WBGT limit curves, shown in Fig. 8, by forcing the correlations of forced convection at lowest air velocity values. To this aim the effect of the air velocity on the WBGT limit curves at values near to the sensitivity of meas-urement devices of C and S classes (see Table 3) has been investigated. In particular, curves displayed in Fig. 9 show a significant sensitivity of iso-WBGT curves to air velocity. This phenomenon appeared to be increased by the difference between the mean radiant temperature and the air temperature. In fact,

at (tr − ta) = 5°C, changing va from 0.01 to 0.02 m s−1 resulted in a quite negligible effect on the limit air temperature (less than 0.5°C), consistent for both WBGT limit values, whereas at higher (tr − ta) val-ues, an even smaller change in air velocity from 0.01 to 0.02 m s−1 resulted in a very significant shift of WBGTlim curves to higher air temperatures (about 2°C at WBGTlim = 29°C as shown in Fig. 9). This phenomenon appeared amplified at higher metabolic rates (WBGTlim = 26°C); in fact, according to the curves in Fig. 9 (bottom), at wa = 10 g kg−1, increas-ing the air velocity from 0.01 to 0.2 m s−1 resulted in a change of the maximum air temperature consist-ent with 8 h of continuous work from about 23° to over 27°C. As a consequence, a reliable smoothing is quite impossible due to the high sensitivity of the

Table 6. Effect of the tnw multiplicity on the WBGT indirect assessment from air temperature, mean radiant temperature, and hygrometric ratio values.

ta (°C) tr (°C) tg (°C) RH (%) tnw (°C) WBGT (°C)

25.0 30.0 28.3 10 14.4 18.6

30 17.5 20.7

50 20.3 22.7

70 22.9 25.1 24.5 26.1

90 25.3 26.2

35.0 31.3 10 15.6 20.3

30 18.7 25.1 22.5 27.0

50 21.5 25.1 24.4 27.0

70 25.2 27.0

90 26.2 27.7

40.0 34.3 10 17.0 25.0 22.2 27.8

30 20.0 25.1 24.3 27.9

50 23.1 25.1 26.5 27.9

70 25.7 28.3

90 27.1 29.3

30.0 35.0 33.4 10 17.2 22.1

30 21.0 24.7

50 24.3 27.0

70 27.3 30.0 29.1 31.0

90 30.3 31.2

40.0 36.4 10 18.2 23.7

30 22.1 26.4

50 25.4 30.0 28.7 31.9

70 28.5 30.1 30.9 32.0

90 30.9 32.6

45.0 39.5 10 19.6 30.0 25.6 32.9

30 23.2 30.0 28.1 32.9

50 26.6 30.1 30.5 32.9

70 30.4 33.1

90 31.7 34.0

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index to air velocity and the arbitrary choice of the air velocity threshold value consistent with the use of forced convection correlations (see Table 4).

CONCLUSIONS

Sixty years after its first formulation, the WBGT index remains the most user-friendly and widespread method for the preliminary assessment of hot ther-mal environments. In any case, as an empirical tool, it should not be used incorrectly or, worse, inappro-priately in conditions different from those for which it was designed. In particular, this work has demon-strated that it is impossible to derive its evaluation

indirectly from the main microclimatic quantities (air temperature, mean radiant temperature, humid-ity, and air velocity) because under natural convec-tion up to three natural wet bulb temperature values can be obtained with the resulting ambiguity in the WBGT calculation. This phenomenon, well known in many engineering systems, is mainly due to the strong non-linear structure of heat and mass transfer equations, characterized by an evaporative term that exhibits a cusp shape under free convection condi-tions. Such an occurrence leads to a saddle-node bifurcation in the presence of even mild differences between the air temperature and the mean radiant temperature of the environment (situations which

Fig. 8. Psychrometric chart provided with iso-WBGT limit curves consistent with 8 h of continuous work under still air for a non-acclimatized subject.

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are very likely in the presence of hot surfaces and/or windows). As a consequence, the same micro-climatic situation characterized by a triplet of air temperature, mean radiant temperature, and humid-ity values can show up to three different tnw values. We cannot exclude a priori that such a phenomenon (predicted in a numerical way) takes place under real situations also (the same heat flow required for the evaporation can be obtained for different values of convective and radiative flows): only a special inves-tigation under controlled conditions could confirm or

exclude such occurrence. In any case, on the basis of these findings, WBGT index should be used under quiet air conditions only when a natural wet bulb thermometer is available. Conversely, under forced convection its indirect evaluation leads to an unam-biguous evaluation of the environment.

Finally, the ISO 7243 Standard should specify more clearly the equations needed to carry out such calculations. This would seem to be essential, espe-cially since neither ISO 7726 nor ISO 7243 provide clear specifications about the air velocity limit value

Fig. 9. Effect of the air velocity value on iso-WBGT limit curves consistent with 8 h of continuous work under forced convection for a non-acclimatized subject. va = 0.01 m s−1 (a); va = 0.02 m s−1 (b); va = 0.05 m s−1 (c); va = 0.2 m s−1 (d).

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consistent with the transition between the natural and the forced convections.

We are aware that the approach of this paper might appear too ‘engineer-like’, such that it seems inaccessible even to specialists in thermal environ-ment assessment (bioclimatologists, physiologists, and military industrial organizations). However, to reduce the risk of an incorrect evaluation of hot envi-ronments, we recommend the index for direct meas-urement using the typical measurement instruments for which WBGT has been designed: a globe, an air temperature, and a natural wet bulb thermometer.

APPENDIX

Under free convection, the heat balance equation at the base of the natural wet bulb thermometer, equa-tion (6), is a scalar equation, which is related to four parameters. Briefly,

f t t t( , , , ) .nw a r RH = 0 (13)

The ‘locus root’ of equation (13), obtained by vary-ing one or more physical parameters, can be obtained by solving a special 2D system of ordinary differen-tial equations, ODE, based on its parametric repre-sentation (Hale and Koçak, 1991). Since under free convection tnw is a function of three different quanti-ties (ta, tr, RH), in principle, three different 2D ODE systems should be solved (Hale and Koçak, 1991). In any case, on the basis of findings discussed above, we restricted the analysis to the sensitivity of tnw to the mean radiant temperature. Thus, the parametric equation of the locus root can be written as follows:

f t t( , ) .nw r = 0 (14)

Now, letting λ be a parameter required for the rep-resentation of the locus root on a tr, tnw plane, the regularity of the curve will imply

¶¶

¶¶

¶¶

¶¶

f

t

t f

t

t

nw

nw

r

r

λ λ0. (15)

Therefore, by arranging in a matrix,

det .

∂∂

∂∂

∂∂

∂∂

f

t

f

t

t tnw r

r nw−=

λ λ

0 (16)

To satisfy this identity, the first vector row in equa-tion (16) must be a constant multiple of the second one, thereby taking into account that f is given by

f t t q t t q t t E t t( , ) ( , ) ( , ) ( , ).nw r c nw r r nw r nw r= + − (17)

According to this choice the 2D ODE system describing the sensitivity of tnw to the mean radiant temperature has been written as follows:

d

dt

tq q E

d

dt

tq q E

λ

λ

rnw

c r

nwr

c r

= − + −

= + −

∂∂

∂∂

( ),

( ),

(18)

with initial conditions

t t

t t

r r

nw nw

( ) ,

( ) .

λ

λ

= =

= =

0

0

0

0

(19)

The 2D ODE system in equation (18) has been solved by means of a numerical 2D ODE solved based on the Runge Kutta Fehlberg algorithm (Press et al., 2007).

NOMENCLATURE

cP Specific heat of dry air at constant pressure, J kg−1 K−1

d Diameter of the wet wick of natural wet bulb thermometer, m

Dg Globe diameter, m

E Evaporative heat flow on the wet bulb surface, W m−2

g Gravity acceleration, m s−2

hc,g Convective heat transfer coefficient between the air in the environment and the globe thermometer, W m−2 K−1

hc,n Convective heat transfer coefficient between the air in the environment and wet bulb thermometer, W m−2 K−1

he Evaporative heat transfer coefficient between the air and the wet bulb thermometer, W m−2 kPa−1

k Air thermal conductivity, W m−1 K−1

kf Air thermal conductivity at film temperature, W m−1 K−1

L Lewis relation, K Pa−1

M Metabolic rate, W m−2

Nu Nusselt number, dimensionless

pas Saturated water vapour pressure, kPa

Pr Prandtl number, dimensionless

qc Convective heat flow on the globe or on the wet wick surface, W m−2

qr Radiative heat flow on the globe or on the wet wick surface, W m−2

Ra Rayleigh number, dimensionless

Re Reynolds number, dimensionless

RH Relative humidity, dimensionless

Re,T Total evaporative resistance of clothing and boundary air layer, m2 kPa W−1

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GREEK SYMBOLS

Acknowledgments—We owe a special thanks to Mrs. Angela Maria Gabriella Izzo for her valuable cooperation. The authors thank Mr. Anthony Polestra for the ‘English polishing’.

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t Temperature, °C

ta Air temperature, °C

tg Globe temperature, °C

tnw Natural wet bulb temperature, °C

tnw,1 Low-temperature solution for the indirect evaluation of the natural wet bulb temperature, °C

tnw,2 Middle-temperature solution for the indirect evaluation of the natural wet bulb temperature, °C

tnw,3 High-temperature solution for the indirect evaluation of the natural wet bulb temperature, °C

tr Mean radiant temperature, °C

tr,crit,1 Minimum critical mean radiant temperature for the onset of multiple tnw values, °C

tr,crit,2 Maximum critical mean radiant temperature for the onset of multiple tnw values, °C

tw Psychrometric wet bulb temperature, °C

va Air velocity, m s−1

wa Humidity ratio, kgwater/kgdry air

WBGT Wet bulb globe temperature, °C

WBGTlim Limit wet bulb globe temperature reference value for 8 h of continuous work, °C.

βf Thermal expansion coefficient at constant pressure at film temperature, K−1

△tr,crit,1 Difference between the minimum and the maximum critical mean radiant temperature for the onset of multiple tnw values, °C

εg Emissivity of the black globe, dimensionless

εn Emissivity of wet wick of the natural wet bulb thermometer, dimensionless

λ Dummy parameter required for the parametric representation of a curve, dimensionless

µ Air viscosity at air temperature, kg m−1 s−1

µf Air viscosity at film temperature, kg m−1 s−1

ρ Air density at air temperature, kg m−3

ρf Air density at film temperature, kg m−3

σ Stefan-Boltzmann constant, W m−2 K−4

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