Poeeing of Clima 2007 WellBeing Inoo Deivation an analyi of the outoo Wet Bulb Globe Tempeatue inex (WBGT) with a human themal engineeing appoah Pat 2. Popetie of the WBGT fomula fo outoo onition with ola aiation Kouhei Kuwabaa, Tohu Mohia an Tomonoi Sakoi 2 Hokkaio Univeity, Japan 2 National Intitute of Avane Inutial Siene an Tehnology, Japan Coeponing email: kuwa@eng.hokuai.a.jp SUMMARY The autho peent a theoetial eivation of the WBGT fomula fo outoo onition that wa oiginally evelope fom the eult of expeiment on human ubjet, bae on a heat balane equation between the human boy an it outoo envionment. The thee oeffiient of wet bulb tempeatue T w, globe tempeatue T g, an ai tempeatue T a wee expee in almot the ame way a in the inoo WBGT fomula, but they ontain a new element haateizing ola aiation. In aition, we alulate the oeffiient in the theoetially eive fomula, hanging the amount of metaboli ativity, othing won, win veloity, an ola aiation. We obtaine the new fomula WBGT = 0.84T w 0.30T g 0.08T a, haateize by a negative oeffiient of ai tempeatue T a, a an altenative to the oiginal outoo fomula WBGT = 0.7T w 0.2T g 0.T a. Finally, we iniate the haateiti of the WBGT a an inex a well a intution fo ue. INTRODUCTION The Wet Bulb Globe Tempeatue (WBGT []) ha been evelope fom expeiment by Yaglou an Mina in the U.S. in 957 fo the pevention of heat toke in olie. The inex i tanaize by the National Intitute fo Oupational Safety an Health (NIOSH) an the Intenational Oganization fo Stanaization (ISO) a ISO-7243. Theefoe the inex i now fequently ue a an evaluation inex fo themal envionment uing wok an outoo pot, an it ue i eommene in many ountie inuing Japan. The WBGT wa evelope bae on expeiment with human, but it bakgoun ha not been iue fom the pepetive of the heat tanfe theoy. The phyial heat exhange between human an the envionment povie phyiologial an pyhologial eation. Hene the popetie of inie, even thoe bae on human expeiment, an be examine fom the viewpoint of phyial heat balane. In thi pape, we ay out theoetial eivation an analyi of the WBGT fo outoo onition employing the heat tanfe theoy bae on the heat balane equation between human boy an an outoo envionment. Then, bae on the eive theoetial fomula, we aify the tutue of the thee ontant oeffiient of wet-bulb tempeatue, globe tempeatue an ai tempeatue that efine the oiginal WBGT fomula fo outoo. In aition, the eive fomula fo outoo i ompae with that fo inoo onition eive in anothe pape [2], an we onie point of iffeene an imilaity between two fomulae. Futhemoe, we pefom an examination uing atual value to peent the haateiti an appliable onition of the WBGT.
Poeeing of Clima 2007 WellBeing Inoo METHODS Yaglou an Mina have uggete the oiginal WBGT fomula () fo outoo onition. WBGT = 0.7T w 0.2T g 0.T a () whee WBGT i wet bulb globe tempeatue [ C], T w i (natual) wet bulb tempeatue [ C], T g i globe tempeatue [ C] an T a i ai tempeatue [ C]. The heat balane equation (2) between the human boy an an outoo envionment i expee a follow: M = (C R) E k E e W S (2) whee M i metaboli ate pe unit boy ufae aea [W/m 2 ], C i onvetive heat lo [W/m 2 ], R i aiative heat lo [W/m 2 ], E k i evapoative heat lo [W/m 2 ], E e i onvetive an evapoative heat lo fom epiation [W/m 2 ], W i extenal mehanial wok [W/m 2 ] an S i ate of heat toage [W/m 2 ]. The heat tanfe equation by onvetion C, aiation R [3], evapoation E k an epiation E e [4] ae given a the following equation. C = h F (T k T a )f (4) R = h F [(T k 273) λ(t g 273)]f f ef F (H H H )f (5) E k = LRh F p (P k P a )f (6) E e = 0.004M(35 T a ) 0.073M(5.87 P a ) (7) whee h an h ae human onvetive an linea aiative heat tanfe oeffiient [W/(m 2 C)], F i themal effiieny fato [N.D.] [5], T k i mean kin tempeatue [ C], T a i ai tempeatue [ C], f i othing aea fato[n.d.] [6], λ i long-wave aiation oeffiient [N.D.] [3], T g i goun tempeatue [ C], f ef i effetive aiant aea fato [N.D.], H, H an H ae iet, attee an efletive ola aiation abobe into boy ufae [W/m 2 ], LR i Lewi elationhip (= 6.5) [ C/kPa], F p i pemeation effiieny fato [N.D.] [7], P k i wate vapo peue at kin tempeatue [kpa] an P a i wate vapo peue in ai [Pa]. λ in Equation (4), phyial quantitie expeing the popetie of aiation peifi to the egion, ate an time, ae efine by the following equation: λ=(ω) 0.25 ( U ) 0.25 (7) Ω=(T ky 273) 4 U /(T g 273) 4 ( U ) (8) whee Ω i atio of atmophei aiation to long-wave aiation fom the goun to the human boy [N.D.], T ky i hypothetial ky tempeatue [ C], U i ky view fato fo the human boy [N.D.]. Alo, Ω in Equation (7) expee a atio of atmophei aiation to long-wave aiation fom the goun to the human boy. U in Equation (7) an (8) enote onfiguation fato between the human boy an the ky, an i efee to a ky view fato fo the human boy [3].In thi tuy, f ef = in equation (4). Even in the outoo envionment with ola aiation, the wet bulb themomete i eentially ue fo meauement in hae aea, o thee i no nee to moify the geneal heat balane equation (9) fo the wet bulb [2]. Beaue the ening pat of the wet bulb themomete i vey mall, on the aumption that h /h i nealy equal to zeo, Equation (9) lea to Equation (0). h (T w T a ) h (T w T ) LR h (P w P a ) = 0 (9) (T w T a ) LR(P w P a ) = 0 (0) whee h an h ae onvetive an linea aiative heat tanfe oeffiient of wet bulb [W/(m 2 C)], T i mean aiant tempeatue [ C], P w i atuate wate vapo peue on wet bulb [kpa]. A fa a a hot envionment uouning human wok i onene, the elationhip of the atuate wate vapo peue P w to the wet bulb tempeatue T w on the pyhometi hat
Poeeing of Clima 2007 WellBeing Inoo i uffiiently linea, an an be appoximate a follow: P w = κ T w ζ () whee κ an ζ ae ontant of linea appoximation of atuate wate vapo peue to wet bulb tempeatue [kpa/ C]. Duing wok in a hot envionment, mean kin tempeatue i within a ange of 35-40 C. Theefoe, by teating kin tempeatue in the ame way a the wet bulb tempeatue, the elationhip of atuate wate vapo peue on kin P k to kin tempeatue T k an be linealy appoximate by Equation (2). P k = κ T k ζ (2) whee κ an ζ ae ontant of linea appoximation of atuate wate vapo peue to mean kin tempeatue [kpa/ C]. In the foth oming numeial examination, we will aopt the following value hown in Refeene [2]: κ = κ = 0.279 kpa/ C an ζ = ζ = 4.03 kpa. Wate vapo peue on kin ufae P k i expee by Equation (2), aompanie with µ that iniate the egee of atuation on kin ufae. Hee we expan equation with µ peeve, but in the late tage, we will uppoe µ = beaue the WBGT inex i eigne fo ue within themal limitation whee wate vapo peue on kin ufae i almot atuate in onequene of pomote weating. P k = µ (κ T k ζ ) (2) whee µ i atuate atio of wate vapo peue to kin tempeatue [N.D.]. The globe themomete i baially intene fo meauing heat aiation, an teay-tate heat balane afte uffiient expoue of the globe themomete to an outoo envionment with ola aiation fo atual meauement i expee, with efeene to heat tanfe equation by aiation fo the human boy (4), a follow: h (T g T a ) h F [(T g 273) λ (T g 273)] f F (H H H ) f = 0 (3) whee h an h ae onvetive an linea aiative heat tanfe oeffiient of globe [W/(m 2 C)], λ i long-wave aiation oeffiient of globe [N.D.], F i themal effiieny fato of globe [N.D.], f i othing aea fato of globe [N.D.], H, H an H ae iet, attee an eflete ola aiation abobe into globe [W/m 2 ]. Fo the globe themomete, F = an f =. In thi tuy, λ =λ. RESULTS Subtituting the elevant equation inuing Equation (3) ~ (8) an (0) ~ (3) into heat balane equation of the human boy (2) give the outoo WBGT fomula (4) in onfomity with the inoo WBGT fomula [2]. λh [( h h ) Ff µκ LRh Fpf ] Tk Ff ( H H H ) Ff ( H H H ) " " λ h µζ ζ LRh F f M 0.8494 0.073ζ W S ( ) ( ) ] p = h Fpf ( κ LR) 0.073M κ Tw LR λ h h Ff T " g λ h λh 0.073 Ff h h h " " Fpf M 0.004 Ta h λ LR (4)
Poeeing of Clima 2007 WellBeing Inoo Witing the oeffiient of T k an the ontant tem on the left-han ie of Equation (4) a ξ an η, epetively, an the oeffiient of T w, T g an T a a α, β an γ, we obtain equation (5) ~ (20). [( h h ) F f µκ LRh F f ] ξ (5) Ff p λh ( H H H ) F f ( H H H ) " λ h ( µζ ζ) LRh F f M( 0.8494 0.073ζ) W S] η p " (6) h Fpf( κ LR) 0.073M κ α LR (7) λ h h Ff β " λ h (8) λh 0.073 F f h h h F f M 0.004 γ " " p λ h LR (9) ξ T η = αt βt γt (20) k w g a Diviing both ie of Equation (20) by ξ yiel Equation (2), a fom bae on the kin tempeatue T k. η α β γ Tk = Tw Tg Ta (2) ξ ξ ξ ξ Equation (2), on emoving the phyial quantity elate to ola aiation, oepon to the WBGT fomula fo inoo ue [2]. We examine the vaying popetie of the vaiable oeffiient in Equation (2) that oepon to the ontant oeffiient in the oiginal WBGT fomula () fo outoo onition with ola aiation, ubtituting onete value. We alulate the value of η/ξ, α/ξ, β/ξ an γ/ξ, hanging metaboli ate fom to 4 met, o unit fom 0.2 to.0 o an outoo ai veloity fom.0 to 5.0 m/ with global ola aiation et at 600 W/m 2. The alulate value of η/ξ ange fom 0.0 to 2.3. If ai veloity, o unit an metaboli ate wee low, the η/ξ value wa poitive. Figue to 3 how the eult of alulation of the α/ξ, δ/ξ an γ/ξ value. Figue how the vaiation of eah oeffiient in epone to the vaiation of metaboli ate M with o unit an ai veloity hel ontant. The hoizontal axi iniate metaboli ate, an the vetial axi iniate the α/ξ, β/ξ, γ/ξ an α/ξ β/ξ γ/ξ value. Reponing to metaboli ate of to 4 met, α/ξ ange fom 0.8 to 0.86, β/ξ i ontant at 0.33 an γ/ξ i appoximately 0.2. Thu, the value of α/ξ, β/ξ an γ/ξ ae oughly 0.84, 0.33 an 0.2 athe than 0.7, 0.2 an 0., the oeffiient in the oiginal fomula (). The um of α/ξ, β/ξ an γ/ξ ange fom.02 to.07, lightly exeeing.0. It i emakable in the numeial examination in Figue, alo in the following examination in Figue 2 an 3, that the mean value of the oeffiient γ/ξ of T a i negative. The value hift to the poitive ie when ai veloity exee a etain value. Figue 2 how the vaiation of the oeffiient in epone to the vaiation of ai veloity (hoizontal axi) with metaboli ate an o unit hel ontant. The value of α/ξ, β/ξ an γ/ξ ae, epetively, 0.84 ~ 0.82, 0.33 ~ 0.24 an 0.2 ~ 0.02, a in Figue. In Figue 2, unlike Figue, the α/ξ an β/ξ value eeae, when the um of the thee oeffiient appoahe.0. Similaly, Figue 3 how the vaiation of the oeffiient in epone to the vaiation of o unit with metaboli ate an ai veloity hel ontant. The value of α/ξ, β/ξ an γ/ξ ae 0.86
Poeeing of Clima 2007 WellBeing Inoo ~ 0.84, 0.24 ~ 0.29 an 0.08 ~ 0.05, epetively. A othing inulation ineae, the α/ξ value eeae an the β/ξ an γ/ξ value ineae. The um of α/ξ, β/ξ an γ/ξ ineae gaually fom.02 to.07. (α/ξ), (β/ξ), (γ/ξ), (α/ξ β/ξ γ/ξ) [N.D.].2 0.8 0.6 0.4 0.2 0-0.2 U = U = 0.5, Ω = Ω = 0.74, λ = λ = 0.97, v =.0 m/, 0.6 o, h = 6.3 W/(m 2 C), W = 0, S = 0.08 0.809-0.20.036 0.827-0.9.055 0.844-0.8.073 0.86-0.7 0 00 200 300 Metabli ate M [W/m 2 ] (α/ξβ/ξγ/ξ) (α/ξ) (β/ξ) (γ/ξ) Figue The vaiation of eah oeffiient in eq.(2) in ae of vaiable metaboli ate with ontant o unit an ai veloity. (α/ξ), (β/ξ), (γ/ξ), (α/ξ β/ξ γ/ξ) [N.D.] U = U = 0.5, Ω = Ω = 0.74, λ = λ = 0.97, 0.6 o, M =74 W/m 2, h = 6.3 W/(m 2 C), W = 0, S = 0.2 0.8 0.6 0.4 0.2 0-0.2.055.050.047.046.045 0.844-0.8 0.836 0.299-0.085 0.829 0.275-0.057 0.823 0.257-0.034 0.89 0.242-0.05 0 2 3 4 5 6 Ai veloity v [m/] (α/ξβ/ξγ/ξ) (α/ξ) (β/ξ) (γ/ξ) Figue 2 The vaiation of eah oeffiient in eq.(2) in ae of vaiable ai veloity with ontant o unit an metaboli ate. U = U = 0.5, Ω = Ω = 0.74, λ = λ = 0.97, v =.0 m/, M =74 W/m 2, h = 6.3 W/(m 2 C), W = 0, S = 0 (α/ξ), (β/ξ), (γ/ξ), (α/ξ β/ξ γ/ξ) [N.D.].2 0.8 0.6 0.4 0.2 0-0.2.022.035.047.060.073 0.858 0.834 0.829 0.83 0.836 0.24 0.264 0.275 0.282 0.286-0.078-0.064-0.057-0.052-0.049 0 0.2 0.4 0.6 0.8.2 Clo unit I [o] (α/ξβ/ξγ/ξ) (α/ξ) (β/ξ) (γ/ξ) Figue 3 The vaiation of eah oeffiient in eq.(2) in ae of vaiable o unit with ontant metaboli ate unit an ai veloity.
Poeeing of Clima 2007 WellBeing Inoo DISCUSSION We onie point of iffeene an imilaity between the eive fomula fo outoo onition an that fo inoo onition [2]. The eive fomula fo inoo onition in Refeene [2] i hown a follow: h h F f µκ LRh F f T [( ) ] [( µζ ζ) LR h F f M( 0.8494 0.073ζ) W S] = h F p f p ( κ LR) [ Ff ( h h )] Tg 0.073 Ff ( h h ) h Fpf M 0.004 Ta p k 0.073M κ LR LR Compaing the oeffiient of T w, T g an T a in Equation (4) with thoe in Equation (22), the oeffiient of fit tem on the left- an ight-han ie i the ame between two equation. The oeffiient of eon tem on the left-han ie i the ame ontituent element between two equation exept ola aiation tem. Meanwhile, the eon tem on the ight-han ie in equation (4) inue the long-wave aiation oeffiient, an thi tem in equation (4) inue the element with epet to globe themomete. Othe element of the oeffiient in two equation ae equally ommon. Examination uing onete value in Equation (22) povie that oeffiient value in two equation wee nealy equal. Examination of the eive fomula onfime that the oeffiient of T w, T g an T a in the eive WBGT fomula, onieing thei tutue, ae not titly ontant but vaiable epening on metaboli ate, othing inulation, ai veloity an othe uh fato. We alulate the onete value of eah oeffiient in Equation (4), hanging metaboli ate, othing inulation an ai veloity. The alulate value of η/ξ ange fom 0.0 to 2.3. We onie the effet of ola aiation on the η/ξ qualitatively. Auming that µ = in Equation (6), the inequality, whih tun int o η 0, i expee a follow: " " " Ff[ ( H H H ) ( H H H )] W S 0. 78 (23) M Equation (23) how that η 0 i obtaine if the um of the ola aiation H, extenal wok W an heat toage S fo the metaboli ate M i geate than 0.78. The iffeene between the fit an eon tem in the left-han ie of Equation (23) epeent the iffeene between ola aiation abobe by the globe themomete an that abobe by the human boy. Thi iffeene, epenent on ola aboptane, pojete aea fato an onfiguation fato, i poitive beaue ola aboptane of the human boy i geneally le than that of the globe themomete. Thi iniate that in an outoo envionment with ola aiation, the η/ξ value may be poitive even without extenal wok o heat aumulation, an alo that in an envionment without ola aiation, the η/ξ value annot be poitive without them. The alulation howe the oeffiient α/ξ, β/ξ an γ/ξ of T w, T g an T a to be 0.8 ~ 0.86, 0.25 ~ 0.33 an -0.2 ~ 0.02, epetively, oughly 0.84, 0.30 an 0.08 athe than 0.7, 0.2 an 0. the oeffiient in the oiginal fomula. The oeffiient γ/ξ of T a i negative value fo low ai veloitie an poitive fo high ai veloitie, an that lowe ai veloitie eulte in lage negative value. Negative value, een fo low ai veloitie, mean a elatively mall influx of y heat by onvetion an aiation fom an envionment an the ooling of kin ue to weat evapoation, wheea poitive value, een fo high ai veloitie, iniate a elatively lage influx of y heat fom the envionment whih, negating the effet of weat evapoation, aie the kin tempeatue T k. T w (22)
Poeeing of Clima 2007 WellBeing Inoo The meteoologial onition inue in the outoo WBGT fomula epen on the egion, ate an time. In patiula, ola aiation an ai veloity vay ignifiantly epening on the plae an hou of wok o exeie. Conequently, it i afe fo the ue of the WBGT to aopt the oeffiient of T w, T g an T a peifi to the egion, ate an time than to aopt the ontant oeffiient uggete by Yaglou et al. CONCLUSIONS We eive theoetially the outoo WBGT fomula, whih wa oiginally evelope empiially, bae on the heat balane equation between the human boy an an outoo envionment. We aify the inne tutue of the thee ontant oeffiient of wet-bulb tempeatue, globe tempeatue an ai tempeatue, an we pefom an examination uing eal value. Examination of the outoo WBGT fomula uing onete value povie the new fomula WBGT = 0.84T w 0.30T g 0.08T a, ompae to the oiginal fomula WBGT = 0.7T w 0.2T g 0.T a of Yaglou an Mina. Bae on the obtaine eult, we peent the haateiti of the WBGT a an inex a well a intution fo ue. REFERENCES. Yaglou, C P, an Mina, D. 957. Contol of heat aualtie at militay taining ente, Ameian Meial Aoiation Ahive of Inutial Health. Vol.6, pp 302 36. 2. Mohia, T, Sakoi, T, an Kuwabaa K. 2007. Deivation an analyi of the inoo Wet Bulb Globe Tempeatue inex (WBGT) with a human themal engineeing appoah Pat. Popetie of the WBGT fomula fo inoo onition with no ola aiation. Clima2007. 3. Kuwabaa, K, Nagano, K, Mohia, T, et al. 2007. Expeion of the aiative heat exhange fo the human boy an it appliation to moifying the oiginal WBGT fo outoo envionment. Clima 2007. 4. Fange, P O. 970. Themal Comfot. Danih Tehnial Pe. 5. ASHRAE. 993. Phyiologial Piniple an Themal Comfot, ASHRAE Hanbook of Funamental, Chapte 8. 6. MCullough, EA, Jone, BW, an Huk, J. 985. A ompehenive ata bae fo etimating othing inulation, ASHRAE Tanation. Vol. 9(II), pp 29 47. 7. MCullough, EA, Jone, B. an Tamua, T. 989. A ata bae fo etemining the evapoative eitane of othing, ASHRAE Tanation, Vol. 95(II), pp 36 328.