Principles of Humidity Dalton s law

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1 Principls of Humidity Dalton s law Air is a mixtur of diffrnt gass. Th main gas componnts ar: Gas componnt volum [%] wight [%] Nitrogn N 2 78,03 75,47 Oxygn O 2 20,99 23,20 Argon Ar 0,93 1,28 Carbon dioxid CO 2 0,03 0,04 Rst: H 2, N, H, Kr 0,02 0,01 On normal physical conditions ths gass bhav idal which mans that th gas molculs act indpndnt from ach othr which lads to Dalton s law : Th total prssur of a gas is th sum of th prssurs of ach componnt Th prssur of a singl componnt is calld partial prssur, so th total prssur of air is th sum of th partial prssurs of its componnts i.. p = p N2 + p O2 + p Ar +... dimnsion of p: [mbar] or [hpa] Watr in its gasous phas (vapour) is an additional idal gas componnt of air which appars in Dalton s law : p = p N2 + p O2 + p Ar = p da + p da partial prssur of (watr) vapour [mbar] partial prssur of dry air

2 Principls of Humidity - Vapour prssur Vapour prssur abov watr Vapour prssur cannot tak on any valu in air. Thr is a maximum partial prssur of vapour which dpnds only on air tmpratur. At high tmpraturs air can hold mor vapour than at low tmpraturs. This bhaviour can b xplaind in th following way : Th molculs, boundd in a liquid lik watr, ar moving with diffrnt vlocitis and th avrag kintic nrgy is proportional to th tmpratur of th liquid. Th nrgy distribution of th molculs is statistical as shown in Fig.1. numbr of molculs tmpratur 1 tmpratur 2 > tmpratur 1 Enrgy Fig. 1: Statistical nrgy distribution of molculs in a liquid at diffrnt tmpraturs. Th ovrall avrag kintic nrgy of th molculs is dpnding on tmpratur whras th nrgy of a singl molcul can b lowr or highr with a statistical probability. Molculs with nrgis blow th binding nrgy of th liquid cannot lav th watr surfac, but som hav nough nrgy to xcd th binding forc. Ths molculs can lav th liquid, thy vaporat from th watr surfac and incras th vapour concntration in th room abov th watr surfac. Th molculs in th vapour bhav similar but for thm th cas to hav lss nrgy than th binding nrgy of watr is xcptional: thy ar absorbd from th watr surfac by condnsation and dcras th vapour concntration. A closd box partly filld with watr at tmpratur t (Fig.2) will stabiliz in quilibrium btwn vaporation and condnsation. If thr is a lack of watr molculs in th moist rgion, mor vaporation will occur and th vapour concntration will incras. In th opposit cas mor molculs will condns than vaporat and th vapour concntration will dcras.

3 Vapour partial prssur w vaporation condnsation Watr Fig. 2 : Box partly filld with watr at constant tmpratur Th balanc btwn vaporation and condnsation lads to a vapour prssur, which dpnds only on tmpratur. Incrasing th tmpratur mor molculs hav highr nrgis (s Fig.1), will lav th watr surfac and shift th quilibrium to highr vapour concntrations. On normal nvironmntal conditions vapour is an idal gas with no intraction with othr prsnt idal gass. So th vapour concntration is practically indpndnt from othr xisting gass in th rgion abov th watr surfac. In th box at tmpratur t th balancd vapour prssur is a maximum at this tmpratur and is calld saturation vapour prssur abov watr w at tmpratur t. Th saturation prssur abov watr w dpnds approximatly xponntial on tmpratur t. Valus for w for diffrnt tmpraturs ar givn in Tab.1 [1].

4 t [dgc] w [mbar] t [dgc] w [mbar] E E E E-05 Tab. 1: Saturation vapour prssur valus w abov watr Vapour prssur abov ic Blow t=0.01 dgc (tripl point of watr) watr can xist in th liquid phas as wll as in th solid phas (ic) but th liquid phas is usually not stabl. Th physical intrprtation of vaporation abov ic is qual to watr. t [dgc] i [mbar] E E E E-05 Tab. 2 : Saturation vapour prssur valus i abov ic

5 According to th xistnc of a solid and liquid phas thr ar two saturation curvs blow t=0.01 dgc as shown in Fig.3 on a logarithmic scal. Mind that btwn -100 dgc and 100 dgc th saturation vapour prssur is changing ovr 8 ordrs of magnitud! 1.E+04 saturation vapour prssur [mbar] 1.E+03 1.E+02 1.E+01 1.E E-01 1.E-02 1.E-03 1.E-04 1.E-05 tmpratur [dgc] watr ic Fig. 3 : Vapour saturation curvs abov ic and watr. Blow th tripl point (t=0.01 dgc) th curv splits into two graphs. Ral gas corrction As so far watr vapour was tratd as an idal gas i.. watr molculs act indpndnt from th surrounding air. In rality thr is an intraction btwn watr molculs and th air which lads to a small incras of saturation vapour prssur undr prsnc of air. This fact is takn into account by th nhancmnt factor f(p,t). Th saturation vapour prssur undr prsnc of air w is givn by w = w (t) f(p,t) At normal prssur (p<1100 mbar) th nhancmnt factor is clos to on and can usually b nglctd. This mans that watr vapour mostly bhavs lik an idal gas. If you wish highr accuracis you should corrct th saturation watr prssur using th nhancmnt factor f, which dpnds on air prssur p and tmpratur t (for valus s Tab.3 [2] ).

6 t [dgc] p [bar] Tab. 3 : Enhancmnt factor f(p,t)

7 Principls of Humidity - Magnus formula Th saturation vapour prssur abov ic and watr can b calculatd with good accuracy with th Magnus formula [1] : w, i m t = A xp Tn + t Th paramtrs A, m, T n ar diffrnt for ic and watr and ar givn in Tab. 4. Tmpratur rang t [dgc] A m T n abov ic : -80 to abov watr : -45 to Tab. 4 : Magnus formula paramtr [1]

8 Principls of Humidity Rlativ humidity Rlativ humidity U w [%] Tabl 1 and 2 (in sction Vapour prssur ) giv valus for th saturation watr vapour prssur as a function of tmpratur. Ths valus ar maximum valus, in practic partial vapour prssurs ar usually lowr. Rlativ humidity U w is dfind as th ratio btwn th actual partial vapour prssur and th saturation vapour prssur abov watr w U w = w 100 [%] Sinc th partial vapour prssur cannot xcd th saturation vapour prssur, th maximum valu of rlativ humidity is U w =100%. Rlativ humidity blow th tripl point Th dfinition of rlativ humidity blow th tripl point of watr t< 0.01 dgc rfrs again to th saturation prssur abov watr. In this xcptional tmpratur rgim you can ithr hav watr with saturation vapour prssur w or ic with a smallr valu i < w. In most applications howvr you will find ic sinc this is th stabl stat for tmpraturs t < 0.01 dgc. So th cas = i givs you an uppr limit for th rlativ humidity: t < 0.01 C: Uw max i, = w 100 [%] t [dgc] i [mbar] w [mbar] U max [%] 100% 95% 91% 87% 83% 79% 76% 73% 70% Tab. 5 : Maximum rlativ humidity valus abov ic

9 Principls of Humidity - Dw Point, Frost Point Whn you cool down air containing vapour blow th saturation concntration th actual partial prssur initially stays constant whil th rlativ humidity is incrasing du to th dcras of th saturation vapour prssur with tmpratur t. U w = w 100 [%] t dcrasing w (t) dcrasing U w incrasing At th dw point tmpratur t d th saturation vapour prssur quals th actual vapour prssur i.. w (t d )= and th rlativ humidity rachs its maximum valu of U w =100%. Th dw point tmpratur t d is thrfor th tmpratur to which you hav to cool down moist air at constant prssur for bginning condnsation. It can b calculatd from th tmpratur and rlativ humidity using th Magnus formula : t [ C] saturation vapour prssur w w [mbar], U w [%] w U = w 100 t d = T n ln A m ln A Th Magnus paramtrs A, m, T n ar givn in Tabl 4 in th sction Magnus Formula. Dcrasing th tmpratur blow th dw point tmpratur th partial vapour prssur xcds th saturation valu, so condnsation occurs until th balanc is rachd again. Blow th tripl point of watr t < 0.01 C you will usually find ic ( i ) in your application instad of watr ( w ). Th tmpratur at which th partial vapour prssur rachs th saturation vapour prssur i and frosting starts is calld frost point t f > t d.

10 Principls of Humidity Absolut Humidity Absolut humidity d v [g/m³] Givs th mass of watr in 1 m³ moist air and can b calculatd from th tmpratur t [dgc] and th partial vapour prssur [mbar] : t [dgc] saturation vapour prssur w = w U w d v = t [ g / m 3 ]

11 Principls of Humidity Mixing ratio Mixing ratio r [g/kg] Givs th mass of watr you hav to vaporat and mix with 1 kg dry air to prform a crtain rlativ humidity U w or partial vapour prssur t [dgc] saturation vapour prssur w = w U w r 622 = ( p ) [ g / kg] p air prssur [mbar]

12 Principls of Humidity Spcific Enthalpy Spcific nthalpy h [kj/kg] Th spcific nthalpi of air with tmpratur t, rlativ humidity U w and corrsponding mixing ratio r is th sum of th nrgis you nd to crat this stat in th following way : a) warming up 1 kg dry air from 0 dgc to t b) vaporating th vapour insid th moist air c) warming up th vapour from 0 dgc to t fi Spcific nthalpi pr 1 kg dry air: h = [c pa t + (l w + c pv t) r] [kj/kg] c pa = kj/kg spcific hat capacity of dry air at constant prssur c pv = kj/kg spcific hat capacity of vapour at constant prssur l w = kj/kg latnt hat of watr Th spcific nthalpy is a rlativ quantity i.. only diffrncs ar significant. Mor gnrally nthalpy givs you th amount of nrgy which you nd to bring moist air from a thrmal stat 1 into a stat 2. Exampl 1 : To warm up air from 20 to 25 dgc and humidify th air from 40 to 60 % rlativ humidity you will nd h=20.2 kj/kg. Exampl 2 : Warming up air from 20 to 25 dgc and kping th rlativ humidity constant dissipats only h=10.3 kj/kg. Exampl 3 : Whn warming up air from 20 to 25 dgc and kping th partial vapour prssur constant (r = constant, t d = constant) th rlativ humidity dcrass down to U w =29.5 % which wasts only h=5.1 kj/kg. Ths xampls ar summarizd in Tabl 6 and ar drawn as procss 1, 2, 3 in Fig. 4 in th sction Mollir diagram.

13 Exampl 1 t [dgc] U w [%] h [kj/kg] stat stat Exampl 2 t [dgc] diffrnc 20.2 U w [%] h [kj/kg] stat stat Exampl 3 t [dgc] diffrnc 10.3 U w [%] h [kj/kg] stat stat diffrnc 5.1 Tab. 6: Enthalpy diffrncs with diffrnt changs of stat

14 Principls of Humidity - Mollir diagram Tabl 7 summarizs humidity function valus at diffrnt tmpraturs. A Mollir diagram srvs for solving problms in air conditioning tchnology graphically. It summarizs diffrnt humidity functions in on chart. Using th mixing ratio r thr is a rlation btwn r and tmpratur t with rlativ humidity U w as fr paramtr : r 622 p r w( t) U w = = = p r 100 Using th Magnus formula you can draw a band of curvs of constant rlativ humidity U w as functions t(r) which is calld Mollir diagram. It is convnint to add curvs of constant nthalpy to th Mollir diagram (as in Fig. 4) : h = [c pa t + (l w + c pv t) r] t = (h - l w r) / (c pa + c pv r) In this way you can dscrib graphically thrmodynamical procsss such as xampls 1, 2, 3 in th sction Spcific Enthalpy. In a profssional Mollir diagram usually furthr humidity functions ar includd. tmpratur t [dgc] %rh 2%rh 5%rh 10%rh 20%rh h = 0 kj/kg procss 1 procss 2 procss 3 h = 100 kj/kg 40%rh 60%rh 80%rh 100%rh fog rgion mixing ratio r [g/kg] h = 50 kj/kg Fig. 4 : Mollir diagram : curvs of constant rlativ humidity and nthalpy. Exampls 1, 2, 3 from th sction Spcific Enthalpy ar drawn as procsss 1, 2, 3.

15 Tab. 7: Humidity function valus at diffrnt tmpraturs (p = mbar, ral gas corrctions ar takn into account) : -20 dgc U w [%] t d [dgc] [mbar] d v [g/m³] r [g/kg] h [kj/kg] dgc Uw [%] t d [dgc] [mbar] d v [g/m³] r [g/kg] h [kj/kg]

16 20 dgc U w [%] t d [dgc] [mbar] d v [g/m³] r [g/kg] h [kj/kg] dgc U w [%] t d [dgc] [mbar] d v [g/m³] r [g/kg] h [kj/kg]

17 60 dgc U w [%] t d [dgc] [mbar] d v [g/m³] r [g/kg] h [kj/kg]

18 Principls of Humidity - Litratur [1] Sonntag D.: Important Nw Valus of Physical Constants of 1986, Vapour Prssur Formulations basd on th ITS-90 and Psychromtr Formula; Z.Mtorol.70 (1990) 5, [2] Hyland R.W.: A Corrlation for th scond Intraction Virial Cofficints and Enhancmnt Factors for Moist air; J.Rsarch NBS, A.Physics and Chmistry 79A (1975)

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