Chapter 7 – Moist Air

1. Chapter 7 – Moist Air • The Dew (Frost) Point • If saturation is reached with respect to ice we are at the frost point temperature, Tf. • Dew point temperature is a new variable that can be used to characterize the humidity of the air. • Additional assumption – pressure can change (rising or subsiding air) AND humidity may change (e.g., turbulent diffusion of vapor from a water source or rain falling through the air mass). • To find the relation between Td, w, and p, we have to apply the C-C equation for the equilibrium curve.

2. Chapter 7 – Moist Air • The Dew (Frost) Point • By definition, at Tdew, e = esw(Tdew). • We can use the C-C equation in the following way • where e is the vapor pressure of the air mass at T, and Tdew corresponds to e over the saturation curve. • Tdew and e are humidity parameters giving the same info.

3. Chapter 7 – Moist Air • The Dew (Frost) Point • If we solve for Tdew, we get • Expressing this as a relative variation we have • where we have used Tdew = 270K indicating that the relative increase in Tdew is ~5% of the sum of the relative increases in w and p.

4. Chapter 7 – Moist Air • The Dew (Frost) Point • If we integrate the C-C between Tdew and T, we get • If we solve for (T – Tdew), and substitute for constants, we get (using lv = 2.501  106 J kg-1) • For the frost point we get (using lf = 2.8345  106 J kg-1)

5. Chapter 7 – Moist Air • The Dew (Frost) Point • The figure shows the relationships between Tand e during a process. • The process starts at P at temperature Tand vapor pressure e. • Isobarically cool the air to Q where T = Tdewand e is on the saturation curve. • The integration we performed was between points Q and R.

6. Chapter 7 – Moist Air • The Dew (Frost) Point • This figure shows the relation between Tdew, Tf(frost point temperature), and triple point. • Starting at P and isobarically cooling the air, we pass thru Tfbefore reaching Tdew. • Once the Tdew is reached, condensation begins (requires solid surface or CN). • Without surface or CN no condensation occurs and the air becomes supersaturated. liquid vapor solid

7. Chapter 7 – Moist Air • The Dew (Frost) Point • As isobaric cooling proceeds from P, sublimation will not generally occur at F. • Between F and D, air will be supersaturated with respect to ice, but only condense water at D. • Tfand Tdew only indicate the point where condensation or sublimation can occur. They do not guarantee that such will occur. liquid vapor solid

8. Chapter 7 – Moist Air • The Dew (Frost) Point • The determining factor is the CN availability. • Atmosphere has abundant CN, so condensation is not a problem (only small supersaturations). • With respect to ice, if a suitable surface is present freezing or sublimation will proceed as soon as the water or the vapor reaches the equilibrium curve. • Ice Nuclei favor the appearance of ice crystals, but only activate at temperatures well below the equilibrium curve. • Spontaneous nucleation of ice does not take place with either small supercooling of water or supersaturation of vapor.

9. Chapter 7 – Moist Air • Equivalent and Wet-Bulb Temperatures Equivalent temperature is the temperature of an air parcel from which all the water vapor has been extracted by an adiabatic process and p=const. Wet-Bulb Temperature is associated with the moisture content of the air: it is the temperature a sample of air would have if cooled adiabatically to saturation at constant pressure by evaporation of water into it, all latent heat being supplied by the sample of air.

10. Chapter 7 – Moist Air • Equivalent and Wet-Bulb Temperatures • Consider a closed system of dry air, water vapor (moist air), and water (or ice). • The enthalpy of the system can be written as • where we have made use of lv(T) = hv – hwand mt = mv + mw. We can substitute h = cpTfor vapor and liquid phase.

11. Chapter 7 – Moist Air • Equivalent and Wet-Bulb Temperatures • Consider two states of the system linked by an isenthalpic process (H = 0). Each state may be represented by a form of the previous expression. • where md, mt, and const are the same in both states. Then, since H = 0,

12. Chapter 7 – Moist Air • Equivalent and Wet-Bulb Temperatures • We can rewrite this as • The denominator on each side is a constant for any (closed) system. Each of the 2 sides of the equation is a function of the state of the system only, i.e., the expression on either side of the equation is an invariant for an isenthalpic process.

13. Chapter 7 – Moist Air • Equivalent and Wet-Bulb Temperatures • Divide both numerator and denominator in the quotients by mdand use the fact that wt = (mv +mw)/md = mt/mdto get • Note that (cpd + wtcw) is constant for a given system, but will vary for different systems according to their total water value (liquid + vapor).

14. Chapter 7 – Moist Air • Equivalent and Wet-Bulb Temperatures • We can simplify this expression first by neglecting the heat capacity of the water (wt  w) yielding, • Now the denominator is no longer constant. Consider wcw as small compared with cpd and make lva constant, we get

15. Chapter 7 – Moist Air • Equivalent and Wet-Bulb Temperatures • In this expression cp can be taken as cpd, or approximated, to a better degree of accuracy, as where the mixing ratio is an average value of w. • Consider the physical process that links 2 specific states (T’, w’) – unsaturated moist air + water – and (T, w) – saturated or unsaturated moist air without water – of the system to which our approximate equation applies. • We have dry air with w gm of water vapor and (w’ – w) gm of liquid water (may or may not be droplets in suspension).

16. Chapter 7 – Moist Air • Equivalent and Wet-Bulb Temperatures • Assume w’ > wand that saturation is not reached at any time (except eventually when the final state is achieved). • Liquid water evaporates so the mixing ratio increases from w to w’. • As water evaporates, it takes up heat of vaporization from moist air and water (system is adiabatically isolated). • Cooling results reducing temperature from Tto T’. • At any instant, the system state differs finitely from saturation – process is spontaneous and irreversible.

17. Chapter 7 – Moist Air • Equivalent and Wet-Bulb Temperatures • Equivalent temperature(isobaric equivalent temperature) Tei, is defined as the temperature moist air would reach if it were completely dried by condensation of all its water vapor. • Water is withdrawn in a continuous fashion, process is performed isobarically, and system is thermally isolated. • The formulation assumes no liquid water initially and starts from an infinitesimal variation dH, condensing

18. Chapter 7 – Moist Air • Equivalent and Wet-Bulb Temperatures • In this expression, dmv is considered negative, i.e., we are condensing a mass dmv of vapor to liquid. • Having condensed this infinitesimal amount of liquid, we remove it before condensing any more vapor. • The enthalpy of the system will decrease by hwdmv, but T is not affected. • For the next infinitesimal condensation, the equation is valid with a new value of mv. • Our equation describes the process with mv as a variable.

19. Chapter 7 – Moist Air • Equivalent and Wet-Bulb Temperatures • Dividing through by md, and rewriting the differentials as differences, we get • Assuming the latent heat to be constant, we get • Assuming Ti = T, Tf= Tei, wi = w, and wf = 0, we get • Leading to equivalent temperature