R,Rw dR = - T,Tw (Cpd + R Cpv)/Lv dT [4]
Among other things, last week's feature included criticism of moisture equations. Now here's your chance to criticize me. Continuing with a similar topic, I explain a few things regarding a wet-bulb equation. Among the most common usenet weather questions concern such things as "If I know temperature and dew point, how can I calculate relative humidity ?" I'm not gonna write an ONA (often needed answers page) because one already exists, but below I offer some explanations it does not include, for satisfaction of curiosity & usefulness.
During a wet-bulb process, air and water vapor coexist. Some of the vapor is condensed, or water vapor is evaporated in air, saturating it. During these isobaric processes, water vapor content changes, and air volume adjusts accordingly. Vaporization latent heat is responsible for all temperature change (no ice crystals assumed). The governing equation for this process is :
(Md Cpd + Mv Cpv) dT = - Lv dMv [1] Md : dry air mass Mv : water vapor mass Cpd : dry air specific heat, constant pressure Cpv : water vapor specific heat, constant pressure Lv : vaporization latent heat T : temperature
The right term represents latent heat change (i.e., heat loss because of evaporation) and the left term air temperature change caused (i.e., dry air & water vapor components). This process occurs when measuring wet-bulb temperature using a psychrometer (evaporation causes wet-bulb cooling). (Standard definitions use liquid water specific heat rather than water vapor specific heat (and neglect this small term). If anyone can convincingly explain why this is so, I'll change my formula. I don't think the air around the wet bulb contains significant liquid water, and thermal equilibrium between it & the surrounding air is achieved via conduction, allowed to occur as long as necessary.) Dividing with Md produces :
(Cpd + R Cpv) dT = -Lv dR [2]
for which water vapor mixing ratio (R) is :
R = Mv/Md [3]
Hopefully, using R is not confusing (typically used for gas constants). Given dry and wet-bulb readings, T and Tw, this equation can be integrated to calculate mixing ratio :
You may notice that because both R and Lv are functions of T, directly solving the integral is difficult if not impossible. It can be solved as a series of sums, or using a representative value for R & T during the wet-bulb process. The latter using an average is not a bad approximation. Doing so produces :
R = Rw + ((Cpd + R~ Cpv)/Lv)(Tw - T) [5] R~ = (R + Rw)/2 T~ = (T + Tw)/2 [6] Lv = 2500800 - 2370 T~ [7]
You may notice that Rw refers to saturation mixing ratio for Tw. During this discussion and in equations, all temperatures are °C, all pressures mb, and all other units MKS unless noted else. (Thus Lv is expressed as J/kg/°K, etc.)
Equations for water vapor pressure are useful for solving the wet-bulb equation. Experiments have indicated that a specific amount of water vapor can exist at a specific temperature. Such values are tabulated, and several equations have been developed to express this. One such equation is Wexler's :
Es = 6.112 e(17.67 T/(243.5 + T)) [8]
If Es (saturation vapor pressure) is known, T can be determined using the following equations :
X = ln(Es/6.112) [9a] T = 243.5 X/(17.67 - X) [9b]
Mixing ratio relates with vapor pressure as follows (derivation not shown for brevity) :
R = z E / (P - E) [10] E = R P / (z + R) [11] z = .62197 : water vapor molecular mass / dry air molecular mass
Distinction between (existing) vapor pressure (E) and saturation vapor pressure (Es) should be noted.
Saturation vapor pressure defines the dew point temperature (Td), I.e., if air cools isobarically until saturation occurs, temperature attained is the dew point.
Relative humidity (H) is defined as :
H = E / Es [12]
E represents vapor pressure existing in air, and Es is evaluated for this with temperature T (i.e., humidity relative with air as if it were saturated at existing temperature).
Although weather reports often include temperature and relative humidity, such is a rather after the fact statement. Hygrometers can measure relative humidity (but must be calibrated), and dew cells dewpoint, but commonly dry & wet-bulb thermometers are used. Thus, dew point and relative humidity are often obtained from such measurements. Above are equations necessary for such calculations.
Calculating dew point and relative humidity using (1) involves the following procedures :
1. Calculate Lv using Tw & [7] 2. Calculate Rw using [8] & [10] with Tw & Ew (= Es with T = Tw) 3. Calculate R~ using [6] (use R~ = Rw/2 initially) 4. Estimate R using [5] 5. Repeat 3 & 4 until R converges (should only require 2 or 3 iterations) 6. Calculate E using [11] 7. Calculate Td using [9] 8. Calculate H using [12]
Example : Suppose T = 25.0 °C (77.0 °F), Tw = 16.0 °C (60.8 °F), and P = 1000.0 mb
1) Lv = 2500800 - 2370(16.0) = 2462880
(last 2 digits insignificant)
2) Ew = 6.112 e(17.67 (16.0)/(243.5 + 16.0)) = 18.169
Rw = (.62197)(18.169)/(1000.0 - 18.169) = .011510
3) R~ = .011510/2 = .005755
4) R = .011510 + ((1006.3 + (.005755)(1850))/2462880)
(16.0 - 25.0) = .007794
5) R~ = (.007794 + .011510)/2 = .009652
R = .007767
R~ = .009639
R = .007768
R~ = .009639
R = .007768 (converged)
6) E = (.007768)(1000.0)/(.62197 + .007768) = 12.335
7) X = ln(12.335/6.112) = .70215
Td = (243.5)(.70215)/(17.67 - .70215) = 10.076
8) Es = 6.112 e(17.67 (25.0)/(243.5 + 25.0)) = 31.674
H = E / Es = 12.335/31.674 = .38942
Because of imprecise equations and measurement difficulties, answers above should be expressed as Td = 10.1 °C (50.1 °F) & H = .389 (38.9 %), if that accurate, though keeping more significant figures during calculation is fine.
You may notice similarity between (5) and an equation for calculating water vapor pressure in the ONA :
E = Ew - (.00066 (1 + .00155 Tw))(P)(T - Tw) [13]
Ew representing saturation vapor pressure for Tw.
An advantage of the method illustrated is that variation of R & Lv with temperature is included (and small variation of Cpd & Cpv also, which is done for tabulated values shown below). For [13], coefficients are chosen which best fit typical meteorological data. For the example, the ONA's equation produces Td = 9.77 °C. The ONA contains much info regarding wet-bulb measurements and errors involved.
An interesting consequence of the wet-bulb process is a minimum wet bulb temperature for a specific dry bulb temperature (i.e., with no water vapor initially present - Td = 0 °K = -273.15 °C), determined using R = 0 in [5]. I won't show required calculations here, but these temperatures for a wet-bulb temperature 0 °C (32 °F) are :
P (mb) T (°C) T (°F) 1050 8.99 48.2 1000 9.44 49.0 950 9.94 49.9 900 10.49 50.9 800 11.81 53.2 700 13.50 56.3 600 15.76 60.3
These temperatures are sometimes thought of as a quasi-upper limit for snow, the idea being that as snow falls, evaporation causes local temperature (i.e., around the snow) to approach the wet-bulb temperature. That requires melting and evaporation and sublimation though, which I plan discussion of next week.
Text is copyright of Joseph Bartlo, though may be used with proper crediting.