Thermodynamics

cen84959_ch03.qxd 4/11/05 12:23 PM Page 152152 | ThermodynamicsTAir25°Cφ = 10%P vLakeFIGURE 3–65Schematic for Example 3–14.P v = P sat @ TEXAMPLE 3–14Temperature Drop of a Lake Due to EvaporationOn a summer day, the air temperature over a lake is measured to be 25°C.Determine water temperature of the lake when phase equilibrium conditionsare established between the water in the lake and the vapor in the air for relativehumidities of 10, 80, and 100 percent for the air (Fig. 3–65).Solution Air at a specified temperature is blowing over a lake. The equilibriumtemperatures of water for three different cases are to be determined.Analysis The saturation pressure of water at 25°C, from Table 3–1, is 3.17kPa. Then the vapor pressures at relative humidities of 10, 80, and 100 percentare determined from Eq. 3–29 to beRelative humidity 10%:P v1 f 1 P sat @ 25°C 0.1 13.17 kPa2 0.317 kPaRelative humidity 80%:P v 2 f 2 P sat @ 25°C 0.8 13.17 kPa2 2.536 kPaRelative humidity 100%:P v3 f 3 P sat @25°C 1.0 13.17 kPa2 3.17 kPaThe saturation temperatures corresponding to these pressures are determinedfrom Table 3–1 (or Table A–5) by interpolation to beT 1 8.0°CT 2 21.2°CandT 3 25°CTherefore, water will freeze in the first case even though the surrounding airis hot. In the last case the water temperature will be the same as the surroundingair temperature.Discussion You are probably skeptical about the lake freezing when the airis at 25°C, and you are right. The water temperature drops to 8°C in thelimiting case of no heat transfer to the water surface. In practice the watertemperature drops below the air temperature, but it does not drop to 8°Cbecause (1) it is very unlikely for the air over the lake to be so dry (a relativehumidity of just 10 percent) and (2) as the water temperature near the surfacedrops, heat transfer from the air and the lower parts of the water bodywill tend to make up for this heat loss and prevent the water temperaturefrom dropping too much. The water temperature stabilizes when the heatgain from the surrounding air and the water body equals the heat loss byevaporation, that is, when a dynamic balance is established between heatand mass transfer instead of phase equilibrium. If you try this experimentusing a shallow layer of water in a well-insulated pan, you can actually freezethe water if the air is very dry and relatively cool.