One way of increasing heat transfer from the head on a hot summer day is to wet it. This is especially effective in windy weather, as you may have noticed. Approximating the head as a 30-cm-diameter sphere at 30°C with an emissivity of 0.95, determine the total rate of heat loss from the head at ambient air conditions of 1 atm, 25°C, 30 percent relative humidity, and 25 km/h winds if the head is (a) dry and (b) wet. Take the surrounding temperature to be 25°C.
A person is standing outdoors in windy weather. The rates of heat loss from the head by radiation, forced convection, and evaporation are to be determined for the cases of the head being wet and dry.
Assumptions 1 The low mass flux conditions exist so that the Chilton-Colburn analogy between heat and mass transfer is applicable since the mass fraction of vapor in the air is low (about 2 percent for saturated air at 300 K). 2 Both air and water vapor at specified conditions are ideal gases (the error involved in this assumption is less than 1 percent). 3 The head can be approximated as a sphere of 30 cm diameter maintained at a uniform temperature of 30°C. 4 The surrounding surfaces are at the same temperature as the ambient air.
Properties The air-water vapor mixture is assumed to be dilute, and thus we can use dry air properties for the mixture. The properties of air at the free stream temperature of 25°C and 1 atm are, from Table A-15,
The mass diffusivity of water vapor in air at the average temperature of (25 + 30)/2 = 27.5°C = 300.5 K is, from Eq. 14-15,
The saturation pressure of water at 25°C is 
Properties of water at 30°C are 
The gas constants of dry air and water are Rair = 0.287 kPa.m3/kg.K and Rwater = 0.4615 kPa.m3/kg.K (Table A-1). Also, the emissivity of the head is given to be 0.95.
Analysis (a) When the head is dry, heat transfer from the head is by forced convection and radiation only. The radiation heat transfer is
The Reynolds number for flow over the head is
Then the Nusselt number and the heat transfer coefficient become
Then the rate of convection heat transfer from the head becomes
Therefore,
(b) When the head is wet, there is additional heat transfer mechanism by evaporation. The Schmidt number is
The Sherwood number and the mass transfer coefficients are determined to be
The air at the water surface is saturated, and thus the vapor pressure at the surface is simply the saturation pressure of water at the surface temperature (4.246 kPa at 30?C). The vapor pressure of air far from the water surface is determined from
Treating the water vapor and the air as ideal gases, the vapor densities at the water-air interface and far from the surface are determined to be
Then the evaporation rate and the rate of heat transfer by evaporation become
Then the total rate of heat loss from the wet head to the surrounding air and surfaces becomes
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