|MadSci Network: Physics|
Robert, Unfortunately there isn't any one general equation to describe how to design a hair dryer. As for whether dryers are designed by science or economics, it's a little bit of both: the design has to do the job (dry your hair quickly), do it safely (you don't get burned, electrocuted or set fire to the house) and do it in a way the people who want one can afford it (which is why hair dryers can be bought for under twenty-five dollars). The economics come in with the choice of shape, materials, and manufacturing process: there wouldn't be hair dryers if, for example, each one had to be built completely by hand. On the science side, a designer needs to look at what's going on both inside the case and when the heated air comes in contact with your hair. The limitations at each of these places have an effect on what happens at the other. First, let's look at the hair end of things, because this is where we want the dryer to do its work for us. In general, wet hair would dry on its own just in open, still air. This is because air can hold a certain amount of water vapor at any given temperature, and it is rare that air holds its full capacity of water at any one time. Relative humidity is the measure of what percentage of this capacity is being held in the air: at anything under 100% there is room for more water, and the moisture from a person's hair simply evaporates away. Depending on conditions, this usually takes a while. If we want to increase the amount of water that air can hold,and thereby increase the drying rate of the hair, we need to increase the amount of energy the air has. This increases the average space between air molecules and makes more room for water vapor between them. The two easiest ways to do this are to make the air move faster and to increase the temperature of the air, which are exactly what a hair dryer does. Generally you can take the rate of drying as inversely proportional to the relative humidity, directly proportional to the air temperature, and directly proportional to the speed the air moves past the hair. Hair thickness and length will affect drying time because longer and thicker hair hold more water which takes longer to completely evaporate, but the evaporation rate is the same for a given air temperature, speed and humidity. Now let's look at the hair dryer. The optimum temperature the air drying someone's hair is determined by the human body, rather than an equation. There is a limit to how hot you want to make a person's head (burns happen above approximately 140 degrees Farenheit, or 60 degrees Centigrade), so we don't want to make the air coming out of a dryer any warmer than that when it reaches the person's head. Most hand-held dryers count on turbulent mixing with the surrounding air as the heated air leaves the dryer's nozzle to cool it down to an acceptable temperature a few inches from the nozzle. The calculations to exactly determine the degree of mixing that happens are very complicated: most development engineers would make a fairly close approximation based on the ratio of temperatures of the heated and outside air, then adjust the nozzle shape and amount of heating inside the dryer and measure the resulting temperature until getting it where it's wanted. The temperature of the air inside the hair dryer can be found from the power of the heater (typically around 1000 watts), the amount of time the air spends in the dryer as it passes through the coils (we'll assume one-half second for a hand-held dryer) and the volume of air the fan pushes in that half-second (for a typical small fan, around 1/12 of a cubic foot). Using the relationship T2=(Q/mc)+T1 where: Q is the amount of energy put into the air ( 1/2 of 1000 watts, because the air is in the dryer for 1/2 second) m is the mass of the air (about .0056 kilograms [5.6 grams] for 1/12 of a cubic foot of air) c is the specific heat (energy capacity per unit weight) for air (around 1010 Joules/mole Kelvin, taken from a table of values) T1 is the temperature of the air the fan sucks into the hair dryer ( around 70 degrees F, or 20 degrees C) and T2 is the final temperature. Plugging in our values, we have T2 = (500/(.0056 x 1010)) + 20 = 107 degrees C = 225 degrees F This is very hot: to get the temperature back down below 140 degrees F, you'd need to mix the hot air with at least as much air as you're heating every second!
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