Question 1133585: Evaporating water into air cools the air; a swamp cooler works by evaporation. The wet-bulb temperature of air is the temperature that the air would have if it were cooled to saturation (100% humidity) by evaporating water into it. The amount of cooling possible with a swamp cooler is proportional to the difference between the actual air temperature, T and the wet-bulb temperature, W. A particular swamp cooler can cool air at a temperature of 40 degrees celsius to 23 degrees celsius if the wet-bulb temperature is 20 degrees celsius. Determine the lowest temperature that this swamp cooler could achieve:
a) Actual temperature is 48 degrees and the wet bulb temp is 22
b) Actual temp is 32 and wet bulb is 26
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! i think this is a direct variation type problem.
the general formula for direct variation is y = k * x
let A equal the amount of cooling and T equal the actual temperature and W equal the wet bulb temperature and the formula becomes:
A = k * (T - W)
let C equal the cooled temperature and you get:
C = T - A
since A = k * (T - W), then you get:
C = T - k * (T - W)
the first thing you need to do is find k.
when T = 40 and W = 20, you are given that C = 23
the formula of C = T - k * (T - W) becomes:
23 = 40 - k * (40 - 20)
simplify this to get 23 = 40 - k * 20
add k * 20 to both sides and subtract 23 from both sides to get:
k * 20 = 40 - 23
simplify to get k * 20 = 17
solve for k to get k = 17/20.
test this out to see if it works.
when k = 17/20 and T = 40 and W = 20, the formula of C = T - k * (T - W) becomes:
C = 40 - 17/20 * (40 - 20) which results in C = 23.
you are asked to find the lowest temperature that the cooler can achieve when:
a) Actual temperature is 48 degrees and the wet bulb temp is 22.
b) Actual temp is 32 and wet bulb is 26.
the same formula is used with k = 17/20
for a), ...
the formula of C = T - k * (T - W) becomes:
C = 48 - 17/20 * (48 - 22) which results in C = 25.9
for b), ...
the formula of C = T - k * (T - W) becomes:
C = 32 - 17/20 * (32 - 26) which results in C = 26.9
the constant of variation (k) is specific to a particular type of cooler.
a different type of cooler might have a different value for k.
the particular value of k would have to be generated from the base case, as it was for this problem.
it appeared that in all cases, the maximum drop in temperature was equal to 85% of the difference between the actual temperature and the wet bulb temperature.
this was based on the calculation of k = 17/20 = .85 = 85% which was derived from the base case for the particular type of cooler that was used.
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