SOLUTION: 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 satur

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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