One evening, a few friends come over for a soak, and you discover that the water

Question

One evening, a few friends come over for a soak, and you discover that the water in the hot tub is 115degreeF (46 degree C) to hot to use. As your friends turn on the cold water to cool down the tub, the engineer in you wants to know how long this is going to take, so you write an energy balance on a well-mixed tank (ignore heat losses to the air). You end up with the following differential equation relation the temperature in the tank. T, to the temperature of the cold water flowing into the tank. T_in, the volume of the tank. V, and the volumetric flow rate of the cold water- integrating, you get an equation for the temperature in the tank as a function of time :- If the initial temperature T_init is 115degreeF, the cold water temperature is 35 degree F (l.7degreeC), and the volume and volumetric flow rate are 3,000 liters and 30 liters per minute, respectively. calculate the expected water temperature 5 minute intervals for the first 60 minute after the flow cold water is established. Plot the water Temperature in the hot tub as a function of time. Calculate how long it should take for the water in the tub to cool to 100 degree F (37.8degreeC). Explain whether a hot tub is really a well mixed tank, if it is not, will your equation predict a time that is too short or too long? Explain your reasoning

Explanation / Answer

a) The expected water temprature can be found by using the above given formula in exponential form.

T = (Tinp - Tiniti) e- vot / v

T = (1.7-46) e30litre / min(60min) / 3000litre

   T= -44.3 e0.6

= -80.7 C

Negative sign shown that water is getting cool.

C) Applying the equation

T = (Tin - Tinti) e-Vot / V

37.8 = (1.7-46) e 30t / 3000

taking ln on both sides

ln 37.6 = 44.4 (0.01t)

3.62 = 44.84t

t = 0.080min

t = 4.84sec

D) The tub is really a well mixed tank , because it took too short to cool water in a tank which is clear cut evidence of its excellency.

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