Math problem...need help

Need a formula if one can provide it

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Math problem...need help

#1  Postby ackmanben1988 » Jul 08, 2019 12:29 am

Hey everyone,

So, I work at a spa, and my job is to empty, clean, and refill tubs. We have to make sure they're at a minimum of 98 degrees fahrenheit when they are refilled to make sure they're good to go for the rest of the day.

So, here it is: It's a 800 gallon tub, and when I hook up the hoses to the spigot it comes out at about 13.1 gallons per minute. Starting off, when I'm first filling the tub, the temperature of the water heater is around 127-129 degrees fahrenheit. After 20 minutes of filling the tub, at an overall loss of heat at 2 degrees/minute (approximately), the tub was at about 260 gallons and its temperature at 98.5. Then, I have to wait for the water heat to heat the water back up to around 110, 115 degrees, at a rate of approximately 1.6 degrees/minute, and then do it over again. I'm trying to see how long I can keep the hose on and how fast I can fill this tub without the water heater cooling down to such a degree that is less than the ideal 98. Anyone have thoughts on what kind of formula I can use that takes into account different temperatures of water mixed together (127 to 126 to 125 and so on in a certain time span)?
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Re: Math problem...need help

#2  Postby Thommo » Jul 08, 2019 1:17 am

If you have two bodies of water V1 and V2, at temperatures f1 and f2 and you mix them together the resultant temperature (ignoring heat losses over time to the outside environment) is simply the weighted average of the two:

(V1xf1 + V2xf2) / (V1 + V2)

If you have a water source that is delivering water at a fixed rate r with a linear decline in the temperature of water provided, from a starting temperature fstart to a finishing temperature fend, over a period of time t, then the result is a volume of water

rt

at a temperature of

(fstart + fend)/2

from the information you've given us you can fill in r as ~13.1 (using the time interval as minutes) and estimate fend as fstart - 2t

Putting that into the previous formula gives a volume of water at time t of:

13.1t

at a temperature of

fstart - t

The problem with all that is you're dealing with a long period of time, so neglecting losses of heat to the outside environment is going to cause some error, and calculating those will depend on more complex stuff like the surface area of the tub, its insulation, the outside temperature etc.
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Re: Math problem...need help

#3  Postby ackmanben1988 » Jul 08, 2019 1:27 am

This is extremely helpful. Thank you.
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Re: Math problem...need help

#4  Postby Thommo » Jul 08, 2019 1:34 am

I'm not sure it is really, you've lost so much heat to the environment in the numbers you gave (10f in less than half the time you put that volume of water in the tub and let your water source return to its original temperature) I'm not sure you can really get to the answer this way.

Maybe a starting point, I don't know. I wasn't really sure what you were after, or the exact specification of the problem, sorry.
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Re: Math problem...need help

#5  Postby ackmanben1988 » Jul 08, 2019 1:36 am

"If you have a water source that is delivering water at a fixed rate r with a linear decline in the temperature of water provided, from a starting temperature fstart to a finishing temperature fend, over a period of time t, then the result is a volume of water

rt

at a temperature of

(fstart + fend)/2

from the information you've given us you can fill in r as ~13.1 (using the time interval as minutes) and estimate fend as fstart - 2t

Putting that into the previous formula gives a volume of water at time t of:

13.1t

at a temperature of

fstart - t "

What would this formula look like?
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Re: Math problem...need help

#6  Postby ackmanben1988 » Jul 08, 2019 1:40 am

I'll work with the numbers and do some testing. Either way, it's helpful.
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Re: Math problem...need help

#7  Postby Fenrir » Jul 08, 2019 3:59 am

Remember r is a variable.

You can always reduce the flow to a level which allows the heater to maintain temp.
Religion: it only fails when you test it.-Thunderf00t.
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