Temperature Problem

[acewert]

Hello!

 

I have a problem, but since this isn't for school I figured I wouldn't put it in the homework section. Okay, so here it is:

 

I need to calculate the resultant temperature of two bodies of water mixed together. The issue is that this is not a classic mixing two liquids in a beaker scenario. The complication is that the two bodies of water are the bottom of a reservoir and an input pipe. Because this is an open system, any water added by the input pipe will mix with the bottom of the lake (and yes, only the bottom because we need to maintain the natural stratification of the lake) without changing the volume of the lake because any water added to the system will be displaced and sent out of the reservoir over the dam. So I think I am dealing with mixing two bodies of water with different temperatures but a fixed final volume(The volume of the bottom layer of the lake). I know the classic heat formula Q=c*m*(T2-T1) and the resulting temperature formula for mixing two liquids Tf=(Va*Ta)+(Vb*TB)/(Va+Vb) but am confused by what to do about the volumes.

 

I suppose theoretically the masses will fully exchange and equalize their heats at the bottom of the reservoir before any water is displaced over the dam, so I could use the classic Tf formula, but I am unsure.

 

Right now I have done the calculations by calculating the heat in the bottom of the reservoir using an assumed reference start temp of 0 degrees and the final temp as the known average temperature of the bottom layer of the reservoir. Then I calculate the heat present in the pipe outflow for a certain volume of outflow by again using 0 degrees as a reference temperature. I then use the Tf=(Va*Ta)+(Vb*TB)/(Va+Vb) formula to calculate the resultant temperature of the mixed water.

 

Is this correct? What do you think???

 

Anne

[CaptainPanic]

Hello and welcome to the forum.

 

First of all, it would have been a lot easier with a picture. That's not criticism, because I know it's much harder to make a good picture with a computer than with a pen and paper. Still, I find that even for myself, I can solve problems easier with a picture.

 

So, I understand the problem that we have a lake. At the bottom you want to add hot water. And at the top you get an overflow as a result of the additional water added. And you say that the top doesn't mix with the bottom.

 

Is that last assumption correct? If the flow of water into the lake is negligible compared to the volume of the lake, you shouldn't worry about it heating up? So, I assume that the flow into the lake is significant. If the flow is quite large, then the bottom will slowly fill up with hot water, and the boundary between the heated water and the cool top water will move up... The boundary will move up because the volume of hot water keeps growing, but also because hot water is lighter than cold water, and it will have a buoyancy effect. Just like hot air goes up, hot water also rises.

 

So, I would think that the hot and cold water will actually mix... and you should approach the problem as a non-stationary heat balance (which is btw something I only learned at university, engineering levels, so it's not a shame not to know this).

 

In a heat balance, we are looking for an accumulation of energy, in this case in the lake. For a balance, you write down:

 

Accumulation = IN - OUT

The accumulation is the bit we're trying to find out. The IN - OUT is what's known. It will result in an equation looking like this:

 

Accumulation = IN - OUT

[math]\frac{d(C_P\cdot{V}\cdot{T})}{dt}=F\cdot{C_P\cdot{T_{in}}}-F\cdot{C_P\cdot{T}}[/math]

 

Assume V, F, Cp, Tin constant, then you can rather easily solve the differential equation.

Tin is the temperature of the water you feed into the lake. To will be the temperature of the lake at t=0. And T is a function of t (Temperature is a function of time).

 

I got:

[math]T=(T_0-T_{in})\cdot{e^{\frac{-F\cdot{t}}{V}}}+T_{in}[/math]

 

Hope I didn't make a mistake, but a plot looks what I would expect.

 

If you play with those numbers, then you find out soon that what matters is the ratio F/V. When the volume of the water that has flowed into the lake is equal to the volume of the lake itself, the temperature will be approximately 0.63*(Tin-T0).

 

Please note that this equation assumes no evaporation... and that assumption will lead to quite large errors (compared to reality) when you do have evaporation.