Heat transfer rate in uninsulated barrel

[durcon]

Hello All:

I have three liquid materials.
I know specific heat of all(say cP1, cP2, cP3).
I know change in cPs with temp. (dcP1/dT, dcP2/dT, dcP3/dT).
I know weight of all materials (say W1, W2, and W3).
I need to mix these materials (all at T deg C) in a barrel in a room (no air movement). Barrel is at room temp (RT)and I know properties of steel (cP, dcP/dT, k, weight W4)
Assuming:
No heat generated during mixing.
Barrel is not insulated
No change in total weight
I want to know the final temp of mix at different time. I want to know the equation in terms of parameters so I can insert different values (weight, cP, T, RT, etc) and in case if I have to use other materials.
I know I have to solve time dependent equation and integrate over a specific time in question. I am looking for this equation.

Thanks.

 

[borgwiser]

Sounds like a good multi-part problem with a good set of assumptions given in an intro thermo course.

Biggest question is what are the initial temperatures of the three liquids? With the initial temperatures you can solve this. Use the following equations:
Q1 = m1*$dcP1|(Tref,T1)
(read as Q1 times m1 times the integral of dcP1 from Tref to T1).
You should have dcP1/dT = something.
That something can be a constant, like dcP1/dT = 1, or a function, like dcP1/dT = T + 2T^2 + 9T^-4. Simply multiply both sides of the equation by dT then you can integrate to get the equation from above. If dcP1/dT is a constant, you will get:
Q2 = m2*Cp2*(T2-Tref)
Q3 = m3*Cp3*(T3-Tref)
for each species. Use the same Tref for each. A good rule is use a temperature common to all species that will not cause a phase change.
Then, find the total Q of the system as:
Q = Q1 + Q2 + Q3

Next, find the contribution each species makes to the system. You can use:
X1 = m1 / (m1 + m2 + m3)
X2 = m2 / (m1 + m2 + m3)
X3 = m3 / (m1 + m2 + m3)

Next, you can approximate the new Cp of the system as:
Cp = X1*Cp1 + X2*Cp2 + X3*Cp3

Use the Q from way above, Cp from directly above, and the same Tref to solve for Ts:
Q = (m1+m2+m3)*Cp*(Ts - Tref)
You can figure out the Ts on it's own.
Now you can use the Q from this liquid system and the other properties like Cp and Ts in conjunction with the information you know of the barrel (T, cP and weight).

That's the end of the first part. Assume the temperature changes between the fluids is instantaneous...otherwise you will have to account for mixing and density variations.

Second part:
You will need to do a time-based heat loss from the solution in the barrel to the room at Tamb. Use the equation:
dT/dt = -k*(T - Tamb)
You will be integrating the temperature from the initial temperature of the system (as calculated above) to the final temperature (your T(t)) and the time interval will be from t = 0 to t = t. Tamb and k are constants (presumably)
Your final equation (after integrating) will be an exponential equation resembling:
T(t) = Tamb + (T(0) - Tamb)e^(-kt)
and you can tell T at any time t. Note that there are some gross assumptions here. For example, you aren't taking radiative properties into account, convective properties, etc. Also, the core of the fluid will not change temperature as fast as the surface contacting the metal barrel. But it should suffice for the work you're doing.

 

[borgwiser]

Oops...the fourth line above should read:
Q1 = m1*$dcP1|(Tref,T1)
Q1 EQUALS m1 times the integral of dcP1 from Tref to T1

 

[durcon]

Thanks borgwiser.
I was looking for these equations.