Temp to raise fluid temp inside a container - Heat transfer & thermodynamics

[abe_cooldude]

Homework Statement

Hi all, I have this problem statement with variables:

20" Diameter, 2000mm (length, L) cylindrical container with 8mm thickness made of plastic
Container 3/4 filled with gasoline.

Question: What is the immediately surrounding temperature required outside the container to raise the temperature of gasoline, initially at 50 F, 6 F per minute.

How will the pressure change/minute inside the tank as temperature of gasoline rises.

Known: Thermal conductivity, k, of plastic (for conduction heat transfer)
Heat capacity of gasoline

Assumptions: Equal heat transfer from outside to inside through all sides
At atmospheric pressure outside
Tank fully closed (no vents, etc.)

Homework Equations

I know this equation q = (mass) (temp change) (Cp)
But I don't know how to incorporate heat transfer through the thickness of the container to the fluid inside the tank.

I also know that the heat transfer rate through cylindrical shell is Q(dot)= 2k(pi)L(T1-T2)/ln(r2/r1)
But how do I take into account heat transfer through circular end caps on each end of the cylinder?

The Attempt at a Solution

I have no idea how to combine heat transfer and thermodynamic aspects of this problem together to solve for what the statement is asking for, outside temperature and pressure change

Any ideas or direction will be greatly appreciated!

Thanks,
Abe

 

[abe_cooldude]

Any ideas? Any help or direction will be appreciated.

UPDATE: In addition to the information above, I can figure out the mass of the gasoline from the volume and density, and use q=m*cp*(T2-T1), and this will give me heat in Joules, but I don't know how to take into account the temperature raise by 6 F (3.33K) PER minute. That throws me off as well.

I guess for the end caps, I can use 2*q(conduction through wall) and for the shell, I can use q(conduction through cylinder).

Since I am looking for temperature immediately surrounding the tank, is it safe to ignore ambient air convection? Pretty much this tank will be engulfed in flames. So then my unknown would be the outside surface temperature of the tank, assuming evenly distributed flame, thus evenly distributed outside surface temperature.

My other question is since the tank is only 3/4 full with gasoline initial temp at 50 F (283.15K), what would my inside surface temp be?

If the density of gasoline is 840 Kg/m^3 and volume of 3/4 full tank is 0.426m^3, mass is 358kg.

Using q=m*cp*(detla T), 358kg*1750J/kg-K*3.33K = 2086.245KJ.

 

[abe_cooldude]

Anyone?

 

[abe_cooldude]

Hello? Please, anyone?

 

[abe_cooldude]

Any one?

 

[LawrenceC]

You will have to determine the heat transfer through the wall of the container. You will have natural convection on both the inside (gasoline to wall) as well as natural convection on the outside (air to wall). You will need the specific correlations for these. You also have the resistance to heat flow due to the conductivity of the container wall.

As for the vapor pressure, you can look that up in tables based on the gasoline bulk temperature.

I would solve the problem using a lumped mass for the gasoline. Set up a differential equation that equates the rate of change in internal energy of the gasoline to the heat reaching the gasoline. You have 3 resistances. Two are natural convections while the third is conduction through the plastic.

You'll have to make an assumption regarding the flame temperature. The end caps can be accounted for by adding their contribution to the overall heat flux. Flames emit radiation. That contributes to the heat flux but non-linearizes the problem. I don't know how much in detail you want to go with this.

 

[LawrenceC]

Since the tank is 8mm thick, almost 1/3 inch and plastic, the tank wall offers significant resistance to heat flow. Also since it is relatively thick, if it is immersed in flame wouldn't it begin to melt? When you apply a strong heat flux, such as flame, to a thick piece of material that is a relatively poor conductor, the skin temperature at the flame side is quit high. That is why melting might be a problem.

For starters, you could do a simple 1-D transient of a piece of plastic. Fix one side at a low temperature and apply a convection boundary condition at the other side with the environmental temperature set to the flame temperature. See how the surface temperature reacts. The thicker it is, the higher the outside skin temperature will get.

 

[abe_cooldude]

THANK YOU! I kind of gave up expecting a reply, but you rock! I will start on this on, and post my progress to see if I am on the right path.