Calculation of time for temperature rise in pipe 2

[jschae8636]

I need a quick and dirty method for calculating the time it takes for the temperature of a meat mixture (of which the properties I do not know, and they would change with every load) to rise from 40 to 45 degrees F or any temperature I'm interested in.

The meat mixture is in a pipe 170ft in length, I would be happy with an approximation for a straight pipe. Stainless steel 6" diameter 10S? Not 100% sure on the schedule.

Air temp outside the pipe is 90 degrees F and the meat mixture is not moving.

I want to know how long it could sit in the pipe and we could still use it ie at a given temperature.

Thanks for any help
JC

 

[25362]

A (very) rough estimate assuming the contents have water thermophysical properties and the surrounding air is still, using a Gurney-Lurie diagram resulted in about 2 hours for the center of the pipe to reach 45 deg F.
Kindly tell me whether this result is in the ballpark, so to say.

 

[friartuck]

Thats one hell of a sausage.
ASHRAE have properties of various meats in. I you want a value post a reply..

Friar Tuck of Sherwood

 

[jschae8636]

25362,

This is in the ballpark. We had assumed 2 hours. I don't know the actual calc...........if it's not too long can you post it?

Thanks!!

 

[25362]

It's not too long, but is based on a semi-log G-L diagram I found in an old book for heating and cooling an infinitely long cylinder. I don't have the book with me now.

The ordinate (on a logarithmic scale) is ([θ]_o-[θ])/([θ]_o-[θ]₁) in this case: (90-45)/(90-40)=0.9.

From the series of curves one selects the group of straight lines corresponding to m=k/(h.r). In SI units for water:
k=0.58 W/(m.K), h(still air)=100 W/(m².K), r=0.076 m. Resulting in m~0.08.

At the center of the pipe it would give an x (abscisa in a linear scale) value of 0.2.

Where (0.2)([ρ])(Cp)(r²)/k = time

[ρ]=density (water)~1000 kg/m³
Cp=4200 J/(kg.K)=4200 (W.s)/(kg.K)
k=0.58 W/(m.K)
r=0.076 m

time= (0.2)(1000)(4200)(0.076²)/0.58=8365 s ~ 2.3 hours.

This exercise shows an indicative result that should be checked in practice. Please note this is an estimate for the center of the pipe. In fact, the more one moves towards the periphery the (exponentially) shorter is the time it takes to reach 45 deg F, and when one deals with food quality one should be careful and possibly consider taking safety factors.

I'm sorry that I'm unable to send you the G-L diagram. I assume you'd be able to find it in books dedicated to heat transfer, in general, and to unsteady state heat transfer, in particular.

 

[jschae8636]

25362,

Thank you!! This is very helpful. I have many books with these diagrams. We also hope to determine the time experimentally due to the fact that the pipe is not straight, and initially the mixture will be flowing and as we know going around 90 degree bends can add heat to the mixture.......leaving an non-uniform temperature mixture.

Ahhh the old unit operations book.......memories.

Hope this will satisfy those whom I work with as far as an estimate.

Thanks again

 

[25362]

There is still another simple formula to estimate the heating time of the whole mass M, knowing Cp, U and A, when the heat supplier is at a constant temperature, air in this case:

time=(MCp/UA)ln[([θ]_air-[θ]₁)/([θ]_air-[θ]₂)]​

Use consistent units.

Assume M=1000 kg, Cp=4200 J/(kg.K), U~5 W/(m².K), A=25 m²

time ~ [(1000)(4200)/(5*25)]ln(50/45) = 3540 s ~ 1 h.

Which comes to show how uncertain are these calculations and how they depend on the assumptions made for the properties of the fluid inside the pipe, or reactor, and the U value.

The formula gets more complicated when the temperature of the heating or cooling medium isn't constant.

 

[jschae8636]

25362,

Hmmmm....interesting. Think making experimental observations will have to be the best bet when it comes to this. Thank you for your help

JC