Batch tank cooling coil with rising inlet temperature of coolant 2

[SNORGY]

I am looking for modified equations such as those derived in Perry's for unsteady state batch cooling, except that the temperature of the coolant entering the cooling coil rises with time, since there is not adequate heat rejection after the coil outlet. Is this something that someone has seen and derived equations for in their past, or could someone point me in the right direction towards a reference?

I am not especially strong with differential equations...if it gets too much more complicated than Newton's Law Of Cooling, I'll need a bit of help from a reference of some sort.

This isn't homework...I am looking at replacing an air cooler that would only be used for a very short time with an uninsulated, rented tank that sheds as much heat as it can naturally to ambient, while enthalpy is withdrawn from the batch tank via coolant pumped through its cooling coil. I am trying to quantify if adequate cooling can be achieved within one operating shift (12 hours).

I can probably take a pretty good flyer at it by making some simplifying assumptions or going about it quasi-theoretically / quasi-graphically, but the "elegant" solution would be nice to have.

Thanks in advance for any suggestions.

Regards,

SNORGY.

 

[speco]

Snorgy,

I have encountered a similar problem recently. My solution was to write a program. I used BASIC because it's fast and easy for me, but any good programing language should work OK. In my program, I broke down the batch cooling range into small increments. Then for each increment, I added the heat load to the mass of the incoming coolant, resulting in a new inlet temperature. The rest of the math works the same then, as long as you some approach temperature to work with.

Regards,

Speco

 

[Dave41A]

If the tank is "well mixed," you can probably apply the lumped-parameter approximation (also called "lumped capacitance"). In this case, the tank will cool according to the relation T=Tinf-(T(0)-Tinf)*e^(-t/Tau), where "T" is the temperature as a function of time, T(0) is its starting temp, Tinf is the ambient temp, and "Tau" is the time constant for your system. This assumes constant ambient temperatures.

Lienhard (link provided) or any good heat tranfser textbook will describe how to compute Tau. In general it is Tau=m*c_p/(h*A). "h" is the hardest to approximate.

http://web.mit.edu/lienhard/

http://www/ahtt.html

If that doesn't work for you (or apply to your situation), the numerical approach suggested by speco is an excellent one and will always work. I have done this numerous times. It is easy to develop the timesteps in a spreadsheet (and just "drag" down the formulas) if you don't like to program.

Good luck,
Dave

 

[SNORGY]

Thanks, Dave41A and speco.

I am involved in a myriad of simultaneous things - stress analysis, P/L integrity assessments, etc. - on other projects right now, and the project to which this problem relates in in a temporary hold status, so it's been on my back-burner for the past week.

I will try both approaches in addition to my own and see how they all compare.

Regards,

SNORGY.