# Need some help in understanding heat transfer application

I am in the process of redesigning a cooling loop for a expanded polystyrene extruder. I would like to develop a better understanding of
the cooling loop especially under differing process variables.
Here is a brief description. There are two loops, the process loop and the cooling loop.
The process loop is water circulated through the extruder to control the extruder's temperature. It consists of a reservoir (open to atmosphere) a pump, a flat brazed heat exchanger and control valves to meter the process water through different extruder zones as needed.
The cooling loop will supply cooled water to one side of the heat exchanger to remove heat from the process water running through the other side. The cooling loop consists of a water to air fluid cooling unit with a pump and temperature controls.
The design assumes a max steady state heat transfer of 3.5 tons (42000 BTU/hr). The lower temp process water leaving the heat exchanger and traveling to the extruder is required to be held to 110°F. The process water flow is 10 gpm. This equates to a 8.4°F delta T on the process side.
My first goal is to determine what happens as I change the cooling loop flow rate.
Let's simplify the effects of the heat exchange by assuming it is oversized and requires a minimal log mean temperature difference (LMTD)between the process and the cooling water. Assume under 1°F. The heat exchanger is plumbed for counter flow. The hot process water enters on same end as the "cold" cooling water. The heat exchanger is so oversized that the lower temp process water leaving the heat exchanger is almost the same temperature as the "cold" cooling water entering it. Let's say its 1°F warmer. By the same token the "warm" chilled water leaving the heat exchanger is almost the same temperature as the hot process water entering the heat exchanger. Again let's say 1°F.
In my prior calculations it appeared that the LMTD is pretty close to the difference of the average temperature of the process flow and cooling flow across the heat exchanger. We will use this average difference to keep things simple.
Here are one set of steady state conditions: Heat transfer 42000 BTU/Hr Cooled Cooling Water entering heat exchanger 110°F Requirement Cooled Process Water leaving heat exchanger 111°F Assumed with oversized heat exchanger Process Water flow 10 gpm Typical actual value Delta T for Process Water 8.4°F Calculated from previous values. Hot Process Water temperature 119.4°F 111+8.4
"Warm" cooling water temperature 118.4°F Assumed with oversized heat exchanger Cooling Water flow 10 gpm Same heat transfer and delta T so same flow
Any problems yet with what I have done so far?
NOW INCREASE COOLING WATER FLOW RATE WHILE STILL MAINTAINING 110°F AT "COLD" COOLING WATER INLET AT HEAT EXCHANGER.
Cooling Water flow increased. Same heat transfer, initially at least. "Cold" Cooling Water temp maintained at same 110°F Delta T on cooling loop must decrease. This decreases "warm" cooling water temperature. This decreases average of cooling water temperatures and increases LMTD. I would expect that increasing LMTD will increase heat transfer. BUT The Cooled Process Water temp is still 111°F as the "Cold" cooling water is held at 110°F and heat exchanger is oversized. The heat load on the process side has not changed. It is still 42000BTU/Hr. The process loop flow is not changed. Still 10 gpm. Therefore the "Hot" Process Water temperature must remain the same.
The apparent paradox is that increasing cooling loop flow should transfer more heat at least for a period of time and alter the process temperatures. But the process temps are set by load, flow rate, and the assumed 111°F temp at process exit from heat exchanger.
I appreciate all who persevered through this long post. I have faith that someone will see the flaw(s) in my analysis.
Dave Miller snipped-for-privacy@rochester.rr.com
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Dear D.Miller:

I don't think it was etiquette. I think it was, punctuation.
David A. Smith
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I'm with Mr. Smith.......I was having real trouble figuring out what you were asking.
Bob
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Dave -
The problem here is the "assumed 111F temp at process exit" -- it's not necessarily true. Some experts can correct me if I'm wrong, but I don't think a true counter-current heat exchanger can be pinched in the way you describe. So the process (hot) fluid outlet temperature *will* decrease when the cooling flow is increased.
Let's throw some numbers at it. I'll use the subscript "c" for the cold fluid, "h" for hot; also "1" refers to the hot side of the exchanger, and "2" to the cold side. The log mean exchanger delta T is "LMTD" and is defined as [(Th1-Tc1) - (Th2-Tc2)] / ln[(Th1-Tc1) / (Th2-Tc2)]. For your initial conditions, the delta T is constant across the whole HX so LMTD is undefined, and the temperature difference to use is the constant delta T of 1F.
The (steady-state) heat balance equations are:
Qh = Mh * Ch * (Th1 - Th2) Qc = Mc * Cc * (Tc1 - Tc2) Qx = UA * LMTD
Plugging in the numbers for your starting point gives: Mh*Ch = McCc = 5,000 Btu/hr-F UA = 42,000 Btu/hr-F
Now increase the flow of cooling water by, say, 10%. Then McCc becomes 5,500 Btu/hr-F; MhCh is still 5,000 Btu/hr-F. I'll assume that the exchanger UA is not affected -- although if it does change, it would probably increase, amplifying the effect below.
You stated that the cold cooling water (Tc2) will be held 110F. In order to solve the problem, I'll assume that the hot process fluid (Th1) will remain at 119.4F. The problem is to find the cooled process outlet (Th2), which will also give us the hot cooling water outlet (Tc1). You'll see that it must change.
The heat balance now gives:
Qh = (5,000 Btu/hr-F) * (119.4 - Th2)F Qc = (5,500 Btu/hr-F) * (Tc1 - 110)F Qx = (42,000 Btu/hr-F) * LMTD
In the end there are two independent equations and two unknowns. Solving them simultaneously gives:
Th2 = 110.69 F Tc1 = 117.92 F
So the process return temperature did indeed decrease when the coolant flow increased. The amount of heat transferred increased by about 3.6% with that 10% increase in coolant flow.
But, as I mentioned before, this only holds for a true counter-current exchanger. A different type of exchanger -- like a shell and tube HX with multiple tube passes -- probably would become temperature pinched such that increasing the coolant flow did nothing. I'll bet some examples can be found in texts dealing with process heat exchange. (My guess is someone watching this can probably point to a couple quickly.) But no process-industry exchanger would ever be designed at such a tiny LMTD for a host of reasons.
By the way, I cross-posted to sci.engr.chem since some folks there will know a lot more about this than I do.
Rob