# EMEM Thermal Fluids Lab Addendum to the Guidelines Thermal

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```					EMEM 551: Thermal-Fluids Lab 2                                                                2008-3

Addendum to the Guidelines: Thermal Performance Analysis
of Chiller Heat Exchanger
As outlined in the original Guidelines for the Thermal Performance Analysis, the chiller heat
exchanger is most appropriately analyzed as a parallel combination of three finned-tube circuits,
each of which consists of three rows of four parallel tubes. However, consistent with the
assumptions made, each row of four tubes actually behaves as if it were one single tube pass four
times as long as the actual tubes. Thus, when the effectiveness ε for the heat exchanger is to be
determined, the most appropriate correlation to use is that for cross-flow over a single tube pass,
with both fluids unmixed—provided that we somehow account for the existence of multiple
rows. Because the heat transfer coefficient is assumed to be uniform over the entire heat
exchanger, the effectiveness is also uniform for all rows by this approach, but the inlet
temperature difference changes between rows.

The “regular” effectiveness is typically used to characterize the total heat transfer that occurs,
based upon the inlet temperature difference of the entire heat exchanger. However, because of
the special configuration of the chiller heat exchanger compared to the simpler designs discussed
in the textbook, each row in this analysis has its own inlet air and water temperatures, and
therefore an inlet temperature difference that decreases from row to row. Thus, in this case, it is
actually more appropriate to determine an overall effectiveness εoverall, which relates the total
heat transfer across all rows (per circuit) to the true overall temperature difference TWi - TAi. The
“regular” effectiveness ε must be applied to each row, and then combined somehow to arrive at
εoverall.

Recall that in the standard scenario where the two fluids enter a heat exchanger at a temperature
difference Thi - Tci (where Thi is the inlet temperature of the hotter stream, and Tci is the inlet
temperature of the cold stream), the heat transfer that occurs between the two fluids while in the
heat exchanger is given by:
q = ε Cmin ⋅ ( Thi − Tci )                                                                       (1)

In this case, we need to apply the standard idea to each row individually, and then combine to
arrive at the heat transfer qcircuit for the circuit as a whole:
q circuit = q row1 + q row 2 + q row 3                                                           (2)

Where qrow1, qrow2, and qrow3 are the heat transfer rates of each row, respectively. Now assuming
that the “regular” effectiveness ε —determined from correlations for cross-flow, single pass, both
fluids unmixed—is the same for each row, and noting that Cmin must be uniform by assumption,
we have:
q circuit = ε C min ⋅ ( TWi − T Ai + TW 1 − T A1 + TW 2 − T A 2 )                                (3)

Where the standard definition in (1) has been applied to each term in (2) to arrive at (3). The
result in (3) is inconvenient, because we do not know the intermediate fluid temperatures.

Guidelines Addendum - Chiller HEX Thermal Analysis 20083.doc        Page 1 of 2        John D. Wellin
EMEM 551: Thermal-Fluids Lab 2                                                                                 2008-3

However, we can eliminate all but the overall inlet temperatures by using the following
definitions that arise from a series of simple energy balances:
q row1 = ε C min ⋅ ( TWi − T Ai ) = CW ⋅ ( TWi − TW 1 ) = C A ⋅ ( T A1 − T Ai )                                   (4)

q row 2 = ε C min ⋅ ( TW 1 − T A1 ) = CW ⋅ ( TW 1 − TW 2 ) = C A ⋅ ( T A 2 − T A1 )                               (5)

Where CW is the heat capacity on the water side, and CA is the heat capacity on the air side. (One
of these two values will, in application, become Cmin as well.) Using (4) and (5) to eliminate
from (3), there results after much substitution:

                                                       1
2

q circuit = ( TWi   − T Ai ) ⋅  3ε C min − 3ε 2 C min
2
⋅  1 + 1  + ε 3 C min
3
⋅    + 1  
                 (6)
                           C                          C      
                            W CA                       W CA    

As mentioned previously, if we define an overall effectiveness εoverall according to:
q circuit = ε overall ⋅ C min ⋅ ( TWi − T Ai )                                                                    (7)

Then we can use (6) in (7) to find:
2
                                  
ε overall = 3ε − 3ε 2 C min    ⋅  1 + 1  + ε 3 C min
2
⋅ 1 + 1                                            (8)
C   CA                     C      
 W                          W CA 
Recall that ε itself is the “regular” effectiveness determined for one row of the heat exchanger at
a time. Also, note that (6) leads easily to an expression for the ratio of total heat transfer qtotal to
inlet temperature difference TWi - TAi, precisely as we need in order to follow Lytron’s example
for presenting the predicted thermal performance of our heat exchanger. Just remember that qtotal
(the heat transfer rate for the entire heat exchanger) is equal to 3 · qcircuit as derived above.
Finally, it can be shown that (8) guarantees εoverall is less than 3ε, which emphasizes that if we
did not account for the changing inlet temperatures between rows, and instead multiplied the
standard approach by three, then we would over predict the heat transfer rate for the entire heat
exchanger.

Guidelines Addendum - Chiller HEX Thermal Analysis 20083.doc                    Page 2 of 2             John D. Wellin

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