Sensible cooling
Not any of the external surface of the cooler coil may be at a temperature less than the dew point of the entering air if dehumidification is to be avoided and sensible cooling achieved. In equation (10.11), for the (/value of a cooler coil, the component that is known with the greatest certainty is the value of the resistance through the water film, Rw, supposing that the flow is turbulent within the tubes (normally the case under design load conditions). The internal surface temperature of the tubes can be calculated using equations (10.11) and
(10.16) if the sensible heat flow rate is known. This may then be regarded as the same as
the external surface temperature of the tubes, for all practical purposes, the thermal resistance of the tube wall being insignificant in this respect.
EXAMPLE 10.3
If the coil in example 10.2 is used to cool 4.75 m3 s1 from 28°C drybulb, 19.5°C wetbulb (sling) to 21°C drybulb, without dehumidification taking place, what is the minimum permissible chilled water flow temperature, if the chilled water flow rate is unaltered?
Answer
Although the coil is the same, the [/value is different because the external surface is not wet with condensate. The new [/value must be determined.
The sensible cooling duty is given by equation (6.6):
<2S = 4.75 x (28 — 21) x 358/(273 + 28) = 39.547 kW
As in example 10.2, by equation (10.12):
1/Aa = Ra = 0.016 77 m2 KW“1 when dry
As before, the fin efficiency is 0.98 and by equation (10.14) the effectiveness of the surface, r, is 0.98.
By equation (10.13) the resistance of the dry fins is:
R{ = —Qjg~8) x 0016 77 = 0000 342 2 m2 KW_1
The resistance of the metal of the tubes’ walls is Rt = 0.000 036 2 m2 KW1
Thus
Rm = 0.000 342 2 + 0.000 036 3 = 0.000 378 5 m2 KW“1
Since the mean velocity of waterflow through the tubes is unchanged, the resistance of the water film, is
R„ = 0.004 358 m2 KW“1
Hence, by equation (10.11):
Ut = 1/(0.016 77 + 0.000 378 5 + 0.004 358)
= 1/0.021 51 = 46.5 Wm"2 K1
If the inner surface of the tubes must not be at a temperature less than the dew point (td) of the entering airstream, then the minimum chilled water temperature, fwmin, can be determined. Denoting the overall thermal resistance of the cooler coil by R and the leaving air temperature by t2, we have
R 
•W 
Whence
(10.17) 
(Rtd — Rwt2) (R — *w)
See Figure 10.6.
13.4°C t, t3 21 °C 28°C = 14.9°C Fig. 10.6 The psychrometry for example 10.3. 
The dew point of the airstream is 14.9°C and hence, for a sixrow coil,
0.02151 x 14.90.004 358 9 x 21 Km in — 0.02151 — 0.004 358 “
This is when the coil has six rows. However, the value of the minimum chilled water flow temperature will alter if the number of rows is changed, because so doing influences the effectiveness of the fins and hence changes the overall {/value, which is the reciprocal of R.
It does not necessarily follow that this chilled water flow temperature will give the sensible cooling duty. The duty obtained depends on the number of rows of the coil and, in turn, this has an effect on the fin efficiency and the effectiveness of the fins, which influence the thermal resistance of the fins, Rf. Since the ratio of the external to internal surface areas is unchanged the resistance of the tubes is not affected (see equation (10.15)).
EXAMPLE 10.4
Determine the number of rows required to achieve the sensible cooling duty for the cooler air forming the subject of example 10.3.
Answer
Successive assumptions are made for the number of rows and the corresponding [/values, minimum permissible chilled water flow temperatures and sensible cooling duties are determined.
The duty is 39.547 kW and, from example 10.2, the chilled water flow rate is 5.454 kg s’1, hence the chilled water temperature rise is 39.457/(4.19 x 5.454) = 1.73 K, for the specified duty. The leaving chilled water temperature is thus? wmin + 1.73. Hence airtowater logarithmic mean temperature differences can be calculated for the various numbers of rows assumed, using equation (10.10). (There is a small error in doing this because, for the various duties determined, the chilled water temperature rise will not necessarily be 1.73 K.)
The calculations lead to the following tabulations:
Rows 
^fi 
4> 
<)i4fl 
At 
N 

6 
224.3 
0.98 
219.8 
11.82 
236.1 
0.98 
5 
186.9 
0.96 
179.4 
9.85 
197.8 
0.96 
4 
149.5 
0.92 
137.5 
7.88 
157.4 
0.92 
3 
112.15 
0.86 
96.4 
5.91 
118.1 
0.87 
2 
74.8 
0.77 
57.6 
3.94 
78.74 
0.78 
Interpolations have been made in Table 10.2 to obtain values of <j>, for odd numbers of rows.
Equation (10.14) is then used to determine r.
The resistance of the airside film, when the coil is dry, has already been established as
0. 016 77 m2 KW1, the resistance of the tubes has been determined as 0.000 036 2 m2 KW1 and the resistance of the water film has been calculated as 0.004 358 m2 KW1. Hence the sum of these, Ra + Rt + Rw, is 0.021 164 2 m2 KW1. The resistance of the fins is found from equation (10.13) and the total resistance and [/value calculated. The following tabulation results.
Rows 
6 
5 
4 
3 
2 
Rf 
0.000 342 2 
0.000 698 8 
0.001 458 3 
0.002 505 9 
0.004 730 0 
R 
0.021 506 4 
0.021 863 0 
0.022 622 5 
0.023 670 1 
0.025 894 2 
Ut 
46.5 
45.74 
44.20 
42.25 
38.62 
The dew point is 14.9°C, the leaving air temperature is 21°C and the values of Rw and

Although the temperature rise of the chilled water is only 1.73 K for the design duty, it is reasonable to assume this is also the rise for all the duties since the calculations are aimed at deciding the number of rows to achieve the design duty. The leaving chilled water temperature, /wb is then rwmin + 1.73°C. The total external surface area, At, is proportional to the number of rows and the (LMTD)av/ can be established by equation (10.10). The sensible cooling duty, Qs, is then calculated using equation (10.9), as the following tabulation shows.
Rows 
6 
5 
4 
3 
2 
^wrnin 
13.35 
13.39 
13.45 
13.53 
13.66 
^wb 
15.08 
15.12 
15.18 
15.26 
15.39 
(28 — ?wb) 
12.92 
12.88 
12.82 
12.74 
12.61 
^2 — ^wmin 
7.65 
7.61 
7.55 
7.47 
7.34 
(.LMTD)aw 
10.06 
10.01 
9.95 
9.87 
9.74 
A, 
236.1 
197.8 
157.4 
118.1 
78.7 
Ut 
46.50 
45.74 
44.20 
42.25 
38.62 
(2S (kW) 
110.4 
90.6 
69.2 
49.2 
29.6 
The required duty is 39.457 kW. Hence three rows of tubes are required.
The uncertainties, mentioned earlier, in the analysis of cooler coil performance imply that a small margin is prudent in the value of fwmin. It is suggested that half a degree be added to the values calculated.
Alternatively, more involved procedures from ASHRAE (1996) and Kays and London (1985) using transfer functions are possible.
Posted in Engineering Fifth Edition