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19-1. Another rating point from the cooling tower catalog from which the data in Example 19-1 are taken specifies a reduction in water temperature from 33 to 27 C when the entering-air enthalpy is 61.6 kJ/kg. The water flow rate is 18.8 kg/s, and the air flow rate is 15.6 kg/s. Using a stepwise integration with 0.5-K increments of change in water temperature, compute hcA/cpm for the tower.
Solution: Eq. 19-4.
pm i a m
c h h
4.19L t c
hA
L = 18.8 kg/s G = 15.6 kg/s t (^) in = 33 C t (^) out = 27 C
Use 12-section, 0.5 K water drop in each section. Eq. 19- dq = gdha = L (4.19 kJ/kg.K)dt kW
Entering air enthalpy = 61.6 kJ/kg
For section 0-1, 27 to 27.5 C
4.19 ( 0.5K)
G
L
h (^) a,1 −ha,0=
ha,0 = 61.6 kJ/kg
4.19 (0.5 K) 2.
ha,1 61.6 =
ha,1 = 64.13 kJ/kg
Average ha = (1/2)(ha,0 + ha,1) = (1/2)(61.6 + 64.13) = 62.86 kJ/kg
Mean water temperature = 27.25 C From Table A-2, Average hi = 86.44 kJ/kg
(hi - ha)m = 86.44 kJ/kg - 62.86 kj/kg = 23.58 kJ/kg
Table
Section Mean Water Av erage ha, Av erage hi, (hi-ha)m 1/(hi-ha)m Temp., C kJ/kg kJ/kg kJ/kg 0-1 27.25 62.86 86.44 23.58 0. 1-2 27.75 65.39 88.78 23.39 0. 2-3 28.25 67.92 91.18 23.26 0. 3-4 28.75 70.45 93.63 23.18 0. 4-5 29.25 72.98 96.13 23.15 0. 5-6 29.75 75.51 98.7 23.19 0. 6-7 30.25 78.04 101.32 23.28 0. 7-8 30.75 80.57 104 23.43 0. 8-9 31.25 83.1 106.74 23.64 0. 9-10 31.75 85.63 109.54 23.91 0. 10-11 32.25 88.16 112.41 24.25 0. 11-12 32.75 90.69 115.35 24.66 0.
h h
i a m
Eq. 19-4.
c
hA
pm
c (^) =
= 20.0 kW/(kJ/kg of enthalpy difference) - - - Ans.
19-2. Solve Prob. 19-1 using a compute program and 0.1-K increments of change of water temperature.
Solution:
Formula: n = 0 to 60 mean water temperature = (1/2)(to + ti)
or = (1/2)(tn + tn+1) - - Eq. 1
Mean air enthalpy ha,1 - ha,0 = (L/G)(4.19)(0.1 K) = (18.0 / 15.6)(4.19)(0.1) = 7.542 / 15.
ha,0 = 61.6 kJ/kg
ha,1 = ha,0 + 7.542/15.
ha = ha,o + 3.771/15.
ha = ha,n + 3.771/16.5 - - Eq. 2
Mean hi Equation 19-
hi = 4.7926 + 2.568t - 0.029834t
2
3
where t = mean water temperature
Water Temp.
C Mean Water Temp. C Mean Air Enthalpy, kJ/kg Mean hi, kJ/kg
- tn tn+ (hi - ha)m 1/(hi - ha)m
- 27 27.1 27.05 61.841731 85.395843 23.554112 0.
- 27.1 27.2 27.15 62.325192 85.857935 23.532742 0.
- 27.2 27.3 27.25 62.808654 86.322144 23.51349 0.
- 27.3 27.4 27.35 63.292115 86.788479 23.496364 0.
- 27.4 27.5 27.45 63.775577 87.256952 23.481375 0.
- 27.5 27.6 27.55 64.259038 87.727571 23.468532 0.
- 27.6 27.7 27.65 64.7425 88.200347 23.457847 0.
- 27.7 27.8 27.75 65.225962 88.675289 23.449328 0.
- 27.8 27.9 27.85 65.709423 89.152408 23.442985 0.
- 27.9 28 27.95 66.192885 89.631714 23.43883 0.
- 28 28.1 28.05 66.676346 90.113217 23.436871 0.
- 28.1 28.2 28.15 67.159808 90.596926 23.437119 0.
- 28.2 28.3 28.25 67.643269 91.082852 23.439583 0.
- 28.3 28.4 28.35 68.126731 91.571005 23.444274 0.
- 28.4 28.5 28.45 68.610192 92.061394 23.451202 0.
- 28.5 28.6 28.55 69.093654 92.554031 23.460377 0.
- 28.6 28.7 28.65 69.577115 93.048923 23.471808 0.
- 28.7 28.8 28.75 70.060577 93.546083 23.485506 0.
- 28.8 28.9 28.85 70.544038 94.045519 23.50148 0.
- 28.9 29 28.95 71.0275 94.547241 23.519741 0.
- 29 29.1 29.05 71.510962 95.051261 23.540299 0.
- 29.1 29.2 29.15 71.994423 95.557587 23.563164 0.
- 29.2 29.3 29.25 72.477885 96.066229 23.588345 0.
- 29.3 29.4 29.35 72.961346 96.577198 23.615852 0.
- 29.4 29.5 29.45 73.444808 97.090504 23.645697 0.
- 29.5 29.6 29.55 73.928269 97.606157 23.677887 0.
- 29.6 29.7 29.65 74.411731 98.124166 23.712435 0.
- 29.7 29.8 29.75 74.895192 98.644541 23.749349 0.
- 29.8 29.9 29.85 75.378654 99.167294 23.78864 0.
- 29.9 30 29.95 75.862115 99.692432 23.830317 0.
- 30 30.1 30.05 76.345577 100.21997 23.874391 0.
- 30.1 30.2 30.15 76.829038 100.74991 23.920871 0.
- 30.2 30.3 30.25 77.3125 101.28227 23.969768 0.
- 30.3 30.4 30.35 77.795962 101.81705 24.021092 0.
- 30.4 30.5 30.45 78.279423 102.35428 24.074852 0.
- 30.5 30.6 30.55 78.762885 102.89394 24.131059 0.
- 30.6 30.7 30.65 79.246346 103.43607 24.189722 0.
- 30.7 30.8 30.75 79.729808 103.98066 24.250852 0.
- 30.8 30.9 30.85 80.213269 104.52773 24.314458 0.
- 30.9 31 30.95 80.696731 105.07728 24.380551 0.
- 31 31.1 31.05 81.180192 105.62933 24.449141 0.
- 31.1 31.2 31.15 81.663654 106.18389 24.520236 0.
- 31.2 31.3 31.25 82.147115 106.74096 24.593849 0.
- 31.3 31.4 31.35 82.630577 107.30056 24.669988 0.
- 31.4 31.5 31.45 83.114038 107.8627 24.748663 0.
- 31.5 31.6 31.55 83.5975 108.42739 24.829885 0.
- 31.6 31.7 31.65 84.080962 108.99463 24.913664 0.
- 31.7 31.8 31.75 84.564423 109.56443 25.000008 0.
- 31.8 31.9 31.85 85.047885 110.13681 25.08893 0.
- 31.9 32 31.95 85.531346 110.71178 25.180438 0.
- 32 32.1 32.05 86.014808 111.28935 25.274542 0.
- 32.1 32.2 32.15 86.498269 111.86952 25.371253 0.
- 32.2 32.3 32.25 86.981731 112.45231 25.47058 0.
- 32.3 32.4 32.35 87.465192 113.03773 25.572534 0.
- 32.4 32.5 32.45 87.948654 113.62578 25.677124 0.
- 32.5 32.6 32.55 88.432115 114.21648 25.78436 0.
- 32.6 32.7 32.65 88.915577 114.80983 25.894253 0.
- 32.7 32.8 32.75 89.399038 115.40585 26.006813 0.
- 32.8 32.9 32.85 89.8825 116.00455 26.122049 0.
- 32.9 33 32.95 90.365962 116.60593 26.239971 0. - SUM = 2.
h h
i a m
Eq. 19-4.
4.19^ (18.8^ )(0.1^ )(2.481^ )
c
hA
pm
c (^) =
= 19.54 kW/(kJ/kg of enthalpy difference) - - - Ans.
19-3. If air enters the cooling tower in Prob. 19-1 with a dry-bulb temperature of 32 C, compute the dry-bulb temperatures as the air passes thorough the tower. For the stepwise calculation choose a change in water temperature of 0.5 K, for which the values of 1/(hi-ha)m starting at the bottom section are, respecitively, 0.04241, 0.04274, 0.04299, 0.04314, 0.04320, 0.04312, 0.04296, 0.04268, 0.04230, 0.04182, 0.04124, and 0.04055.
Solution: ta,0 = 32 C
For section 0-
h h
i a m
Dividing Eq. 19-7 by 2G.
2Gc
h A
pm
c ∆ = =
From Eq. 19-6.
31.36C
t (^) a,1 =
Tabulation:
n section
(h i ha)m
− (^) pm
c 2Gc
h ∆A
ta.n+
0 0-1 0.04241 0.05354 31. 1 1-2 0.04274 0.05395 30. 2 2-3 0.04299 0.05427 30. 3 3-4 0.04314 0.05446 30. 4 4-5 0.04320 0.05453 30. 5 5-6 0.04312 0.05443 30. 6 6-7 0.04296 0.05423 30. 7 7-8 0.04268 0.05388 30. 8 8-9 0.04230 0.05340 30. 9 9-10 0.04182 0.05279 30. 10 10-11 0.04124 0.05206 30. 11 11-12 0.04055 0.05119 30.
ta,12 = 30.94 C - - - Ans.
19-4. A crossflow cooling tower operating with a water flow rate of 45 kg/s and an airflow rate of 40 kg/s has a value of hcA/cpm = 48 kW/(kJ/kg of enthalpy difference). The enthalpy of the entering air is 80 kJ/kg, and the temperature of entering water is 36 C. Develop a computer program to predict the outlet water temperature when the tower is divided into 12 sections, as illustrated in Fig. 19-8.
Solution: Refer to Fig. 19-8.
41.4613 1.7392h t 1 36 i,out
t 1 = 36.8796 - 0.03690hi, out
t 1 = 36.8796 - 0.03690 (4.7926 + 2.568t 1 - 0.029834t 1
2
3 )
0.000061464t 13 - 0.001109t 12 + 1.09476t 1 - 36.7028 = 0
Try t 1 = 32 C
f(t 1 ) = 0.000061464t 13 - 0.001109t 12 + 1.09476t 1 - 36.7028 = -0.7920 < 0.
Try t 1 = 33 C f(t 1 ) = 0.4254 > 0.
Try t 1 = 32.5 C, f(t 1 ) = -0.1845 < 0. Try t 1 = 32.6 C, f(t 1 ) = -0.0628 < 0. Try t 1 = 32.7 C, f(t 1 ) = 0.0591 > 0. Try t 1 = 32.65 C, f(t 1 ) = 0.0000 = 0.
Then t 1 = 32.65 C
For computer program (Spreadsheet)
Table Data:
- Section No.
- Entering Water Temperature
- Entering Air Enthalpy
- Entering Enthalpy of Saturated Air
- Leaving Water Temperature (Trial Value)
- Leaving Air Enthalpy
- Leaving Enthalpy of Saturated Air
- Leaving Water Temperature (Actual Value)
Formula:
Section 1 Entering water temperature = tin
Entering Air Enthalpy = hin
( )
(h Ac ) 2 G
Gh h Ac 2 h h t h h c pm
in c pm i,in i,ou in (^1) ∆ +
q =G (h 1 -hin))
( )
= (^) in c pm
in c pm i,in i,ou in h h Ac 2 G
Gh h Ac 2 h h t h q G
4.19L
q t 1 =tin−
( )
c pm in
in c pm i,in i,out in c pm
1 in h Ac 2 Gh
Gh h Ac 2 h h h
4.19Lh Ac 2 G
G
t t
( (^) i,in i,out in) c pm
c pm 1 in h h h 4.19Lh Ac 2 G
Gh A c 2 t t + − 2 ∆ +
Eq. 19- 3 in
2 h (^) i,in =4.7926+2.568tin−0.029834tin +0.0016657t 3 out
2 h (^) i,out =4.7926+2.568tout−0.029834tout +0.0016657t
Entering Values:
1 in^ (h^ i,in hi,out 2hin)
3
t t + −
1 in^ (h^ i,in hi,out 2hin)
t t + −
Subscript “in” is replaced in any section by subscript of its entering conditions. Subscript “1” is replaced in any section by subscript of its leaving conditions.
Programming by spreadsheet: Note.
- Trial value should equal actual leaving water temperature in the Table by trial and error.
- For sections 1, 2, 3 and 4, tin = 36 C.
- For section 1,5 and 9, hin = 80 kJ/kg.
PROGRAM:
Row 1 A1 = “Section No.” B1 = “Entering Water Temp., C” C1 = “Entering Air Enthalpy, kJ/kg” D1 = “Entering Enthalpy of Saturated Air, kJ/kg” E1 = ”Leaving WaterTemp.,C (Trial)” F1 = “Leaving Air Enthalpy, kJ/kg” G1 = “Leaving Enthalpy of Saturated Air, kJ/lg” H1 = “Leaving Water Temp., C (Actual)” Row 2 A2 = 1 B2 = 36 C2 = 80 D2 = 4.7926 + 2.568B2 - 0.029834B2^2 + 0.0016657B2^ E2 = INPUT (Trial Value) G2 = 4.7926 + 2.568E2 - 0.029834E2^2 + 0.0016657E2^ H2 = B2 - (80/2168.325)(G2 + D2 - 2C2) Row 3 A3 = A2 + 1 B3 = B C3 = F D3 = 4.7926 + 2.568B3 - 0.029834B3^2 + 0.0016657B3^ E3 = INPUT (Trial Value) G3 = 4.7926 + 2.568E3 - 0.029834E3^2 + 0.0016657E3^ H3 = B3 - (80/2168.325)(G3 + D3 - 2C3) Row 4 A4 = A3 + 1 B4 = B
C11 = F
D11 = 4.7926 + 2.568B11 - 0.029834B11^2 + 0.0016657*B11^
E11 = INPUT (Trial Value) G11 = 4.7926 + 2.568E11 - 0.029834E11^2 + 0.0016657E11^ H11 = B11 - (80/2168.325)(G11 + D11 - 2C11) Row 12 A12 = A11 + 1 B12 = H C12 = F D12 = 4.7926 + 2.568B12 - 0.029834B12^2 + 0.0016657B12^ E12 = INPUT (Trial Value) G12 = 4.7926 + 2.568E12 - 0.029834E12^2 + 0.0016657E12^ H12 = B12 - (80/2168.325)(G12 + D12 - 2C12) Row 13 A13 = A12 + 1 B13 = H C13 = F D13 = 4.7926 + 2.568B13 - 0.029834B13^2 + 0.0016657B13^ E13 = INPUT (Trial Value) G13 = 4.7926 + 2.568E13 - 0.029834E13^2 + 0.0016657E13^ H13 = B13 - (80/2168.325)(G13 + D13 - 2C13)
Output: Section No.
Entering Water Temp., C
Entering Air Enthalpy, kJ/kg
Entering Enthalpy of Saturated Air, kJ/kg
Leaving Water Temp., C (Trial)
Laeaving Air Enthalpy, kJ/kg
Leaving Enthalpy of Saturated Air, kJ/kg
Leaving Water Temp., C (Actual) 1 36.0000 80.0000 136.2906 32.6409 91.8755 114.7557 32. 2 36.0000 91.8755 136.2906 33.3574 101.2179 119.0839 33. 3 36.0000 101.2179 136.2906 33.9182 108.5777 122.5695 33. 4 36.0000 108.5777 136.2906 34.3581 114.3823 125.3649 34. 5 32.6409 80.0000 114.7555 30.5199 87.4983 102.7312 30. 6 33.3575 87.4983 119.0843 31.4423 94.2689 107.8193 31. 7 33.9182 94.2689 122.5697 32.2114 100.3031 112.2270 32. 8 34.3582 100.3031 125.3654 32.8534 105.6230 116.0250 32. 9 30.5199 80.0000 102.7312 29.114 84.9703 95.3750 29. 10 31.4424 84.9703 107.8196 30.0389 89.9319 100.1613 30. 11 32.2115 89.9319 112.2277 30.8503 94.7443 104.5294 30. 12 32.8534 94.7443 116.0252 31.5611 99.3131 108.4902 31.
