The article \"Adiabatic Humidification of Air with Water in a Packed Tower\" ( C

Question

The article "Adiabatic Humidification of Air with Water in a Packed Tower" (Chem. Eng. Prog., 1952: 362-370) reports data on gas film heat transfer coefficient (Btu/hr ft2 on °F) as a function of gas rate (factor A) and liquid rate (factor B).

(a) After constructing an ANOVA table, test at level .01 both the hypothesis of no gas-rate effect against the appropriate alternative and the hypothesis of no liquid-rate effect against the appropriate alternative. (Give answers accurate to 1 decimal place.)

critical F-value = 6.991917222   (for gas-rate effect)

Conclusion

Reject the null hypothesis, there is a gas-rate effect.

(b) Use Tukey's procedure to investigate differences in expected heat transfer coefficient due to different gas rates. (Give answers accurate to 2 decimal places.)
w =  

Select the gas rates (factor A levels) that differ significantly. (Use = 0.01.)

1,2

1,3

1,4

2,3

2,4

3,4

(c) Repeat part (b) for liquid rates. (Give answers accurate to 2 decimal places.)

Select the liquid rates (factor B levels) that differ significantly. (Use = 0.01.)

1,2

1,3

1,4

2,3

2,4

3,4

Source Df SS MS F A 3 324082.1875 108027.3958 105.3119563 B 3 39934.1875 13311.39583 12.97679283 Error 9 9232.0625 1025.784722 Total 15 373248.4375

Explanation / Answer

Data frame created in R :

data A B
[1,] 200 1 1
[2,] 226 1 2
[3,] 240 1 3
[4,] 261 1 4
[5,] 278 2 1
[6,] 312 2 2
[7,] 330 2 3
[8,] 381 2 4
[9,] 369 3 1
[10,] 416 3 2
[11,] 462 3 3
[12,] 517 3 4
[13,] 500 4 1
[14,] 575 4 2
[15,] 645 4 3
[16,] 733 4 4

> model=aov(data~A+B)
> summary(model)
Df Sum Sq Mean Sq F value Pr(>F)   
A 3 324082 108027 105.31 2.5e-07 ***
B 3 39934 13311 12.98 0.00128 **
Residuals 9 9232 1026   
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Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> TukeyHSD(model)
Tukey multiple comparisons of means
95% family-wise confidence level

Fit: aov(formula = data ~ A + B)

$A
diff lwr upr p adj
2-1 93.50 22.80023 164.1998 0.0112706
3-1 209.25 138.55023 279.9498 0.0000332
4-1 381.50 310.80023 452.1998 0.0000002
3-2 115.75 45.05023 186.4498 0.0029047
4-2 288.00 217.30023 358.6998 0.0000023
4-3 172.25 101.55023 242.9498 0.0001577

b) Among the pair of factor A levels, all the pairs except pair (1,2) are significantly different, since p-values < 0.01.

$B
diff lwr upr p adj
2-1 45.50 -25.19977 116.1998 0.2535346
3-1 82.50 11.80023 153.1998 0.0229218
4-1 136.25 65.55023 206.9498 0.0009269
3-2 37.00 -33.69977 107.6998 0.4084725
4-2 90.75 20.05023 161.4498 0.0134309
4-3 53.75 -16.94977 124.4498 0.1521835

c) Among all the pairs of levels of factor B, only the pair (1,4) differ significantly, since p-value < 0.01.

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