Question

4.34 Cooling method for gas turbines. Refer to the Journal of Engineering for Gas Turbines and Power (January 2005) study of a high-pressure inlet fogging method for a gas turbine engine, Exercise 4.15 (p. 190). Recall that you fit a first-order model for heat rate (y) as a function of speed (x1), inlet temperature (x2), exhaust temperature (x3), cycle pressure ratio (x4), and air flow rate (x5) to data saved in the GASTURBINE file.

a, Researchers hypothesize that the linear relationship between heat rate (y) and temperature (both inlet and exhaust) depends on air flow rate. Write a model for heat rate that incorporates the researchers’ theories.

b. Use statistical software to fit the interaction model, part a, to the data in the GASTURBINE file. Give the least squares prediction equation.

c. Conduct a test (at α = .05) to determine whether inlet temperature and air flow rate interact to affect heat rate.

d, Conduct a test (at α = .05) to determine whether exhaust temperature and air flow rate interact to affect heat rate.

e. Practically interpret the results of the tests, parts c and d.

Answer 1

(a) The model for heat rate is:

Heart rate = 16,569.8271 - 14.1293*inlet temperature + 20.6848*exhaust temperature

	R²	0.795
	Adjusted R²	0.788
	R	0.891
	Std. Error	733.823
	n	67
	k	2
	Dep. Var.	HeatRate
ANOVA table
Source	SS	df	MS	F	p-value
Regression	13,34,33,406.1993	2	6,67,16,703.0997	123.89	9.87E-23
Residual	3,44,63,802.2484	64	5,38,496.9101
Total	16,78,97,208.4478	66
Regression output				confidence interval
variables	coefficients	std. error	t (df=64)	p-value	95% lower	95% upper
Intercept	16,569.8271
Inlettemp	-14.1293	0.9592	-14.730	3.10E-22	-16.0456	-12.2131
exhtemp	20.6848	2.9865	6.926	2.51E-09	14.7185	26.6510

(b) The model for heat rate is:

Heart rate = 13,696.6672 - 11.7055*inlet temperature + 26.0407*exhaust temperature - 0.0045*inlet temperature*exhaust temperature

	R²	0.795
	Adjusted R²	0.785
	R	0.892
	Std. Error	739.111
	n	67
	k	3
	Dep. Var.	HeatRate
ANOVA table
Source	SS	df	MS	F	p-value
Regression	13,34,81,260.9118	3	4,44,93,753.6373	81.45	1.20E-21
Residual	3,44,15,947.5360	63	5,46,284.8815
Total	16,78,97,208.4478	66
Regression output				confidence interval
variables	coefficients	std. error	t (df=63)	p-value	95% lower	95% upper
Intercept	13,696.6672
Inlettemp	-11.7055	8.2462	-1.419	.1607	-28.1843	4.7733
exhtemp	26.0407	18.3441	1.420	.1607	-10.6172	62.6985
Inlettemp*exhtemp	-0.0045	0.0152	-0.296	.7682	-0.0348	0.0258

(c) The hypothesis being tested is:

H0: β3 = 0

H1: β3 ≠ 0

The p-value from the output is 0.0002.

Since the p-value (0.0002) is less than the significance level (0.05), we can reject the null hypothesis.

Therefore, we can conclude that inlet temperature and air flow rate interact to affect heat rate.

	R²	0.751
	Adjusted R²	0.739
	R	0.866
	Std. Error	815.350
	n	67
	k	3
	Dep. Var.	HeatRate
ANOVA table
Source	SS	df	MS	F	p-value
Regression	12,60,15,126.8500	3	4,20,05,042.2833	63.18	5.64E-19
Residual	4,18,82,081.5977	63	6,64,794.9460
Total	16,78,97,208.4478	66
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