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Extracting water from oil. In the oil industry, water that
Chapter 12, Problem 20E(choose chapter or problem)
Extracting water from oil. In the oil industry, water that mixes with crude oil during production and transportation must be removed. Chemists have found that the oil can be extracted from the water/oil mix electrically. Researchers at the University of Bergen (Norway) conducted a series of experiments to study the factors that influence the voltage (y) required to separate the water from the oil (Journal of Colloid and Interface Science, Aug. 1995). The seven independent variables investigated in the study are listed in the table below. (Each variable was measured at two levels—a “low” level and a “high” level.) Sixteen water/oil mixtures were prepared using different combinations of the independent variables; then each emulsion was exposed to a high electric field. In addition, three mixtures were tested when all independent variables were set to 0. The data for all 19 experiments are saved in the file (selected observations are shown in the table at the bottom of the page).
a. Propose a first-order model for y as a function of all seven independent variables.
b. Use a statistical software package to fit the model to the data in the table.
c. Fully interpret the \(\beta\) estimates.
d. Evaluate the overall utility of the model at \(\alpha=.10\).
Questions & Answers
QUESTION:
Extracting water from oil. In the oil industry, water that mixes with crude oil during production and transportation must be removed. Chemists have found that the oil can be extracted from the water/oil mix electrically. Researchers at the University of Bergen (Norway) conducted a series of experiments to study the factors that influence the voltage (y) required to separate the water from the oil (Journal of Colloid and Interface Science, Aug. 1995). The seven independent variables investigated in the study are listed in the table below. (Each variable was measured at two levels—a “low” level and a “high” level.) Sixteen water/oil mixtures were prepared using different combinations of the independent variables; then each emulsion was exposed to a high electric field. In addition, three mixtures were tested when all independent variables were set to 0. The data for all 19 experiments are saved in the file (selected observations are shown in the table at the bottom of the page).
a. Propose a first-order model for y as a function of all seven independent variables.
b. Use a statistical software package to fit the model to the data in the table.
c. Fully interpret the \(\beta\) estimates.
d. Evaluate the overall utility of the model at \(\alpha=.10\).
ANSWER:Step 1 of 6
a) A first-order model for mean voltage (y) as a function of disperse phase
volume (\(x_{1}\)), salinity (\(x_{2}\)) , temperature (\(x_{3}\)) , time delay (\(x_{4}\)) , surfactant
concentration (\(x_{5}\)), span: triton (\(x_{6}\)) and solid particles (\(x_{7}\)) is
\(E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{3}+\beta_{4} x_{4}+\beta_{5} x_{5}+\beta_{6} x_{6}+\beta_{7} x_{7}\)
Where \(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}\) are all quantitative variables that are not functions of other independent variables and \(\beta_{2}(i=1,2,3,4,5,6,7)\) represent the slope of the line relating y to \(x_{i}\) when all the other x’s are held fixed.