This study investigates how four different operating factors affect the pressure drop in a screen plate bubble column.Pressure drop in a screen plate bubble column plays a critical role in system efficiency and performance. This study examines how four operating variable superficial gas velocity , kinematic velocity ,mesh opening , and a dimensionless gas–fluid ratio influence pressure drop.
Multiple linear regression modeling was used to analyze the relationship between these predictors and the response variable. Both main effects and two-factor interactions were considered to determine whether combined effects of variables contribute significantly to pressure drop.
we have 62 observation in this study:
Y : response variable(pressure drop),that is what we are trying to predict.
x1 : Superficial fluid velocity of the gas
x2 : Kinematic velocity
x3 : Mesh opening
x4 : Dimensionless number relating gas velocity to fluid properties
df <- read.csv(file.choose())
rmarkdown::paged_table(df)we use multiple linear regression model for this :
\[ y= \beta_{0}+\beta_{1}x_{1}+\beta_{2}x_{2}+\beta_{3}x_{3}+\beta_{4}x_{4}+\beta_{12}x_{1}x_{2}+\beta_{13}x_{1}x_{3}+\beta_{14}x_{1}x_{4}+\beta_{23} x_{2}x_{3}+\beta_{24}x_{2}x_{4}+\beta_{34}x_{3}x_{4}+\epsilon \]
When fitting the first-order model with all two-factor interactions, R reported that the interaction terms x1:x3 and x2:x3 had coefficients listed as NA. These terms were removed due to singularities in the model matrix. This occurred because the mesh opening variable x3 is nearly constant for most observations, making the interaction terms involving x3 nearly linear combinations of the main effects. As a result, the model could not estimate these coefficients uniquely.
fullmodel <- lm(y~(x1+x2+x3+x4)^2,data = df)
summary(fullmodel)##
## Call:
## lm(formula = y ~ (x1 + x2 + x3 + x4)^2, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.4804 -3.0766 -0.6635 2.9625 12.2221
##
## Coefficients: (2 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 15.88376 23.17863 0.685 0.49616
## x1 0.18696 0.78447 0.238 0.81255
## x2 0.37921 0.06332 5.989 1.89e-07 ***
## x3 -11.99940 67.31148 -0.178 0.85919
## x4 -8.86442 35.62553 -0.249 0.80446
## x1:x2 0.01155 0.00869 1.329 0.18955
## x1:x3 NA NA NA NA
## x1:x4 -1.11525 1.14847 -0.971 0.33592
## x2:x3 NA NA NA NA
## x2:x4 -0.38547 0.11962 -3.222 0.00218 **
## x3:x4 72.85976 103.15353 0.706 0.48308
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.683 on 53 degrees of freedom
## Multiple R-squared: 0.7496, Adjusted R-squared: 0.7118
## F-statistic: 19.83 on 8 and 53 DF, p-value: 1.947e-13The coefficient of determination (R²) for the model is 0.7496. This suggests that the model explains 74.96% of the total variation in pressure drop observed in the dataset.
The F-statistic for the full regression model was 19.83 with 8 and 53 degree of freedom , we reject null hypothesis indicating that the model is significant.
A partial F-test was used to compare the reduced main effects model to the full first-order interaction model. The test produced F = 3.08 and the p-value of 0.0235 indicating that adding the two factor interaction terms provides a statistically meaningful improvement in explaining pressure drop (y) compared to using main effects alone.
reducedmodel <- lm(y~(x1+x2+x3+x4),data = df)
summary(reducedmodel)##
## Call:
## lm(formula = y ~ (x1 + x2 + x3 + x4), data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.9958 -3.3092 -0.2419 3.3924 10.5668
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.89453 4.32508 1.363 0.17828
## x1 -0.47790 0.34002 -1.406 0.16530
## x2 0.18271 0.01718 10.633 3.78e-15 ***
## x3 35.40284 11.09960 3.190 0.00232 **
## x4 5.84391 2.90978 2.008 0.04935 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.014 on 57 degrees of freedom
## Multiple R-squared: 0.6914, Adjusted R-squared: 0.6697
## F-statistic: 31.92 on 4 and 57 DF, p-value: 5.818e-14 anova(reducedmodel,fullmodel)## Analysis of Variance Table
##
## Model 1: y ~ (x1 + x2 + x3 + x4)
## Model 2: y ~ (x1 + x2 + x3 + x4)^2
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 57 1432.8
## 2 53 1162.4 4 270.37 3.0819 0.02352 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Backward stepwise regression using AIC was performed starting from the full first-order interaction model. The final selected model retained the following terms: x2,x3,x4,x2:x4 . These predictors were justified by improved AIC and statistical significance of the regression coefficients.
fullmodel <- lm(y~(x1+x2+x3+x4)^2,data = df)
stepmodel <- step(fullmodel,direction = "backward")## Start: AIC=199.73
## y ~ (x1 + x2 + x3 + x4)^2
##
##
## Step: AIC=199.73
## y ~ x1 + x2 + x3 + x4 + x1:x2 + x1:x3 + x1:x4 + x2:x4 + x3:x4
##
##
## Step: AIC=199.73
## y ~ x1 + x2 + x3 + x4 + x1:x2 + x1:x4 + x2:x4 + x3:x4
##
## Df Sum of Sq RSS AIC
## - x3:x4 1 10.942 1173.4 198.31
## - x1:x4 1 20.682 1183.1 198.82
## <none> 1162.4 199.73
## - x1:x2 1 38.737 1201.2 199.76
## - x2:x4 1 227.751 1390.2 208.82
##
## Step: AIC=198.31
## y ~ x1 + x2 + x3 + x4 + x1:x2 + x1:x4 + x2:x4
##
## Df Sum of Sq RSS AIC
## - x1:x4 1 19.837 1193.2 197.35
## <none> 1173.4 198.31
## - x1:x2 1 38.709 1212.1 198.32
## - x2:x4 1 228.394 1401.8 207.34
## - x3 1 249.320 1422.7 208.26
##
## Step: AIC=197.35
## y ~ x1 + x2 + x3 + x4 + x1:x2 + x2:x4
##
## Df Sum of Sq RSS AIC
## - x1:x2 1 32.307 1225.5 197.01
## <none> 1193.2 197.35
## - x2:x4 1 220.026 1413.2 205.84
## - x3 1 252.209 1445.4 207.24
##
## Step: AIC=197.01
## y ~ x1 + x2 + x3 + x4 + x2:x4
##
## Df Sum of Sq RSS AIC
## - x1 1 11.262 1236.8 195.57
## <none> 1225.5 197.01
## - x2:x4 1 207.286 1432.8 204.69
## - x3 1 248.430 1473.9 206.45
##
## Step: AIC=195.57
## y ~ x2 + x3 + x4 + x2:x4
##
## Df Sum of Sq RSS AIC
## <none> 1236.8 195.57
## - x3 1 243.60 1480.4 204.72
## - x2:x4 1 245.68 1482.4 204.81summary(stepmodel)##
## Call:
## lm(formula = y ~ x2 + x3 + x4 + x2:x4, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.959 -3.358 -1.131 3.040 11.646
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.52261 4.03964 0.377 0.70763
## x2 0.38056 0.06084 6.255 5.47e-08 ***
## x3 34.51062 10.29961 3.351 0.00144 **
## x4 9.52471 2.96093 3.217 0.00214 **
## x2:x4 -0.30472 0.09056 -3.365 0.00137 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.658 on 57 degrees of freedom
## Multiple R-squared: 0.7336, Adjusted R-squared: 0.7149
## F-statistic: 39.24 on 4 and 57 DF, p-value: 9.297e-16The best-fitting model selected using backward stepwise regression was bestmodel. The model produced an adjusted R squared of 0.7149, indicating that approximately 71.49% of the variability in pressure drop is explained by the included predictors. The estimated coefficient for the interaction x2:x4 was −0.305.
bestmodel <- lm(y~x2+x3+x4+x2:x4,data = df)
summary(bestmodel)##
## Call:
## lm(formula = y ~ x2 + x3 + x4 + x2:x4, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.959 -3.358 -1.131 3.040 11.646
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.52261 4.03964 0.377 0.70763
## x2 0.38056 0.06084 6.255 5.47e-08 ***
## x3 34.51062 10.29961 3.351 0.00144 **
## x4 9.52471 2.96093 3.217 0.00214 **
## x2:x4 -0.30472 0.09056 -3.365 0.00137 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.658 on 57 degrees of freedom
## Multiple R-squared: 0.7336, Adjusted R-squared: 0.7149
## F-statistic: 39.24 on 4 and 57 DF, p-value: 9.297e-16bestmodel <- lm(y~x2+x3+x4+x2:x4,data = df)
predict(bestmodel,data.frame(x2=10,x3=0.5,x4=0.75))## 1
## 27.44168predict(bestmodel,data.frame(x2=10,x3=0.5,x4=0.75),interval = "confidence")## fit lwr upr
## 1 27.44168 24.00618 30.87717predict(bestmodel,data.frame(x2=3,x3=0.25,x4=0.85),interval = "prediction")## fit lwr upr
## 1 18.61092 8.860183 28.36165