# An Improvement in Biodiesel Production from Waste Cooking Oil by Applying Thought Multi-Response Surface Methodology Using Desirability Functions

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{3}) [8]. The transesterification process is influenced by several process variables. The type and dosage of catalyst, process temperature, agitation speed, agitation time, water content and impurities, are the main variables that affect biodiesel yield [9,10]. Thus, the amount and type of products that are formed during frying affect either the biodiesel properties or the transesterification reaction. For example, the water in waste cooking oil affects the methyl ester yield by favoring a saponification reaction [11,12,13]. To remove the undesirable compounds in waste cooking oil, pretreatment is necessary prior to transesterification.

## 2. Materials and Methods

#### 2.1. Materials

#### 2.2. Response Surface Method for Optimizing Biodiesel

_{1}, X

_{2}, X

_{3}…, X

_{k}are the input vectors and e is an error. The quadratic model (second-order) is a commonly used polynomial function. It is seen in Equation (2):

_{0}, b

_{i}, b

_{ii}and b

_{ij}are determined by regression analysis, although these functions do not always provide good results for complex problems that involve many nonlinearities and inputs. The reason is that they cannot be adjusted when the data are sparse as they are continuous functions that are defined by polynomials. The p-value (or Prob. > F) is defined as the probability of obtaining a result that is equal to or greater than what was actually observed. This assumes that the model is accurate. The p-value can be computed by analyzing the variance (ANOVA). If it exceeds the model’s F and the model has no term that exceeds the significance level (e.g., α = 0.05), the model will suffice within a confidence interval of (1–α). Some researchers have used ANOVA to determine the effect of the process variables or inputs on the process outputs [19,20]. If there is more than one output for a problem, the latter is described as MRS. It causes and suggests that outputs are in conflict, because the optimal configuration may differ greatly from one output to output. Harrington [15] suggested a compromise. It provides desirability functions for each output, Equations (3) and (4), as well as an overall desirability, namely the geometric mean of the D (desirability) of each output (Equation (5)):

_{r}is the model for prediction. It is desirable to use a second higher degree polynomial to optimize responses [21]. In the desirability approach, each estimated response is transformed into a unit-less utility that is bounded by 0 < d

_{r}< 1. A higher value of d

_{r}indicates a more desirable response value. The optimization aspect of the R package v.1.6 [22] looks for a combination of importance factors (or weights from 1 to 3) that satisfies the process criteria of each response and input.

## 3. Experiments Design

## 4. Using Response Surface Methodology to Optimize Biodiesel Variables

_{2}catalyst.

## 5. Results

#### 5.1. Experimental Results

#### 5.2. Analysis of Variance

^{2}+ 13.61993·M·C − 53.42693·C

^{2}+ 7.60976·T − 0.12404·T

^{2}− 29.99115·H + 10.00732·C·H + 4.98413·H

^{2}

^{2}− 9.20669·M·C + 15.08333·C

^{2}+ 2.59409·t − 0.04323·t

^{2}− 0.01376·M·S + 0.07336·C·S − 0.70959·C·T − 46.81611·H + 4.85389·M·H − 74.14871·I + 4.99833·M·I + 0.59234·T·I + 5.46056·H·I + 2.56599·I

^{2}

^{2}− 0.00537·t + 0.00055·M·t – 1 × 10

^{-5}·M·S + 1 × 10

^{-5}·t·S + 0.00028·M·T − 0.00037·C·T − 0.00016·t·T − 0.04724·H + 0.00306·M·H + 3 × 10

^{-5}·S·H + 0.00078·t·I – 3 × 10

^{-5}·S·I

^{2}+ 0.3601·M·H − 1.32117·C·H + 0.08747·t·H − 0.00473·S·H + 0.25198·H

^{2}+ 0.26241·M·I − 0.06006·T·I

^{2}− 0.003·M·t − 0.00875·M·T − 6 × 10

^{-5}·S·T + 0.00217·T

^{2}+ 0.77049·H − 0.33924·C·H + 0.01583·t·H − 0.00122·S·H + 0.09299·H

^{2}+ 0.06526·M·I − 0.01564·T·I

^{2}) is a measure of the variation about the mean of the values provided by the regression model. All values of R

^{2}in the following results are close to 1. This indicates that these models have good predictive capability.

_{k}

_{Experiment}are the experimentally-obtained responses, and Y

_{k}

_{Model}are responses from the quadratic models that RSM and m experiments produced. Prediction errors are shown in Table 9. The maximum error corresponds to η (MAE equal to 10.445 and RMSE equal to 12.803), and the minimum error corresponds to ρ (MAE equal to 0.009 and RMSE equal to 0.012).

#### 5.3. Multi-Response Optimization

_{k,norm}are the normalized outputs of the models that were developed with RSM and of those that were obtained experimentally. The error that appears in the last two columns represents the MAE and RMSE that were normalized for each variable in each of the nine biodiesel optimization scenarios that were studied. However, the normalized MAE and RMSE in the last two rows correspond to the errors in each of the outputs that were studied. For example, when minimize the turbidity is considered to be an optimization variable for biodiesel production, the errors obtained are the smallest (MAE = 0.04 and RMSE = 0.02), but when minimize the viscosity and density are considered, the error is the largest (MAE = 0.12 and RMSE = 0.08). In contrast, the maximum errors obtained for each of the outputs are lower when predicting viscosity (MAE = 0 and RMSE = 0.01) and greater when predicting turbidity (MAE = 0.03 and RMSE = 0.09). In addition, the values of MAE and RMSE that were obtained for each of the different outputs are all in acceptable agreement according to the texting error.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Scatter diagram of: (

**a**) Yield (η); (

**b**) turbidity (Turb); (

**c**) density (ρ); (

**d**) viscosity (µ); and (

**e**) high heating value (HHV).

**Table 1.**Independent variables and experimental design levels used with the Box–Behnken design (BBD) method.

Inputs | Notation | Magnitude | Levels | ||
---|---|---|---|---|---|

−1 | 0 | +1 | |||

Ratio oil | M | 6/1 | 7.5/1 | 9/1 | |

Catalyst | C | wt. % | 1 | 1.5 | 2 |

Time | t | min | 20 | 30 | 40 |

Speed | S | rpm | 500 | 750 | 1000 |

Temp | T | °C | 20 | 30 | 40 |

Humidity | H | wt. % | 0 | 1.5 | 3 |

Impurity | I | wt. % | 0 | 1.5 | 3 |

Sample | Inputs | ||||||
---|---|---|---|---|---|---|---|

Molar Ratio | Catalyst (wt. %) | Time (min) | Speed (rpm) | Temp (°C) | Humidity (wt. %) | Impurity (wt. %) | |

25 | 6 | 1 | 30 | 500 | 30 | 1.5 | 1.5 |

29 | 6 | 1 | 30 | 1000 | 30 | 1.5 | 1.5 |

27 | 6 | 2 | 30 | 500 | 30 | 1.5 | 1.5 |

31 | 6 | 2 | 30 | 1000 | 30 | 1.5 | 1.5 |

41 | 6 | 1.5 | 20 | 750 | 20 | 1.5 | 1.5 |

45 | 6 | 1.5 | 20 | 750 | 40 | 1.5 | 1.5 |

9 | 6 | 1.5 | 30 | 750 | 30 | 0 | 0 |

13 | 6 | 1.5 | 30 | 750 | 30 | 0 | 3 |

11 | 6 | 1.5 | 30 | 750 | 30 | 3 | 0 |

15 | 6 | 1.5 | 30 | 750 | 30 | 3 | 3 |

43 | 6 | 1.5 | 40 | 750 | 20 | 1.5 | 1.5 |

47 | 6 | 1.5 | 40 | 750 | 40 | 1.5 | 1.5 |

26 | 9 | 1 | 30 | 500 | 30 | 1.5 | 1.5 |

30 | 9 | 1 | 30 | 1000 | 30 | 1.5 | 1.5 |

28 | 9 | 2 | 30 | 500 | 30 | 1.5 | 1.5 |

32 | 9 | 2 | 30 | 1000 | 30 | 1.5 | 1.5 |

42 | 9 | 1.5 | 20 | 750 | 20 | 1.5 | 1.5 |

46 | 9 | 1.5 | 20 | 750 | 40 | 1.5 | 1.5 |

10 | 9 | 1.5 | 30 | 750 | 30 | 0 | 0 |

14 | 9 | 1.5 | 30 | 750 | 30 | 0 | 3 |

12 | 9 | 1.5 | 30 | 750 | 30 | 3 | 0 |

16 | 9 | 1.5 | 30 | 750 | 30 | 3 | 3 |

44 | 9 | 1.5 | 40 | 750 | 20 | 1.5 | 1.5 |

48 | 9 | 1.5 | 40 | 750 | 40 | 1.5 | 1.5 |

49 | 7.5 | 1 | 20 | 750 | 30 | 0 | 1.5 |

53 | 7.5 | 1 | 20 | 750 | 30 | 3 | 1.5 |

17 | 7.5 | 1 | 30 | 750 | 20 | 1.5 | 0 |

21 | 7.5 | 1 | 30 | 750 | 20 | 1.5 | 3 |

19 | 7.5 | 1 | 30 | 750 | 40 | 1.5 | 0 |

23 | 7.5 | 1 | 30 | 750 | 40 | 1.5 | 3 |

51 | 7.5 | 1 | 40 | 750 | 30 | 0 | 1.5 |

55 | 7.5 | 1 | 40 | 750 | 30 | 3 | 1.5 |

50 | 7.5 | 2 | 20 | 750 | 30 | 0 | 1.5 |

54 | 7.5 | 2 | 20 | 750 | 30 | 3 | 1.5 |

18 | 7.5 | 2 | 30 | 750 | 20 | 1.5 | 0 |

22 | 7.5 | 2 | 30 | 750 | 20 | 1.5 | 3 |

20 | 7.5 | 2 | 30 | 750 | 40 | 1.5 | 0 |

24 | 7.5 | 2 | 30 | 750 | 40 | 1.5 | 3 |

52 | 7.5 | 2 | 40 | 750 | 30 | 0 | 1.5 |

56 | 7.5 | 2 | 40 | 750 | 30 | 3 | 1.5 |

33 | 7.5 | 1.5 | 20 | 500 | 30 | 1.5 | 0 |

37 | 7.5 | 1.5 | 20 | 500 | 30 | 1.5 | 3 |

35 | 7.5 | 1.5 | 20 | 1000 | 30 | 1.5 | 0 |

39 | 7.5 | 1.5 | 20 | 1000 | 30 | 1.5 | 3 |

1 | 7.5 | 1.5 | 30 | 500 | 20 | 0 | 1.5 |

5 | 7.5 | 1.5 | 30 | 500 | 20 | 3 | 1.5 |

3 | 7.5 | 1.5 | 30 | 500 | 40 | 0 | 1.5 |

7 | 7.5 | 1.5 | 30 | 500 | 40 | 3 | 1.5 |

2 | 7.5 | 1.5 | 30 | 1000 | 20 | 0 | 1.5 |

6 | 7.5 | 1.5 | 30 | 1000 | 20 | 3 | 1.5 |

4 | 7.5 | 1.5 | 30 | 1000 | 40 | 0 | 1.5 |

8 | 7.5 | 1.5 | 30 | 1000 | 40 | 3 | 1.5 |

34 | 7.5 | 1.5 | 40 | 500 | 30 | 1.5 | 0 |

38 | 7.5 | 1.5 | 40 | 500 | 30 | 1.5 | 3 |

36 | 7.5 | 1.5 | 40 | 1000 | 30 | 1.5 | 0 |

40 | 7.5 | 1.5 | 40 | 1000 | 30 | 1.5 | 3 |

Sample | Outputs | ||||
---|---|---|---|---|---|

η | Turb (NTU) | ρ (g/mL) | µ (mm^{2}/s) | HHV (MJ/Kg) | |

25 | 93 | 0.65 | 0.83 | 7.03 | 42.70 |

29 | 93 | 1.05 | 0.85 | 5.15 | 41.83 |

27 | 40 | 1.78 | 0.822 | 5.65 | 42.06 |

31 | 9.5 | 58.5 | 0.83 | 5.05 | 41.78 |

41 | 29 | 8.48 | 0.84 | 5.40 | 41.94 |

45 | 29 | 0.21 | 0.85 | 5.93 | 42.19 |

9 | 87 | 86 | 0.83 | 6.00 | 42.22 |

13 | 76 | 1 | 0.86 | 5.40 | 41.94 |

11 | 75 | 1.89 | 0.85 | 8.05 | 43.17 |

15 | 77 | 1.18 | 0.83 | 5.39 | 41.94 |

43 | 31 | 2.44 | 0.85 | 4.67 | 41.61 |

47 | 57 | 2.94 | 0.79 | 7.40 | 42.87 |

26 | 88 | 6 | 0.83 | 9.83 | 43.99 |

30 | 55 | 3.52 | 0.80 | 6.07 | 42.26 |

28 | 50 | 3.81 | 0.79 | 9.24 | 43.72 |

32 | 23 | 1.66 | 0.79 | 6.38 | 42.40 |

42 | 59 | 1.09 | 0.81 | 6.16 | 42.30 |

46 | 91 | 2.45 | 0.83 | 7.97 | 43.13 |

10 | 83 | 6.29 | 0.82 | 5.67 | 42.07 |

14 | 90 | 1.42 | 0.80 | 5.65 | 42.06 |

12 | 90 | 1.01 | 0.82 | 7.81 | 43.06 |

16 | 71 | 10.14 | 0.85 | 10.15 | 44.14 |

44 | 85 | 0.67 | 0.84 | 6.15 | 42.29 |

48 | 82 | 1.01 | 0.81 | 6.17 | 42.30 |

49 | 93 | 1.47 | 0.80 | 8.96 | 43.59 |

53 | 89 | 1.62 | 0.82 | 8.20 | 43.24 |

17 | 92 | 1.06 | 0.81 | 6.47 | 42.44 |

21 | 95 | 0.69 | 0.82 | 9.40 | 43.79 |

19 | 91 | 1.53 | 0.83 | 9.43 | 43.81 |

23 | 86 | 1.61 | 0.81 | 7.24 | 42.80 |

51 | 94 | 0.22 | 0.84 | 6.13 | 42.28 |

55 | 89 | 0.42 | 0.83 | 11.99 | 44.99 |

50 | 63 | 0.34 | 0.82 | 0 | 39.45 |

54 | 87 | 0.36 | 0.81 | 4.48 | 41.52 |

18 | 45.5 | 53 | 0.82 | 5.97 | 42.21 |

22 | 23 | 1.09 | 0.82 | 5.32 | 41.91 |

20 | 24.6 | 0.64 | 0.80 | 6.07 | 42.25 |

24 | 22 | 18.03 | 0.81 | 4.67 | 41.61 |

52 | 66 | 0.31 | 0.81 | 4.08 | 41.33 |

56 | 90 | 1.01 | 0.84 | 5.77 | 42.12 |

33 | 93 | 1.18 | 0.82 | 5.65 | 42.06 |

37 | 83 | 1.34 | 0.84 | 8.22 | 43.25 |

35 | 100 | 0.56 | 0.81 | 6.10 | 42.27 |

39 | 93 | 0.72 | 0.73 | 6.36 | 42.39 |

1 | 98 | 0.48 | 0.84 | 3.78 | 41.20 |

5 | 75 | 1.85 | 0.79 | 11.98 | 44.99 |

3 | 86 | 2 | 0.81 | 4.82 | 41.68 |

7 | 86 | 1.59 | 0.77 | 7.15 | 42.75 |

2 | 83 | 2.71 | 0.79 | 12.14 | 45.06 |

6 | 82 | 0.9 | 0.82 | 8.16 | 43.22 |

4 | 88 | 0.63 | 0.83 | 6.03 | 42.23 |

8 | 95 | 0.94 | 0.81 | 6.35 | 42.38 |

34 | 78 | 0.74 | 0.81 | 5.83 | 42.14 |

38 | 77 | 7.42 | 0.82 | 8.59 | 43.42 |

36 | 93 | 0.78 | 0.83 | 5.54 | 42.01 |

40 | 85 | 1.96 | 0.82 | 7.81 | 43.06 |

Variable | Degrees of freedom | Sum of Square | Mean Square | F Value | p-Value | Significance Code |
---|---|---|---|---|---|---|

M | 1 | 1211.3 | 1211.3 | 6.0928 | 0.017267 | * |

M^{2} | 1 | 2992.5 | 2992.5 | 15.0525 | 0.000325 | *** |

M × C | 1 | 10,324.9 | 10,324.9 | 51.9357 | 3.98 × 10^{-9} | *** |

C^{2} | 1 | 4004.9 | 4004.9 | 20.1451 | 4.63 × 10^{-5} | *** |

T | 1 | 67 | 67 | 0.337 | 0.564327 | |

T^{2} | 1 | 2819 | 2819 | 14.1802 | 0.000462 | *** |

H | 1 | 0 | 0 | 0.0002 | 0.988511 | |

C × H | 1 | 498.1 | 498.1 | 2.5057 | 0.120145 | |

H^{2} | 1 | 1610.2 | 1610.2 | 8.0996 | 0.006541 | * |

Residuals | 47 | 9343.7 | 198.8 | |||

R^{2} | 0.846 | - | - | - | - |

Variable | Degrees of freedom | Sum of Square | Mean Square | F Value | p-Value | Significance Codes |
---|---|---|---|---|---|---|

M | 1 | 672.6 | 672.57 | 6.8629 | 1.24 × 10^{-2} | * |

M^{2} | 1 | 379.6 | 379.58 | 3.8733 | 5.60 × 10^{-2} | . |

M·C | 1 | 481.1 | 481.06 | 4.9088 | 3.25 × 10^{-2} | * |

C^{2} | 1 | 601.6 | 601.62 | 6.1389 | 1.75 × 10^{-2} | * |

t | 1 | 0 | 0 | 0 | 9.98 × 10^{-1} | |

t^{2} | 1 | 366.4 | 366.44 | 3.7392 | 6.02 × 10^{-2} | . |

M·S | 1 | 44.1 | 44.08 | 0.4497 | 5.06 × 10^{-1} | |

C·S | 1 | 1011.1 | 1011.13 | 10.3175 | 2.60 × 10^{-3} | ** |

T | 1 | 108.6 | 108.56 | 1.1077 | 2.99 × 10^{-1} | |

C·T | 1 | 266.4 | 266.4 | 2.7183 | 1.07 × 10^{-1} | |

H | 1 | 954.2 | 954.19 | 9.7365 | 3.35 × 10^{-1} | ** |

M·H | 1 | 486.7 | 486.72 | 4.9665 | 3.15 × 10^{-2} | * |

I | 1 | 1011.8 | 1011.83 | 10.3246 | 2.60 × 10^{-3} | ** |

M·I | 1 | 691.1 | 691.14 | 7.0524 | 1.13 × 10^{-2} | * |

T·I | 1 | 1207.6 | 1207.62 | 12.3225 | 1.12 × 10^{-3} | ** |

H·I | 1 | 425.6 | 425.63 | 4.3431 | 4.36 × 10^{-2} | * |

I^{2} | 1 | 672.6 | 672.57 | 6.8629 | 1.24 × 10^{-2} | * |

Residuals | 1 | 379.6 | 379.58 | 3.8733 | 5.60 × 10^{-2} | . |

R^{2} | 0.8304 | - | - | - | - | - |

Variable | Degrees of freedom | Sum of Square | Mean Square | F Value | p-Value | Significance Codes |
---|---|---|---|---|---|---|

M | 1 | 0.002408 | 0.002408 | 11.0961 | 1.87 × 10^{-3} | ** |

M^{2} | 1 | 0.002043 | 0.002043 | 9.4134 | 3.86 × 10^{-3} | ** |

t | 1 | 0.000626 | 0.000625 | 2.8818 | 9.74 × 10^{-2} | . |

M·t | 1 | 0.000535 | 0.000535 | 2.4668 | 1.24 × 10^{-1} | |

M·S | 1 | 0.000229 | 0.000229 | 1.0563 | 3.10 × 10^{-1} | |

t·S | 1 | 0.002631 | 0.002631 | 12.1215 | 1.22 × 10^{-3} | ** |

S^{2} | 1 | 0.001614 | 0.001614 | 7.4355 | 9.45 × 10^{-3} | ** |

M·T | 1 | 0.000399 | 0.000399 | 1.8386 | 1.83 × 10^{-1} | |

C·T | 1 | 0.000831 | 0.000831 | 3.8272 | 5.74 × 10^{-2} | . |

t·T | 1 | 0.002083 | 0.002083 | 9.5978 | 3.56 × 10^{-3} | ** |

S·T | 1 | 0.001085 | 0.001085 | 4.9986 | 3.10 × 10^{-2} | * |

H | 1 | 1.6E-06 | 1.62E-06 | 0.0075 | 9.32 × 10^{-1} | |

M·H | 1 | 0.000379 | 0.000379 | 1.7466 | 1.94 × 10^{-1} | |

S·H | 1 | 0.001178 | 0.001178 | 5.4282 | 2.49 × 10^{-2} | * |

t·I | 1 | 1.63E-05 | 1.63E-05 | 0.075 | 7.86 × 10^{-1} | |

S·I | 1 | 0.002431 | 0.002431 | 11.203 | 1.79 × 10^{-3} | ** |

Residuals | 40 | 0.008681 | 0.000217 | |||

R^{2} | 0.8249 | - | - | - | - | - |

Variable | Degrees of freedom | Sum of Square | Mean Square | F Value | p-Value | Significance Codes |
---|---|---|---|---|---|---|

M·t | 1 | 2.524 | 2.524 | 1.3462 | 2.52 × 10^{-1} | |

S^{2} | 1 | 1.383 | 1.383 | 0.7375 | 3.95 × 10^{-1} | |

T^{2} | 1 | 1.613 | 1.613 | 0.8601 | 3.59 × 10^{-1} | |

M·H | 1 | 21.914 | 21.914 | 11.6874 | 1.33 × 10^{-3} | ** |

C·H | 1 | 37.86 | 37.86 | 20.1916 | 4.70 × 10^{-5} | *** |

T·H | 1 | 5.704 | 5.704 | 3.0419 | 8.78 × 10^{-2} | . |

S·H | 1 | 32.831 | 32.831 | 17.5093 | 1.27 × 10^{-4} | *** |

H^{2} | 1 | 3.889 | 3.889 | 2.074 | 1.57 × 10^{-1} | |

M·I | 1 | 2.316 | 2.316 | 1.2353 | 2.72 × 10^{-1} | |

T·I | 1 | 10.54 | 10.54 | 5.6212 | 2.20 × 10^{-2} | * |

Residuals | 46 | 86.252 | 1.875 | |||

R^{2} | 0.7635 | - | - | - | - | - |

Variable | Degrees of freedom | Sum of Square | Mean Square | F Value | p-Value | Significance Codes |
---|---|---|---|---|---|---|

M | 1 | 1.3244 | 1.3244 | 10.9006 | 2.00 × 10^{-3} | ** |

C | 1 | 1.912 | 1.91204 | 15.7373 | 2.86 × 10^{-4} | *** |

C^{2} | 1 | 0.0526 | 0.05259 | 0.4329 | 5.14 × 10^{-1} | |

M·t | 1 | 0.002 | 0.00199 | 0.0164 | 8.99 × 10^{-1} | |

S^{2} | 1 | 0.1591 | 0.15914 | 1.3098 | 2.59 × 10^{-1} | |

M·T | 1 | 0.1132 | 0.11318 | 0.9315 | 3.40 × 10^{-1} | |

S·T | 1 | 0.1062 | 0.10621 | 0.8741 | 3.55 × 10^{-1} | |

T^{2} | 1 | 0.6665 | 0.66652 | 5.4859 | 2.41 × 10^{-2} | * |

H | 1 | 0.517 | 0.51699 | 4.2551 | 4.55 × 10^{-2} | * |

C·H | 1 | 0.5179 | 0.51788 | 4.2625 | 4.53 × 10^{-2} | * |

t·H | 1 | 0.4687 | 0.46874 | 3.858 | 5.63 × 10^{-2} | . |

S·H | 1 | 1.6756 | 1.67561 | 13.7913 | 6.09 × 10^{-4} | *** |

H^{2} | 1 | 0.5589 | 0.55887 | 4.5999 | 3.79 × 10^{-2} | * |

M·I | 1 | 0.0383 | 0.03825 | 0.3148 | 5.78 × 10^{-1} | |

T·I | 1 | 0.6052 | 0.60519 | 4.9811 | 3.12 × 10^{-2} | * |

Residuals | 41 | 4.9814 | 0.1215 | |||

R^{2} | 0.7977 | - | - | - | - | - |

**Table 9.**Results of the predicted error process criteria for yield (η), turbidity (Turb), density (ρ), viscosity (µ), and high heating value (HHV) using the quadratic models.

Errors | η | Turb (NTU) | ρ (g/mL) | µ (mm^{2}/s) | HHV (MJ/Kg) |
---|---|---|---|---|---|

MAE | 10.445 | 6.305 | 0.009 | 0.980 | 0.231 |

RMSE | 12.803 | 8.293 | 0.012 | 1.230 | 0.296 |

Sample | Inputs | ||||||
---|---|---|---|---|---|---|---|

Molar Ratio | Catalyst (wt. %) | Time (min) | Speed (rpm) | T (°C) | Humidity (wt. %) | Impurity (wt. %) | |

1 | 6.35 | 1.16 | 21.79 | 531.17 | 38.04 | 1.37 | 2.66 |

2 | 8.77 | 1.49 | 39.99 | 804.66 | 26.08 | 2.07 | 1.84 |

3 | 6.98 | 1.7 | 22.05 | 587.12 | 27.44 | 2.98 | 1.8 |

4 | 7.48 | 1.57 | 21.41 | 881.61 | 37.85 | 2.86 | 0.6 |

5 | 6.3 | 1.57 | 23.74 | 531.29 | 36.59 | 1.77 | 0.9 |

6 | 6.2 | 1.56 | 25.97 | 808.46 | 20.3 | 0.58 | 1.39 |

7 | 6.64 | 1.27 | 32.54 | 747.92 | 38.14 | 1.58 | 1.54 |

8 | 7.95 | 1.91 | 25.67 | 772.08 | 29.35 | 1.51 | 0.12 |

9 | 7.36 | 1.03 | 37.08 | 526.82 | 39.83 | 2.39 | 2.32 |

10 | 7.31 | 1.71 | 32.93 | 528.27 | 34.67 | 1.02 | 1.98 |

11 | 6.75 | 1.2 | 29.07 | 711.16 | 34.27 | 2.57 | 2.17 |

12 | 8.76 | 1.35 | 25.69 | 929.16 | 26.92 | 0.33 | 1.34 |

13 | 7.9 | 1.69 | 24.33 | 855.05 | 34.23 | 0.3 | 0.94 |

14 | 8.49 | 1.86 | 28.35 | 830.26 | 31.21 | 2.17 | 0.11 |

15 | 7.31 | 1.71 | 32.93 | 528.27 | 34.67 | 1.02 | 1.98 |

**Table 11.**Results of the errors in yield (η), turbidity (Turb), density (ρ), viscosity (µ), and HHV using the quadratic models.

Errors | η | Turb (NTU) | ρ (g/mL) | µ (mm^{2}/s) | HHV (MJ/Kg) |
---|---|---|---|---|---|

MAE | 15.081 | 9.130 | 0.063 | 1.941 | 0.449 |

RMSE | 22.853 | 14.648 | 0.066 | 2.202 | 0.556 |

**Table 12.**The first biodiesel optimization scenario: all variables considered to be equally important.

Goal | Importance | Value | Desirability | |
---|---|---|---|---|

M | Minimize → 6 | 1.0 | 6.24 | 0.921 |

C | Minimize → 1 | 1.0 | 0.75 | 1.000 |

t | Minimize → 20 | 1.0 | 20.00 | 1.000 |

S | Minimize → 500 | 1.0 | 499.99 | 1.000 |

T | In range → 30 | 1.0 | 20.40 | 1.000 |

H | In range → 1.5 | 1.0 | 0.20 | 1.000 |

I | In range → 1.5 | 1.0 | 0.15 | 1.000 |

η | Maximize → 9.50 | 1.0 | 99.99 | 1.000 |

Turb | Minimize → 0.21 | 1.0 | 0.00 | 1.000 |

ρ | Minimize → 0.74 | 1.0 | 0.83 | 0.232 |

µ | Minimize → 3.79 | 1.0 | 8.00 | 0.496 |

HHV | Maximize → 20.52 | 1.0 | 21.55 | 0.431 |

Overall Desirability | 0.710 |

Goal | Importance | Value | Desirability | |
---|---|---|---|---|

M | Minimize → 6 | 1.0 | 6.52 | 0.827 |

C | Minimize → 1 | 1.0 | 0.78 | 1.000 |

t | Minimize → 20 | 1.0 | 19.99 | 1.000 |

S | Minimize → 500 | 1.0 | 499.67 | 1.000 |

T | In range → 30 | 1.0 | 24.44 | 1.000 |

H | In range → 1.5 | 1.0 | 0.21 | 1.000 |

I | In range → 1.5 | 1.0 | 0.17 | 1.000 |

η | Maximize → 9.50 | 3.0 | 100.00 | 1.000 |

Turb | Minimize → 0.21 | 1.0 | 0.00 | 1.000 |

ρ | Minimize → 0.74 | 1.0 | 0.83 | 0.247 |

µ | Minimize → 3.79 | 1.0 | 7.78 | 0.523 |

HHV | Maximize → 20.52 | 1.0 | 21.53 | 0.422 |

Overall Desirability | 0.709 |

Goal | Importance | Value | Desirability | |
---|---|---|---|---|

M | Minimize → 6 | 1.0 | 6.24 | 0.921 |

C | Minimize → 1 | 1.0 | 0.75 | 1.000 |

t | Minimize → 20 | 1.0 | 20.00 | 1.000 |

S | Minimize → 500 | 1.0 | 499.99 | 1.000 |

T | In range → 30 | 1.0 | 20.40 | 1.000 |

H | In range → 1.5 | 1.0 | 0.20 | 1.000 |

I | In range → 1.5 | 1.0 | 0.15 | 1.000 |

η | Maximize → 9.50 | 1.0 | 99.99 | 1.000 |

Turb | Minimize → 0.21 | 3.0 | 0.00 | 1.000 |

ρ | Minimize → 0.74 | 1.0 | 0.83 | 0.232 |

µ | Minimize → 3.79 | 1.0 | 8.00 | 0.496 |

HHV | Maximize → 20.52 | 1.0 | 21.55 | 0.431 |

Overall Desirability | 0.710 |

Goal | Importance | Value | Desirability | |
---|---|---|---|---|

M | Minimize → 6 | 1.0 | 7.35 | 0.551 |

C | Minimize → 1 | 1.0 | 1.00 | 1.000 |

t | Minimize → 20 | 1.0 | 20.00 | 1.000 |

S | Minimize → 500 | 1.0 | 500.01 | 1.000 |

T | In range → 30 | 1.0 | 26.79 | 1.000 |

H | In range → 1.5 | 1.0 | 0.00 | 1.000 |

I | In range → 1.5 | 1.0 | 0.10 | 1.000 |

η | Maximize → 9.50 | 1.0 | 94.35 | 0.938 |

Turb | Minimize → 0.21 | 1.0 | 3.23 | 0.965 |

ρ | Minimize → 0.74 | 3.0 | 0.82 | 0.025 |

µ | Minimize → 3.79 | 3.0 | 7.04 | 0.228 |

HHV | Maximize → 20.52 | 1.0 | 21.43 | 0.378 |

Overall Desirability | 0.467 |

Goal | Importance | Value | Desirability | |
---|---|---|---|---|

M | Minimize → 6 | 1.0 | 6.40 | 0.868 |

C | Minimize → 1 | 1.0 | 1.00 | 1.000 |

t | Minimize → 20 | 1.0 | 20.00 | 1.000 |

S | Minimize → 500 | 1.0 | 499.60 | 1.000 |

T | In range → 30 | 1.0 | 33.70 | 1.000 |

H | In range → 1.5 | 1.0 | 0.21 | 1.000 |

I | In range → 1.5 | 1.0 | 0.16 | 1.000 |

η | Maximize → 9.50 | 1.0 | 136.97 | 1.000 |

Turb | Minimize → 0.21 | 1.0 | 0.00 | 1.000 |

ρ | Minimize → 0.74 | 1.0 | 0.83 | 0.241 |

µ | Minimize → 3.79 | 1.0 | 9.86 | 0.274 |

HHV | Maximize → 20.52 | 3.0 | 22.04 | 0.256 |

Overall Desirability | 0.626 |

Goal | Importance | Value | Desirability | |
---|---|---|---|---|

M | Minimize → 6 | 1.0 | 6.09 | 0.972 |

C | Minimize → 1 | 3.0 | 0.73 | 1.000 |

t | Minimize → 20 | 1.0 | 20.00 | 1.000 |

S | Minimize → 500 | 1.0 | 499.96 | 1.000 |

T | In range → 30 | 1.0 | 29.78 | 1.000 |

H | In range → 1.5 | 1.0 | 0.20 | 1.000 |

I | In range → 1.5 | 1.0 | 0.17 | 1.000 |

η | Maximize → 9.50 | 1.0 | 100.00 | 1.000 |

Turb | Minimize → 0.21 | 1.0 | 0.00 | 1.000 |

ρ | Minimize → 0.74 | 1.0 | 0.83 | 0.224 |

µ | Minimize → 3.79 | 1.0 | 8.03 | 0.493 |

HHV | Maximize → 20.52 | 1.0 | 21.55 | 0.430 |

Overall Desirability | 0.710 |

Goal | Importance | Value | Desirability | |
---|---|---|---|---|

M | Minimize → 6 | 1.0 | 6.24 | 0.921 |

C | Minimize → 1 | 1.0 | 0.75 | 1.000 |

t | Minimize → 20 | 1.0 | 20.00 | 1.000 |

S | Minimize → 500 | 3.0 | 499.99 | 1.000 |

T | In range → 30 | 1.0 | 20.40 | 1.000 |

H | In range → 1.5 | 1.0 | 0.21 | 1.000 |

I | In range → 1.5 | 1.0 | 0.15 | 1.000 |

η | Maximize → 9.50 | 1.0 | 99.99 | 1.000 |

Turb | Minimize → 0.21 | 1.0 | 0.00 | 1.000 |

ρ | Minimize → 0.74 | 1.0 | 0.83 | 0.232 |

µ | Minimize → 3.79 | 1.0 | 8.00 | 0.496 |

HHV | Maximize → 20.52 | 1.0 | 21.55 | 0.431 |

Overall Desirability | 0.710 |

Goal | Importance | Value | Desirability | |
---|---|---|---|---|

M | Minimize → 6 | 1.0 | 6.09 | 0.972 |

C | Minimize → 1 | 1.0 | 0.73 | 1.000 |

t | Minimize → 20 | 3.0 | 19.98 | 1.000 |

S | Minimize → 500 | 1.0 | 499.96 | 1.000 |

T | In range → 30 | 1.0 | 29.78 | 1.000 |

H | In range → 1.5 | 1.0 | 0.20 | 1.000 |

I | In range → 1.5 | 1.0 | 0.16 | 1.000 |

η | Maximize → 9.50 | 1.0 | 100.00 | 1.000 |

Turb | Minimize → 0.21 | 1.0 | 0.00 | 1.000 |

ρ | Minimize → 0.74 | 1.0 | 0.83 | 0.224 |

µ | Minimize → 3.79 | 1.0 | 8.03 | 0.493 |

HHV | Maximize → 20.52 | 1.0 | 21.55 | 0.430 |

Overall Desirability | 0.710 |

Goal | Importance | Value | Desirability | |
---|---|---|---|---|

M | Minimize → 6 | 3.0 | 6.00 | 1.000 |

C | Minimize → 1 | 1.0 | 0.72 | 1.000 |

t | Minimize → 20 | 1.0 | 20.00 | 1.000 |

S | Minimize → 500 | 1.0 | 499.96 | 1.000 |

T | In range → 30 | 1.0 | 29.78 | 1.000 |

H | In range → 1.5 | 1.0 | 0.19 | 1.000 |

I | In range → 1.5 | 1.0 | 0.17 | 1.000 |

η | Maximize → 9.50 | 1.0 | 100.00 | 1.000 |

Turb | Minimize → 0.21 | 1.0 | 0.00 | 1.000 |

ρ | Minimize → 0.74 | 1.0 | 0.83 | 0.219 |

µ | Minimize → 3.79 | 1.0 | 8.07 | 0.488 |

HHV | Maximize → 20.52 | 1.0 | 21.55 | 0.431 |

Overall Desirability | 0.710 |

**Table 21.**Experimental outputs that were obtained according to the nine biodiesel optimization scenarios.

Optimization Scenarios | Experimental Values Obtained | ||||||
---|---|---|---|---|---|---|---|

η | Turb (NTU) | ρ (g/mL) | µ (mm^{2}/s) | HHV (MJ/Kg) | MAE | RMSE | |

1st Scenario | 0.98 | 0.03 | 0.89 | 0.34 | 0.20 | 0.05 | 0.03 |

2nd Scenario | 0.99 | 0.02 | 0.63 | 0.27 | 0.16 | 0.09 | 0.05 |

3rd Scenario | 0.98 | 0.01 | 0.89 | 0.34 | 0.21 | 0.04 | 0.02 |

4th Scenario | 0.00 | 0.92 | 0.00 | 0.00 | 0.00 | 0.12 | 0.08 |

5th Scenario | 0.98 | 0.02 | 0.77 | 1.00 | 0.99 | 0.06 | 0.03 |

6th Scenario | 0.98 | 0.02 | 0.98 | 0.35 | 0.20 | 0.07 | 0.04 |

7th Scenario | 0.97 | 0.01 | 0.89 | 0.34 | 0.20 | 0.04 | 0.03 |

8th Scenario | 0.98 | 0.03 | 0.94 | 0.35 | 0.20 | 0.10 | 0.07 |

9th Scenario | 0.98 | 0.02 | 0.99 | 0.37 | 0.20 | 0.05 | 0.03 |

MAE | 0.02 | 0.03 | 0.02 | 0.00 | 0.01 | 0.07 | 0.04 |

RMSE | 0.06 | 0.09 | 0.08 | 0.01 | 0.02 | 0.25 | 0.14 |

© 2017 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Corral Bobadilla, M.; Lostado Lorza, R.; Escribano García, R.; Somovilla Gómez, F.; Vergara González, E.P.
An Improvement in Biodiesel Production from Waste Cooking Oil by Applying Thought Multi-Response Surface Methodology Using Desirability Functions. *Energies* **2017**, *10*, 130.
https://doi.org/10.3390/en10010130

**AMA Style**

Corral Bobadilla M, Lostado Lorza R, Escribano García R, Somovilla Gómez F, Vergara González EP.
An Improvement in Biodiesel Production from Waste Cooking Oil by Applying Thought Multi-Response Surface Methodology Using Desirability Functions. *Energies*. 2017; 10(1):130.
https://doi.org/10.3390/en10010130

**Chicago/Turabian Style**

Corral Bobadilla, Marina, Rubén Lostado Lorza, Rubén Escribano García, Fátima Somovilla Gómez, and Eliseo P. Vergara González.
2017. "An Improvement in Biodiesel Production from Waste Cooking Oil by Applying Thought Multi-Response Surface Methodology Using Desirability Functions" *Energies* 10, no. 1: 130.
https://doi.org/10.3390/en10010130