# Splitting Triglycerides with a Counter-Current Liquid–Liquid Spray Column: Modeling, Global Sensitivity Analysis, Parameter Estimation and Optimization

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## Abstract

**:**

## 1. Introduction

#### 1.1. Hydrolysis Reaction

#### 1.2. Spray Column

#### 1.2.1. Research by Jeffreys et al.

#### 1.2.2. Research by Rifai et al.

- The hydrolysis of triglycerides with water to fatty acids and glycerol follows the first order reaction to validate the model in this work with the experimental data set from Jeffreys et al.
- Constant mass flow rates are assumed for the continuous and dispersed phases in case of validating the model by Jeffreys et al.
- Variable mass flow rate is assumed for the continuous and dispersed phase and the model is then re-parameterized in respect to the mass-transfer rates, reaction rate and the backmixing coefficients.
- The finite volume model also takes backmixing into account as van Egmond and Goossens [20] showed that they obtain better results when considering axial dispersion.

## 2. Methodology

#### 2.1. Process Model of a Counter-Current Spray Column

#### 2.1.1. Material Balance

#### 2.1.2. Phase Equilibrium

#### 2.1.3. Solving the System of Equations

#### 2.2. Parameter Estimation via Differential Evolution (DE)

- Specify population size, number of generations, crossover probability, mutation factor.
- Initialize vector population where parameters are uniformly distributed within their bounds.
- Evaluate the objective (cost) function for all individuals (vectors) and store in the fitness variable.
- Generation loop until number of generations or fitness of cost function is reached:
- 4.1.
- Mutation (Parameter mixing): Select a target vector, choose randomly three other vectors and create mutant vector $m={v}_{1}+{m}_{factor}\ast ({v}_{2}-{v}_{3})$ where ${m}_{factor}$ is called the mutant factor or differential weight.
- 4.2.
- Recombination: Generate trial vector by a probabilistic swapping (crossover) of elements from current target vector with mutant vector.
- 4.3.
- Replacement: Evaluate cost function and replace target vector with trial vector if the cost function is lower with the parameters from the trial vector.

- Parameter vector is returned with best fitness.

#### 2.3. Multi-Criteria Optimization via Differential Evolution

## 3. Results

#### 3.1. Model Validation

#### 3.2. Global Sensitivity Analysis

#### 3.3. Parameter Estimation

#### 3.4. Multi-Criteria Optimization

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

DE | Differential evolution |

FA | Fatty acid |

FVM | Finite volume model |

G${}_{k}$ | Internal flow rate of aqueous phase at stage k [lb/h] |

G${}_{median}$ | [lb/h] |

GLY | Glycerol |

H | Column height [ft] |

h | Height of stage [ft] h = H/N |

HTU | Height to transfer unit [ft] |

Ka${}_{i}$ | Mass-transfer rate constant for species i [lb/(ft${}^{2}$*h)] |

K | Reaction equilibrium constant [−] |

k${}_{i}$ | Forward reaction rate constant of species i [1/h] |

L${}_{k}$ | Internal flow rate of oil phase at stage k [lb/h] |

L${}_{median}$ | [lb/h] |

m${}_{k}$ | Total mass of finite volume element system at stage k [lb] |

N | Number of elements [−] |

PDE | Partial differential equation |

R | Fatty acid sidechain of triglyceride |

r${}_{i}$ | Reaction rate of species i [1/h] |

S | Cross-sectional column area [ft${}^{2}$] |

TAC | Total annual cost [$/a] |

TG | Triglyceride |

DG | Diglyceride |

MG | Monoglyceride |

W | Water |

w${}_{FA}$ | Ration between the required pounds of TG to produce one pound of FA |

w${}_{GLY}$ | Ratio between the required pounds of TG to produce one pound of GLY |

x${}_{i,k}$ | Mass fraction in oil phase of species i on stage k [−] |

x${}_{i,k}^{*}$ | Mass fraction at oil interphase of species i on stage k [−] |

y${}_{i,k}$ | Mass fraction in aqueous phase of species i on stage k [−] |

y${}_{i,k}^{*}$ | Mass fraction at aqueous interphase of species i on stage k [−] |

${\alpha}_{b,k}$ | Eco${}_{99}$ indicator points for impact k and resource flow b |

${\alpha}_{x}$ | Backmixing coefficient for oil phase [−] |

${\alpha}_{y}$ | Backmixing coefficient for aqueous phase [−] |

${\beta}_{b}$ | Resource (e.g., material, energy) flows |

$\u03f5$ | Fraction of column occupied by the continuous phase [−] |

${\varphi}_{F,k}$ | Fraction of oil feed fed to column at stage k [−] |

${\varphi}_{{F}_{S},k}$ | Fraction of fed steam via column inlet at stage k [−] |

${\rho}_{Oil}$ | Density of oil feed [lb/ft${}^{3}$] |

${\upsilon}_{S}$ | Slip velocity [ft/h] |

${\omega}_{d}$ | weighting factors in impact category d |

## Appendix A. Data Study

**Table A1.**Distribution ratio, reaction equilibrium constant and reaction rate constant values from different literature sources.

Type of Oil/Fat & Correlations | Type of Reaction | Unit | 225 ${}^{\xb0}$C | 280 ${}^{\xb0}$C |
---|---|---|---|---|

Coconut oil | ||||

$m=exp(-6.69+4976.22/T)$ [12,13] | - | - | 27 | 10 |

$m=exp(-9.6+6470/T)$ [15] | - | - | 29.6 | 8.14 |

${k}_{e}$ [12] | reversible 1st order | - | 0.458 | 1.160 |

${k}_{e}=exp(9.604-4913.01/T)$ [13] | reversible 1st order | - | 0.7725 | 2.0596 |

${k}_{e}=2.22$ [15] | reversible 2nd order | - | 2.22 | 2.22 |

${k}_{1}={10}^{5.062-3367/T}$ [9] | irreversible 1st order | 1/min | 0.0201 | 0.0944 |

${k}_{1}=exp(12.116-8089.2437/T)$ [13] | reversible 1st order | 1.8605 × 10^{−4} | 0.0120 | |

${k}_{1}=exp(7.1-5750/T)$ [15] | reversible 2nd order | kmol/(m${}^{3}$min) | 0.0118 | 0.0371 |

Beef tallow fat | ||||

$m=exp(-12.0062+7473.2363/T)$ [12,13] | - | - | 20 | 4.5 |

$m=exp(-10.25+6565/T)$ [15] | - | - | 18.7 | 5.04 |

${k}_{e}=exp(12.987-6206.7356/T)$ [13] | reversible 1st order | - | 1.6795 | 5.7971 |

${k}_{e}=2.22$ [15] | reversible 2nd order | - | 2.22 | 2.22 |

${k}_{1}={10}^{4.663-3170/T}$ [9] | irreversible 1st order | 1/min | 0.0199 | 0.0855 |

${k}_{1}=exp(12.0207-7924.5695/T)$ [13] | reversible 1st order | 0.0205 | 0.0997 | |

${k}_{1}=exp(10.34-6825/T)$ [15] | reversible 2nd order | kmol/(m${}^{3}$min) | 0.0347 | 0.1355 |

Peanut oil | ||||

$m=exp(-8.0+5680/T)$ [15] | - | - | 30.0 | 9.7 |

${k}_{e}=2.22$ [15] | reversible 2nd order | - | 2.22 | 2.22 |

${k}_{1}={10}^{5.025-3410/T}$ [9] | irreversible 1st order | 1/min | 0.0151 | 0.0725 |

${k}_{1}=exp(5.83-4505/T)$ [15] | reversible 2nd order | kmol/(m${}^{3}$min) | 0.0402 | 0.0988 |

Type of TG | Unit | 225 ${}^{\xb0}$C | 276 ${}^{\xb0}$C | 280 ${}^{\xb0}$C |
---|---|---|---|---|

C8:0 (Caprylic) | kg/m${}^{3}$ | 807.93 | 763.86 | 760.49 |

lb/ft${}^{3}$ | 50.44 | 47.69 | 47.48 | |

C12:0 (Lauric) | kg/m${}^{3}$ | 763.32 | 721.63 | 717.85 |

lb/ft${}^{3}$ | 47.65 | 45.05 | 44.81 | |

C18:0 (Stearic) | kg/m${}^{3}$ | 762.42 | 722.25 | 719.31 |

lb/ft${}^{3}$ | 47.60 | 45.09 | 44.91 |

**Table A3.**Liquid density for vegetable oils [43].

Type of Oil | Unit | 200 ${}^{\xb0}$C |
---|---|---|

Canola oil | kg/m${}^{3}$ | 806.6 |

lb/ft${}^{3}$ | 50.4 | |

Corn oil | kg/m${}^{3}$ | 807.8 |

lb/ft${}^{3}$ | 50.4 | |

Peanut oil | kg/m${}^{3}$ | 801.5 |

lb/ft${}^{3}$ | 50.0 | |

Soybean oil | kg/m${}^{3}$ | 807.4 |

lb/ft${}^{3}$ | 50.4 |

**Table A4.**Liquid density for coconut oil and water [13].

Substance | Unit | 225 ${}^{\xb0}$C | 280 ${}^{\xb0}$C |
---|---|---|---|

Coconut oil | |||

kg/m${}^{3}$ | 738.0025 | 690.5853 | |

lb/ft${}^{3}$ | 46.0720 | 43.1118 | |

Water | |||

kg/m${}^{3}$ | 833.7878 | 747.7287 | |

lb/ft${}^{3}$ | 52.0517 | 46.6792 |

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**Figure 4.**Properties and phenomena interaction in respect to a counter-current liquid–liquid spray column under subcritical conditions adapted from [21].

**Figure 6.**Validation of finite volume model (constant internal flows) with analytical model from Jeffreys et al.

**Figure 8.**Response surface for glycerol fraction in bottom product with variable total steam flowrate and two inlets for the steam injection.

**Figure 9.**Response surface for fatty acid fraction in top product with variable water flowrate and two inlets for steam.

**Figure 10.**Pareto frontier for fatty acid fraction in product stream in respect to steam flowrate and inlet fraction (red point is the final solution of the differential evolution procedure; green triangle is the starting point).

**Figure 11.**Pareto frontier for glycerol fraction at bottom of the column in respect to steam flowrate and inlet fraction (red point is the final solution of the differential evolution procedure, green triangle is the starting point).

First Author | Reactor Type | Reaction Type and Order | Catalyst and Process Conditions | Model and Studied Parameters |
---|---|---|---|---|

Patil [5] | CSTR | Reversible & Pseudo-1st Order | None | Algebraic Equations |

Forero-Hernandez [6] | Batch | None | ||

Lascaray, 1949 [7] | Review | Review | Review; 100–220 ${}^{\xb0}$C | - |

Lascaray, 1952 [8] | Review | Review | Review; 100–220 ${}^{\xb0}$C | - |

Sturzenegger and Sturm [9] | Batch | Irreversible & Pseudo-1st Order | ZnO | |

Jeffreys [10] | Spray column | Irreversible & Pseudo-1st Order | ZnO | Algebraic Equations |

Rifai [11] | Spray Column | Reversible & 2nd Order? | ||

Namdev [12] | Review | Reversible & Pseudo-1st Order | ||

Attarakih [13] | Spray column | Reversible & Pseudo-1st Order | Reduced Population Balance Model |

Aspect | Jeffreys et al. | Rifai et al. |
---|---|---|

Reaction kinetics | irreversible first order | reversible second order: |

${r}_{i}=({k}_{i}S{\rho}_{Oil}/{w}_{i}){x}_{T{G}_{i},k}$dh | ${r}_{i}=({k}_{i}S{\rho}_{Oil}^{2}/{w}_{i})({x}_{W,k}{x}_{T{G}_{i},k}-\frac{1}{K}{x}_{GLY,k}{x}_{F{A}_{i},k})$dh | |

Internal flowrates | assumed constant over column height | changes over column height |

Water solubility | assumed constant over column height | changes over column height |

Hydrodynamic model | - | Beyaert et al. [19] |

${v}_{s}=\frac{G}{S(1-\u03f5)}+\frac{L}{S\u03f5}$ | ||

Solution formulation | Analytical | System of non-linear differential equations |

Experimental Run | Input | Output | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

${\mathit{L}}_{\mathit{in}}$ [lb/h] | ${\mathit{G}}_{\mathit{in}}$ [lb/h] | ${\mathit{\rho}}_{\mathit{Oil}}$ [lb/ft${}^{3}$] | m [−] | ${\mathit{y}}_{\mathit{GLY}}$ [−] | ${\mathit{x}}_{\mathit{GLY}}$ [−] | ${\mathit{L}}_{\mathit{out}}$ [lb/h] | ${\mathit{G}}_{\mathit{out}}$ [lb/h] | ${\mathit{G}}_{\mathit{median}}$ [lb/h] | ${\mathit{L}}_{\mathit{median}}$ [lb/h] | |

#1 | 7260 | 4600 | 45 | 10.32 | 0.1605 | 0.03 | 8050 | 3810 | 4205 | 7655 |

#2 | 6490 | 4440 | 45.05 | 9.56 | 0.1705 | 0.037 | 7180 | 3750 | 4095 | 6835 |

#3 | 6905 | 4300 | 45 | 11.38 | 0.189 | 0.027 | 7370 | 3835 | 4070 | 7140 |

#4 | 7400 | 3980 | 45.1 | 11.67 | 0.182 | 0.019 | 7770 | 3610 | 3795 | 7585 |

#5 | 6570 | 4480 | 44.9 | 8.32 | 0.227 | 0.027 | 7340 | 3710 | 4095 | 6955 |

#6 | 8175 | 4120 | 45.05 | 10.32 | 0.188 | 0.024 | 8900 | 3395 | 3760 | 8540 |

**Table 4.**Fixed and variable cost for high pressure (HP) steam production [29].

Cost | Unit | per 1000 lb Steam |
---|---|---|

Average boiler fuel | MMBtu | 1.56 |

Fresh water | $ | 0.02 |

Water treatment cost | $ | 0.74 |

Water preheating and pumping | $ | 0.62 |

Deareation steam | $ | 1.10 |

FD fan | $ | 0.05 |

C${}_{var}$ (variable cost) | $ | 11.9 |

Boiler capital | MM$ | 20 |

R depraciation factor | % of capital | 15 |

Maintenance cost | % of capital | 2 |

Two employees | $/a | 120,000 |

Employee cost factor | - | 3 |

C${}_{fix}$ (fixed cost) | $ | 1.7 |

C${}_{ST}$ = C${}_{var}$ + C${}_{fix}$ | $ | 13.6 |

**Table 5.**Impact categories for the Eco${}_{99}$-indicator and normalized data for steel, steam and electricity [33].

Impact Category | Steel [Points/lb] | Steam [Points/lb] | Electricity [Points/kWh] |
---|---|---|---|

Human health (d = 1) | |||

Carcinogenics | $2.867\times {10}^{-3}$ | $5.352\times {10}^{-5}$ | $4.360\times {10}^{-4}$ |

Climate change | $5.942\times {10}^{-3}$ | $7.257\times {10}^{-4}$ | $3.610\times {10}^{-6}$ |

Ionizing radiation | $2.046\times {10}^{-4}$ | $5.126\times {10}^{-4}$ | $8.240\times {10}^{-4}$ |

Ozone layer depletion | $2.064\times {10}^{-6}$ | $9.525\times {10}^{-7}$ | $1.210\times {10}^{-4}$ |

Respiratory effects | $3.633\times {10}^{-2}$ | $3.570\times {10}^{-7}$ | $1.350\times {10}^{-6}$ |

Ecosystem (d = 2) | |||

Acidification | $1.229\times {10}^{-3}$ | $5.488\times {10}^{-3}$ | $2.810\times {10}^{-4}$ |

Ecotoxicity | $3.379\times {10}^{-2}$ | $1.270\times {10}^{-3}$ | $1.670\times {10}^{-4}$ |

Resources (d = 3) | |||

Land occupation | $1.692\times {10}^{-3}$ | $3.892\times {10}^{-5}$ | $4.680\times {10}^{-4}$ |

Fossil fuels | $2.690\times {10}^{-2}$ | $5.670\times {10}^{-2}$ | $1.200\times {10}^{-3}$ |

Mineral extraction | $3.366\times {10}^{-2}$ | $4.001\times {10}^{-6}$ | $5.7\times {10}^{-6}$ |

Section | Model Assumptions | Significant Variables and/or Parameters |
---|---|---|

3.1. Model validation | - Constant internal flowrates | Process parameters by Jeffreys et al. |

3.2. Global sensitivity analysis | - Constant internal flowrates | Phenomena-based parameters |

3.3. Parameter estimation | - Variable internal flowrates | All relevant parameters (Table 8) |

- Solubility and mass transfer | ||

of water in and to oil phase | ||

3.4. Multi-criteria optimization | - Variable internal flowrates | Process operation variables |

- No solubility and mass transfer | ||

of water in and to oil phase |

**Table 7.**Parameters for the counter-current oil-splitting column in English units and mass-based; experimental run number 6 by Jeffreys et al.

Parameter | Symbol | Nominal Value | Unit |
---|---|---|---|

Overall mass-transfer coefficient for glycerol | $Ka$ | 14.21 | [lb/(ft${}^{3}$h)] |

Cross-sectional area of tower | S | 3.688 | [ft${}^{2}$] |

Mass flow of extract (aqueous phase) | G | 3760 | [lb/h] |

Mass flow of raffinate (oil phase) | L | 8540 | [lb/h] |

Glycerol distribution ratio/coefficient | ${\psi}_{GLY}$ | 10.32 | [−] |

Forward reaction rate coefficient | k | 10.2 | [1/h] |

Height of column | H | 73.5 | [ft] |

Glycerol content in fat | ${z}_{0}/{w}_{GLY}$ | 0.0853 | [−] |

Liquid density of fat | ${\rho}_{Oil}$ | 45.05 | [lb/ft${}^{3}$] |

Backmixing coefficient of cont. phase (oil) | ${\alpha}_{x}$ | 0.0 | [−] |

Backmixing coefficient of disp. phase (water) | ${\alpha}_{y}$ | 0.0 | [−] |

**Table 8.**Results of parameter estimation via DE for the FVM with variable internal water and oil flowrates.

Parameter | Re-Parameterized Model | Jeffreys et al. | ||||
---|---|---|---|---|---|---|

$K{a}_{GLY}$ | 43.64 | 19.06 | ||||

$K{a}_{W}$ | 0.19 | - | ||||

k | 77.14 | 10.2 | ||||

${\alpha}_{x}$ | 0.37 | 0.9 | ||||

${\alpha}_{y}$ | 0.19 | 0.1 | ||||

${\psi}_{GLY}$ | 64.17 | 10.32 | ||||

${\psi}_{W}$ | 27.40 | - | ||||

Experiment | ${y}_{GLY}^{sim}$ [−] | ${y}_{GLY}^{exp}$ [−] | Deviation from exp. [%] | ${G}_{out}^{sim}$ [lb/h] | ${G}_{out}^{exp}$ [lb/h] | Deviation from exp. [%] |

1 | 0.1708 | 0.1605 | 6.4174 | 3879 | 3810 | 1.8110 |

2 | 0.1832 | 0.1705 | 7.4487 | 3741 | 3750 | −0.24 |

3 | 0.2072 | 0.189 | 9.6296 | 3686 | 3835 | −3.8853 |

4 | 0.2076 | 0.182 | 14.0659 | 3294 | 3610 | −8.7535 |

5 | 0.2265 | 0.227 | −0.2203 | 3983 | 3710 | 7.3585 |

6 | 0.2030 | 0.188 | 7.9787 | 3436 | 3395 | 1.2077 |

${x}_{GLY}^{sim}$ [−] | ${x}_{GLY}^{exp}$ [−] | ${L}_{out}^{sim}$ [lb/h] | ${L}_{out}^{exp}$ [lb/h] | |||

1 | 0.0000 | 0.03 | −100 | 7981 | 8050 | -0.8571 |

2 | 0.0000 | 0.037 | −100 | 7189 | 7180 | 0.1253 |

3 | 0.0000 | 0.027 | −100 | 7519 | 7370 | 2.0217 |

4 | 0.0000 | 0.019 | −100 | 8086 | 7770 | 4.0669 |

5 | 0.0000 | 0.027 | −100 | 7067 | 7340 | −3.7193 |

6 | 0.0000 | 0.024 | −100 | 8859 | 8900 | −0.4607 |

${G}_{median}^{sim}$ [lb/h] | ${G}_{median}^{exp}$ [lb/h] | ${L}_{median}^{sim}$ [lb/h] | ${L}_{median}^{exp}$ [lb/h] | |||

1 | 4236 | 4205 | 0.7372 | 7586 | 7655 | −0.9014 |

2 | 4088 | 4095 | −0.1709 | 6798 | 6835 | −0.5413 |

3 | 3990 | 4070 | −1.9656 | 7169 | 7140 | 0.4062 |

4 | 3634 | 3795 | −4.2424 | 7709 | 7585 | 1.6348 |

5 | 4228 | 4095 | 3.2479 | 6764 | 6955 | −2.7462 |

6 | 3775 | 3760 | 0.3989 | 8487 | 8540 | −0.6206 |

Input and Objective | Unit | Value | Input Bounds |
---|---|---|---|

Input | |||

Steam flowrate | lb/h | 786.55 | [50, 5000] |

First steam inlet fraction | - | 0.27 | [0.1, 1.0] |

Objective | |||

Revenue | $/a | 47,410,513 | |

Total annual cost (TAC) | $/a | 94,076 | |

Raw material cost | $/a | 17,452,458 | |

Profit | $/a | 29,863,978 | |

Eco99 indicator | Points | 22,316 |

© 2019 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**

Jones, M.N.; Forero-Hernandez, H.; Zubov, A.; Sarup, B.; Sin, G.
Splitting Triglycerides with a Counter-Current Liquid–Liquid Spray Column: Modeling, Global Sensitivity Analysis, Parameter Estimation and Optimization. *Processes* **2019**, *7*, 881.
https://doi.org/10.3390/pr7120881

**AMA Style**

Jones MN, Forero-Hernandez H, Zubov A, Sarup B, Sin G.
Splitting Triglycerides with a Counter-Current Liquid–Liquid Spray Column: Modeling, Global Sensitivity Analysis, Parameter Estimation and Optimization. *Processes*. 2019; 7(12):881.
https://doi.org/10.3390/pr7120881

**Chicago/Turabian Style**

Jones, Mark Nicholas, Hector Forero-Hernandez, Alexandr Zubov, Bent Sarup, and Gürkan Sin.
2019. "Splitting Triglycerides with a Counter-Current Liquid–Liquid Spray Column: Modeling, Global Sensitivity Analysis, Parameter Estimation and Optimization" *Processes* 7, no. 12: 881.
https://doi.org/10.3390/pr7120881