Abstract

Chemical refining converts free fatty acids in vegetable oil into sodium soaps by contacting the oil with an aqueous sodium hydroxide solution. The acid base reaction is fast after the reactants meet. The observable rate is therefore governed mainly by interfacial area, oil film transfer, internal droplet transport, soap accumulation, and reactor hydraulics. Smaller droplets increase interfacial area and shorten the calculated residence time, but they also create a finer soap rich dispersion that is harder to separate. Industrial systems consequently benefit from staged hydrodynamics: intense dispersion for contact, controlled retention for completion, and gentle conditioning before centrifugal separation. [1, 2, 3, 16]

Effect of Sodium Hydroxide Solution Droplet Size on Mass Transfer and Free Fatty Acid Neutralization in Vegetable Oil

Mathematical model and calculated residence time for soybean oil containing 500 ppm free fatty acids

Calculation basis: 15 wt% sodium hydroxide, exact stoichiometric dose, 80 °C, interfacial tension 0.3 mN/m

Prepared for subsequent technical editing

23 August 2026

Executive Summary

Chemical refining converts free fatty acids in vegetable oil into sodium soaps by contacting the oil with an aqueous sodium hydroxide solution. The acid base reaction is fast after the reactants meet. The observable rate is therefore governed mainly by interfacial area, oil film transfer, internal droplet transport, soap accumulation, and reactor hydraulics. Smaller droplets increase interfacial area and shorten the calculated residence time, but they also create a finer soap rich dispersion that is harder to separate. Industrial systems consequently benefit from staged hydrodynamics: intense dispersion for contact, controlled retention for completion, and gentle conditioning before centrifugal separation. [1, 2, 3, 16]

Principal calculated result

Using the stated assumptions and the conservative Ranz Marshall mass transfer analogy, the 99% neutralization time varies with the square of droplet diameter and inversely with the effective Sherwood number. At effective Re = 50, the calculated 99% times are 9.63 s for 10 µm droplets, 4.01 min for 50 µm, 16.05 min for 100 µm, and 2.41 h for 300 µm. The corresponding 300 µm times decrease to 14.57 min at Re = 5,000 and 4.61 min at Re = 50,000.

The phrase neutralization completion requires a numerical definition. Exact 100% conversion is asymptotic in the first order model. This report therefore uses 99% conversion as the principal completion criterion and also reports 90% and 99.9%. For a 500 ppm feed, 99% conversion corresponds to 5 ppm residual FFA, while 99.9% corresponds to 0.5 ppm.

A major interpretation limit applies to the Reynolds numbers. The user supplied Re values do not include a mixer diameter, speed, pipe diameter, or local velocity. The numerical calculation therefore treats them as effective droplet scale Reynolds numbers in a sphere mass transfer analogy. When interpreted literally, the implied droplet slip velocities and Weber numbers are physically impossible at an interfacial tension of 0.3 mN/m. The Re = 5,000 and 50,000 results should be read as aggressive mathematical sensitivity cases, not as validated equipment performance. If the supplied Re values are mixer or pipe Reynolds numbers, mixer geometry, power input, energy dissipation, and droplet size distribution are required before residence time can be predicted.

• Stoichiometric 15 wt% NaOH solution demand: 0.472 kg per tonne of oil, equal to 0.418 L per tonne.

• Initial caustic dispersed phase volume: 0.0368% of the oil and lye mixture.

• Theoretical sodium soap generation: 0.539 kg per tonne, equal to 538.9 ppm on the oil basis.

• The base diffusivity is an engineering estimate. Residence time must be verified by pilot or plant data because the prediction is sensitive to diffusivity, interfacial mobility, soap film development, coalescence, and residence time distribution.

• A plug flow or staged retention section is strongly preferred. A single completely mixed tank requires about 21.5 times the plug flow mean residence time for the same 99% conversion in this ideal first order model.

Figure ES1. Calculated 99% neutralization residence time for all requested droplet sizes and effective Reynolds numbers. Source: Author calculation using Equations (12) to (20).

Contents

1. Scope and calculation basis

2. Alkali refining process and neutralization mechanism

3. Stoichiometric material balance

4. Mathematical mass transfer model

5. Numerical results

6. Physical consistency of the requested Reynolds numbers

7. Engineering interpretation and process design

8. Model limitations and validation plan

9. Conclusions

Appendix A. Detailed calculated results

Appendix B. Worked calculation

Appendix C. Sensitivity and reactor hydraulics

Bibliography

Nomenclature

Calculation Status and Intended Use

This report is a desktop engineering model for process screening, comparison of droplet size scenarios, preliminary residence time selection, and definition of a validation program. It is not a performance guarantee for a mixer, retention vessel, or centrifuge. The calculations use a representative Sauter mean diameter and ideal plug flow unless another hydraulic model is explicitly stated.

Critical use restriction

Do not set plant retention time from the Re = 5,000 or 50,000 curves alone. First determine whether the stated Reynolds number is based on the mixer, pipe, rotor, or droplet. Then measure the actual droplet size distribution and calibrate an effective kL a from FFA decay data.

1. Scope and Calculation Basis

The assignment is to quantify how sodium hydroxide solution droplet size affects mass transfer and free fatty acid neutralization in soybean oil. Droplet diameters of 1, 5, 10, 30, 50, 100, 150, 200, 250, and 300 µm are evaluated for effective Reynolds numbers of 50, 5,000, and 50,000. The oil temperature is 80 °C, the initial FFA concentration is 500 ppm by mass, the sodium hydroxide concentration is 15 wt%, and the oil and aqueous interfacial tension is 0.3 mN/m.

The calculations use a basis of 1,000 kg oil and represent the free fatty acids as oleic acid. The lye is dosed at the exact stoichiometric requirement for the FFA only. No caustic excess, phosphoric acid demand, residual phosphatide demand, trace metal demand, or neutral oil saponification is included. Industrial alkali refining normally applies a controlled caustic surplus and may include conditioning acid, retention, soapstock centrifugation, water washing, and vacuum drying. [1, 2, 3, 16]

Table 1. Calculation basis and model status

Parameter	Value	Status or interpretation
Oil mass basis	1,000 kg	Selected calculation basis
Initial FFA	500 ppm = 0.500 kg	Oleic acid equivalent
NaOH solution	15 wt%	Exact stoichiometric dose
Oil temperature	80 °C	Specified
Oil density	881.7 kg/m³	Published vegetable oil property basis and prior project model [7, 18]
Oil dynamic viscosity	0.0114 Pa s	Published vegetable oil property basis and prior project model [7, 18]
15 wt% lye density	1,130.5 kg/m³	Engineering interpolation from caustic soda property data [6, 18]
FFA diffusivity in oil	1.0 × 10⁻¹⁰ m²/s	Conservative engineering estimate, requires validation
Internal aqueous diffusivity	1.0 × 10⁻⁹ m²/s	Screening estimate
Interfacial tension	0.0003 N/m	Specified soap conditioned value
Droplet diameter definition	d = d₃₂	Representative Sauter mean in the main model
Hydraulic model	Ideal plug flow	CSTR sensitivity presented separately

Figure 1. Specific gravity chart for aqueous caustic soda solutions. Source: Occidental Chemical Caustic Soda Handbook [6].

Figure 2. Cutaway illustration of an inline high shear rotor stator mixer. Source: Silverson Machines [5].

The OxyChem chart places 15 wt% sodium hydroxide near a specific gravity of approximately 1.13 at 80 °C, which is consistent with the engineering density adopted here. The mixer image illustrates the type of intense inline dispersion stage that can generate small aqueous droplets. It does not establish the droplet size or kL a for the present case; vendor testing and measured droplet size are still required. [5, 6]

2. Alkali Refining Process and Neutralization Mechanism

2.1 Industrial process sequence

In chemical refining, caustic soda converts free fatty acids to sodium soaps, and the soap rich aqueous phase is then separated from the neutral oil. AOCS process guidance describes caustic addition before an intensive mixer, followed by either direct separation or retention and hydration before a disc separator. Long mix processing of crude soybean oil uses diluted caustic, several minutes of retention, soapstock separation, water washing, and vacuum drying. Alfa Laval describes the same functional sequence with dosing pumps, mixers, retention equipment, and disc stack centrifuges. [1, 3]

Figure 3. Recommended staged hydrodynamic sequence for caustic neutralization and soap removal. Source: Author schematic based on AOCS and Alfa Laval process descriptions [1, 3].

Figure 4. Commercial process arrangement showing mixing, retention, centrifugal separation, and drying. Source: Alfa Laval application brochure [4].

Figure 5. Simplified disc stack centrifuge flow path. Source: Alfa Laval [3].

The commercial process layout is shown to emphasize that the reaction and the separation functions should not be assigned to one hydrodynamic zone. Continued intense shear after soap formation may keep the heavy phase finely dispersed and increase neutral oil entrainment. Gentle conditioning before the separator can increase the effective droplet size and improve separation.

2.2 Reaction and controlling transport steps

$${\mathrm{RCOOH}}_{(\mathrm{oil})}+{\mathrm{NaOH}}_{(\mathrm{aq})}\to {\mathrm{RCOO}}^{-}{\mathrm{Na}}^{+}+{\mathrm{H}}_2\mathrm{O}$$

(1)

The intrinsic acid base reaction is rapid after the FFA molecule and hydroxide ion meet. The observed process rate can be controlled by transport of FFA through the continuous oil film, mixing and diffusion within the aqueous droplet, and formation or restructuring of a soap rich interfacial layer. The soap is amphiphilic, so it can lower interfacial tension, change interfacial mobility, retard coalescence, and stabilize a fine dispersion. These effects are not fully represented by a clean sphere correlation. [8, 14, 15, 17, 19]

Figure 6. Conceptual oil film transfer and neutralization at a sodium hydroxide solution droplet. Source: Author schematic based on transport theory and the prior capillary oscillation project report [14, 15, 19].

2.3 Why droplet size is central

• Smaller droplets provide more interfacial area per unit volume of caustic solution.

• Smaller droplets shorten characteristic oil film and aqueous diffusion distances.

• Large droplets carry more hydroxide per droplet but supply less interfacial area per unit reactor volume.

• Fine droplets and soap rich interfacial films can form persistent emulsions that are difficult to remove by centrifugation.

• Larger droplets and agglomerates migrate faster in a centrifuge, approximately with d² in the Stokes regime.

• At very low interfacial tension, high local deformation stresses can cause rapid breakup, making a prescribed fixed droplet diameter difficult to maintain.

Figure 7. Geometric effect of droplet breakup at fixed dispersed phase volume. Source: Author illustration.

3. Stoichiometric Material Balance

3.1 Free fatty acid and caustic demand

$$m_{\mathrm{FFA}}=w_{\mathrm{FFA}}{m}_{\mathrm{oil}}$$

(2)

$$n_{\mathrm{FFA}}=\frac{m_{\mathrm{FFA}}}{M_{\mathrm{FFA}}}$$

(3)

$$m_{\mathrm{NaOH}}=n_{\mathrm{FFA}}{M}_{\mathrm{NaOH}}$$

(4)

$$m_{\mathrm{lye}}=\frac{m_{\mathrm{NaOH}}}{w_{\mathrm{NaOH}}}$$

(5)

$$m_{\mathrm{soap}}=n_{\mathrm{FFA}}{M}_{\mathrm{soap}}$$

(6)

$$m_{{\mathrm{H}}_2\mathrm{O},\mathrm{r}\mathrm{x}\mathrm{n}}=n_{\mathrm{FFA}}{M}_{{\mathrm{H}}_2\mathrm{O}}$$

(7)

For 1,000 kg of oil, 500 ppm FFA is 0.500 kg. On an oleic acid basis, the FFA inventory is 0.001770 kmol. Exact stoichiometry requires 0.07080 kg pure NaOH, which is supplied as 0.472 kg of 15 wt% solution. Complete neutralization generates 0.539 kg sodium oleate equivalent and 0.03189 kg reaction water.

Figure 8. Stoichiometric material balance on the 1,000 kg oil basis. Source: Author calculation using Equations (2) to (7).

Table 2. Stoichiometric and phase volume results

Quantity	Value	Unit	Comment
Free fatty acids in oil	0.500000	kg/t oil	Reactant
Pure NaOH required	0.070804	kg/t oil	Exact stoichiometry
15 wt% NaOH solution	0.472027	kg/t oil	Aqueous feed
Water in lye	0.401223	kg/t oil	Before reaction
Sodium soap formed	0.538889	kg/t oil	538.9 ppm on oil basis
Reaction water formed	0.031889	kg/t oil	Stoichiometric product
Initial lye volume	0.417538	L/t oil	Using 1,130.5 kg/m³
Initial dispersed volume	0.036801	vol%	Oil plus lye basis
Lye NaOH molarity	4.239	mol/L	Initial droplet concentration
FFA molarity in oil	1.561	mol/m³ oil	Initial continuous phase

Important exact stoichiometry limitation

The base case has no chemical margin. Any caustic demand from conditioning acid, phosphatides, trace metals, measurement error, or neutral oil saponification changes the available hydroxide inventory. Industrial controls commonly use a validated excess rather than exact theoretical stoichiometry [1, 2].

4. Mathematical Mass Transfer Model

4.1 Oil and lye phase volumes

$$V_{\mathrm{oil}}=\frac{m_{\mathrm{oil}}}{{\rho }_o}$$

(8)

$$V_{\mathrm{lye}}=\frac{m_{\mathrm{lye}}}{{\rho }_{\mathrm{lye}}}$$

(9)

$$\varphi =\frac{V_{\mathrm{lye}}}{V_{\mathrm{oil}}+V_{\mathrm{lye}}}$$

(10)

The oil volume is 1.134173 m³ and the initial lye volume is 0.000417538 m³. The dispersed phase fraction is therefore φ = 0.00036801, or 0.03680%. Because the lye fraction is very small, the injection pattern and micromixing determine whether the caustic is distributed through the full oil flow or concentrated in local streaks.

4.2 Droplet population and interfacial area

$$N_d=\frac{V_{\mathrm{lye}}}{\pi d^3/6}$$

(11)

$$a=\frac{6\varphi }{d_{32}}$$

(12)

For monodisperse spheres, d₃₂ equals the droplet diameter. For a measured polydisperse population, the Sauter mean d₃₂ should be used because it preserves the ratio of dispersed volume to interfacial area. At fixed lye volume, the total area varies as 1/d and the number of droplets varies as 1/d³.

Figure 9. Calculated total interfacial area and number of caustic droplets per tonne of oil. Source: Author calculation using Equations (11) and (12).

Table 3. Geometric effect of droplet size

d₃₂ (µm)	Droplets per tonne	Total area (m²/t)	Area density (m²/m³)
1	7.974e+14	2,505.23	2,208.05
5	6.380e+12	501.05	441.61
10	7.974e+11	250.52	220.80
30	2.953e+10	83.51	73.60
50	6.380e+09	50.10	44.16
100	7.974e+08	25.05	22.08
150	2.363e+08	16.70	14.72
200	9.968e+07	12.53	11.04
250	5.104e+07	10.02	8.83
300	2.953e+07	8.35	7.36

4.3 Reynolds, Schmidt, and Sherwood numbers

$${\mathrm{Re}}_d=\frac{{\rho }_o{u}_{\mathrm{rel}}d}{{\mu }_o}$$

(13)

$$\mathrm{Sc}=\frac{{\mu }_o}{{\rho }_o{D}_{\mathrm{FFA}}}$$

(14)

$$\mathrm{Sh}=2+0.6{\mathrm{Re}}_d^{1/2}{\mathrm{Sc}}^{1/3}$$

(15)

The calculated Schmidt number is 129,296, reflecting slow molecular diffusion in a relatively viscous oil. Equation (15) is the Ranz Marshall sphere analogy. It was developed for conventional transfer around drops or particles and is used here as an engineering screening relation. Its extrapolation to Re = 5,000 and 50,000 and to a soap covered deformable interface is uncertain. A Whitaker type relation is shown as a sensitivity comparison, but its published validation range also does not cover the present very high Schmidt number. [9, 10, 14]

Figure 10. Sherwood number sensitivity to effective droplet Reynolds number. Source: Author calculation based on Ranz and Marshall [9] and Whitaker [10].

Table 4. Sherwood numbers used in the calculations

Re or reference	Sh	Interpretation	Status
50	216.535	Requested case	Extrapolated correlation input
5,000	2,147.347	Requested case	Extrapolated correlation input
50,000	6,786.184	Requested case	Extrapolated correlation input
Reference	2.000	Stagnant sphere	Diffusion limit
Reference	6.000	Prior project screening	Intensive mixing calibration value [17]

4.4 Film coefficient, volumetric coefficient, and conversion

$$k_L=\frac{\mathrm{Sh}{D}_{\mathrm{FFA}}}{d}$$

(16)

$$k_{L}a=\frac{6\varphi \mathrm{Sh}{D}_{\mathrm{FFA}}}{d^2}$$

(17)

$$\frac{d{C}_{\mathrm{FFA}}}{dt}=-k_{L}a{C}_{\mathrm{FFA}}$$

(18)

$$\frac{C_{\mathrm{FFA}}}{C_{\mathrm{FFA},0}}=\exp (-k_{L}at)$$

(19)

$$t_X=\frac{-\ln (1-X)}{k_{L}a}$$

(20)

Equation (17) combines the two principal size effects: kL varies as 1/d and interfacial area varies as 1/d. Consequently, kL a varies approximately as 1/d² at fixed φ, Sh, and diffusivity. The calculated time for a selected conversion therefore varies approximately with d².

$$t_X=\frac{-\ln (1-X)d^2}{6\varphi \mathrm{Sh}{D}_{\mathrm{FFA}}}$$

(21)

Figure 11. Calculated oil side mass transfer coefficient for the requested Re cases. Source: Author calculation using Equation (16).

Figure 12. Calculated volumetric mass transfer coefficient. Source: Author calculation using Equation (17).

4.5 Perfect sink and internal droplet assumptions

The base model assumes that hydroxide is sufficiently concentrated to maintain a near zero FFA concentration at the oil side interface until the target conversion is approached. The initial lye concentration is approximately 4.24 mol/L, while the FFA concentration in the oil is only about 1.56 mol/m³. At exact total stoichiometry, hydroxide inventory decreases together with FFA inventory, but the local aqueous concentration remains high through most of the conversion. The approximation becomes less reliable at the final fraction of a percent and when soap gels or interfacial films restrict access.

$$\frac{1}{K_o}=\frac{1}{k_o}+\frac{m}{k_a}+R_{\mathrm{int}}$$

(22)

Equation (22) is the more general resistance form. The base calculation neglects the aqueous side and interfacial resistances by setting K_o ≈ k_o. The distribution coefficient m and the interfacial resistance Rint are not known for the actual soybean oil, lye, soap, phosphatide, and temperature combination.

$$t_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f},\mathrm{a}\mathrm{q}}\approx \frac{{(d/2)}^2}{{\pi }^2{D}_{\mathrm{aq}}}$$

(23)

The internal diffusion estimate is short for clean aqueous droplets at Daq = 1 × 10⁻⁹ m²/s, but it can become important in large soap rich droplets if the effective diffusivity decreases by one or two orders of magnitude.

Figure 13. Internal aqueous diffusion time estimate and soap rich sensitivity. Source: Author calculation using Equation (23).

4.6 Capillary oscillation and interfacial tension

$${\omega }_2^2=\frac{24\sigma }{R^3(3{\rho }_d+2{\rho }_o)}$$

(24)

$$f_2=\frac{{\omega }_2}{2\pi }$$

(25)

The user specified an interfacial tension of 0.3 mN/m. This is much lower than the approximately 23 to 26 mN/m reported for clean commercial vegetable oils against water and is consistent with a strongly soap conditioned interface. The low value substantially reduces capillary restoring stress and makes breakup and coalescence behavior highly sensitive to interfacial composition. [8, 18, 19]

Figure 14. Estimated n = 2 capillary oscillation frequency at the specified interfacial tension. Source: Author calculation using Equations (24) and (25), based on Rayleigh and two fluid droplet theory [12, 13, 20].

4.7 Reactor hydraulics

$$\frac{C_{\mathrm{out}}}{C_{\mathrm{in}}}=\exp (-k_{L}a\tau )$$

(26) (ideal plug flow)

$$\frac{C_{\mathrm{out}}}{C_{\mathrm{in}}}=\frac{1}{1+k_{L}a\tau }$$

(27) (single CSTR)

$$\frac{C_{\mathrm{out}}}{C_{\mathrm{in}}}={(1+\frac{k_{L}a\tau }{N})}^{-N}$$

(28) (N equal CSTRs)

The main results assume an ideal plug flow parcel. A completely mixed tank has a broad residence time distribution and allows some material to leave early. At 99% conversion, a single CSTR requires τ = 99/(kL a), while plug flow requires τ = 4.605/(kL a). The ideal CSTR mean residence time is therefore about 21.5 times larger.

Figure 15. Residence time penalty caused by backmixing for 99% conversion. Source: Author calculation using Equations (26) to (28).

5. Numerical Results

5.1 Principal 99% residence time results

Table 5 reports the principal completion time. Each diameter is treated as the representative d₃₂ maintained during the active transfer period. The values do not include a design margin for injection maldistribution, nonideal residence time distribution, soap film resistance, coalescence, property uncertainty, or analytical sampling lag.

Table 5. Calculated ideal plug flow residence time for 99% neutralization

d₃₂ (µm)	Re = 50	Re = 5,000	Re = 50,000
1	0.096 s	9.71 ms	3.07 ms
5	2.41 s	0.243 s	0.077 s
10	9.63 s	0.971 s	0.307 s
30	1.44 min	8.74 s	2.77 s
50	4.01 min	24.28 s	7.68 s
100	16.05 min	1.62 min	30.73 s
150	36.12 min	3.64 min	1.15 min
200	1.07 h	6.48 min	2.05 min
250	1.67 h	10.12 min	3.20 min
300	2.41 h	14.57 min	4.61 min

Figure 16. Calculated 99% neutralization residence time, principal result. Source: Author calculation using Equation (20).

5.2 Completion criterion sensitivity

The 90%, 99%, and 99.9% times differ only by the logarithmic conversion factor. For a constant kL a, t99 is twice t90, and t99.9 is 1.5 times t99. Exact 100% cannot be reached in the exponential model.

Figure 17. Completion criterion at Re = 50. Source: Author calculation.

Figure 18. Completion criterion at Re = 5,000. Source: Author calculation.

Figure 19. Completion criterion at Re = 50,000. Source: Author calculation.

5.3 Residual FFA versus time

The decay curves show that coarse droplets retain a long kinetic tail. At each Re case, reducing d from 300 µm to 100 µm reduces the required time by a factor of nine, and reducing from 100 µm to 50 µm reduces it by a factor of four. These ratios follow the d² scaling in Equation (21).

Figure 20. Predicted residual FFA versus time at Re = 50. Source: Author calculation.

Figure 21. Predicted residual FFA at Re = 5,000. Source: Author calculation.

Figure 22. Predicted residual FFA at Re = 50,000. Source: Author calculation.

5.4 Maximum diameter for an available contact time

$$d_{\max ,99}={\left(\frac{6\varphi \mathrm{Sh}{D}_{\mathrm{FFA}}t}{\ln (100)}\right)}^{1/2}$$

(29)

Equation (29) is useful for preliminary reactor selection. For example, a 60 s ideal plug flow contact permits a maximum diameter of approximately 25.0 µm at Re = 50, 78.6 µm at Re = 5,000, and 139.7 µm at Re = 50,000 under the same extrapolated correlation assumptions.

Figure 23. Maximum representative droplet diameter that meets 99% conversion at a selected plug flow contact time. Source: Author calculation using Equation (29).

Table 6. Largest d₃₂ meeting 99% neutralization

Contact time (s)	Re = 50 (µm)	Re = 5,000 (µm)	Re = 50,000 (µm)
10	10.19	32.09	57.04
30	17.65	55.58	98.80
60	24.96	78.60	139.72
120	35.30	111.15	197.60
300	55.81	175.75	312.43
600	78.93	248.55	441.84

5.5 Compact result heat map

Figure 24. Heat map of the calculated 99% residence time in minutes. Source: Author calculation. Numbers in cells are minutes.

6. Physical Consistency of the Requested Reynolds Numbers

6.1 Reynolds number definition ambiguity

A Reynolds number is not a complete operating condition unless its characteristic velocity and length are defined. In an agitated vessel, mixer Reynolds number is commonly based on impeller speed and diameter. In a pipe or static mixer, it is based on bulk velocity and hydraulic diameter. In the mass transfer correlation used here, Re is based on droplet diameter and relative velocity. These three definitions are not interchangeable.

$${\mathrm{Re}}_{\mathrm{mixer}}=\frac{\rho N{D}_i^2}{\mu }\mathrm{or}{\mathrm{Re}}_{\mathrm{pipe}}=\frac{\rho U{D}_h}{\mu }$$

(30)

If 50, 5,000, and 50,000 are mixer or pipe Reynolds numbers, they describe the bulk flow regime but do not directly provide the local droplet Re, energy dissipation, droplet size, or Sherwood number. A mixer geometry and power model are required.

6.2 Implied relative velocity

$$u_{\mathrm{rel}}=\frac{{\mathrm{Re}}_d{\mu }_o}{{\rho }_{o}d}$$

(31)

Applying Equation (31) literally produces very large velocities, especially for micron scale droplets. For example, Re = 50 at d = 100 µm requires a relative velocity of approximately 6.47 m/s. Re = 5,000 at the same diameter requires about 647 m/s. Such velocities cannot represent stable sodium hydroxide droplets in hot oil.

Figure 25. Relative velocity implied by a literal droplet Reynolds number interpretation. Source: Author calculation using Equation (31).

6.3 Weber number and breakup

$$\mathrm{We}=\frac{{\rho }_o{u}_{\mathrm{rel}}^{2}d}{\sigma }$$

(32)

$$\mathrm{We}=\frac{{\mathrm{Re}}_d^2{\mu }_o^2}{{\rho }_o\sigma d}$$

(33)

$${\mathrm{Re}}_{\mathrm{We}=1}=\frac{{({\rho }_o\sigma d)}^{1/2}}{{\mu }_o}$$

(34)

At the specified interfacial tension, the Re corresponding to We = 1 is only about 0.045 for a 1 µm droplet and 0.781 for a 300 µm droplet. All requested Re values give We far above unity. The literal interpretation therefore violates the assumption that each specified droplet remains intact and fixed in size.

Figure 26. Weber number for the literal droplet Re interpretation. Source: Author calculation using Equation (33).

6.4 Energy dissipation and droplet breakup

$$d_{\max }=C_H{\left(\frac{\sigma }{{\rho }_o}\right)}^{3/5}{ϵ}^{-2/5}$$

(35)

$$ϵ={\left[\frac{C_H{(\sigma /{\rho }_o)}^{3/5}}{d_{\max }}\right]}^{5/2}$$

(36)

The Hinze relation connects the maximum stable droplet size to local turbulent energy dissipation. It shows why the mixer Reynolds number alone cannot determine d. The local energy dissipation, interfacial tension, density, residence in the high shear zone, viscosity ratio, and coalescence kinetics are all required. The low specified interfacial tension makes breakup to tens of micrometres possible at modest local dissipation, but 1 µm droplets still require an extremely intense local field. [11, 18]

Figure 27. Indicative local energy dissipation required for a selected maximum stable droplet size. Source: Author calculation using Hinze scaling [11]. Values are order of magnitude screening estimates.

Correct interpretation of the Re curves

The residence time curves answer the mathematical sensitivity question posed by the user. They do not establish that a mixer with bulk Re = 50,000 will deliver the Re = 50,000 curve. A validated workflow must predict or measure d₃₂ and kL a at the actual mixer operating point.

7. Engineering Interpretation and Process Design

7.1 Droplet size selection

The optimum reaction droplet size is not the optimum separator droplet size. Fine droplets improve contact and reduce neutralization time. Coarser droplets and agglomerates improve centrifugal removal. A practical process should therefore create a controlled dispersion in the caustic mixer, avoid unnecessary high shear during the reaction tail, and deliberately promote hydration, collision, and coalescence before the separator. [1, 3, 5, 18]

Table 7. Qualitative droplet size tradeoff

Representative size	Reaction effect	Separation and operating effect	Engineering interpretation
1 to 5 µm	Very high area and very short theoretical transfer time	Extremely fine soap rich tail, difficult separation, high local breakup energy for 1 µm	Use only if downstream coalescence and separation are demonstrated
10 to 30 µm	High area and practical reaction time in many staged systems	Requires controlled retention and measured separator feed distribution	Strong screening range for pilot trials
50 to 100 µm	Moderate area, residence time becomes important	Easier separation after reaction, but coarse tail can miss FFA target	Suitable only with adequate retention and verified kL a
150 to 300 µm	Low area, long transfer time	Separation favorable but localized caustic and incomplete reaction risk	Generally not preferred as the active reaction dispersion

7.2 Recommended equipment sequence

1. Meter the 15 wt% NaOH solution by mass ratio to oil flow. Use a multiport injection device or equivalent distribution method because the lye flow is only about 0.0368 vol% of the combined stream.

2. Use an adjustable inline rotor stator mixer, controlled static mixer, or other characterized dispersion device. Measure the outlet droplet distribution rather than inferring it from motor speed or pressure drop alone.

3. Provide a baffled tubular retention section, static retention mixer, or multiple compartments with a measured residence time distribution. Avoid a single large completely mixed vessel when short completion time is required.

4. Reduce shear after the primary contact period. Add any conditioning water in a separate low shear zone so that hydration and coalescence are not immediately reversed by renewed breakup.

5. Use a disc stack centrifuge to remove the bulk soapstock. Wash the neutral oil with hot water when required to reduce residual soap, then separate and dry.

6. Trend residual FFA, soap, moisture, phosphorus, neutral oil loss, droplet size before the separator, and heavy phase flow.

7.3 Practical residence time selection

A design residence time should exceed the ideal plug flow value. The margin must cover the measured residence time distribution, FFA assay uncertainty, caustic metering error, diffusion uncertainty, interfacial aging, and the coarse droplet tail. A reasonable preliminary design procedure is to select the measured d₉₅ or an area based d₃₂, calculate the ideal time, convert the actual reactor to an equivalent number of mixed stages, and then add a commissioning adjustment range.

For illustration, a measured d₃₂ of 50 µm gives an ideal 99% time of 4.01 min at effective Re = 50. A six stage equivalent reactor requires approximately 1.50 times the plug flow value, or about 6.0 min, before any property or operating margin. The same calculation at a calibrated effective Sh = 6 instead of the extrapolated Re = 50 correlation gives a much longer ideal time of approximately 145 min, demonstrating why direct kL a calibration is essential.

Table 8. 99% residence time sensitivity to effective Sherwood number

d₃₂ (µm)	Sh = 2	Sh = 6	Requested Re = 50 curve
10	17.38 min	5.79 min	9.63 s
30	2.61 h	52.14 min	1.44 min
50	7.24 h	2.41 h	4.01 min
100	28.97 h	9.66 h	16.05 min

7.4 Interfacial tension interpretation

The stated 0.3 mN/m is likely a dynamic or soap conditioned value rather than the clean oil and fresh lye value at the instant of injection. Clean vegetable oil and water interfaces are commonly tens of mN/m. Since the Hinze breakup size scales with σ³ᐟ⁵, the early contact zone may require substantially more power than a calculation using 0.3 mN/m. Dynamic interfacial tension should be measured over the millisecond to second time range relevant to the mixer. [8, 11, 18]

8. Model Limitations and Validation Plan

8.1 Material limitations

• The FFA is represented by oleic acid. Real soybean oil contains a distribution of fatty acids and polar minor components.

• The FFA diffusivity is not a measured property for this exact oil, temperature, and composition. The base value is an engineering estimate.

• The interfacial tension is held constant at 0.3 mN/m even though it changes as soap forms and as phosphatides and other surfactants adsorb.

• The model does not calculate interfacial rheology, Marangoni effects, spontaneous emulsification, soap gel formation, or liquid crystalline soap phases.

• The lye density is an engineering interpolation. The final soapstock density and viscosity are not calculated from phase equilibrium data.

8.2 Hydrodynamic limitations

• The requested Reynolds numbers are not tied to a defined geometry. Their use as droplet Re is a sensitivity assumption.

• The Ranz Marshall correlation is extrapolated far beyond its normal low to moderate Re use for two requested cases and far beyond clean interface conditions.

• The droplet size is treated as a constant representative d₃₂. Real droplets break, coalesce, grow by FFA transfer, and develop a broad distribution.

• The calculation assumes a dilute dispersion and neglects droplet interactions.

• The main time values assume ideal plug flow. Real retention vessels have backmixing, dead zones, and short circuiting.

8.3 Chemical and separation limitations

• Exact stoichiometry provides no margin for acid conditioning demand, gums, metals, or dosing error.

• The model does not calculate neutral oil saponification by excess NaOH.

• Neutralization completion and soap removal are separate constraints. A rapid reaction can still produce unacceptable residual soap or oil loss.

• The centrifuge is not sized in this report. Disc geometry, bowl speed, equivalent area, feed zone, density difference, viscosity, and separator residence time are required.

Figure 28. Residence time sensitivity to the assumed FFA diffusivity. Source: Author calculation. The selected 50 µm case is shown.

8.4 Recommended validation program

Figure 29. Measurement and calibration program required before equipment guarantees. Source: Author recommendation.

1. Measure soybean oil density and viscosity at the operating temperature and actual feed quality.

2. Measure dynamic oil and lye interfacial tension before reaction and during controlled soap formation.

3. Measure the droplet distribution immediately after the high shear mixer, during retention, and immediately before the centrifuge. Report d₁₀, d₃₂, d₅₀, d₉₀ or d₉₅, and the fine volume fraction.

4. Perform timed neutralization tests and fit kL a from ln(C/C0) versus time. Repeat at several mixer settings and caustic concentrations.

5. Run a salt or temperature tracer test to determine the retention time distribution and equivalent number of tanks in series.

6. Measure residual FFA, soap, moisture, phosphorus, heavy phase flow, and neutral oil loss during the same tests.

7. Provide the calibrated droplet distribution and kL a data to mixer and centrifuge vendors for final equipment selection.

9. Conclusions

1. Droplet size has a dominant effect on the calculated mass transfer rate. Under the base model, kL a varies approximately as d⁻² and the required residence time varies as d².

2. For the requested Re = 50 curve, the 99% time increases from 0.096 s at 1 µm to 9.63 s at 10 µm, 4.01 min at 50 µm, 16.05 min at 100 µm, and 2.41 h at 300 µm.

3. For Re = 5,000, the 99% times are 0.0097 s at 1 µm, 0.971 s at 10 µm, 24.28 s at 50 µm, 1.62 min at 100 µm, and 14.57 min at 300 µm.

4. For Re = 50,000, the 99% times are 0.00307 s at 1 µm, 0.307 s at 10 µm, 7.68 s at 50 µm, 30.73 s at 100 µm, and 4.61 min at 300 µm.

5. The literal droplet Re interpretation is not physically consistent with stable droplets at 0.3 mN/m. The high Re curves are therefore mathematical sensitivity cases.

6. If the Reynolds numbers are mixer or pipe Reynolds numbers, they cannot be converted to reaction time without geometry, power, local energy dissipation, and measured droplet size.

7. The recommended process uses high shear only to generate controlled contact, plug flow or staged retention to complete reaction, and low shear conditioning before disc stack separation.

8. Final residence time and equipment settings must be established from measured d₃₂, dynamic interfacial tension, residual FFA decay, and residence time distribution.

Recommended next engineering step

Run a controlled pilot matrix at three measured d₃₂ targets, for example 10, 30, and 50 µm, and at three retention times. Fit kL a directly from FFA decay, then use the measured separator feed distribution to select the best reaction and separation compromise.

Appendix A. Detailed Calculated Results

The tables below provide all requested droplet sizes. Times are reported in seconds to preserve calculation precision. For engineering selection, the formatted values in Table 5 are easier to interpret.

Table A1. Detailed mass transfer results at effective Re = 50

d (µm)	Sh	kL (m/s)	a (m²/m³)	kL a (1/s)	t90 (s)	t99 (s)	t99.9 (s)
1	216.535	2.1653e-02	2,208.046	4.7812e+01	0.0481593	0.0963186	0.144478
5	216.535	4.3307e-03	441.609	1.9125e+00	1.20398	2.40796	3.61195
10	216.535	2.1653e-03	220.805	4.7812e-01	4.81593	9.63186	14.4478
30	216.535	7.2178e-04	73.602	5.3124e-02	43.3434	86.6867	130.03
50	216.535	4.3307e-04	44.161	1.9125e-02	120.398	240.796	361.195
100	216.535	2.1653e-04	22.080	4.7812e-03	481.593	963.186	1444.78
150	216.535	1.4436e-04	14.720	2.1250e-03	1083.58	2167.17	3250.75
200	216.535	1.0827e-04	11.040	1.1953e-03	1926.37	3852.74	5779.11
250	216.535	8.6614e-05	8.832	7.6499e-04	3009.96	6019.91	9029.87
300	216.535	7.2178e-05	7.360	5.3124e-04	4334.34	8668.67	13003

Table A2. Detailed mass transfer results at effective Re = 5,000

d (µm)	Sh	kL (m/s)	a (m²/m³)	kL a (1/s)	t90 (s)	t99 (s)	t99.9 (s)
1	2,147.347	2.1473e-01	2,208.046	4.7414e+02	0.0048563	0.0097126	0.0145689
5	2,147.347	4.2947e-02	441.609	1.8966e+01	0.121407	0.242815	0.364222
10	2,147.347	2.1473e-02	220.805	4.7414e+00	0.48563	0.97126	1.45689
30	2,147.347	7.1578e-03	73.602	5.2683e-01	4.37067	8.74134	13.112
50	2,147.347	4.2947e-03	44.161	1.8966e-01	12.1407	24.2815	36.4222
100	2,147.347	2.1473e-03	22.080	4.7414e-02	48.563	97.126	145.689
150	2,147.347	1.4316e-03	14.720	2.1073e-02	109.267	218.533	327.8
200	2,147.347	1.0737e-03	11.040	1.1854e-02	194.252	388.504	582.756
250	2,147.347	8.5894e-04	8.832	7.5863e-03	303.519	607.037	910.556
300	2,147.347	7.1578e-04	7.360	5.2683e-03	437.067	874.134	1311.2

Table A3. Detailed mass transfer results at effective Re = 50,000

d (µm)	Sh	kL (m/s)	a (m²/m³)	kL a (1/s)	t90 (s)	t99 (s)	t99.9 (s)
1	6,786.184	6.7862e-01	2,208.046	1.4984e+03	0.00153667	0.00307335	0.00461002
5	6,786.184	1.3572e-01	441.609	5.9937e+01	0.0384169	0.0768337	0.115251
10	6,786.184	6.7862e-02	220.805	1.4984e+01	0.153667	0.307335	0.461002
30	6,786.184	2.2621e-02	73.602	1.6649e+00	1.38301	2.76601	4.14902
50	6,786.184	1.3572e-02	44.161	5.9937e-01	3.84169	7.68337	11.5251
100	6,786.184	6.7862e-03	22.080	1.4984e-01	15.3667	30.7335	46.1002
150	6,786.184	4.5241e-03	14.720	6.6596e-02	34.5752	69.1504	103.726
200	6,786.184	3.3931e-03	11.040	3.7461e-02	61.467	122.934	184.401
250	6,786.184	2.7145e-03	8.832	2.3975e-02	96.0422	192.084	288.127
300	6,786.184	2.2621e-03	7.360	1.6649e-02	138.301	276.601	414.902

Appendix B. Worked Calculation for 50 µm Droplets at Re = 50

This example shows the numerical substitutions used in the principal model.

Table B1. Numerical worked example

Step	Substitution	Result
1. FFA mass	mFFA = 1,000(500 × 10⁻⁶)	0.500 kg
2. Pure NaOH	mNaOH = 0.500(40.00/282.47)	0.070804 kg
3. 15 wt% lye	mlye = mNaOH/0.15	0.472027 kg
4. Lye volume	Vlye = mlye/1,130.5	0.000417538 m³
5. Dispersed fraction	φ = Vlye/(Voil + Vlye)	0.000368008
6. Schmidt number	Sc = 0.0114/[881.7(1 × 10⁻¹⁰)]	129,295.7
7. Sherwood number	Sh = 2 + 0.6(50)¹ᐟ²Sc¹ᐟ³	216.535
8. Film coefficient	kL = ShD/d	4.330694e-04 m/s
9. Area density	a = 6φ/d	44.160916 m²/m³
10. Volumetric coefficient	kL a = kL(a)	1.912474e-02 s⁻¹
11. 99% time	t99 = ln(100)/(kL a)	240.796 s = 4.013 min

Appendix B.1 Check of internal diffusion

For d = 50 µm and Daq = 1 × 10⁻⁹ m²/s, Equation (23) gives tdiff,aq = 0.0633 s. This is much shorter than the calculated oil side 99% time of 240.8 s. The base model therefore treats oil side transport as controlling. If a soap rich structure lowers the internal diffusivity to 1 × 10⁻¹¹ m²/s, the internal estimate becomes 6.33 s and is no longer negligible.

Appendix B.2 Check of physical Reynolds number interpretation

For d = 50 µm, Re = 50 implies urel = 12.93 m/s and We = 24,566. The very large Weber number confirms that the Re input cannot describe a stable 50 µm droplet at σ = 0.3 mN/m. The calculated residence time remains a correlation sensitivity result only.

Appendix C. Sensitivity and Reactor Hydraulics

Appendix C.1 Diffusivity scaling

Because Sh also contains Sc, the Ranz Marshall model does not give exactly t proportional to 1/D. At high Sc and a large convective contribution, Sh is approximately proportional to D^(-1/3), so kL a is approximately proportional to D^(2/3), and t approximately follows D^(-2/3). This still creates substantial uncertainty when D is not measured.

Figure C1. Diffusivity sensitivity at d = 50 µm. Source: Author calculation.

Appendix C.2 Tanks in series formula

$${\tau }_N=\frac{N[{(C_{\mathrm{in}}/C_{\mathrm{out}})}^{1/N}-1]}{k_{L}a}$$

(37)

Table C1. Hydraulic residence time penalty

Equal mixed stages	τ/τPFR for 99%	Relative reactor volume
1	21.498	2149.8%
2	3.909	390.9%
4	1.878	187.8%
6	1.504	150.4%
8	1.352	135.2%
10	1.270	127.0%
16	1.159	115.9%
32	1.076	107.6%
Ideal plug flow	1.000	100.0%

Bibliography

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Nomenclature

Symbols are listed in the order used in the model. Units are SI unless otherwise stated.

Table N1. Roman symbols

Symbol	Definition	Unit
a	Interfacial area per unit total mixture volume	m²/m³
A	Total interfacial area	m²
C_FFA	Free fatty acid concentration in the oil phase	ppm, kg/m³, or mol/m³
C_H	Hinze correlation constant	dimensionless
D_FFA	Diffusion coefficient of FFA in oil	m²/s
D_aq	Effective internal diffusivity in the aqueous or soap rich droplet	m²/s
d	Representative droplet diameter	m or µm
d₃₂	Sauter mean droplet diameter	m or µm
d_max	Maximum stable or allowable droplet diameter	m or µm
f₂	Natural capillary frequency for the n = 2 mode	Hz
k_L	Oil side liquid film mass transfer coefficient	m/s
k_L a	Volumetric mass transfer coefficient	s⁻¹
K_o	Overall mass transfer coefficient based on the oil phase	m/s
m	Phase distribution coefficient in the resistance model	dimensionless
m_FFA	Mass of free fatty acids	kg
m_NaOH	Mass of pure sodium hydroxide	kg
m_lye	Mass of sodium hydroxide solution	kg
M	Molar mass	kg/kmol
N	Number of equal completely mixed stages	dimensionless
N_d	Number of droplets	dimensionless
N	Impeller rotational speed when used in Re_mixer	s⁻¹
R	Droplet radius	m
R_int	Interfacial mass transfer resistance	s/m
Re_d	Droplet Reynolds number	dimensionless
Re_mixer	Mixer Reynolds number	dimensionless
Re_pipe	Pipe Reynolds number	dimensionless
Sc	Schmidt number	dimensionless
Sh	Sherwood number	dimensionless
t	Time	s or min
t_X	Time to fractional conversion X	s or min
t_diff,aq	Internal aqueous diffusion time estimate	s
u_rel	Droplet relative velocity	m/s
U	Bulk pipe velocity	m/s
V	Volume	m³
We	Weber number	dimensionless
w	Mass fraction	dimensionless
X	Fractional FFA conversion	dimensionless

Table N2. Greek symbols

Greek symbol	Definition	Unit
ε	Local turbulent energy dissipation per unit mass	W/kg
φ	Initial dispersed lye volume fraction	dimensionless
μ_o	Oil dynamic viscosity	Pa s
ρ_o	Oil density	kg/m³
ρ_d	Droplet phase density	kg/m³
σ	Oil and aqueous interfacial tension	N/m
τ	Mean hydraulic residence time	s or min
ω₂	Angular capillary frequency for the n = 2 mode	rad/s
π	Circular constant	dimensionless

Subscripts and Abbreviations

Table N3. Subscripts and abbreviations

Term	Meaning
aq	Aqueous phase
in	Reactor inlet
out	Reactor outlet
o	Oil phase
rxn	Reaction product
FFA	Free fatty acids
PFR	Plug flow reactor
CSTR	Continuous stirred tank reactor
IFT	Interfacial tension
NaOH	Sodium hydroxide
ppm	Parts per million by mass unless noted otherwise