Rate Modelling of Alkali Gelatinization at Low Conversions
Osoka Emmanuel CHIBUIKE
Department of Chemical Engineering, Federal University of Technology P.M.B. 1526, OwerriImo State, Nigeria.
Email: emmaosoka@yahoo.com
Received: 30 March 2010 / Accepted: 20 December 2010 / Published: 30 December 2010
Abstract
The rate of starch gelatinisation under strong alkali conditions was modeled at low conversion (x < 0.4), with the degree of gelatinisation (conversion) defined in terms of sample viscosity. Experimental data at low conversion were fit to eleven rate models based on the mechanism of the unreactedcore model and the rate controlling steps determined. Film diffusion (stokes regime) plus Product layer diffusion steps control the rate of reaction for all sodium hydroxide concentrations and at low conversion (x < 0.4), with the dominance shifting from Film diffusion to Product layer diffusion as sodium hydroxide concentration is increased.
Keywords
Starch; Gelatinisation; Viscosity; Fluidity; Rheological; Equilibrium
Introduction
Starch is one of the most common substances existing in nature and is the major basic constituent of the average diet. The most important practical property of starch is its gelatinisation under the influence of heat and/or certain chemical in an aqueous medium to give starch paste.
The chemical gelatinisation of starchunder strong alkali conditions is the basic industrial reaction in the conversion of starch to adhesives (glue) [14].
The rate of starch gelatinisation under strong alkali conditions has be studied rheologically, with the degree of gelatinisation defined in terms of viscosity and modeled on the mechanism of the unreactedcore model. Two steps at most are rate controlling [59]. The shift in the rate controlling step with conversion makes it needful to study the rate data presented in our earlier work at different conversion levels.
The aim of this research is to develop a model for starch gelatinization under strong alkali conditions that will be useful in predicting viscosity and by extension, bond streght of starch base adhesives.
Material and Method
The following model equations apply as given below:
OneStep Controls
A1 Diffusion through fluid film controls (stokes regime)
(tt_{dA1}) = τ_{F1}(1(1x)^{2/3}) 
(1) 
A2 Diffusion through fluid film controls (nonstokes regime)
(tt_{dA2}) = τ_{F2}(1(1x)^{1/2}) 
(2) 
B Diffusion through Product layer controls
(tt_{dB}) = τ_{p} (13(1x)^{2/3} + 2(1x)) 
(3) 
C Chemical reaction controls
(tt_{dC}) = τ_{C} (1(1x)^{1/3}) 
(4) 
TwoStep Controls
D1 Film diffusion plus Product layer controls (stokes regime)
(tt_{dD1}) = τ_{F1} (1(1x)^{2/3}) + τ_{P} (13(1x)^{2/3}+2(1x)) 
(5) 
D2 Film diffusion plus Product layer controls (nonstokes regime)
(tt_{dD2}) = τ_{F2} (1(1x)^{1/2}) + τ_{P} (13(1x)^{2/3}+2(1x)) 
(6) 
E1 Film diffusion plus Chemical reaction controls (stokes regime)
(tt_{dE1}) = τ_{F1} (1(1x)^{2/3}) + τ_{C} (1(1x)^{1/3}) 
(7) 
E2 Film diffusion plus Chemical reaction controls (nonstokes regime)
(tt_{dE2}) = τ_{F2} (1(1x)^{1/2})^{ }+ τ_{C} (1(1x)^{1/3}) 
(8) 
F Product layer diffusion plus Chemical reaction controls
(tt_{dF}) = τ_{p} (13(1x)^{2/3 }+2(1x)) + τ_{C} (1(1x)^{1/3}) 
(9) 
ThreeStep Controls
G1 Film diffusion plus Product layer plus Chemical reaction controls (stokes regime)
(tt_{dG1}) = τ_{F1} (1(1x)^{2/3})^{ }+τ_{p} (13(1x)^{2/3}+2(1x))+ τ_{C} (1 (1x)^{1/3}) 
(10) 
G2 Film diffusion plus Product layer plus Reaction controls (nonstokes regime)
(tt_{dG2}) = τ_{F2} (1(1x)^{1/2})+τ_{p} (13(1x)^{2/3}+2(1x))+ τ_{C} (1 (1x)^{1/3}) 
(11) 
These rate equations (Eq. 1 – Eq. 11) can be used with the MATLAB curve fitting toolbox without modification, for analysis of experimental data.
Results and Discussion
The experimental results are presented in Table 17. Graphical representations of the models are presented in Figures 13.
Table 1. Numerical fit Results for 17g/dm^{3} NaOH Concentration at low conversion
Model 
t_{d}(secs) 
t_{F1}(secs) 
t_{F2}(secs) 
t_{P}(secs) 
t_{C}(secs) 
R^{2} 
RMSE 
SSE 
A1 
186.7 
272.1 



0.9983 
0.863 
2.981 
A2 
187.8 

344.9 


0.9977 
1.013 
4.108 
B 
209.4 


927 

0.9649 
3.917 
61.37 
C 
188.9 



491.6 
0.9969 
1.168 
5.456 
D1 
186.7 
272.1 

1.21E7 

0.9983 
0.863 
2.981 
D2 
187.8 

344.9 
1.39E7 

0.9977 
1.013 
4.108 
E1 
186.7 
272.1 


3.46E11 
0.9983 
0.863 
2.981 
E2 
187.8 

344.9 

4.33E7 
0.9977 
1.013 
4.108 
F 
188.9 


1.03E7 
491.6 
0.9969 
1.168 
5.456 
G1 
186.7 
272.1 

3.42E13 
1.65E11 
0.9983 
0.863 
2.981 
G2 
187.8 

344.9 
3.69E8 
2.27E14 
0.9977 
1.013 
4.108 
Table 2. Numerical fit Results for 18g/dm^{3} NaOH Concentration at low conversion
Mode1 
t_{d}(secs) 
t_{F1}(secs) 
t_{F2}(secs) 
t_{P}(secs) 
t_{C}(secs) 
R^{2} 
RMSE 
SSE 
A1 
149.4 
304.8 



0.9985 
1.025 
6.31 
A2 
150.2 

389.9 


0.9979 
1.225 
9.005 
B 
168.1 


1231 

0.9386 
6.555 
257.8 
C 
150.9 



560.8 
0.9970 
1.440 
12.45 
D1 
149.4 
304.8 

3.63E11 

0.9985 
1.025 
6.31 
D2 
150.2 

389.9 
1.15E7 

0.9979 
1.225 
9.005 
E1 
149.4 
304.8 


3.46E12 
0.9985 
1.025 
6.31 
E2 
150.2 

389.9 

1.19E7 
0.9979 
1.225 
9.005 
F 
150.9 


5.36E10 
560.8 
0.9970 
1.44 
12.45 
G1 
149.4 
304.8 

1.02E9 
2.47E14 
0.9985 
1.025 
6.31 
G2 
150.2 

389.9 
1.71E8 
6.18E10 
0.9979 
1.225 
9.005 
Table 3. Numerical fit Results for 19g/dm^{3} NaOH Concentration at low conversion
Model 
t_{d}(secs) 
t_{F1}(secs) 
t_{F2}(secs) 
t_{P}(secs) 
t_{C}(secs) 
R^{2} 
RMSE 
SSE 
A1 
107.7 
257.7 



0.9903 
2.327 
27.07 
A2 
108.2 

330.9 


0.9886 
2.529 
31.98 
B 
121 


1128 

0.8924 
7.761 
301.2 
C 
108.7 



477.7 
0.9867 
2.733 
37.35 
D1 
107.7 
257.7 

2.79E12 

0.9903 
2.327 
27.07 
D2 
108.2 

330.9 
2.36E11 

0.9886 
2.529 
31.98 
E1 
107.7 
257.7 


1.25E9 
0.9903 
2.327 
27.07 
E2 
108.2 

330.9 

9.65E5 
0.9886 
2.828 
31.98 
F 
108.7 


5.33E11 
477.7 
0.9867 
2.733 
37.35 
G1 
107.7 
257.7 

8.65E7 
4.88E5 
0.9903 
2.602 
27.07 
G2 
108.2 

330.9 
9.54E8 
2.22E14 
0.9886 
2.529 
31.98 
Table 4. Numerical fit Results for 20g/dm^{3} NaOH Concentration at low conversion
Model 
t_{d}(secs) 
t_{F1}(secs) 
t_{F2}(secs) 
t_{P}(secs) 
t_{C}(secs) 
R^{2} 
RMSE 
SSE 
A1 
80.46 
233.7 



0.9995 
0.4853 
0.9421 
A2 
81.36 

296.4 


0.9998 
0.3289 
0.4328 
B 
99.06 


796.2 

0.9742 
3.359 
45.13 
C 
82.24 



422.6 
0.9999 
0.2492 
0.2484 
D1 
82.42 
208.2 

89.67 

0.9999 
0.2843 
0.2425 
D2 
82.38 

278.6 
49.15 

0.9999 
0.2853 
0.2441 
E1 
82.24 
1.136e8 


422.6 
0.9999 
0.2492 
0.2484 
E2 
82.24 

5.51E8 

422.6 
0.9999 
0.2492 
0.2484 
F 
82.32 


3.845 
420.6 
0.9999 
0.2871 
0.2474 
G1 
82.53 
206.7 

94.71 
5.80E4 
0.9999 
0.3498 
0.2447 
G2 
82.38 

278.6 
49.15 
2.51E6 
0.9999 
0.3494 
0.2441 
Table 5. Numerical fit Results for 21g/dm^{3} NaOH Concentration at low conversion
Model 
t_{d}(secs) 
t_{F1}(secs) 
t_{F2}(secs) 
t_{P}(secs) 
t_{C}(secs) 
R^{2} 
RMSE 
SSE 
A1 
54.49 
235.4 



0.9979 
1.081 
5.839 
A2 
55.06 

300.6 


0.9971 
1.274 
8.116 
B 
68.76 


908.1 

0.9305 
6.237 
194.5 
C 
55.63 



431.5 
0.9961 
1.482 
10.98 
D1 
54.49 
235.4 

7.08E10 

0.9979 
1.081 
5.839 
D2 
55.06 

300.6 
2.07E11 

0.9971 
1.274 
8.116 
E1 
54.49 
235.4 


4.14E11 
0.9979 
1.081 
5.839 
E2 
55.06 

300.6 

7.13E10 
0.9971 
1.274 
8.116 
F 
55.63 


3.63E12 
431.5 
0.9961 
1.482 
10.98 
G1 
54.49 
235.4 

5.55E12 
2.39E10 
0.9979 
1.081 
5.839 
G2 
55.06 

300.6 
8.73E9 
3.08E7 
0.9971 
1.274 
8.116 
Table 6. Numerical fit Results for 22g/dm^{3} NaOH Concentration at low conversion
Model 
t_{d}(secs) 
t_{F1}(secs) 
t_{F2}(secs) 
t_{P}(secs) 
t_{C}(secs) 
R^{2} 
RMSE 
SSE 
A1 
14.97 
148.1 



0.9990 
0.5226 
0.8195 
A2 
15.26 

189.6 


0.9994 
0.4087 
0.5011 
B 
22.87 


608.9 

0.9637 
3.148 
29.73 
C 
15.56 



272.9 
0.9996 
0.3270 
0.3208 
D1 
15.81 
131.1 

73.03 

0.9996 
0.3859 
0.2978 
D2 
15.8 

175.2 
48.13 

0.9996 
0.3829 
0.2933 
E1 
15.56 
7.39E8 


272.9 
0.9996 
0.3270 
0.3208 
E2 
15.56 

1.23E5 

272.9 
0.9996 
0.4005 
0.3208 
F 
15.78 


20.81 
263.9 
0.9997 
0.3777 
0.2853 
G1 
15.63 
8.71E3 

5.78 
270.2 
0.9996 
0.5514 
0.304 
G2 
15.78 

2.22E14 
20.81 
263.9 
0.9997 
0.3777 
0.2853 
Table 7. Numerical fit Results for 23g/dm^{3} NaOH Concentration at low conversion
Model 
t_{d}(secs) 
t_{F1}(secs) 
t_{F2}(secs) 
t_{P}(secs) 
t_{C}(secs) 
R^{2} 
RMSE 
SSE 
A1 
2.854 
124.7 



0.9963 
0.8291 
1.375 
A2 
3.151 

159.5 


0.9972 
0.7227 
1.045 
B 
10.33 


506.7 

0.9860 
1.606 
5.157 
C 
3.444 



229.3 
0.9979 
0.6164 
0.760 
D1 
5.281 
82.88 

173.6 

0.9999 
0.1597 
0.026 
D2 
5.272 

110.8 
157.8 

0.9999 
0.1581 
0.025 
E1 
3.444 
2.16E9 


229.3 
0.9979 
0.6164 
0.760 
E2 
3.444 

1.11E10 

229.3 
0.9979 
0.6164 
0.760 
F 
5.258 


140.5 
166.9 
0.9999 
0.1552 
0.024 
G1 
5.296 
2.861e4 

143.3 
165.7 
0.9999 

0.0244 
G2 
5.258 

7.008e6 
140.5 
166.9 
0.9999 

0.0241 
Figure 1. Low Conversion against time (secs) for all rate models (18g/dm^{3} NaOH Conc.)
Figure 2. Low Conversion against time (secs) for all rate models (20g/dm^{3} NaOH Conc.)
Figure 3. Low Conversion against time (secs) for all rate models (22g/dm^{3} NaOH Conc.)
All the models gave good fit for data obtained using sodium hydroxide concentration of 17g/dm^{3}, except the model for Product layer diffusion control (B). The numerical fits show that the model for Film diffusion control, under stokes regime (A1) gave the best fit (R^{2} = 0.9983). The TwoStep and ThreeStep control models do not improve the numerical fits and values of time for complete conversion show that only OneStep Film diffusion dominates, with other steps having values of time for complete conversion approximately equal to zero.
Fit results for data obtained using sodium hydroxide concentration of 18g/dm^{3} also show that the Film diffusion step alone controls the rate. The model for Film diffusion control, under stokes regime (A1) gives the best fit (R^{2} = 0.9985). TwoStep and ThreeStep control models do not improve the fit results.
Fit results for data obtained using sodium hydroxide concentration of 19g/dm^{3}also show that the Film diffusion step alone controls the rate. The model for Film diffusion control, under stokes regime (A1) gives the best fit (R^{2} = 0.9903) and the TwoStep and ThreeStep models offer no improvement.
All models gave very good fit results for data obtained using sodium hydroxide concentration of 20g/dm^{3} except the model for Product layer diffusion control (B). The OneStep control model that gave the best fit was Chemical reaction control (C), followed closely by film diffusion control under nonstokes regime (A2) and Film diffusion control under stokes regime (A1) with R^{2} values of 0.9999, 0.9998 and 0.9995 respectively.
The TwoStep control models offer some improvement with the Chemical reaction control step dominating, while the ThreeStep control models gave poorer numerical fits than the TwoStep control models. Therefore two steps, at most, will be ratecontrolling.
Fit results from data obtained using sodium hydroxide concentration of 21g/dm^{3} show that Film diffusion alone controls the rate. TwoStep and ThreeStep control models gave no improvement of the numerical fits. All other steps had values of time for complete conversion approximately equal to zero. The model for Film diffusion control, under stokes regime (A1) gave the best fit (R^{2} = 0.9979).
All the models gave good fit results for data obtained using sodium hydroxide concentration of 22g/dm^{3} except the model for Product layer diffusion control (B). The OneStep control model that gave the best fit was Chemical reaction control (C) with R^{2} = 0.9996. The TwoStep control models offer some improvement to the fit with the model for Product layer diffusion plus Chemical reaction control (F) giving the best fit (R^{2} = 0.9997). The ThreeStep control models offer no improvement of the fit results.
For data obtained using sodium hydroxide concentration of 23g/dm^{3}, all the models gave good fit results except the model for Product layer diffusion control (B). The OneStep control model that gave the best fit was Chemical reaction control (C) with R^{2} = 0.9979. The TwoStep control models offer some improvement to the fit with the model for Product layer diffusion plus Chemical reaction control (F) giving the best fit (R^{2} = 0.9999). The ThreeStep control models offer little improvement to the graphical and numerical fit results, but also show that Product layer diffusion and Chemical reaction steps dominate.
No OneStep control model gave the bets fit to data for all concentrations of sodium hydroxide used. The model for Film diffusion, under stokes regime (A1) gave the best fit at low concentrations of sodium hydroxide, with its associated TwoStep control models (D1 and E1) having similar fit results.
At higher values of sodium hydroxide concentration, where Product layer diffusion and Chemical reaction steps play a prominent role, the models for Product layer diffusion plus Chemical reaction control (F) offer the best fit, with fit results very close to those of D1 and D2.
Therefore TwoStep control model for Film diffusion (stokes regime) plus Product layer diffusion control (D1) gives the overall best fit for low conversion data obtained for all sodium hydroxide concentrations. Its fit results show that Film diffusion dominates at low sodium hydroxide concentrations, while Product layer diffusion dominates at higher sodium hydroxide concentrations.
Starch gelatinisation under strong alkali conditions is a heterogeneous (fluidparticle) reaction based on the mechanism of the unreactedcore model. The individual granules adsorb alkali and gelatinise when the amount of adsorbed alkali exceeds a certain threshold concentration [1, 10].
Conclusion
Two steps at most control the rate of the process for all concentrations of sodium hydroxide studied. Film diffusion (stokes regime) plus Product layer diffusion steps control the rate of reaction for all sodium hydroxide concentrations and at low conversion (x < 0.4), with the dominance shifting from Film diffusion to Product layer diffusion as sodium hydroxide concentration is increased.
The model equation for this is given as:
(tt_{dD1}) = τ_{F1}(1(1x)^{2/3})+τ(13(1x)^{2/3} + 2(1x))
It is recommended that the model from this work be used to effectively study the effect of variables like: waterstarch ratios and rate of agitation on the kinetics of starch gelatinisation under strong alkali conditions, so as to develop a robust model that can be used to optimize the process of producing adhesives from (cassava) starch.
References
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