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The kinetics of protein enzymatic extraction from the amaranth stalk.

1. INTRODUCTION

Most of the so far published publications deal with amaranth as a food supplement for healthy population or as a possible compound in special diets of diabetic patients or people with protein allergy; new amaranth products are conducive against civilization diseases. On the other hand, amaranth starts to be used also in different areas -medicine, pharmaceutics and cosmetics. (Paredes-Lopez, 1994)

Further research is in progress, for example in amaranth biomass treatment. The rest of the plant after conversion of basic parts contains still high percent of protein, depending on growing season. Therefore amaranth biomass could be successfully used in foodstuff industry or for production of very pure biogas. Hardly ever are published papers which solve production of separate valuable amaranth compounds and optimalization of particular technologies from the process engineering point of view. This is important especially in the processing of high amount of raw material that means for industry requirements. (Kolomaznik, Kupec, 2000)

The purpose of our present experiments is to enrich the solid phase of amaranth stalk with the vegetable protein by hydrolysis. The kinetics of these reactions and modeling of concentration fields are presented in our paper.

2. MATHEMATICAL MODEL OF PROTEIN ENZYMATIC EXTRACTION

2.1 The kinetics of protein enzymatic extraction from the amaranth stalk

In ascertained stadium of plant development, the shoot part contains considerable amount of protein. With regard to the plant weight, the highest volume of protein occurs in the stalk. Hence, we were engaged with the quantitative model, where the footstalk was approximated with an infinite cylinder.

Some of the papers dealt already with concentration field computation according to a model of 'infinite tablet', published in (Crank, 1975) for instance. A work dealing with approximation of infinite cylinder is not common putted out. In our case we applied a diffusion model using cylindrical coordinates, we obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

[partial derivative]c/[partial derivative]r (0, [tau]) = 0 (2)

-SD [partial derivative]c([R.sub.1], [tau])/[partial derivative]r = [V.sub.0] [partial derivative][c.sub.0] / [partial derivative][tau] (3)

c([R.sub.1], [tau]) = [epsilon][c.sub.0] (4)

c(r, 0) = [c.sub.p]; [c.sub.0] (0) = 0 (5)

Equation (1) responds the concentration field in the stalk during the extraction process, (2) presents the axis symmetry of the concentration field, (3) describes the solid-state balance equality of the surface diffusion flow with the accumulation speed of the hydrolysed protein in the aqueous solution. (4) is a complete mixing condition of the aqueous phase and (5) are starting conditions.

Due to generalization and an easier way of solving we vote following dimensionless quantities:

C = c/[c.sub.p] (6)

[C.sub.0] = [epsilon][c.sub.0]/[c.sub.p] (7)

R = r/[R.sub.1] (8)

[F.sub.0] = [D.sub.[tau]]/[R.sub.1.sup.2] (9)

With the application of non-dimensional magnitudes (2) to (5) for the equations (6) to (9) we obtain an extraction model in dimensionless formulation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

[partial derivative]C / [partial derivative]R (1, [F.sub.0]) = 0 (11)

[partial derivative]C / [partial derivative]R (1, [F.sub.0]) = Na / 2[epsilon] [partial derivative][C.sub.0] ([F.sub.0]) / [partial derivative][F.sub.0] (12)

C(1, [F.sub.0]) = [C.sub.0] ([F.sub.0]) (13)

C(R, 0) = 1; [C.sub.0] (0) = 0 (14)

Considering the equation (11) to (14) represent a linear system we compute the solution with the Laplace transform. The Laplace faces gain to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

We obtain:

s[C.sub.L] - 1 = [C.sub.L] + 1 / R [C.sub.L] (17)

[C.sub.L] (0, s) = 0 (18)

[C.sub.L] x (1,s) = Na / [epsilon] s x [C.sub.0L] (19)

[C.sub.L] (1,s) = [C.sub.0L](s) (20)

With respect to the equation (18) the solution of the differential equation (17) is consequent:

[C.sub.L] = [AI.sub.0] (R[square root of s]) + 1/s (21)

where [I.sub.0] is modified Bessel function of the first kind, zero order. By analogy, [I.sub.1] is modified Bessel function of the first kind, first order. The A constant is determined from the equation (22)

A = - Na / [2[epsilon][I.sub.1] ([square root of s]) / [square root of s] + Na[I.sub.0] ([square root of s])] (22)

Including this, the Laplace face of the hydrolyzed protein concentration field is consecutive:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

Equation is symbolical rewritten to

[C.sub.L] - 1 / s = [F.sub.(s) / [sf.sub.(s)] (24)

for s = 0, f(s) [not equal to] 0 count

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

A second account is s[not equal to]0, f(s)=0

2[epsilon][J.sub.1] (i[square root of s]) / i[square root of s] + Na[J.sub.0] (i[square root of s]) = 0 (26)

Here we employ the affinities

[I.sub.0] ([square root of s]) = [J.sub.0] (i[square root of s]) (27)

[I.sub.1] ([square root of s]) = [J.sub.1] (i[square root of s]) / i (28)

where [J.sub.0], [J.sub.1] are the Bessel function of grade, order 0 and 1. By the substitution of i[square root of s] = q we obtain:

[J.sub.1] (q) / [J.sub.0] (q) = - qNa / 2[epsilon] [right arrow] (29)

Transcendental equation (29) has infinite number of roots, using the Heavisides sentence (24), where s sn and by utilization of (27), (28) we get the final relation for the non-dimensional concentration field of hydrolyzed protein in the amaranth stalk:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

where [q.sub.n] are the roots of the equation (29).

[FIGURE 1 OMITTED]

The time dependence of the dimensionless concentration [C.sub.0(F0)] we gain from the relation (30) by substitution for R = 1 and by using (29) and after the modification we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

where [q.sub.n] are again the equation roots (29).

The protein concentration fields in the footstalk of cylindric shape are modeled in program Matlab and shown in figure 1 for non-dimensional solvent consumption.

3. EXPERIMENTAL SET-UP

3.1 Protein enzymatic hydrolysis of the amaranth stalk

Protein liquefaction of the amaranth stalk runs under slight acid hydrolysis which was proposed:

* rate amaranth straw/salt acid = 1/7

* dosing: 100g of straw and 770ml of acid in laboratory conditions

The experiments proceed in a reverse cooler, where 100g of dry straw was dozen into 770ml of salt acid so that the material was plunged. Warmed under the boil point up, where it maintained for 11 hours. During the hydrolysis ten samples each hour were taken off for orientation determination of nitrogen in the sample by the Biuret method. After finishing the experiments, filtrate and filtration cake were dry, out of both samples protein content was analyzed using TKN method.

Effectiveness of the acid hydrolysis was 65% reached for the nitrogen weight in the filtrate to the original nitrogen weight in dry amaranth straw.

4. CONCLUSION

In this paper, the modeling of amaranth stalk is solved using Matlab computer environment. Amaranth stalk was approximated with an infinite cylinder and model was solved using Bessel functions. The model was verified by experimental measurements, fulfilling the condition of stalk permanent preservation after protein extraction we considerably simplify the process of mixture separation which leads to an economical profit when industry used.

5. ACKNOWLEDGEMENTS

The work has been supported by the grant VZ MSM 7088352102 the Ministry of Education of the Czech Republic. This support is very gratefully.

6. REFERENCES

Paredes-Lopez, O., (1994) Amaranth--biology, chemistry and technology. Library of Congress Cataloging in Publication Data

Crank, J, (1975) The mathematics of diffusion. Clarendon Press, Oxford 2nd Edition, 1975

Kolomaznik, K, Taylor, M., Kupec, J. (2000) The valuable products from wastes, In Proc. 5th World Congress R'2000, Recovery, Recycling, Re-integrat, Canada, pp 5
Tab. 1. Chemical composition of the amaranth straw
analyzed by TKN method

 Content in %

 Total residue 92,96
 Ash (in total residue) 13,07
Nitrogen (in total residue) 1,19
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Author:Kodrikova, Klara; Kolomaznik, Karel
Publication:Annals of DAAAM & Proceedings
Article Type:Report
Geographic Code:4EUAU
Date:Jan 1, 2009
Words:1371
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