Optimal synthesis and design of the number of cycles in the leaching process for surimi production

Abstract

Water consumption required during the leaching stage in the surimi manufacturing process strongly depends on the design and the number and size of stages connected in series for the soluble protein extraction target, and it is considered as the main contributor to the operating costs. Therefore, the optimal synthesis and design of the leaching stage is essential to minimize the total annual cost. In this study, a mathematical optimization model for the optimal design of the leaching operation is presented. Precisely, a detailed Mixed Integer Nonlinear Programming (MINLP) model including operating and geometric constraints was developed based on our previous optimization model (NLP model). Aspects about quality, water consumption and main operating parameters were considered. The minimization of total annual costs, which considered a trade-off between investment and operating costs, led to an optimal solution with lesser number of stages (2 instead of 3 stages) and higher volumes of the leaching tanks comparing with previous results. An analysis was performed in order to investigate how the optimal solution was influenced by the variations of the unitary cost of fresh water, waste treatment and capital investment.

Appendix

The kinetic model of soluble protein extraction from spherical particles of minced fish is described as follows:

$$ (1 - \varepsilon_{w,c} ) \cdot \frac{1}{{D_{\beta \gamma } }}\frac{{\partial c_{\beta ,c} \left( {r,t} \right)}}{\partial t} = \frac{{\partial^{2} c_{\beta ,c} }}{{\partial r^{2} }} + \frac{2}{r}\frac{{\partial c_{\beta ,c} }}{\partial r},\quad 0 < r < R $$

(30)

$$ c_{\beta } = \left\langle {c_{\beta \;inl} } \right\rangle_{c} ,\quad t = 0,\quad \forall \;0 \le r \le R $$

(31)

$$ \frac{{\partial c_{\beta ,c} \left( {r,t} \right)}}{\partial r} = 0,\quad r = 0,\quad \forall t > 0 $$

(32)

$$ - D_{\beta \gamma } \frac{{\partial c_{\beta ,c} }}{\partial r} = k_{c\gamma ,c} \left( {c_{\gamma i,c} - c_{\gamma \;out,c} } \right)\,,\quad r = R,\quad \forall t > 0 $$

(33)

Equations (30), (32) and (33) were discretized using the central finite difference method (CFDM) using the second-order accurate method in both space and time. The number of discretization nodes used for the time-domain and spatial-domain were, respectively 50 and 10.

The semi-empirical equation of Polson (1950) was used to estimate the protein diffusion coefficient, D _βγ , which is recommended for biological solutes:

$$ D_{\beta \gamma } = \frac{9.40e - 15 \cdot T}{{\mu_{w} \cdot \left( {Mw_{\gamma } } \right)^{1/3} }} $$

(34)

where Mw _γ is the protein’s molecular weight, T is the system’s temperature and µ _w is the fluid viscosity.

The overall mass transfer coefficient was calculated using the correlation proposed by Geankoplis (1993) for fixed beds and also valid for fluidized beds of spheres in the Reynolds number range of 10–4000:

$$ J_{D} = \frac{0.4548}{{\varepsilon_{c} }} \cdot \text{Re}^{ - 0.4069} $$

(35)

$$ kc_{\gamma ,c} = \frac{{J_{D} \cdot v_{c} }}{{Sc^{2/3} }} $$

(36)

where

$$ \text{Re}_{c} = \frac{{Dp \cdot \rho_{\gamma } \cdot v_{c} }}{{\mu_{\gamma } }} $$

(37)

and

$$ Sc = \frac{{\mu_{\gamma } }}{{D_{\beta \gamma } \cdot \rho_{\gamma } }} $$

(38)

The equilibrium of soluble proteins concentration under diluted assumption is expressed as:

$$ c_{\gamma i} = K \cdot c_{\beta i} $$

(39)

The average concentration of total proteins in phase β, after the leaching process, is computed as follows:

$$ \left\langle {c_{\beta \;out} } \right\rangle_{c} = {{3A\int_{0}^{R} {c_{\beta } } } \mathord{\left/ {\vphantom {{3A\int_{0}^{R} {c_{\beta } } } {(AR)}}} \right. \kern-0pt} {(AR)}}\;,\quad t = \theta_{c} $$

(40)

The initial concentration of protein in the washed minced fish is equal to the final protein concentration of the previous cycle (Fig. 3). Then, the following constraints are considered:

$$ \left\langle {c_{\beta \;out} } \right\rangle_{c} = \left\langle {c_{\beta \;inl} } \right\rangle_{c + 1} \;,\quad c = 1,2, \ldots (c_{\hbox{max} } - 1) $$

(41)

The maximum percentage of extraction [Y _EP ] is defined as the ratio of the amount of proteins extracted after washing and the maximum amount of proteins that can be extracted, according to the following constraints:

$$ Y_{EP} \% = \frac{{\left\langle {c_{\beta \;inl} } \right\rangle_{EP,1} - \left\langle {c_{\beta \;out} } \right\rangle_{EP,c} }}{{\left\langle {c_{\beta \;inl} } \right\rangle_{EP,1} }}\% $$

(42)

where:

$$ \left\langle {c_{\beta 0} } \right\rangle_{EP,1} = 0.25 \cdot \left\langle {c_{\beta 0} } \right\rangle_{1} $$

(43)

The leaching process model in the countercurrent configuration is described as follows:

It is considered that the operative volume of the leaching tanks is 70 percentage of the total volume:

$$ V_{op,c} = 0.7 \cdot V_{TK,c} $$

(44)

where the total volume (V _TK,T ) is given by:

$$ V_{TK,T} = \sum\limits_{c = 1}^{3} {V_{TK,c} } $$

(45)

During the washing process in each cycle, the operative volume of the tank is filled by minced fish and washing water and is computed as follows:

$$ V_{op,c} = V_{f,c} + V_{w,c} $$

(46)

The residence time of the minced fish during each washing stage is calculated as:

$$ \theta_{c} = \frac{{V_{f,c} \cdot \rho_{f} }}{{M_{c} }} $$

(47)

The volume fractions of minced fish and solvent are calculated as follows:

$$ \varepsilon_{f,c} = \frac{{V_{f,c} }}{{V_{op,c} }} $$

(48)

$$ \varepsilon_{w,c} = \frac{{V_{w,c} }}{{V_{op,c} }} $$

(49)

where:

$$ \varepsilon_{f,c} + \varepsilon_{w,c} = 1 $$

(50)

The global and protein mass balances in each leaching stage (leaching tank and rotary sieve, as control volume) is given by:

$$ Qr(c + 1) = Q_{inl} (c),\quad c = 1,2, \ldots (c_{\hbox{max} } - 1) $$

(51)

$$ FWT = Q_{inl} (c),\quad c = c_{\hbox{max} } $$

(52)

$$ Qr_{{}} (c + 1) \cdot c_{\gamma } (c + 1) = Q_{inl} (c) \cdot c_{\gamma \;inl} $$

(53)

where

$$ Qout(c) = Qr_{{}} (c),\quad c > 1 $$

(54)

$$ Qout(c) = QwasteT,\quad c = 1 $$

(55)

$$ Qout(c) = E(c),\quad \forall c $$

(56)

The mass balance at each leaching tank is given by:

$$ M(c)_{{}} \cdot \left\langle {c_{\beta \;inl} } \right\rangle_{c} + Q_{inl} (c) \cdot c_{\gamma \;inl,c} = Q_{out} (c) \cdot c_{\gamma \;out,c} + M_{{}} (c + 1) \cdot \left\langle {c_{\beta \;out} } \right\rangle_{c + 1} + S(c) \cdot \left\langle {c_{\beta \;out} } \right\rangle_{c} $$

(57)

where:

$$ M(c + 1)_{{}} = 0.98 \cdot M(c),\quad c = 1 , $$

(58)

$$ M(c + 1)_{{}} = 0.99 \cdot M(c),\quad c > 1 $$

(59)

and

$$ M_{{}} (c) = M(c + 1)_{{}} + S(c),\quad c < c_{\hbox{max} } $$

(60)