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Modeling of Gel Layer-Controlled Fruit Juice Microfiltration in a Radial Cross Flow Cell

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Abstract

An analysis of gel layer-controlled microfiltration in a radial cross flow cell is presented in this study. Clarification of a real fruit juice, i.e., cactus pear juice is considered. An expression of Sherwood number is derived using an integral method under the framework of boundary layer theory. The effects of viscosity and temperature are incorporated in the Sherwood number relationship through the Sieder–Tate type correction factor and Stokes–Einstein equation, respectively. The transient flux behavior is modeled successfully both for the total recycle mode and batch concentration mode of operation. The model parameters are evaluated from the total recycle mode and are used in the predictive calculation of the batch concentration mode. In batch concentration mode of filtration, the model predicted results match excellently with the experimental data.

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Abbreviations

A :

Constant in Eq. 8 (=3Re ∙ Sc ∙ h/R)

a 1,a 2,a 3 :

Constants in Eq. 9

A m :

Effective membrane area (square meter)

C :

Bulk concentration in batch recycle mode (kilogram per cubic meter)

C * :

Non-dimensional concentration

C 0 :

Initial bulk concentration (kilogram per cubic meter)

C b :

Bulk concentration in batch mode (kilogram per cubic meter)

C g :

Gel concentration (kilogram per cubic meter)

C g * :

Non-dimensional gel layer concentration

C m :

Mean concentration in the boundary layer (kilogram per cubic meter)

C m * :

Non-dimensional mean concentration

C p :

Permeate concentration (kilogram per cubic meter)

D :

Diffusivity of solute (square meter per second)

D 0 :

Diffusivity at the reference temperature T 0 (square meter per second)

h :

Half height of channel (meter)

h 0 :

Initial channel thickness (meter)

H b :

Gel layer thickness in batch mode (meter)

H t :

Thickness of gel layer in total recycle mode (meter)

k :

Mass transfer coefficient (meter per second)

m :

Geometric factor in Eq. 42

n :

Exponent in viscosity–temperature relation \( \mu\;\alpha \frac{1}{{{T^n}}} \)

P ew :

Non-dimensional flux

Q :

Volumetric flow rate (cubic meter per second)

r :

Radial coordinate (meter)

r * :

Non-dimensional radial coordinate (=r/R)

Re:

Reynolds number \( \left( {\frac{{\rho uh}}{\mu }} \right) \)

R gb :

Gel layer resistance defined by Eq. 36 (per meter)

R gt :

Gel layer resistance in total recycle mode (per meter)

R m :

Membrane hydraulic resistance (per meter)

Sc:

Schmidt number \( \left( {\frac{\mu }{{\rho D}}} \right) \)

Sh:

Sherwood number (kr/D 0 )

\( \overline{\mathrm{Sh}} \) :

Length averaged Sherwood number (Eq. 15)

T :

Temperature in absolute scale (kelvin)

t :

Time of operation (second)

T 0 :

Reference temperature (=298 K)

u :

Effective transverse velocity (meter per second) defined in Eq. 2

u 0 :

Initial linear velocity (meter per second)

V :

Retentate volume (cubic meter)

V 0 :

Feed volume in batch mode (cubic meter)

v :

y-component velocity (meter per second)

v w :

Permeate flux (cubic meter per square meter per second)

y :

y-coordinate direction (meter)

y * :

Non-dimensional y-coordinate (=y/h)

α :

Parameter in viscosity–temperature relation μ = μ 0 e αC

β :

Specific gel resistance in Eq. 26, meter per kg

δ :

Concentration boundary layer thickness (meter)

δ * :

Non-dimensional concentration boundary layer thickness

∆P :

Transmembrane pressure drop (pascal)

ε g :

Gel layer porosity

ξ :

Parameter in Eq. 27 (per square meter)

μ :

Viscosity (pascal-second)

μ 0 :

Reference bulk viscosity when concentration of suspended solids is zero (pascal-second)

μ m :

Viscosity corresponding to mean concentration C m in the boundary layer (pascal-second)

μ w :

Viscosity at the wall (pascal-second)

ρ f :

Density of feed (kilogram per cubic meter)

ρ g :

Gel layer density (kilogram per cubic meter)

ρ p :

Density of permeate (kilogram per cubic meter)

τ :

Non-dimensional time \( \left( {=\frac{tD }{{{h^2}}}} \right) \)

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Authors and Affiliations

  1. Department of Chemical Engineering, Indian Institute of Technology Kharagpur, Kharagpur, 721302, India

    Sourav Mondal & Sirshendu De

  2. Institute on Membrane Technology, ITM-CNR, c/o University of Calabria, via P. Bucci, 17/C, 87030, Rende, Cosenza, Italy

    Alfredo Cassano

Authors
  1. Sourav Mondal
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  2. Alfredo Cassano
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  3. Sirshendu De
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Corresponding author

Correspondence to Sirshendu De.

Appendices

Appendix A

The derivatives, \( \frac{{\partial {C^{*}}}}{{\partial {r^{*}}}} \), \( \frac{{\partial {C^{*}}}}{{\partial {y^{*}}}} \) and \( \frac{{{\partial^2}{C^{*}}}}{{\partial {y^{*}}^2}} \) in Eq. 8 are evaluated using Eq. 10. These partial derivatives are inserted in Eq. 8 and after simplification the following equation is obtained,

$$ \frac{A}{{{r^{*}}}}\left( {\frac{{{y^{*2 }}}}{{{\delta^{*2 }}}}-\frac{{{y^{*3 }}}}{{{\delta^{*3 }}}}} \right)\frac{{d{\delta^{*}}}}{{d{r^{*}}}}-P{e_w}\left( {\frac{{{y^{*}}}}{{{\delta^{*2 }}}}-\frac{1}{{{\delta^{*}}}}} \right)=\frac{1}{{{\delta^{*2 }}}} $$
(A.1)

Taking the zeroth moment of above equation by multiplying both sides by dy * and integrating across the boundary layer thickness from 0 to δ *, the following equation is obtained.

$$ \frac{A}{{{r^{*}}}}\left( {\frac{{d{\delta^{*}}}}{{d{r^{*}}}}} \right)\int\limits_0^{{{\delta^{*}}}} {\left( {\frac{{{y^{*2 }}}}{{{\delta^{*2 }}}}-\frac{{{y^{*3 }}}}{{{\delta^{*3 }}}}} \right)} d{y^{*}}-{P_{ew }}\int\limits_0^{{{\delta^{*}}}} {\left( {\frac{{{y^{*}}}}{{{\delta^{*2 }}}}-\frac{1}{{{\delta^{*}}}}} \right)} d{y^{*}}=\frac{1}{{{\delta^{*2 }}}}\int\limits_0^{{{\delta^{*}}}} {d{y^{*}}} $$
(A.2)

On solving the above integral the final equation is arrived

$$ \frac{A}{12}\frac{{{\delta^{*2 }}}}{{{r^{*}}}}\left( {\frac{{d{\delta^{*}}}}{{d{r^{*}}}}} \right)+\frac{{{P_{ew }}{\delta^{*}}}}{2}=1 $$
(A.3)

Substituting \( \frac{{\partial {C^{*}}}}{{\partial {y^{*}}}} \) from Eq. 10 in Eq. 8c, the following expression is obtained.

$$ {P_{ew }}{\delta^{*}}=2\left( {\frac{{C_g^{*}-1}}{{C_g^{*}}}} \right) $$
(A.4)

Replacing this value of P ew δ * in Eq. A.3, results to the following governing equation of concentration boundary layer thickness,

$$ \frac{A}{12}\frac{{{\delta^{*2 }}}}{{{r^{*}}}}\left( {\frac{{d{\delta^{*}}}}{{d{r^{*}}}}} \right)=\frac{1}{{C_g^{*}}} $$
(A.5)

Integration of the above equation leads to the profile of concentration boundary layer thickness with r * as,

$$ {\delta^{*}}={{\left( {\frac{18 }{{AC_g^{*}}}} \right)}^{1/3 }}{r^{*2/3 }} $$
(A.6)

Appendix B

The non-dimensional solute balance equation within concentration boundary layer can be written as,

$$ \mathop{{\frac{{\partial {C^{*}}}}{{\partial \tau }}}}\limits_{{\left( {{T_1}} \right)}}+\mathop{{A\frac{{{y^{*}}}}{{{r^{*}}}}\frac{{\partial {C^{*}}}}{{\partial {r^{*}}}}}}\limits_{{\left( {{T_2}} \right)}}-\mathop{{{P_{ew }}\frac{{\partial {C^{*}}}}{{\partial {y^{*}}}}}}\limits_{{\left( {{T_3}} \right)}}=\mathop{{\frac{{{\partial^2}{C^{*}}}}{{\partial {y^{*}}^2}}}}\limits_{{\left( {{T_4}} \right)}} $$
(B.1)

where, the non-dimensional time is defined as, τ = tD/h 2. Next, an order of magnitude analysis of Eq. B.1 is carried out term wise. O(x *) is 1; order of y is same as that of thickness of concentration boundary layer, \( \delta \approx \frac{D}{k}=\frac{{{10^{-11 }}}}{{{10^{-6 }}}}={10^{-5 }} \). Thus, \( \mathrm{O}\left( {\frac{{{y^{*}}}}{{{r^{*}}}}} \right) \) is \( \frac{{{{{{10^{-5 }}}} \left/ {{{10^{-3 }}}} \right.}}}{{{{{{10^{-1 }}}} \left/ {{{10^{-1 }}}} \right.}}}={10^{-2 }} \). O(A) is \( \frac{{{u_0}{h^2}}}{DR }=\frac{{1\times {10^{-6 }}}}{{{10^{-11 }}\times {10^{-1 }}}}={10^6} \). O(Pe w ) is \( \frac{{{v_w}h}}{R}=\frac{{{10^{-6 }}\times {10^{-3 }}}}{{{10^{-11 }}}}={10^2} \). Therefore, order of the terms, T 2, T 3, and T 4 is 104. Thus, it may be noted that T 1 has significant magnitude compared to other three terms up to a time of operation of 100 s. Beyond 100 s, it is reduced in order of magnitude. Hence, comparing the full operation time in this experiment (360 min), T 1 is small enough to be ignored. Therefore, we can take recourse to a quasi-steady-state analysis for estimation of concentration boundary layer profile. The governing equation of solute mass balance is same as Eq. 8. The concentration profile can be approximated as described in Appendix A, with the following boundary conditions,

$$ \mathrm{at}\quad {y^{*}}={\delta^{*}},\quad C={C_b};\quad {C^{*}}=C_b^{*} $$
(B.2)

The concentration profile within the boundary layer now becomes,

$$ {C^{*}}=C_g^{*}-2\left( {C_g^{*}-C_b^{*}} \right)\left( {\frac{{{y^{*}}}}{{{\delta^{*}}}}} \right)+\left( {C_g^{*}-C_b^{*}} \right){{\left( {\frac{{{y^{*}}}}{{{\delta^{*}}}}} \right)}^2} $$
(B.3)

The mean concentration within boundary layer is (Mondal et al. 2011a),

$$ C_m^{*}=\frac{1}{3}\left( {C_g^{*}+2C_b^{*}} \right) $$
(B.4)

and,

$$ \frac{{{\mu_m}}}{{{\mu_w}}}={e^{{-\frac{2}{3}\alpha {C_0}\left( {C_g^{*}-C_b^{*}} \right)}}} $$
(B.5)

The Sherwood number relation is modified as,

$$ \overline{\mathrm{Sh}}=1.65{{\left( {\operatorname{Re}\cdot \mathrm{Sc}\frac{h}{R}} \right)}^{{\frac{1}{3}}}}{{\left( {{e^{{-\frac{2}{3}\alpha {C_0}\left( {C_g^{*}-C_b^{*}} \right)}}}} \right)}^{0.14 }}{{\left( {\frac{{C_g^{*}}}{{C_b^{*}}}} \right)}^{{\frac{1}{3}}}} $$
(B.6)

Including temperature correction, the average Sherwood number is,

$$ \overline{\mathrm{Sh}}=1.65{{\left( {\operatorname{Re}\cdot \mathrm{Sc}\frac{h}{R}} \right)}^{{\frac{1}{3}}}}{{\left( {{e^{{-\frac{2}{3}\alpha {C_0}\left( {C_g^{*}-C_b^{*}} \right)}}}} \right)}^{0.14 }}{{\left( {\frac{{{T_0}}}{T}} \right)}^{{\frac{n+1 }{3}}}}{{\left( {\frac{{C_g^{*}}}{{C_b^{*}}}} \right)}^{{\frac{1}{3}}}} $$
(B.7)

Average dimensionless permeate flux becomes,

$$ \overline{{{P_{ew }}}}=1.65{{\left( {\operatorname{Re}\cdot \mathrm{Sc}\frac{h}{R}} \right)}^{{\frac{1}{3}}}}{{\left( {{e^{{-\frac{2}{3}\alpha {C_0}\left( {C_g^{*}-C_b^{*}} \right)}}}} \right)}^{0.14 }}{{\left( {\frac{{{T_0}}}{T}} \right)}^{{\frac{n+1 }{3}}}}\left[ {{{{\left( {\frac{{C_g^{*}}}{{C_b^{*}}}} \right)}}^{{\frac{1}{3}}}}-{{{\left( {\frac{{C_g^{*}}}{{C_b^{*}}}} \right)}}^{{-\frac{2}{3}}}}} \right] $$
(B.8)

For \( \frac{{C_g^{*}}}{{C_b^{*}}} < <20 \), \( {{\left( {\frac{{C_g^{*}}}{{C_b^{*}}}} \right)}^{{\frac{1}{3}}}}-{{\left( {\frac{{C_g^{*}}}{{C_b^{*}}}} \right)}^{-2/3 }} \), is reduced to \( \ln \left( {\frac{{C_g^{*}}}{{C_b^{*}}}} \right) \). Under this condition, the final expression of length averaged permeate flux is presented in Eq. 35.

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Mondal, S., Cassano, A. & De, S. Modeling of Gel Layer-Controlled Fruit Juice Microfiltration in a Radial Cross Flow Cell. Food Bioprocess Technol 7, 355–370 (2014). https://doi.org/10.1007/s11947-013-1077-9

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