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1. Introduction

Nowadays membrane based processes are the main area of interests growing among the researchers.

Especially in the field of water treatment, membrane applications are the most coveted technologies that are

rarely to be replaced in near future. Cross-flow and dead end processes are the two modes of membrane

applications in generic and extensively applied in different industrial applications. Mainly cross-flow

filtration is an effective separation technology that finds application in a wide range of areas including water

treatment, clean environment technologies, and recovery of worthy components from the waste stream [1].

Main limitations associated with any of the membrane applications are the continuous reduction in its

throughput due to gradual accumulation of solutes on the surface and thereby initiation of the concentration

polarization effect. To alleviate this problem various high shear devices have been used so far. Among

different high shear application cross-flow membrane filtration, frequently termed low shear tangential

filtration is used to clean fluids that are difficult to filter and to separate fine matter such as cells, proteins,

enzymes and viruses [2]. Rotating disk membrane module (RDMM) is another type of high sheared device

which was used previously by many researchers, mainly in the separation of protein [3, 4] from milk or in

any biological separation [5]. Bhattacharjee and Bhattacharya [6] performed ultrafiltration (UF) of Kraft

black liquor (KBL) using RDMM. In case of RDMM mainly the rotation of membrane creates a high

turbulence on membrane surface which ultimately reduces the retention period of solutes on the surface and

thus too some extent alleviate the chances of reversible as well as irreversible fouling of the membrane. To

understand the efficacy of the process, study on hydrodynamics and thereby development of some model

equations describing the actual scenario within the system is inevitable.

Transport modelling in membrane systems has been so far the most complicated element of research

work that were being carried out by several researchers hitherto. Sarkar and Bhattacharjee [5] have already

established an analytical solution for describing the actual hydrodynamics within RDMM based on the back

transport flux mechanism. The main complexity arises during the formulation of these models are the

occurring of different uncertainties associated with the membrane separation processes. Especially in case of

RDMM the inner hydrodynamics is so complicated to understand analytically and thereby limits the

development of mathematical model. Computational fluid dynamics (CFD) could be an alternative tool to

overcome this limitation. CFD is a theoretical way to investigate and predict the performance of the process

without creating much complexity from the design aspect for complicated geometrical modules [7]. Torras

et al. [8] made a similar study on RDMM, using CFD simulation of RDMM. In this model they had

compared the CFD simulated shear stress with that one calculated from the analytical correlation and found

a good agreement between the two results. A CFD simulation for a two-dimensional transport in a

membrane module was carried out by Pellerin et al. [9]. They found that the pressure contours for the

laminar and turbulent models are quite similar near the control axis of the module but they are different near

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the wall region. In membrane separation process the main areas of concern are the concentration polarization

and the flow distribution within the membrane module. The former is responsible for continuous decline in

the permeate flux and the later controls the maximum utilization of the membrane area. In order to reduce

the concentration polarization, present work employs a membrane and a propeller rotating in opposite to

each other within the module. Thus creating high turbulence adverse effect on the permeation through

membrane could be reduced, but at the same this may affect the non-uniform flow distribution within the

module after creating vortex. CFD simulation helps to understand the effect of rotation and provide an

effective way-out to design the module in such a fashion so that both concentration polarization and non-

uniformity in the flow could be reduced.

The true insight of the actual phenomena including the behavioural aspects of the governing

principles was not available in the literature so far. In the present work, RDMM was employed with BSA

solution as a feed to the system. The operational parameters include transmembrane pressure (TMP),

concentration of the solution, rotation of the membrane and propeller. The use of CFD in the analysis of

concentration boundary layer during the steady and unsteady time has been attempted greatly for the purpose

of theoretical interest. The complete simulation to predict the concentration distribution within the module

was conducted with precise level of accuracy using Ansys FLUENT®

2. Theory

6.3.26.

Ultrafiltration (UF) is one of the mostly adopted technologies in membrane separation field,

especially in the separation of protein. Equation 1 is the typical Darcy equation, relating the permeate flux

with the TMP and resistances occurred on the membrane surface.

( )m c

PJR R

πµ∆ − ∆

=+

(1)

However turbulent rotational flow developed within RDMM promotes the appearances of eddies having a

wide range of length scale, which could be a cumbersome task to handle without applying CFD. The most

attractive part to solve the problem is to represent turbulent flow by the mean values of flow properties and

the statistical properties of their fluctuations. Introducing the time-averaged properties for the flow (mean

velocities, mean pressures and mean stresses) to the time dependent Navier–Stokes equations, is led to time-

average Navier–Stokes equations as follows [10]:

Continuity Equation:

( ) 0div Utρ ρ∂+ =

∂ (2)

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Momentum Equation:

( ) ( ) ( )( ) ( ) ( )2' ' ' ' '

Mx

u u v u wu Pdiv uU div u St x x y z

ρ ρ ρρρ µ

∂ ∂ ∂∂ ∂ + = − + ∇ + − − − + ∂ ∂ ∂ ∂ ∂

(3)

( ) ( ) ( )( ) ( ) ( )2' ' ' ' '

My

u v v v wv Pdiv vU div v St y x y z

ρ ρ ρρρ µ

∂ ∂ ∂∂ ∂ + = − + ∇ + − − − + ∂ ∂ ∂ ∂ ∂

(4)

( ) ( ) ( )( ) ( ) ( )2' ' ' ' '

Mz

u w v w ww Pdiv wU div w St z x y z

ρ ρ ρρρ µ

∂ ∂ ∂∂ ∂ + = − + ∇ + − − − + ∂ ∂ ∂ ∂ ∂

(5)

where,

j' ' ii j ij T

j i

uuu ux x

ρ τ µ ∂∂

− = = + ∂ ∂ (6)

Here to solve the flow domain within RDMM Renormalization Group (RNG) [11] k-ε model was used

which performs better than the normal k-ε model in case of rotational flow [12] by eliminating the small

scale motion from the turbulence model equation, expressing their effects in terms of larger scale motion and

a modified viscosity. The model provides the following equations for k and ε [13]:

eff Tµ µ µ= + (7)

2

TkCµµ ρε

= (8)

( ) ( ) eff ji i iT

i i k i j i j

uk ku u ukt x x x x x x

µρ ρµ ρε

σ ∂∂ ∂ ∂ ∂∂ ∂

+ = + + − ∂ ∂ ∂ ∂ ∂ ∂ ∂ (9)

( ) ( ) 2 2

1 2eff ji i i

Ti i i j i j

uu u uC Ct x x x k u u u k kε ε

ε

µρε ρε ε ε ε εµ ρ αρσ

∂∂ ∂ ∂ ∂∂ ∂+ = + + − − ∂ ∂ ∂ ∂ ∂ ∂ ∂

(10)

where,

3 03

1

1Cµ

ηη

α ηβη

−=

+, kEη

ε= , 2 2 ij ijE E E= and 0 5 ji

ijj i

uuE .x x

∂∂= + ∂ ∂

(11)

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3. Experimental setup:

The module, made of SS316, was manufactured by Gurpreet Engineering Works (Kanpur, India) as

per the specified design. The module (Figure 1) was equipped with two motors with speed-controllers to

provide rotation of the stirrer (not used in this study) and membrane housing. A digital tachometer was used

to measure the rotational speed of the membrane. The set-up was equipped with an arrangement for

recycling the permeate to the feed cell to run it in continuous mode with constant feed composition. The later

mode was not investigated in this study. The module was also equipped with an outside water-jacket,

through which provision for water circulation was available. An adequate mechanical sealing mechanism

was provided to prevent leakage from the rotating membrane assembly up to a pressure of 1 MPa. An air

compressor was used to provide compressed air for pressurization of the cell. An intermediate air reservoir

fitted with on-off controller based on pressure sensor was provided which maintains the pressure within

reservoir between 1–1.2 MPa. A differential pressure regulator was used to set the pressure at the desired

level within the module. The complete schematic diagram of the rotating disk module set-up is given in Fig

2. The PES membranes (flat disk of 0.076 m diameter) of 30 kg/kmol molecular weight cut-off (MWCO)

were imported from Millipore (Bedford, USA) through its Indian counterpart (Millipore India). The flat-

sheet membrane operable in a pH range of 1–14 has an actual diameter of 0.076 m, whereas the effective

diameter was 0.056 m.

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4. Methodology:

Experiments were carried out with 500 mg/l of Bovine-Albumin-Serum (BSA) solution, batch wise for

different combinations of feed concentration (Cb), membrane speed (ωm), stirrer speed and trans-membrane

pressure (TMP) (∆P) in RDMM. After each experiment, the membrane was thoroughly washed for 1200 s

under running water, then soaked in washing solution, as mentioned in section 2.3, for 3600 – 5400 s and

then again thoroughly washed for 1200 s. In each case, the water flux regained by about 98% of its original

value, suggesting the cause of flux decline to be either osmotic pressure limited or due to a reversible fouling

layer.

4.1 The grid generation of the rotating disc membrane module experimental module in GAMBIT.

4.1.1 The propeller design

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4.1.2 The experimental setup

4.1.3 The grid generation

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The meshing above generated in GAMBIT is exported as a *.msh file.

4.2 The steps in analysis in FLUENT are as follows

4.2.1 File Read Case

Read the mesh file (*.msh) from the specified folder. Internally, FLUENT stores the computational grid in

meters, the SI unit of length. When grid information is read into the solver, it is assumed that the grid was

generated in units of meters. So if the read mesh file is in other unit (say cm) using scale button we can

adjust the dimensions in m unit. So scale button does not change the unit from meter to any other units. It

just converts the other dimensions into meter by multiplying it with proper scale factor. But if we want to do

our work in other units instead of meter we have to click on “Change Length Units”. So the subsequent unit

in Fluent will be only the other unit.

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In the next step we have to check the grid. During grid check negative volume should not come.

Grid Check

Domain Extents:

x-coordinate: min (m) = -5.600000e-002, max (m) = 5.600000e-002

y-coordinate: min (m) = -5.599999e-002, max (m) = 5.599999e-002

z-coordinate: min (m) = 0.000000e+000, max (m) = 8.000000e-002

Volume statistics:

minimum volume (m3): 3.487422e-011

maximum volume (m3): 2.234592e-008

total volume (m3): 7.866323e-004

Face area statistics:

minimum face area (m2): 1.315645e-007

maximum face area (m2): 1.650526e-005

Checking number of nodes per cell.

Checking number of faces per cell.

Checking thread pointers.

Checking number of cells per face.

Checking face cells.

Checking bridge faces.

Checking right-handed cells.

Checking face handedness.

Checking face node order.

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Checking element type consistency.

Checking boundary types:

Checking face pairs.

Checking periodic boundaries.

Checking node count.

Checking nosolve cell count.

Checking nosolve face count.

Checking face children.

Checking cell children.

Checking storage.

Done.

4.2.2 Define Models Solver

Now we have to select the Solver for the subsequent calculation.. Solver is basically used to define how the

hydrodynamic parameters are going to be set during the solution of the problem (Eg. N-S Equation, Energy

equation etc.). In the present study we have coded the solver as per the following screenshot.

In both methods the velocity field is obtained from the momentum equations. In the Density-Based Approach, the

continuity equation is used to obtain the density field while the pressure field is determined from the equation of

state.

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On the other hand, in the Pressure-Based Approach, the pressure field is extracted by solving a pressure or

pressure correction equation which is obtained by manipulating continuity and momentum equations. In both the

cases the Finite-Volume approach is taken.

In case of Pressure-Based Approach for the incompressible flow the velocity components are solved for Navier-

Stokes equation using staggered grid technique. Along with the N-S equation the following pressure correction was

used.

0soundP u vat x y

∂ ∂ ∂+ + = ∂ ∂ ∂

(12)

where asound

The absolute velocity formulation is preferred in applications where the flow in most of the domain is not rotating

(e.g., a fan in a large room). The relative velocity formulation is appropriate when most of the fluid in the domain is

rotating, as in the case of a large impeller in a mixing tank. In the present study we are simulating Rotating disk

membrane module where most of the fluid domain is rotating. So Relative was opted in the solver.

Here the discretization is mainly Temporal i.e. time based. For the pressure based solver the scheme is Implicit.

Following methods are showing the implicit temporal discretization.

1

is the speed of sound in this pseudo continuity equation.

st

( )1

1n n

nFt tφ φ φ φ

++∂ −

= =∂ ∆

order Implicit:

(13)

2nd

( )1 1

13 42

n n nnF

t tφ φ φ φ φ

+ −+∂ − +

= =∂ ∆

order Implicit:

(14)

where, n+1 is at the time step t+∆t, n-1 is at the time step t-∆t. Equation (2) and (3) use the following iterative

scheme.

1. Take initial guess for1n

oldφ+

.

2. Evaluate ( )1n

old oldF φ+

.

3. Evaluate 1n

newφ+

using ( )1n

old oldF φ+

and nφ .

4. Check for convergence 1n

newφ+

and 1n

oldφ+

.

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1φ

5φ 4φ

2φ

3φ

2 32 2F

φ φφ

+=

5. If convergence criterion was met iteration will be stopped, otherwise go to step 2 after assigning 1n

oldφ+

=1n

newφ+

.

For Non-Iterative Time-Advancement Scheme each equation was solved using the temporal implicit method for

each equation (Pseudo continuity equation, N-S equation, k-equation, ε-equation. Once the entire convergence

criterion was met we will move to next time step.

In the Gradient Option we have used Green-Gauss Node Based. Here the property value at a face of a cell was

evaluated using the weighted average of the node values on that face. The nodal values using this scheme are

evaluated from surrounding cell-centered values on arbitrary unstructured meshes by solving a constrained

minimization problem, preserving a second-order spatial accuracy.

The node-based averaging scheme is known to be more accurate than the default cell-based scheme for unstructured

meshes, most notably for triangular and tetrahedral meshes.

Following diagram and equation explain the process.

1

1 NodesN

Face NodenNodesN

φ φ=

= ∑ (15)

For Porous Formulation Superficial Velocity will be retained for the entire simulation. It is mainly valid for

porous medium.

4.2.3 Viscous Model

In the viscous model we have chosen two equations (one for kinetic energy [k] and the other one for dissipation

rate [ε]) k-ε mod el. As ou r system is highly strained flo w with swirl we have chosen RNG k-ε model which was

derived from instantaneous N-S equation using ‘Renormalization Group’ (RNG) method. It has an additional term of

strain in the ε eq u ation and also it accou nts for the swirling flo w. So it is far su p erior to stan d ard k-ε model. In

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realizable k-ε model the only problem is the chances of having multiple reference frame (both rotating and stationary)

on which realizable gives poor performance.

Here the flow is swirl dominated flow. So in fluent we have checked the Swirl Dominated Flow. By default we have

Swirl Factor = 0.07. For highly swirled flow we have to set slightly higher value for this factor. Presently we are

working with the default value. As we have the stationary zones within the system so in some of the zones consist of

low value of Reynold’s Number. To account that we have to check the Differential Viscosity Model (Because by

default Fluent uses high Reynold’s Number). We have chosen here Enhanced Wall Treatment with checking

Pressure Gradient Effects.

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4.2.4 Define Materials

4.2.5 Operating pressure

Pabsolute – Poperating = Pgauge

Here as far as concerned with the normal flow pressure we know that operating pressure given in equation 5

resembles atmospheric pressure. By default it was set for atmospheric pressure i.e. 101325 Pa.

Keep the Reference Pressure Location values to (0, 0, 0). As our system is not affected by the Buoyancy

so we don’t need to activate the Gravity.

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4.2.6 Setting Boundary Conditions:

/define/boundary-conditions> list-zones

id name type material kind

---- ------------------------- ------------------ -------------------- ----

2 bsa_solution mixture bsa and water cell

3 cell_body wall aluminum face

4 propeller_blade wall aluminum face

5 pressure_top pressure-inlet face

6 membrane wall aluminum face

8 default-interior interior face

BC for bsa solution (defined in Continuum in GAMBIT as Fluid):

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To define the axis, set the Rotation-Axis Direction and Rotation-Axis Origin. This axis is independent of

the axis of rotation used by any adjacent wall zones or any other cell zones. For 3D problems, the axis of

rotation is the vector from the Rotation-Axis Origin in the direction of the vector given by your Rotation-

Axis Direction inputs. In our study the origin of the axis is (0, 0, 0) and as the fluid motion is in the

anticlockwise direction so we have set the direction (0, 0, -1) (as the rotation was followed by Right hand

thumb rule). As our system is having multiple rotating reference frame so we are using here as sliding mesh

concept activating Moving Mesh scheme.

BC for cell_body (defined in Boundary in GAMBIT as Wall):

The species mass fraction is to be specified at the inlet which is considered to be the pressure top.

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BC for propeller_blade (defined in Boundary in GAMBIT as Wall):

For one experimental run propeller was fixed, so we have made it stationary in the definition of boundary

condition for the propeller blade.

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BC for membrane (defined in Boundary in GAMBIT as Wall):

For membrane we have used Moving Wall and Rotational motion. If the cell zone adjacent to the wall is

moving (e.g., if you are using a moving reference frame or a sliding mesh), you can choose to specify

velocities relative to the zone motion by enabling the Relative to Adjacent Cell Zone option. If you choose

to specify relative velocities, a velocity of zero means that the wall is stationary in the relative frame, and

therefore moving at the speed of the adjacent cell zone in the absolute frame. If you choose to specify

absolute velocities (by enabling the Absolute option), a velocity of zero means that the wall is stationary in

the absolute frame, and therefore moving at the speed of the adjacent cell zone--but in the opposite direction

in the relative reference frame. Here we have assumed that the membrane wall is fixed with respect to the

rotating reference frame and having the same rotational speed of the bsa_solution (Adjacent Cell Zone).

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BC for pressure_top (defined in Boundary in GAMBIT as Wall):

Gauge Total Pressure is the static operating pressure with respect to the operating pressure set at

Operating Conditions panel. In case of rotating reference frame static pressure is calculated on the basis

of relative velocity if the velocity formulation will be given as Relative in the Solver panel. Now the

total pressure value is given by the following equation,

212tot statP P vρ= +

(16)

and Ptot = Pabsolute – Poperating

The total pressure and flow direction at a pressure inlet must be specified in the absolute frame if the

absolute velocity formulation is used. For calculations using relative velocities, the total pressure and

flow direction must be specified with respect to the rotating frame. For problems involving multiple

(17)

In our case it is (294300 – 101325 = 192975 Pa).

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zones (e.g., multiple reference frames or sliding meshes), the coordinate system is defined by the rotation

axis specified in the Fluid (or Solid) panel for the fluid (or solid) zone that is adjacent to the inlet.

For the Turbulence we have adopted the rotational Reynolds Number,

2m mN DRe ρµ

= (18)

Turbulent Intensity was calculated using the following equation,

( )1

80 16I . Re −= (19)

Turbulent length scale is calculated using the following equation,

0 07l . L= (20)

Turbulent K.E. is calculated as follows,

( )232 m mk N D I= (21)

Turbulent dissipation rate is calculated according to the following equation,

1 5

0 157.k.

lε = (22)

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4.2.7 Solve Controls Solution

The Pressure-Implicit with Splitting of Operators (PISO) pressure-velocity coupling scheme with neighbor

correction is highly recommended for all transient flow calculations, especially when you want to use a large

time step. When you use PISO neighbor correction, under-relaxation factors of 1.0 or near 1.0 are

recommended for all equations. If you use just the PISO skewness correction for highly-distorted meshes

(without neighbor correction), set the under-relaxation factors for momentum and pressure so that they sum

to 1 (e.g., 0.3 for pressure and 0.7 for momentum). If you use both PISO methods, follow the under-

relaxation recommendations for PISO neighbor correction, above.

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4.2.8 Solve Monitors Residual

4.2.9 Solve Initialize Initialize

Select pressure_top in the Compute From. Automatically it will take the initial values.

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4.2.10 Solve Iterate

We have taken the time step size 5 seconds for 2500 s. Therefore Number of time steps = (2500/5).

Click on Iterate.

Solution Strategies for a Single Rotating Reference Frame

The difficulties associated with solving flows in rotating reference frames are similar to those discussed in Section 9.5.5. The primary issue you

must confront is the high degree of coupling between the momentum equations when the influence of the rotational terms is large. A high degree

of rotation introduces a large radial pressure gradient which drives the flow in the axial and radial directions, thereby setting up a distribution

of the swirl or rotation in the field. This coupling may lead to instabilities in the solution process, and hence require special solution techniques

to obtain a converged solution. Some techniques that may be beneficial include the following:

• (Pressure-based solver only) Consider switching the frame in which velocities are solved by changing the velocity formulation setting in

the Solver panel. (See Section 10.7.1 for details.)

• (Pressure-based segregated solver only) Use the PRESTO! scheme (enabled in the Solution Controls panel), which is well-suited for the

steep pressure gradients involved in rotating flows.

• Ensure that the mesh is sufficiently refined to resolve large gradients in pressure and swirl velocity.

• (Pressure-based solver only) Reduce the under-relaxation factors for the velocities, perhaps to 0.3-0.5 or lower, if necessary.

24 | P a g e

• Begin the calculations using a low rotational speed, increasing the rotational speed gradually in order to reach the final desired

operating condition (see below).

Gradual Increase of the Rotational Speed to Improve Solution Stability

• Because the rotation of the reference frame and the rotation defined via boundary conditions can lead to large complex forces in the flow,

your FLUENT calculations may be less stable as the speed of rotation (and hence the magnitude of these forces) increases. One of the most

effective controls you can exert on the solution is to start with a low rotational speed and then slowly increase the rotation up to the desired

level. The procedure you use to accomplish this is as follows:

• 1. Set up the problem using a low rotational speed in your inputs for boundary conditions and for the angular velocity of the reference

frame. The rotational speed in this first attempt might be selected as 10% of the actual operating condition.

• 2. Solve the problem at these conditions.

• 3. Save this initial solution data.

• 4. Modify your inputs (i.e., boundary conditions and angular velocity of the reference frame). Increase the speed of rotation, perhaps

doubling it.

• 5. Restart or continue the calculation using the solution data saved in Step 3 as the initial guess for the new calculation. Save the new

data.

• 6. Continue to increment the rotational speed, following Steps 4 and 5, until you reach the desired operating condition.

The Cylindrical coordinate system uses the axial, radial, and tangential components based on the following coordinate systems:

• For problems involving a single cell zone, the coordinate system is defined by the rotation axis and origin specified in the Fluid panel.

• For problems involving multiple zones (e.g., multiple reference frames or sliding meshes), the coordinate system is defined by the rotation

axis specified in the Fluid (or Solid) panel for the fluid (or solid) zone that is adjacent to the inlet.

Under Non-Iterative Solver Controls, there are several parameters that control the sub-iterations for the individual equations.

The sub-iterations for an equation stop when the total number of sub-iterations exceeds the value specified for Max. Corrections, regardless of

whether or not the convergence criteria (described below) are met.

The sub-iterations for an equation end when the ratio of the residuals at the current sub-iteration and the first sub-iteration is less than the value

specified in the Correction Tolerance field. You can monitor the details of the sub-iteration convergence by looking at the AMG solver

performance (i.e., setting the Verbosity field in the Multigrid Controls panel to 1). Be sure to pay attention to the residuals for the current sub-

iteration (i.e., the residual for the 0-th AMG cycle at the current sub-iteration) and the initial residual of the time step (i.e., the residual for the 0-

th AMG cycle of the first sub-iteration). The ratio of these two residuals is what is controlled by the Correction Tolerance field. These two

residuals are also the residuals plotted when using the Residual Monitor panel and reported in the FLUENT console at the end of a time step.

Note that the residuals reported at the end of a time step can be scaled or unscaled, depending on the settings in the Residual Monitor panel.

The residuals reported when monitoring the AMG solver performance are always unscaled.

For each interim sub-iteration, the AMG cycles continue until the usual AMG termination criteria (0.1 by default, and set in the Multigrid

Controls panel) are met. However, for the last sub-iteration (i.e., either when the maximum number of sub-iterations are reached or when the

correction tolerance is satisfied), the AMG cycles continue until the ratio of the residual at the current cycle to the initial residual (the residual

for the 0-th AMG cycle of the first sub-iteration of the time step) drops below the value specified for Residual Tolerance. You may want to adjust

the Residual Tolerance, depending on the time step selected. The default Residual Tolerance should be well suited for moderate time steps (i.e.,

for cell CFL numbers of 1 to 10). Note that you can display the cell CFL numbers for unsteady problems by selecting Cell Courant Number in

the Velocity... category of all postprocessing panels. For very small time steps (cell CFL << 1), the diagonal dominance of the system is very

high and the convergence should be driven further by reducing the Residual Tolerance value. For larger time steps (cell CFL >> 1), it may be

possible that the residual tolerance cannot be reached due to round-off errors, and unless the Residual Tolerance value is increased, AMG

cycles can be wasted. Again, this can be monitored by monitoring the AMG solver performance.

25 | P a g e

The Relaxation Factor field defines the explicit relaxation of variables between sub-iterations. The relaxation factors can be used to prevent the

solution from diverging. They should be left at their default values of 1, unless divergence is detected. If the solution diverges, you should first try

to stabilize the solution by lowering the relaxation factors for pressure to 0.7- 0.8, and by reducing the time step.

The following is a list of models that are compatible with the non-iterative solver:

• Inviscid flow (excluding ideal gas)

• Laminar flow

• All models of turbulence (including LES and DES)

• Heat transfer

• Non-reacting species transport

• General compressible flows (most subsonic and some transonic applications)

• VOF multiphase model (most applications)

• Phase change (solidification and melting)

The following is a list of models that are compatible with the non-iterative solver, but may result in some instabilities and inaccuracies for

certain flow conditions:

• MDM

• Non-Newtonian fluids

• General compressible flows (aerospace supersonic applications)

• Floating operating pressure

• Reacting species and any type of combustion including PDF

The following is a list of models that are not compatible with the non-iterative solver:

• Eulerian multiphase (all non-VOF models)

• Radiation models

• DPM, spark, and crevice models

• UDS transport

• Porous jump

The PRESTO! pressure interpolation scheme, when compared to using the iterative time-advancement, is less stable when using the

non-iterative time-advancement solver. As a consequence, smaller time steps may be required.

As mentioned above, the default control settings are optimally designed to obtain a second-order solution. In order to save CPU time, in

cases where transient accuracy is not a main concern (i.e., first-order integration in time and space), or when NITA is used to converge

toward a steady state solution, you may want to set the Max. Corrections value to 1 in the Solution Controls panel for all transport

equations except pressure.

26 | P a g e

An user defined function has been incorporated to imply the positive aggregative effect of concentration

boundary layer profile inside the rotating disc membrane module setup.

#include "udf.h"

DEFINE_ADJUST(BSAFlux, d)

{

int i,id;

real flux, Cbulk = 1, k = 7.85*pow(10,-6);

real conc_solids, rho;

face_t f;

cell_t c;

Thread *t;

thread_loop_f(t,d)

{

id = THREAD_ID(t);

if (CURRENT_TIME > CURRENT_TIMESTEP)

{

begin_c_loop(c,t)

{

if(id == 8)

{

rho = ((C_R(F_C0(f,t),THREAD_T0(t))+C_R(F_C1(f,t),THREAD_T1(t))) / 2.0);

conc_solids = rho*(C_YI(F_C0(f,t),THREAD_T0(t),i)+C_YI(F_C1(f,t),THREAD_T1(t),i));

flux = k*log(conc_solids/Cbulk);

}

}

end_c_loop(c,t)

}

}

printf("id = %d:Current Time = %g:density = %g:Flux = %g:conc_solids=%g",id,CURRENT_TIME,rho,flux,conc_solids);

}

27 | P a g e

5. Results

The distribution of static pressure over the membrane surface and inside the RDMM

The dynamic pressure distribution inside the module is

28 | P a g e

The pathlines of velocity magnitude inside the RDMM over the membrane surface

The contours of radial velocity inside the RDMM is

29 | P a g e

The contours of tangential velocity inside the RDMM is portrayed as

The contours of vorticity magnitude over the membrane surface,

30 | P a g e

The contours of helicity inside the RDMM is displayed as

The distribution of local Reynolds inside the module is

31 | P a g e

The degree of turbulence is shown by the contours of turbulent kinetic energy (k) (m2/s2)

The contours of turbulent dissipation rate (ε) (m2/s3)

32 | P a g e

The contours of turbulent Reynolds number is shown below

The contours of turbulent viscosity is illustrated below,

33 | P a g e

The contours of density distribution inside the RDMM (almost constant)

The mass fraction of BSA inside the RDMM

34 | P a g e

The plot of mass fraction of BSA with the cartesian co – ordinate position

The effective diffusivity of BSA inside the RDMM is,

35 | P a g e

The wall shear stress across the membrane surface and inside the setup (excluding the propeller blade, since

it has very high and almost constant magnitude)

The local strain rate (s-1) distribution in the different zones inside the RDMM

36 | P a g e

The plots ofPx

∂∂

, Py

∂∂

and Pz

∂∂

with respect to position is displayed as

37 | P a g e

Discussions and Conclusion

In the present work an attempt was made to solve Navier-Stoke's equation within a rotating domain. The

most critical part of this study was to incorporate membrane characteristics with the normal hydrodynamics

within the volume. Deposition on membrane or in other words fouling could be the most critical part that has

to be accounted with the normal CFD application. For that UDF was introduced with the normal finite

element analysis (FEA) to account the deposition on the surface. In the present work we have neglected the

adsorption part or the pore plugging phenomena that could be occurred. Results show that almost the

prediction of concentration on membrane surface was done with 70-75% accuracy. Main advantages with

the CFD modelling are that change in the design parameters for the system in order to have an optimum

model, which can reduce the normal drawbacks with the membrane separation process. Therefore changing

the parameters and solving the domain using FEA to converge to an optimum design could be considered as

a future scope of the work.

Acknowledgement

I am very much grateful to my guide Dr. Chiranjib Bhattacharjee without whose support the successful

completion of this project would only have been a distant dream. I also would like express thanks and

cordial gratitude to Mr. Dwaipayan Sen whose precious suggestion and help has been an invaluable asset to

this project. I also thank for the computer lab facility of our department in providing me technical support.

38 | P a g e

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