# Geometrical Influence on Particle Transport in Cross-Flow Ultrafiltration: Cylindrical and Flat Sheet Membranes

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Dispersion of Brownian Hard Spheres

## 3. Modeling Concentration-Polarization in Ultrafiltration

## 4. Boundary Layer Analysis

#### 4.1. Outer Solution

#### 4.2. Inner Solution

#### 4.3. Asymptotic Matching and Particle Conservation

#### 4.4. Remarks on the Generalized mBLA Method

## 5. Results and Discussions

#### 5.1. CP-Layer and Longitudinal Particle Transport for Reference Conditions

#### 5.2. TMP, Feed Concentration, and Velocity Effects on Global Indicators

#### 5.3. Universal Behavior of Global UF Indicators

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CP | concentration-polarization |

CM | cylindrical membrane |

FMM | flat sheet membranes (top and bottom) |

FMS | flat sheet membrane (top)/substrate (bottom) |

mBLA | modified boundary layer analysis (method) |

TMP | transmembrane pressure |

UF | ultrafiltration |

## List of Symbols (SI Units)

$P{e}_{a}$ | single-particle shear-Pèclet number |

$P{e}_{R}$ | transversal Pèclet number |

$\alpha \phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}$ | concentration factor |

$\beta $ | solvent-recovery indicator |

${\alpha}^{0}$, ${\beta}^{0}$ | the infinite dilution value of $\alpha $, $\beta $ |

${\gamma}^{0}$ | third dimensionless variable characterizing $\beta $ |

${\u03f5}_{\delta}=1/P{e}_{R}$ | perturbation parameter |

${\delta}_{CP}$ | characteristic thickness of CP layer (m) |

R, L, W | half-height, axial length, width of channel (m) |

${R}_{H}$ | hydraulic radius (m) |

h, H | membrane thickness, curvature-corrected thickness (m) |

A, M | channel cross-section, membrane surface area (m^{2}) |

y, z | transversal, longitudinal coordinate (m) |

$\mathbf{V}$ | dispersion-averaged velocity (m/s) |

v, u | transversal, longitudinal velocity (m/s) |

${u}^{0}$ | axial velocity at center of inlet (m/s) |

${U}^{out}$ | longitudinal velocity factor of outer solution |

U | asymptotically matched longitudinal velocity factor |

${v}_{w}$ | permeate flux (m/s) |

${V}^{out}$ | transversal velocity factor |

${\dot{\gamma}}^{*}$ | shear rate at inlet of membrane wall (1/s) |

${\tau}_{w}$ | shear stress at membrane wall (Pa) |

P | dispersion-averaged pressure (Pa) |

${P}^{perm}$, ${P}^{L}$ | pressure at permeate side, at outlet port (Pa) |

$\langle {\Delta}_{T}P\rangle $ | length-averaged transmembrane pressure (Pa) |

$\langle {\Delta}_{T}^{\left(l\right)}P\rangle $ | length-averaged, linearized transmembrane pressure (Pa) |

$\Pi $ | particles osmotic pressure (Pa) |

$\rho $ | dispersion mass density (kg/m^{3}) |

n | particle number density (1/m^{3}) |

a | radius of hard spheres (m) |

${V}_{a}$ | particle volume (m^{3}) |

$\varphi $ | particle volume fraction |

${\varphi}_{b}$, ${\varphi}_{w}$ | particle volume fraction at inlet (feed), at membrane wall |

$\eta $, ${\eta}_{s}$ | suspension-, solvent viscosity (Pa s) |

D | gradient diffusion coefficient (m^{2})/s) |

${D}_{0}$ | Stokes–Einstein diffusion coefficient (m^{2})/s) |

$\kappa \phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}$ | mean Darcy permeability (m^{2}) |

${L}_{p}$ | hydraulic permeability of clean membrane (m/(Pa s)) |

K | dimensionless effective permeability parameter |

${Q}^{0}$, ${Q}^{perm}$, ${Q}^{L}$ | volume flow rate through inlet, membrane, outlet (m^{3})/s) |

${J}_{z}$, ${J}_{z}^{ex}$, ${J}_{z}^{b}$ | longitudinal particle-flux, excess part, bulk part (m/s) |

$\langle \left(\cdots \right)\rangle $ | length-average of $\left(\cdots \right)$ (c.f. Equation (18)) |

$\overline{\left(\cdots \right)}$ | cross-section average of $\left(\cdots \right)$ (c.f. Equation (22)) |

## Appendix A. Simplified mBLA Method

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**Figure 1.**Considered membrane geometries: (

**a**) cylindrical membrane of inner radius R (left), and flat sheets membranes of height $2R$ and width $W\gg R$ (right). (

**b**) Cross-sectional structure of a cylindrical membrane (left, CM), a membrane consisting of an upper and lower flat sheet part at vertical distance $2R$ (middle, FMM), and a flat membrane–flat substrate combination where the bottom substrate sheet is impermeable to particles and solvent (right, FMS). The membrane thickness is h, and L with $L\gg R$ is the axial length of the membrane.

**Figure 2.**(

**a**) Single-particle shear-Pèclet number, $P{e}_{a}\propto {a}^{3}$, as a function of particle radius a and for a shear rate ${\dot{\gamma}}^{*}=270/\mathrm{s}$ typical of UF. (

**b**) Carnahan–Starling-based osmotic pressure at freezing, $\Pi ({\varphi}_{f},a)$, and thermal pressure, ${k}_{B}T/{V}_{a}$, as functions of particle radius a. Open circles indicate the radii $a=3.13$ nm and $a=10$ nm used in this work.

**Figure 3.**Reduced osmotic pressure, $\Pi /\left(n{k}_{B}T\right)$, as a function of particle volume fraction $\varphi $. The solid line is the Carnahan–Starling prediction for hard spheres, compared to experimental data (open symbols) for bovine serum albumin solutions at different pH values as indicated. Experimental data are reproduced from [30]. Dashed lines are empirical fits to the data.

**Figure 4.**(

**a**) Concentration dependence of the short-time sedimentation coefficient ${K}^{sed}\left(\varphi \right)$. Solid line is the prediction by Equation (4), and open symbols are dynamic simulation data by three groups as indicated [32,33,34]. (

**b**) Short-time collective diffusion coefficient, $D\left(\varphi \right)$, according to Equation (3) (solid line), given in units of ${D}_{0}$. (

**c**) Dispersion shear viscosity, $\eta \left(\varphi \right)$, in units of the solvent viscosity ${\eta}_{s}$. Solid line is the prediction by Equations (5)–(7). Open symbols are dynamic simulation and filled symbols experimental viscosity data reproduced from Refs. [35,36] and Refs. [33,37], respectively.

**Figure 5.**(

**a**) mBLA prediction for the longitudinal velocity factor $U(y,z,\left[{\varphi}_{w}\right])$, using ${\varphi}_{w}=0.4$, $a=10$ nm, ${\u03f5}_{\delta}\approx 1.28\times {10}^{-2}$, and ${v}_{w}=3.35\times {10}^{-6}$ m/s. The inset shows the stretched $U/{\u03f5}_{\delta}$ as a function of the stretched distance from the bottom wall, $(y+R)/{\delta}_{CP}$. While the bottom wall is a permeable membrane for FMM, it is impermeable for FMS. (

**b**) Transversal velocity factor $V\left(y\right)={V}^{out}\left(y\right)$ in Equation (31) for CM, FMM, and FMS, respectively.

**Figure 6.**CM, FMM, and FMS concentration profiles at the membrane wall, ${\varphi}_{w}\left(z\right)$, for a dispersion of Brownian hard spheres of radius $a=3.13$ nm (

**a**) and $a=10$ nm (

**b**), respectively. The insets show the according axial velocity profiles, $u(y=0,z)$, at the channel center-line $y=0$, in units of the CM inlet velocity ${u}_{CM}^{0}=u(0,0)$. Black curves are for the same mean inlet velocity ${\overline{u}}^{0}$. Dashed (red) curves represent a second FMM system (referred to as FMM2), having a flow shear rate at the membrane inlet equal to that for the CM geometry.

**Figure 7.**Excess and bulk axial flux parts, ${J}_{z}^{ex}(y,z)$ (solid lines) and ${J}_{z}^{b}(y,z)$ (dashed lines), plotted as functions of the reduced distance, $1-y/R$, from the membrane wall, for systems CM (

**left**) and FMM (

**right**) using $a=3.13$ nm. The fluxes are given in units of ${\overline{{J}_{z}}}^{0}={\varphi}_{b}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{\overline{u}}^{0}$. The arrows mark increasing values of z, with $z/L=0.2,0.4,0.6,0.8$, $1.0$. The dotted vertical lines mark the distance from the membrane wall equal to ${\delta}_{CP}$. System parameters as in Figure 6a.

**Figure 8.**Cross-section averaged axial excess particle-flux, ${\overline{{J}_{z}}}^{\phantom{\rule{0.166667em}{0ex}}ex}\left(z\right)$, in units of ${\overline{{J}_{z}}}^{0}$, as function of reduced axial distance, $z/L$, from the inlet. Two dispersions with particle radius $a=3.13$ nm (

**a**) and $a=10$ nm (

**b**) are considered for input parameters as in Figure 6a,b, respectively.

**Figure 9.**(

**a-1**,

**b-1**): TMP dependence of average permeate flux $\langle {v}_{w}\rangle $, solvent recovery indicator $\beta $, and concentration factor $\alpha $ for particle radius $a=3.13$ nm (panels

**a-1**–

**a-3**) and $a=10$ nm (panels

**b-1**–

**b-3**), respectively. Symbols are mBLA results, and solid lines are pure solvent predictions. Input parameters except for TMP are as in Figure 6.

**Figure 10.**Reduced average permeate flux, $\langle {v}_{w}\rangle (R/{D}_{0})$, (

**left**) and concentration factor, $\alpha $, (

**right**) versus feed concentration, ${\varphi}_{b}$, for fixed $P{e}_{R}\approx 78$. Input parameters are as in Figure 6, except for ${\varphi}_{b}$, which is varied. Open symbols are for $a=3.13$ nm and filled symbols for $a=10$ nm. The inset in the left panel shows the pressure ratio $\langle \Pi \rangle /\mathrm{TMP}$. The plateau values, ${\alpha}^{0}$, of $\alpha $ are marked in the right panel by the horizontal dashed line segments.

**Figure 11.**Solvent recovery indicator, $\beta $, as function of the cross-section averaged inlet velocity ${\overline{u}}^{0}$ in units of $L{D}_{0}/{R}^{2}$. Here, ${\overline{u}}^{0}$ is varied for a fixed $\mathrm{TMP}=16$ kPa for $a=3.13$ nm, and 5 kPa for $a=10$ nm. The feed concentration is ${\varphi}_{b}=0.001$. Remaining input parameters as in Figure 6a,b. Open (filled) symbols are mBLA results for $a=3.13$ nm ($a=10$ nm). Solid, dashed, and dotted lines represent the pure solvent values, ${\beta}^{0}$, for the respective geometries as indicated.

**Figure 12.**Solvent recovery indicator, $\beta $, as a function of $P{e}_{R}$, for ${({\beta}^{0}/P{e}_{R})}_{\mathrm{G}1,\phantom{\rule{4.pt}{0ex}}\mathrm{G}2}\approx 8.08\times {10}^{-3}$, and ${({\gamma}^{0}/P{e}_{R})}_{\mathrm{G}1}\approx 1.55\times {10}^{-4}$ (list G1 in Table 3: open symbols) and ${({\gamma}^{0}/P{e}_{R})}_{\mathrm{G}2}\approx 3.10\times {10}^{-4}$ (list G2: filled symbols), respectively. Identical membrane properties are used for CM, FMM, and FMS. The employed operating parameters are listed in rows G1 and G2 of Table 3.

**Figure 13.**Solvent recovery indicator, $\beta $, as a function of $P{e}_{R}$ for three different variable sets $({\beta}^{0}/P{e}_{R},{\gamma}^{0}/P{e}_{R})$ given in the figure for the respective CM, FMM, and FMS geometries. Symbols are mBLA results for $\beta $ based on the according input parameter lists in Table 3. See text for details.

**Table 1.**Summary of geometry-dependent quantities characterizing the outer solution. Notice that ${\lambda}_{2}={R}_{H}M/\left(LA{\overline{U}}^{out}\right)$. The effective permeability parameter, K, is given in units of ${K}^{*}=\sqrt{{\eta}_{s}{L}_{p}{L}^{2}/{R}^{3}}$ so that $K/{K}^{*}=\sqrt{{\lambda}_{1}{\lambda}_{2}{(R/{R}_{H})}^{3}}$.

Membrane Geometry | Transversal Velocity Boundary Condition | ${\mathit{\lambda}}_{1}$ | ${\mathit{\lambda}}_{2}$ | ${\mathit{R}}_{\mathit{H}}$ | ${\overline{\mathit{U}}}^{\mathit{out}}$ | $\frac{\mathit{M}}{\mathit{A}}$ | $\frac{\mathit{K}}{{\mathit{K}}^{*}}$ | H | ${\mathit{V}}^{\mathit{out}}\left(\mathit{y}\right)$ |
---|---|---|---|---|---|---|---|---|---|

CM | $\phantom{\rule{1.em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}v(R,z)={v}_{w}\left(z\right)$ | 1 | 2 | $\frac{R}{2}$ | $\frac{1}{2}$ | $\frac{2L}{R}$ | 4 | $Rln\left(1+\frac{h}{R}\right)$ | $2\frac{y}{R}-{\left(\frac{y}{R}\right)}^{3}$ |

FMM | $\begin{array}{cc}\hfill v(R,z)& ={v}_{w}\left(z\right)\hfill \\ \hfill v(-R,z)& =-{v}_{w}\left(z\right)\hfill \end{array}$ | 2 | $\frac{3}{2}$ | R | $\frac{2}{3}$ | $\frac{L}{R}$ | $\sqrt{3}$ | h | $\frac{1}{2}\left[3\frac{y}{R}-{\left(\frac{y}{R}\right)}^{3}\right]$ |

FMS | $\begin{array}{cc}\hfill v(R,z)& ={v}_{w}\left(z\right)\hfill \\ \hfill v(-R,z)& =0\hfill \end{array}$ | 2 | $\frac{3}{4}$ | R | $\frac{2}{3}$ | $\frac{L}{2R}$ | $\sqrt{\frac{3}{2}}$ | h | $\frac{1}{4}\left[3\frac{y}{R}-{\left(\frac{y}{R}\right)}^{3}+2\right]$ |

**Table 2.**Summary of conditions for the validity of the mBLA method. The Reynolds number associated with the channel flow was $Re=4\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{R}_{H}\phantom{\rule{0.166667em}{0ex}}{\overline{u}}^{0}\rho /\eta $, where $\rho $ is the dispersion mass density, $\eta $ the effective dispersion viscosity, ${\overline{u}}^{0}$ the cross-section averaged inlet velocity, and ${R}_{H}$ the hydraulic channel radius. The solvent recovery indicator, $\beta $, is defined in Equation (21).

Conditions | Remarks |
---|---|

$\phantom{\rule{1.em}{0ex}}P{e}_{a}\le 0.1$ | Strong Brownian motion |

$R/L\ll 1$ | Small aspect ratio. Note that ${\u03f5}_{\delta}\ge R/L$ |

$\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}Re\lesssim 2000$ | Condition for laminar (non-turbulent) flow |

$Re\phantom{\rule{0.166667em}{0ex}}R/L\ll 1\phantom{\rule{1.em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}$ | No inertial flow effects on length scale L |

$\phantom{\rule{2.em}{0ex}}\phantom{\rule{1.em}{0ex}}\beta \ge \mathcal{O}\left[{\u03f5}_{\delta}\right]$ | Condition for significant permeability effects |

$\phantom{\rule{1.em}{0ex}}{\varphi}_{b}\ll 1$ | Small feed concentration (i.e., ${\varphi}_{b}P{e}_{R}<1$) |

**Table 3.**Input parameters used for the mBLA results for the solvent recovery indicator $\beta $ depicted in Figure 12 and Figure 13, respectively, as indicated. Fixed parameters are particle radius $a=3.13$ nm and mean Darcy permeability $\kappa =1.36\times {10}^{16}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{\mathrm{m}}^{2}$. The employed values for the reference system cross-section averaged inlet velocity and hydraulic permeability are ${\overline{u}}_{REF}^{0}=3.40\times {10}^{-2}$ and ${L}_{p,REF}=6.7\times {10}^{-10}$ m/(Pa s), respectively.

Data(Figure 12) | R(mm) | L(m) | $\mathit{h}/\mathit{R}$ | ${\mathit{L}}_{\mathit{p}}/{\mathit{L}}_{\mathit{p},\mathit{REF}}$CM, FMM/FMS | ${\mathit{\varphi}}_{\mathit{b}}/{10}^{-3}$CM, FMM, FMS | ${\overline{\mathit{u}}}^{0}/{\overline{\mathit{u}}}_{\mathit{REF}}^{0}$CM, FMM, FMS | $({\mathit{\beta}}^{0}/{\mathit{Pe}}_{\mathit{R}})/{10}^{-3}$ | $({\mathit{\gamma}}^{0}/{\mathit{Pe}}_{\mathit{R}})/{10}^{-4}$ |

G1 | 0.5 | 0.5 | - | 1.00, 1.00 | 1.00, 0.375, 0.186 | 1.00, 0.50, 0.25 | 8.08 | 1.55 |

G2 | 0.5 | 0.5 | - | 1.00, 1.00 | 2.00, 0.750, 0.375 | 1.00, 0.50, 0.25 | 8.08 | 3.10 |

(Figure 13) | $\mathit{R}$(mm) | $\mathit{L}$(m) | $\mathit{h}/\mathit{R}$ | ${\mathit{L}}_{\mathit{p}}/{\mathit{L}}_{\mathit{p},\mathit{REF}}$ | ${\mathit{\varphi}}_{\mathit{b}}/{\mathbf{10}}^{-\mathbf{3}}$ | ${\overline{\mathit{u}}}^{\mathbf{0}}/{\overline{\mathit{u}}}_{\mathit{REF}}^{\mathbf{0}}$ | $({\mathbf{\beta}}^{\mathbf{0}}/{\mathit{Pe}}_{\mathit{R}})/{\mathbf{10}}^{-\mathbf{3}}$ | $({\mathbf{\gamma}}^{\mathbf{0}}/{\mathit{Pe}}_{\mathit{R}})/{\mathbf{10}}^{-\mathbf{4}}$ |

CM-1 | 0.5 | 0.5 | - | 1.00 | 1.00 | 1.00 | 8.08 | 1.55 |

CM-2 | 0.5 | 0.5 | 0.5 | 1.00 | 1.00 | 1.00 | 8.08 | 1.55 |

CM-3 | 0.5 | 0.5 | 1 | 0.59 | 1.69 | 1.00 | 8.08 | 1.55 |

CM-4 | 0.25 | 0.5 | 0.5 | 2.00 | 1.00 | 4.00 | 8.08 | 1.55 |

FMM-1 | 0.5 | 0.5 | - | 1.00 | 1.00 | 1.00 | 4.04 | 2.07 |

FMM-2 | 0.5 | 0.5 | 0.5 | 0.81 | 1.24 | 1.00 | 4.04 | 2.08 |

FMM-3 | 0.5 | 0.5 | 1 | 0.41 | 2.44 | 1.00 | 4.04 | 2.07 |

FMM-4 | 0.25 | 0.5 | 0.5 | 1.62 | 1.23 | 4.00 | 4.04 | 2.07 |

FMS-1 | 0.5 | 0.5 | - | 1.00 | 1.00 | 1.00 | 2.02 | 2.07 |

FMS-2 | 0.5 | 0.5 | 0.5 | 0.81 | 1.24 | 1.00 | 2.02 | 2.08 |

FMS-3 | 0.5 | 0.5 | 1 | 0.41 | 2.44 | 1.00 | 2.02 | 2.07 |

FMS-4 | 0.25 | 0.5 | 0.5 | 1.62 | 1.23 | 4.00 | 2.02 | 2.07 |

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**MDPI and ACS Style**

Park, G.W.; Nägele, G.
Geometrical Influence on Particle Transport in Cross-Flow Ultrafiltration: Cylindrical and Flat Sheet Membranes. *Membranes* **2021**, *11*, 960.
https://doi.org/10.3390/membranes11120960

**AMA Style**

Park GW, Nägele G.
Geometrical Influence on Particle Transport in Cross-Flow Ultrafiltration: Cylindrical and Flat Sheet Membranes. *Membranes*. 2021; 11(12):960.
https://doi.org/10.3390/membranes11120960

**Chicago/Turabian Style**

Park, Gun Woo, and Gerhard Nägele.
2021. "Geometrical Influence on Particle Transport in Cross-Flow Ultrafiltration: Cylindrical and Flat Sheet Membranes" *Membranes* 11, no. 12: 960.
https://doi.org/10.3390/membranes11120960