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A reverse osmosis

(\mathrm{RO})

membrane with an area of $10 \mathrm{~cm}(\mat...

Apr 28, 2024

A reverse osmosis

(\mathrm{RO})

membrane with an area of

10 \mathrm{~cm}(\mathrm{~L}) \times 10 \mathrm{~cm}(\mathrm{~W})

, thickness of

300 \mu \mathrm{m}

, water permeability of

1.5 \mathrm{~L} / \mathrm{m}^{2} \cdot \mathrm{h} \cdot

bar and intrinsic rejection of

\mathrm{NaCl}

98 \%

is used to treat

5,000 \mathrm{mg} / \mathrm{L} \mathrm{NaCl}

solution at an operating pressure of 15 bar and temperature of

20^{\circ} \mathrm{C}

. After operating for 24 hours, the membrane is damaged, i.e., a straight through cylindrical pinhole with diameter of

10 \mu \mathrm{m}

is formed on the membrane. Determine the percentage changes in the flux and permeate

\mathrm{NaCl}

concentration of the damaged membrane. Assume negligible concentration polarization and the viscosity of solution is

1.0 \mathrm{mPa} \cdot \mathrm{s}

. State all assumptions made.

Solution

Calculate the initial flux: The initial flux (

J_{\text{initial}}

) can be calculated using the water permeability (

A

), the operating pressure (

\Delta P

), and the membrane area (

A_m

). The formula is

J_{\text{initial}} = A \times \Delta P \times A_m

Calculate the initial permeate flow rate: The initial permeate flow rate (

Q_{\text{initial}}

) is the product of the initial flux and the membrane area,

Q_{\text{initial}} = J_{\text{initial}} \times A_m

Calculate the area of the pinhole: The area of the pinhole (

A_{\text{pinhole}}

) can be calculated using the formula for the area of a circle,

A_{\text{pinhole}} = \pi \times \left(\frac{d}{2}\right)^2

, where

d

is the diameter of the pinhole

Calculate the flux through the pinhole: The flux through the pinhole (

J_{\text{pinhole}}

) can be calculated using the Hagen-Poiseuille equation for laminar flow through a cylindrical tube,

J_{\text{pinhole}} = \frac{\Delta P \times A_{\text{pinhole}}}{8 \times \mu \times l}

, where

\mu

is the viscosity of the solution and

l

is the thickness of the membrane

Calculate the new total flux: The new total flux (

J_{\text{new}}

) is the sum of the initial flux and the flux through the pinhole,

J_{\text{new}} = J_{\text{initial}} + J_{\text{pinhole}}

Calculate the percentage change in flux: The percentage change in flux can be calculated using the formula

\frac{J_{\text{new}} - J_{\text{initial}}}{J_{\text{initial}}} \times 100\%

Calculate the new permeate concentration: Since the intrinsic rejection is 98%, the concentration of

NaCl

in the permeate through the intact membrane is 2% of the feed concentration. However, the pinhole allows salt to pass through without rejection. The new permeate concentration (

C_{\text{new}}

) is a weighted average of the concentration from the intact membrane and the pinhole

Calculate the percentage change in permeate $NaCl$ concentration: The percentage change in permeate

NaCl

concentration can be calculated using the formula

\frac{C_{\text{new}} - C_{\text{initial}}}{C_{\text{initial}}} \times 100\%

Answer

The percentage changes in the flux and permeate

NaCl

concentration of the damaged membrane cannot be provided without the actual calculations.

Key Concept

Reverse osmosis membrane performance is characterized by flux and solute rejection, which can be affected by damage such as pinholes.

Explanation

The flux is a measure of how much water passes through the membrane, and the permeate concentration indicates the quality of the treated water. Damage to the membrane can lead to increased flux and decreased solute rejection, thus affecting the performance of the RO system.