Q.63 In a cross-flow filtration process, the pressure drop (ΔP) driving the fluid flow is 2 atm,
inlet feed pressure (Pi) is 3 atm and filtrate pressure (Pf) is equal to atmospheric pressure.
The average transmembrane pressure drop (ΔPm) is ________ atm.
Solution Summary
Cross-flow filtration uses the average transmembrane pressure drop (ΔPm) calculated as the average of inlet feed pressure (Pi) and retentate pressure (Pr), since filtrate pressure (Pf) is atmospheric (0 atm gauge). Given ΔP = 2 atm (driving fluid flow along the membrane), Pi = 3 atm, and Pf = 0 atm, Pr = Pi – ΔP = 1 atm, so ΔPm = (3 + 1)/2 = 2 atm.
Key Concept
In cross-flow filtration, feed flows parallel to the membrane, creating a pressure drop (ΔP) from inlet (Pi) to retentate outlet (Pr), while transmembrane pressure drives perpendicular filtrate flux. The standard formula is ΔPm = (Pi + Pr)/2, distinguishing flow-driving ΔP from filtration-driving ΔPm.
Step-by-Step Solution
- Start with Pr = Pi – ΔP = 3 atm – 2 atm = 1 atm.
- Then ΔPm = (Pi + Pr)/2 = (3 + 1)/2 = 2 atm.
- This assumes linear pressure decay along the channel, common in engineering calculations.
Understanding Parameters
- ΔP (2 atm): Pressure drop driving axial fluid flow from inlet to retentate.
- Pi (3 atm): Inlet feed pressure on membrane feed side.
- Pf (0 atm): Filtrate side at atmospheric pressure.
- Pr: Retentate pressure = Pi – ΔP = 1 atm.
Formula and Derivation
ΔPm = (Pi + Pr)/2 assumes uniform average across membrane length.
Substitute: ΔPm = (3 + 1)/2 = 2 atm.
For non-linear profiles, logarithmic means apply, but linear is standard here.
Applications in Biotech
Used in protein purification, cell harvesting; monitors flux J = ΔPm / (μ Rm). Higher ΔPm boosts filtrate yield but risks fouling.
Pressure Parameters Summary
| Parameter | Value (atm) | Role |
|---|---|---|
| Pi | 3 | Inlet feed pressure |
| Pr | 1 | Retentate pressure (Pi – ΔP) |
| Pf | 0 | Filtrate (atmospheric) |
| ΔP | 2 | Axial flow driver |
| ΔPm | 2 | Average transmembrane |
