**Rationale**

Internal (or lumen) pressure drop in hollow fiber is one of the dominant parameters controlling the maximum sustainable flux in HF membrane. In this section, the relation among flux, fiber dimension, and internal pressure drop will be reviewed.

As illustrated in Fig. 1, permeate flows through the lumen toward the exit while more permeate joins the way. While permeate flows, pressure decreases due to the friction from the wall. More importantly, the pressure declining rate accelerates toward the exit due to the increasing flow velocity. As a result, the suction pressure (or TMP) is the highest in the fiber exit and the lowest in the dead end of the membrane, which in turn causes higher flux near the fiber exit than in the other end. Meanwhile, the static pressure of the permeate is always cancelled out by the static pressure in shell side.

The high local flux near the exit can exceed the critical flux, thereby accelerated fouling and flux loss can occur in the area. Under a constant flux mode, where operators only observe apparent average flux, the flux in the adjacent upstream area must increase to compensate the flux loss in downstream area to meet the average flux. Since more water travels longer distance, pressure loss (or TMP) increases.

Fig. 1. Internal pressure drop in hollow fiber. X_{L} =effective membrane length (m), P_{f} =suction pressure in the potting entrance (Pa), V_{f} = flow velocity in fiber exit (m/s), P_{0} =suction pressure in the fiber exit (Pa), L_{n} =potting depth (m), and L_{0} =elevation of pressure gauge from the water surface (m).

**Model equation development**

The pressure drop inside a hollow fiber can be written as equation (1), where multiplication of the cross-sectional area of flow channel () and a flow velocity gain (*dv*) in an infinitesimal block of the hollow fiber equals to the amount of the water permeates in the same block that is calculated by multiplying the surface area (*πD _{0}dx*) and flux (

*J)*of the block. This equation can be rearranged to equation (2).

———————————— (1)

———————————— (2)

where

= internal diameter of fiber (m)

= outer diameter of fiber (m)

= liquid velocity in fiber lumen (m/s)

= flux (m/s)

= distance from the dead end or the fiber center (m)

Pressure drop in lumen side can be described as equation (3) based on Hagen-Poiseuille equation, where no slip on the lumen surface is assumed. Internal static pressure is not necessary to consider in this equation because it always matches with external static pressure as long as whole fiber is immersed in the water.

————————————- (3)

where

= pressure in fiber lumen (kg/m/s^{2 }or Pa)

= viscosity of permeate (kg/m/s)

It is convenient to assume flux is proportional to TMP as equation (4) assuming clean water condition. Since and are zero at clean water condition, the permeability constant (m/s/Pa or m^{2}s/kg) corresponds to in resistance in series model. Here, is a pressure outside the hollow fiber, but can be assumed zero because it is always offset by lumen side static pressure, thereby the internal pressure, , becomes TMP.

————————————– (4)

The three equations can be solved simultaneously to obtain internal pressure and flux profiles. The average flux (or the apparent flux measured based on permeate flow rate and membrane surface area) of the hollow fiber is calculated using the following equation.

————————————– (5)

TMP measured by a pressure gauge in Fig. 1 includes the effect of the pressure gauge elevation from the water surface. The TMP reading can be estimated by equation 3-6 after adding the static water pressure in the pipeline between the water surface and the pressure gauge.

———————————-(6)

where

= TMP reading by pressure gauge (kPa)

= true TMP in hollow fiber module exit (kPa)

= pressure gauge elevation from water surface (m)

Practically is estimated by subtracting the static pressure measured during pause mode from the dynamic pressure measured during filtration mode. The elevation of the pressure gauge will be neglected in the future discussion.

The accuracy of the above equations was experimentally verified as shown in Fig. 2 using a single hollow fiber in clean water. The experimentally measured suction pressures in each node of the fiber match very well with the curves obtained theoretically for many different average/apparent fluxes.

Fig. 2. Comparison of experimental and theoretical data in clean water conditions. Dots indicate experimental data and solid lines indicate theoretical data. Effective internal diameter of hollow fiber was 0.84 mm (Yoon, 2008).

**Internal pressure and flux profile as a function of average flux**

The pressure and the flux profiles were calculated for a commonly used commercial hollow fiber membranes produced by GE (former Zenon) assuming an average flux of 30 LMH. The raw data including effective internal diameter, true membrane permeability, *etc*. were adapted from a literature (Yoon, 2008). Here the term, true permeability, is used to distinguish this from the apparent permeability estimated based on the apparent flux and the apparent TMP. Since the actual TMP is higher than the apparent TMP due to the internal pressure drop, apparent permeability is always lower than true permeability.

As shown in Fig. 3, TMP is the highest in the permeate exit at 6.5 kPa and the lowest in the dead end at 4.4 kPa. The local flux in clean water conditions decreases from 38 LMH in the fiber exit to 26 LMH in the dead end, while the average flux maintains at 30 LMH.

If permeability changes under the identical condition, the flux profile along the fiber does not change. Permeability only changes the TMP required to obtain the target average flux.

Fig. 3. Local flux and TMP in hollow fiber membrane. =0.0084 m, =0.0019 m, =1.0 m, =0.05m, =30LMH (or 8.3×10^{-6}m/s), =0.001 kg/m/s, and =1.66 x 10^{-9} m/s/Pa.

The other aspect of the internal pressure drop is that the gap between the highest and the lowest fluxes in one fiber increases as the average flux increases. As shown in Fig. 4, the difference between the highest and lowest fluxes in a fiber is 5 LMH when average flux is 20LMH, but it increases to 15LMH when the average flux is 50LMH. The increasingly uneven flux distributions at high average (or apparent) fluxes can cause an accelerated membrane fouling near the fiber exit. Membrane fouls exponentially faster as flux increases as discussed elsewhere. Once membrane fouling is initiated in the fiber exit, it tends to propagate upstream until the balance between TMP and flux distribution is reached.

Fig. 4. Local flux profiles in hollow fiber membranes as a function of average flux. Identical assumptions were used as for Fig. 3.

**Calculator**

The following spread sheet is designed to calculate internal pressure and flux profiles in clean water.

- Put input values.
- If apparent TMP and average flux in output values are not in desired range, change TMP at dead end until the desired output values are obtained.

© Seong Hoon Yoon