1st order approx of transmembrane pressure in Dean's chip-in-a-beaker system

ByHenry

Hi all,

During last NRG meeting some questions are raised on what is the transmembrane pressure in Dean’s chip-in-a-beaker dialysis system and what is the direction of flow of the whole system.

i gave them some thought and proposed the following scenario. Please feel free to correct me should I made any mistake.  i am simply throwing this out there so we can all think about it (carefully).

Screen Shot 2013-06-22 at 2.46.56 PM

The arrow indicate the direction of the pressure, and the length of the arrow somewhat denote the magnitude.  The white arrow represents atmospheric pressure; the brown arrow represents the pressure given through the syringe pump;  and the red arrow represent the transmembrane pressure given due to the input flow given by the syringe pump.

1st, let us consider the case in which the system is not in the beaker of water, but suspended in the air, and the syringe pump is pushing water into the system.  In this case, there is a flow along the length of the membrane.  If we assume the pressure at any given point in a plane is isotropic (equal in all direction), then technically the transmembrane pressure varies along the length of the membrane, from higher pressure to lower pressure as we examine along the length of the membrane from left to right, because water flow from point of higher pressure (left) to the point of lower pressure (right).  This transmembrane pressure can be calculated from Hagen-Poiseuille equation, which stated that delta_P = mu*LQ/(8*pi*r^4), where delta P is the pressure difference, mu is water’s dynamic viscosity, L is the distance along the tubing from where the membrane is to where the tubing is open to air on the top right.  So the absolute pressure acting on the membrane from inside the system out is 1 atmosphere + delta_P. Luckily, since the membrane is open to air there is 1 atmosphere acting on the membrane from outside in.  So the transmembrane pressure is still just delta P, which after you calculate would be quite small (if we pump really slowly, which happened to be the case).

 

Now let’s consider the case when the system is now submerged in the beaker of water:

Screen Shot 2013-06-22 at 12.27.31 PM

The arrow indicate the direction of the pressure, and the length of the arrow somewhat denote the magnitude.  The white arrow represents atmospheric pressure; the brown arrow represents the pressure given through the syringe pump;  the red arrow represent the transmembrane pressure given due to the input flow given by the syringe pump; and the blue arrows represent hydrostatic pressure/buoyancy force.

1st question is whether or not there would be flow pushing into the system due to hydrostatic pressure/buoyancy force.  Assuming that the inside of the system is already filled with water,  I think there will not be any, because the water pressure from inside the system will  balance out the water pressure outside the system.  However, if the inside of the system is filled with air, then there is a transmembrane pressure  rho*p*h, where h is the distance between the position of the beaker’s water surface and the position of the membrane.  If this transmembrane pressure is greater than the surface tension of the air/water interface, then water will rush into the system, filling it up.

So if we superimpose the two above scenarios together then the transmembrane pressure is just delta_P, and water from inside the system always flows out into the beaker.  I am still a little bit confused though.  It may also be possible that the transmembrane pressure is actually rho*p*h – delta_P, in which case we get water rushing into our system to dilute the “blood” if rho*p*h is greater than delta_P.  I am leaning toward the interpretation that the transmembrane pressure is just delta_P, but i am not too confident about it.

Another thing is that I am still not fully convinced that the transmembrane pressure is just delta_P.  For closed tubing i think it is, but we have a membrane here, and water do flow through the membrane, locally relieving the pressure… do we are getting something less than delta_P.  I think we may need to come up with some physical model or COMSOL.

 

 

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3 Comments

  1. Thanks for helping with this discussion Henry.

    We need to be careful what we are calling the ‘transmembrane’ pressure. This term should be reserved for the pressure drop across the membrane in a perpendicular direction (inside – outside). I think you are using the term to mean the pressure drop down the channel (all inside).

    If Dean’s set-up looked just like your pictures you would be correct in your conclusion that the pressure above the membrane must be everywhere higher than the pressure below the membrane. Unfortunately, that is not how Dean’s set-up looks. The input and output tubes stay horizontal through holes he cut in the beaker. Thus the hydrostatic pressure in the beaker is potentially higher than the exit pressure of the flow channel. We need to use your DeltaP calculation to see if the exit tubing provides enough resistance to elevate the pressure above this hydrostatic pressure.

    By the way, the blog is equipped with a LATEX plug-in and you should be able to quickly beautify post’s like this with formulas that look like real mathematics.

  2. Delta P = frac{8 mu L Q}{ pi r^4}

    We had a good discussion about the experimental setup at Monday’s group lunch. I’m doing more vigorous calculations and will post clarifications when I’m done.

    1. Thanks for the correction.

      I think knowing the pressure distribution in the system is good overall because this way we can make adjustments to make sure that the transmembrane pressure will not burst the membrane (shorter tubings, etc.).

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