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Transmural Pressure in Tube System Splitting and Converging

  1. Feb 4, 2012 #1
    1. The problem statement, all variables and given/known data

    The points a, b and c and mid way along a series of tubes arranged such:

    There is a tube of 4mm for 100 cm
    It splits into 3 tubes of diameter 1mm, length 20 cm
    It converges into a tube of diameter 6mm, length 100 cm

    The fluid flowing through the tubes is 3ml/sec and has viscosity 0.01 poise (dyne*sec*cm^-2) and a density of 1g/cm^3.

    The flow is laminar and the pressure at the end of the series is 0 (or atmospheric).

    Calculate the transmural pressure at the points c, b and a

    2. Relevant equations

    Transmural pressure = Pressure(internal) - Pressure(external)

    Bernoulli's - P + (pu^2)/2 = constant
    Poiseullie's - delta-P = 8QµL/(pie*R^4)
    Resistance = (pie *R^4) / 8QµL
    Flow = velocity * cross sectional area
    Flow = (P1 - P2)(pie*r^4 / 8*viscosity*length)

    3. The attempt at a solution

    I thought, see as we have the end pressure, and the question starts with "calculate c". First I calculated the velocity of the 3 tube systems:

    First Tube - v = Flow/area = 3/0.502 = 5.97 cm/s
    Second Tube - 31.83 cm/s
    Third - 2.65 cm/s

    But I can't really see how that would be useful. I thought I unsure if the viscosity given, 0.01 poise, is "negligible" enough that I can use Bernoulli's. So I started with working out Pressure difference between the end of the tube, and the start of the third tube and the middle of the third tube.

    3 = (P1 - P2) (pie*0.6^4 / 8*0.01*100)
    so Pressure difference = 58.82 dyne*ml/cm^5 = 5.882 Pascals

    Seen as pressure at the end is 0, then i reasoned the pressure at the start of the tube = 5.882 Pa.

    By the same logic, the midpoint, c, will be of pressure 2.947 Pa.

    So I thought the transmural (difference between the inner and outside) = 2.947 at c

    However, I do not know how to translate this into working out the other tube pressures, or indeed if I should have used Bernoulli's. However I don't know:
    1. If I can use Bernoulli's, as there is some degree of friction
    2. How I can use it to work out pressure? Considering I would need to have a pressure value to work out the constant? Unless the constant from Bernoulli's is the same for the entire system.
     
  2. jcsd
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