Abstract
It is shown in this paper that the nearly constant length-to-diameter ratio observed with conducting airways of human bronchial tree can be explained based on the fluid dynamic optimality principle. In any branched tube there are two pressure loss mechanisms, one for wall friction in the tube section and the other for flow division in the branching section, and there exists an optimal length-to-diameter ratio which minimizes the total pressure loss for a branched tube in laminar flow condition. The optimal length-to-diameter ratio predicted by the pressure loss minimization shows an excellent agreement with the length-to-diameter ratios found in the human conducting airways.
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Abbreviations
- A :
-
cross sectional area
- a :
-
constant
- d :
-
tube diameter
- F :
-
proportional constant between pressure loss coefficient and Reynolds number
- K :
-
pressure loss coefficient
- l :
-
tube length
- n :
-
generation number
- P :
-
pressure
- ΔP :
-
pressure drop or loss
- ΔP b :
-
pressure drop due to branching or flow division
- ΔP s :
-
pressure drop due to wall friction
- Q :
-
volumetric flow rate
- Re :
-
Reynolds number, Vd/ν
- V :
-
flow velocity
- μ:
-
dynamic viscosity
- ν:
-
kinematic viscosity
- ρ:
-
density
- Θ:
-
branching angle
- 0:
-
mother branch
- 1:
-
major daughter branch
- 2:
-
minor daughter branch
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Acknowledgment
This work was supported by MOST(KOSEF) through Systems Bio-Dynamics Research Center.
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Lee, J.W., Kang, M.Y., Yang, H.J. et al. Fluid-dynamic optimality in the generation-averaged length-to-diameter ratio of the human bronchial tree. Med Bio Eng Comput 45, 1071–1078 (2007). https://doi.org/10.1007/s11517-007-0232-8
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DOI: https://doi.org/10.1007/s11517-007-0232-8