ps7_solutions

ps7_solutions - Harvard-MIT Division of Health Sciences and...

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Unformatted text preview: Harvard-MIT Division of Health Sciences and Technology HST.542J: Quantitative Physiology: Organ Transport Systems Instructors: Roger Mark and Jose Venegas MASSACHUSETTS INSTITUTE OF TECHNOLOGY Departments of Electrical Engineering, Mechanical Engineering, and the Harvard-MIT Division of Health Sciences and Technology 6.022J/2.792J/BEH.371J/HST542J: Quantitative Physiology: Organ Transport Systems PROBLEM SET 7 SOLUTIONS April 9, 2004 Problem 1 In this problem, consider the different factors which, in combination, give rise to the pressure dif- ferences encountered in the lung during breathing. For this purpose, use a simple model comprised of five generations with the geometry described below. Number of Generation Airways Total Cross-sectional Area (cm 2 ) Length (cm) 1 1 2 10 2 4 3 2 3 16 5 1.5 4 128 10 1.0 5 2000 50 0.5 In the following questions, consider inspiration at a normal breathing rate of 0.5 L/sec. Assume the ow to be steady and the uid to be = . 15 cm 2 /sec and = 2 10 3 gm/cm 3 . (Remember = / , the kinematic viscosity.) A. Compute the mean ow velocity in each generation assuming a uniform distribution of ven- tilation. See Table 1. B. Assume the ow to be inviscid and calculate the total pressure difference between the first and fifth generations. This represents the pressure difference necessary to decelerate a uid particle as it passes through the network. If ow is inviscid and steady, use Bernoulli: 1 2 P 1 + 1 v 1 2 = P 2 + v 2 2 2 = 2 10 3 P 1 = 0 1 P 2 = 2 10 3 ( 250 ) 2 1 2 10 3 10 2 2 2 = 10 3 62 , 500 10 1 = 62 . 5 dynes/cm 2 C. Assume, as a first approximation, that the ow in each airway is laminar (check your Reynolds numbers to see if this is reasonable) and fully developed. Calculate the pressure difference acting across the network (first to fifth generation) due to viscous forces alone. This repre- sents the pressure drop necessary to overcome the tendency of wall friction to impede the ow. (Note that the actual pressure drops in the upper airways will tend to be greater than those estimates due to secondary ow and the boundary layer developing after each branch.) 6.022j2004: Solutions to Problem Set 7 2 6 . 2 2 j 2 4 :...
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This note was uploaded on 11/11/2011 for the course BIO 2.797j taught by Professor Matthewlang during the Fall '06 term at MIT.

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ps7_solutions - Harvard-MIT Division of Health Sciences and...

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