ps6solutions

# ps6solutions - Harvard-MIT Division of Health Sciences and...

This preview shows pages 1–5. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 6 SOLUTIONS April 1, 2004 Problem 1 A pressure wave, P i , incident on an arterial (or bronchial) bifurcation will suffer a reﬂection, P r , of magnitude P r Z L − Z o P i ≡ = Z L + Z o where Z o is the upstream arterial impedance, and Z L is the impedance of the bifurcation. The impedance, Z , of a vessel of radius R may be calculated from the formulæderived in the notes: ρ 2 π R 3 Z 2 = AC u C u ≈ hE where ρ is the ﬂuid density, A is the vessel cross-section, h is the vessel wall thickness, and E is the vessel modulus of elasticity. Figure 1: 2 R 1 2 R 0 U 0 U 1 U 1 h 1 h A. Assuming a symmetric bifurcation, ρ = constant, E 0 E 1 , , and no increase in = R 1 = R 0 total cross-sectional area across the bifurcation, calculate the reﬂection coefficient. 2 π R 3 C u ≈ hE A π R 2 = 1 1 1 Z L = Z l + Z l 2 = Z l ρ same = 6.022j—2004: Solutions to Problem Set 6 2 Z 2 ρ ρ hE = AC u = π R 2 2 π R 3 ρ hE = 2 π 2 R 5 1 / 2 1 / 2 1 h 1 E 1 h 0 E 0 2 R 5 R 5 P r Z L − Z o 1 − 0 1 / 2 1 / 2 P i ≡ = Z L + Z o = 1 h 1 E 1 h 0 E 0 2 R 5 + R 5 1 0 h 1 h If E 1 E 0 and , = R 1 = R 0 P r 1 1 1 − 1 0 1 R R 1 0 2 − 1 2 R 2 R 2 2 P i = 1 1 1 = 1 R 0 2 2 R 1 2 + R 2 2 R 1 + 1 1 R 0 2 For 2 A 1 A , R 1 = = √ 2 R . R 1 = 2 P r 0 P i = B. Suppose the vessels distal to the bifurcation are severely calcified, so that E 1 E . What will the reﬂection coeffecient be? Does this suggest a noninvasive method of detecting the presence of severe arterial disease? P r Z L Z , so > 1 and reﬂection is positive for positive wave in. P i 2004/24 3 6.022j—2004: Solutions to Problem Set 6 Problem 2 This problem deals with the estimation of the pressure drop to be expected across a vascular steno- sis. Figure 2 is a sketch of the cross-section of a stenotic artery with a concentric plaque. An idealized model is shown in Figure 3, demonstrating a narrowed region followed by a sudden ex- pansion where the ﬂuid will generally exhibit turbulent ﬂow. Energy will be lost in two ways: in viscous ﬂow in the narrow region and in turbulent loss in the expansion. In this problem we will concentrate on the latter. Figure 2: Figure 3: u 1 , A 1 , P 1 u 2 , A 2 , P 2 u 3 , A 3 , P 3 1 2 3 The first task will be to estimate the pressure drop from point 2 to point 3. Position 3 is chosen far enough downstream to be in a region of uniform ﬂow, of velocity u 3 . In order to solve this problem we must make use of the “linear momentum theorem”....
View Full Document

{[ snackBarMessage ]}

### Page1 / 15

ps6solutions - Harvard-MIT Division of Health Sciences and...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online