quiz2_solutions

# quiz2_solutions - Harvard-MIT Division of Health Sciences...

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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 QUIZ 2 SOLUTIONS

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Problem 1 (20 min) One method for determining the degree of ﬂow obstruction produced by a stenosed aortic valve is to measure pressures simultaneously inside the left ventricle and at a point several aortic diameters downstream of the valve. In this question you will be asked to develop the relationship between these pressures and the cross-sectional area of the valve. (See Figure 1.1.) Figure 1.1: a a (2) (1) (3) Surface a–a Control volume A s P V v Turbulent mixing zone Valve area v, Ventricle Pressure P Area A , Pressure P 3 A. Find the pressure, P 2 , at the exit of the stenosed valve (2) assuming ﬂow between (1) and (2) to be inviscid and steady . (Although the real ﬂow is unsteady, the effects of unsteadiness in this calculation are small.) You may express P 2 in terms of the instantaneous ﬂow rate through the valve, Q , the pressure in the ventricle, P v (where the velocity can be assumed to be zero), the area of the valve, A s , and the density of blood, ρ . Use continuity and Bernoulli: P v + 1 2 ρ V 2 v = P 2 + 1 2 ρ V 2 2 but V v 0 V 2 = Q A s P 2 = P v 1 2 ρ Q 2 A 2 s B. Using the control volume indicated in the sketch, and assuming that the pressure acting all along the upstream surface (a–a) is equal to the pressure at (2) calculated in (A), obtain an expression for the pressure difference P 3 P 2 . You may assume that both the entering and exit velocity proﬁles are ﬂat as indicated in the sketch, and that there is vigorous turbulent 6.022j—2004: Solutions to Quiz 2 2
mixing between points (2) and (3). Neglect the effects of wall shear stress on the sides of the

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quiz2_solutions - Harvard-MIT Division of Health Sciences...

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