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 Assigned: March 18, 2004 Due: April 1, 2004Problem 1 A pressure wave, Pi , incident on an arterial (or bronchial) bifurcation will suffer a reflection, Pr , of magnitude Pr ZL − Zo≡ � = Pi ZL + Zo where Zo is the upstream arterial impedance, and ZL is the impedance of the bifurcation. The impedance, Z , of a vessel of radius R may be calculated from the formulæderived in the notes: Z 2 ρ 2π R3 = Cu ≈ ACu h E 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: 2R1 2R0 U1 U0 U1 h1 h0A. Assuming a symmetric bifurcation, ρ = constant, E0 = E1, = , and no increase in R1 R0 total cross-sectional area across the bifurcation, calculate the reflection coefficient. B. Suppose the vessels distal to the bifurcation are severely calcified, so that E1 � E0. What will the reflection coeffecient be? Does this suggest a noninvasive method of detecting the presence of severe arterial disease? 2004/24 2 6.022j—2004: 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: 1 2 3 u3, A3, P3u2, A2, P2u1, A1, P1 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 u3. In order to solve this problem we must make use of the “linear momentum theorem”. Consider a flow field of fluid with a superimposed control volume (CV). Newton’s second law for the system included within the CV is d ��M = Fsystem (1) dt This equation states that the rate of change of the system momentum equals the sum of forces acting on the mass inside the control volume. Equation 1 may be written as: d ρU�dV + ρU�U�· n� d A = F�(2) dt VCV ACV The first term is the rate of accumulation of momentum within the control volume; the second term is the net rate of momentum flux out of the CV. 6.022j—2004: Problem Set 6 3Let us establish a CV for our model of the sudden expansion in Figure 3. (The volume within the dashed line in Figure 4.) Figure 4: u3, A3, P3u2, A2, P2 The fluid is incompressible, and there is no net acceleration of the mass within the CV, so the first term of Equation 2 disappears. How do we evaluate the second term? A. Calculate the momentum flux into the CV. (Note that flux is the rate of momentum passing into the CV per unit time.) Do you agree that the answer is: ρu2(u2 A2) = 2ρ A2u2 B. What is the momentum flux out of the CV at point 3? C. Use conservation of mass to relate u3 to u2 and the cross-sectional areas A2 and A3. D. Next we need to calculate the right hand side of equation 2. The forces acting on the CV are pressure forces. Assume that the pressure at the entrance orifice, P2, is equal across the entire face of the CV at position 2. What is the net force acting on the CV? E. What is the pressure drop P2 − P3 as a function of the velocity u2, the areas A2 and A3, and the fluid density? F. Now let us focus on the entrance portion of the stenosis shown below, and consider inviscid flow (Figure 5). Figure 5: u2, A2, P2u1, A1, P1 Use the Bernoulli principle and conservation of mass to relate the pressure drop P1 − P2 to the characteristics of the fluid, the entrance velocity (u1), and the geometry. 6.022j—2004: Problem Set 6 4G. If we neglect viscous losses in the region of the stenosis, what will be the total pressure drop P1 − P3? Your derivation should end up with the following: 1 2 � A1 �2 P1 − P3 = ρu1 − 1 2 A2 H. In a vessel where u1 is 30 cm/sec, and the ratio A2/A1 is 0.1, estimate the pressure drop in mmHg. (Remember that 1 mmHg=1330 dynes/cm2.) What pressure drop would be expected if the lumen diameter is reduced to 25% of its original size? 2004/234 6.022j—2004: Problem Set 6 5Problem 3 Engineers for a medical products company have developed an inexpensive, hand-held device for monitoring peak expiratory flow rates in asthmatics. The device, shown in Figure 6, consists of a rigid cylinder with a long, narrow slit down the side and a close-fitting, spring-loaded piston diaphragm. Expired air enters the chamber of the peak flow meter from the left through the mouth-piece, then exits via the slit at a uniform velocity. The length of the slit available for flow, x , is equal to the displacement of the piston. Gas exits from the slit in the form of a turbulent jet into the atmosphere at pressure pa . Assuming the flow to be quasi-steady and inviscid, that the piston moves without friction, and that the displacement is linearly proportional to the force F acting on it: F = k x answer the following questions. You may take the following as given: • density of expired air, ρ • atmospheric pressure, pa • slit height, h (h � D) • volume flow rate, Q • spring constant, k • piston diameter, D • exposed slit length, x A. Given the volume flow rate entering the device, Q, and assuming the piston to be displaced a distance x but non-accelerating, what is the velocity of gas exiting through the slit? B. What is the pressure inside the chamber of the flowmeter, po? C. What is the relationship between the displacement of the piston, x , and the other given parameters? Figure 6: Spring Slit Piston x Q 2004/448 6 6.022j—2004: Problem Set 6Problem 4 You are conducting experiments in your laboratory on a new artificial lung which consists of two parallel membranes separated by a small distance d. (See Figure 7.) Blood