Figure 1 below shows an early attempt to design, manufacture, and test a Direct Bloodoxygenator. Air, or O2 enriched air, was flown inside a "plastic bag" while blood was released
atthe top of the oxygenator to slowly flow along the plastic bag's walls downward under theinfluence of gravity. Blood and Air were in direct contact, thus supporting a very efficient masstransport of O2 from Air to blood and transport of CO2 from blood to Air. Later, in the mid 20thcentury, scientists and engineers replaced these bags with gas permeable membranes, whichfacilitated gas transport between blood and Air. Nevertheless, the mass transport coefficient inmembrane-supported devices was lower than in the Direct Blood oxygenator. Consider isothermal, steady, unidirectional laminar flow of Blood down the wall of the direct contact oxygenator as illustrated in Figure 2a. The thin film of blood of approximate thickness8 = 165 [um] is flowing due to gravity only. The Shear Stress - Shear Rate data for blood at 25°C is presented in Figure 2b. a) (30 points) Using the data presented in Figure 2b determine the coefficients of the Shear Rate - Shear Stress relationship for Blood. b) (10 points) Apply the continuity equation for this application. What is the conclusion? c) (30 points) Develop a mathematical model [differential equation(s) + boundary conditions]that will represent the flow of Blood in the film along the walls of the plastic bag. Start from the conservation of momentum equations (Navier-Stokes) and show your work for the simplification. d) (50 points) Solve the mathematical model developed in (c) and obtain an algebraic expression that will represent the velocity profile u,(y) of Blood. e) (30 points) Develop the expression for the volumetric flow rate (Q) of blood in the Direct Blood oxygenator. Determine the volumetric flow rate of blood in Q(=)mL/min] if the width of the bag is W = 1.25 [m]. f) (10 points) Make a graph u (y) versus y; use Excel. g) (30 points) If the exponent 'n' in the solutions obtained in parts (d) and (e) is set to n =1,(and n= u) do your solutions reduce to i) velocity profile and ii) volumetric flow rate that could be obtained for a Newtonian fluid. Check and show all your work. Assumptions: The flow is assumed to be fully developed, isothermal, unidirectional and laminar. Momentum transfer between Blood and Air is negligible. One could assume that bloodди,is a non-Newtonian Power-Law fluid: T, =-nдуAlso, ignore entrance and exit effects of Blood flow. State any additional assumption. \begin{aligned} &\text { Momentum equation in } x \text { direction: }\\ &\rho\left[\frac{\partial u_{x}}{\partial t}+u_{x} \frac{\partial u_{x}}{\partial x}+u_{y} \frac{\partial u_{x}}{\partial y}+u_{z} \frac{\partial u_{x}}{\partial z}\right]=-\left[\frac{\partial \tau_{x x}}{\partial x}+\frac{\partial \tau_{x y}}{\partial y}+\frac{\partial \tau_{x z}}{\partial z}\right]+\rho g_{x} \end{aligned} \begin{aligned} &\text { Momentum equation in } y \text { direction: }\\ &\rho\left[\frac{\partial u_{y}}{\partial t}+u_{x} \frac{\partial u_{y}}{\partial x}+u_{y} \frac{\partial u_{y}}{\partial y}+u_{z} \frac{\partial u_{y}}{\partial z}\right]=-\left[\frac{\partial \tau_{y x}}{\partial x}+\frac{\partial \tau_{y y}}{\partial y}+\frac{\partial \tau_{y z}}{\partial z}\right]+\rho g_{y} \end{aligned} \begin{aligned} &\text { Momentum equation in z direction: }\\ &\rho\left[\frac{\partial u_{z}}{\partial t}+u_{x} \frac{\partial u_{z}}{\partial x}+u_{y} \frac{\partial u_{z}}{\partial y}+u_{z} \frac{\partial u_{z}}{\partial z}\right]=-\left[\frac{\partial \tau_{z x}}{\partial x}+\frac{\partial \tau_{z y}}{\partial y}+\frac{\partial \tau_{z z}}{\partial z}\right]+\rho g_{z} \end{aligned}
Fig: 1
Fig: 2
Fig: 3
Fig: 4
Fig: 5
Fig: 6
Fig: 7
Fig: 8
Fig: 9
Fig: 10
Fig: 11
Fig: 12
Fig: 13
Fig: 14
Fig: 15
Most Viewed Questions Of Mechanics
From Gerhart Flow of a viscous fluid over a flat plate surface results in the development of a region of reduced velocity adjacent to the wetted surface as depicted in Fig. P5.25. This region of reduced flow is called a boundary layer. At the leading edge of the plate, the velocity profile may be considered uniformly distributed with a value U. All along the outer edge of the boundary layer, the fluid velocity component parallel to the plate surface is also U. If the x-direction velocity profile at section(2) is \frac{u}{U}=\frac{3}{2}\left(\frac{y}{\delta}\right)-\frac{1}{2}\left(\frac{y}{\delta}\right)^{3} develop an expression for the volume flow rate through the edge of the boundary layer from the leading edge to a location downstream at x where the boundary layer thickness is 6.Figure P5.25 1) 5.25 – assume a width 'l' into the paper
5) 5.63 Water is sprayed radially outward over 180° as indicated in Fig. P5.67. The jet sheet is in the horizontal plane. If the jet velocity at the nozzle exit is 20 ft/s, determine the direction and magnitude of the resultant horizontal anchoring force required to hold the nozzle in place.
Fluids of viscosities µ ₁ = 0.1 N.s/m² and µ ₂ =0.15 N.s/m² are contained between two plates(each plate is 1 m² in area). The thicknesses are h₁ = 0.5 mm and h₂ = 0.3 mm, respectively.Find the force F to make the upper plate move at a speed of 1 m/s. What is the fluid velocity at the interface between the two fluids?
A tank, which is open to the atmosphere, is filled with water to a level h and allowed to drain through an orifice at the bottom, as shown in the figure below. The cross-sectional area of the tank is At and the cross-sectional area of the orifice is Ag. Assume that the cross-sectional area of the tank is much greater than the cross-sectional area of the orifice (AT>>Ao) and that the exit losses are negligible. 6) Use Reynolds Transport Theorem to derive an expression for the variation of h with time as a function of velocity at the outlet.
*5-96. Determine the power delivered to the turbine if the water exits the 400-mm-diameter pipeat 8 m/s. Draw the energy and hydraulic grade lines for the pipe using a datum at point C.Neglect all losses.
3) 5.45 Determine the magnitude and direction of the anchoring force needed to hold the horizontal elbow and nozzle combination shown in Fig. P5.47 in place. Atmospheric pressure is 100 kPa(abs). The gage pressure at section (1) is 100 kPa. At section (2), the water exits to the atmosphere.
Launch Meeting-Zoum5-111. A fire truck supplies 150 gal/min of water to the third story of a building at B. If the frictionloss through the 60-ft-long, 2.5-In.-diameter hose is 12 ft for every 100 ft of hose, determine therequired pressure developed at the outlet A of the pump located within the truck close to theground. Also, what is the average velocity of the water as it is ejected through a 1.25-in.-diameter nozzle at B?
10) A conveyor belt discharges gravel onto a barge at a rate of 50 yd/min. If the gravel weighs 120 Ibf/ft^3, what is the tension in the hawser that secures the barge to the dock?
10.2-39 The following vapor-liquid equilibrium data havebeen reported' for the system water (1) + 1,4-dioxane (2) at 323.15 K. Compute the activity coefficients for thissystem at each of the reported compositions.a. b. Are these data thermodynamically consistent? c. Plot the excess Gibbs energy for this system asa function of composition.
"5-104. The flow of air at A through a 200-mm-diameter duct has an absolute inlet pressure of180 kPa, a temperature of 15°C, and a velocity of 10 m/s. Farther downstream a 2-kW exhaustsystem increases the outlet velacity at B to 25 m/s. Determine the change in enthalpy of the air.Neglect heat transfer through the pipe.B
