Consider a membrane of thickness 10 μm that has a number of tiny cylindrical pores (of radius 10 nm) passing through it. The density of pores in the membrane is such that the porosity (fractions of water-filled space) of the membrane is 0.1%.

1. (a) Find the hydraulic conductivity (Lp, flow rate per unit area per unit pressure drop) of this membrane.

2. (b) Consider a 4 mM solution of a large protein on one side of this membrane and physiologic saline on the other, with the same pressure on both sides of the membrane. Assume that the protein is sufficiently large that it cannot pass through the membrane and that van ’t Hoff’s law holds for this solute. Calculate the initial flow rate of saline through a membrane of area 5 cm2 at a temperature of 300 K.

The graph shown in the figure overleaf is adapted from a 1927 paper [16] in which Landis proved the existence of Starling’s phenomenon by occluding capillaries. The ordinate is the volume of fluid leaking out of (or re-entering) the capillary per unit capillary wall area, j. Although it is not precisely true, for the purposes of this question you may assume that the reflection coefficient of this capillary wall to plasma proteins is unity.

1. (a) Assuming that p – Π for the interstitium is –5 cm H2O, estimate the plasma osmotic pressure (Π) from the figure. Note that the plasma proteins are the main species influencing the osmotic pressure difference across the capillary wall.

2. (b) Estimate the filtration coefficient Lp for this capillary.

3. (c) Consider a capillary 0.05 cm long of diameter 8 µm, for which the arteriolar and venular luminal pressures are 25 and 5 cm H2O, respectively. Assume that Lp and Π are constant and that the pressure drop varies linearly along the capillary. What is the net rate of fluid loss (gain) from the capillary?

Fluid often passes through pores in cell membranes or cell layers. The dimensions are small and the velocities low, so viscous forces dominate (low Reynolds number). Use dimensional analysis, or an approximate method of analysis based on the viscous flow equations, to determine the scaling law that expresses the dependence of the pressure drop across a pore (ΔP) on the flow rate through it (Q). The other parameters that are given include the pore radius, R, and the viscosity of the fluid, μ. The membrane itself should be considered infinitesimally thin so that its thickness does not influence the pressure drop.

Using a stroboscope, it has been observed that freely falling water drops vibrate. The characteristic time for this vibration does not depend on the viscosity of the water (except for very, very small drops). Determine what parameters you expect this vibration time to depend on, and find a relationship between the vibration time and these parameters. Estimate the characteristic vibration time for a water droplet of diameter 2 mm at a temperature of 25° C. (Hint: this time-scale is the same for a droplet inside of a rocket in space as it is for a droplet falling on the Earth.)

When blood is taken out of the body for processing into an extracorporeal device, a majorconcern is that the level of shear stress to which the blood is exposed should be less than acritical level (roughly 1000 dyne/cm2). For exposure to shear-stress levelshigher than this, lysis of the red blood cells can occur, together with platelet activation andinitiation of the clotting process.

Consider the flow of blood through a device that has a set of parallel tubes each with a diameterof 1 mm and a length of 10 cm. What is the maximum pressure drop that should be usedfor such a device if the highest shear levels in the device occur in these tubes? (Blood has aviscosity about five times that of water.) You may neglect entry effects and treat the flow as fullydeveloped.

A parallel-plate flow chamber is to be designed to study the effects of shear stress on adhesionof leukocytes to endothelia. However, endothelial cells can be damaged by shear stress greater than400 dyne/cm2. The width of the flow channel is to be 1 cm and its length 5 cm. The flowis to be driven by gravity, and a fluid column 1 m in height is available. The system must work bothfor saline and for blood. What should be the maximum separation (s ≪ 1 cm) between the two platessuch that the endothelial cells are not damaged? The schematic diagram below is not to scale. Youmay neglect entry effects and treat the flow as fully developed.

A fluorescently labeled molecule (with a diffusion coefficient of 1 × 10−6 cm2/s) is released into the upstream end of a small blood vessel of diameter 100 μm and length 2 mm. The flow rate of blood passing through this vessel is 0.3 μl/min. Roughly estimate how long it should take before the molecule can be detected at the downstream end of the blood vessel.

Transport of nutrients, growth factors, and other molecules to tissues frequently takes place through the capillary wall. In many capillaries, there are tiny gaps between endothelial cells that allow both diffusion and convection of solutes across the vessel wall. Consider a particular endothelium in which the gaps between the cells are characterized by the following dimensions: L = 1 μm long, h = 200 nm high, and W = 10 nm in width (the last dimension is the distance between the two cells; see the figure below). The fluid in this gap is at 37 °C, and has the same properties as physiologic saline.

Nanoparticles can be used to probe the intracellular environment. By tracking their motion one can draw conclusions regarding transport inside a cell.

An investigator has placed a nanoparticle of diameter 100 nm inside of a Xenopus oocyte. The cytoplasm of this cell behaves like a viscous fluid with a viscosity 20 times that of water. Over a period of 20 s (at a temperature of 18 °C), the particle travels (on a somewhat erratic path) over a distance of approximately 3 μm from the periphery of the cell toward the nucleus in the center of the cell.

The investigator concludes that there is a preferential motion or “flow” from the periphery of the cell toward the nucleus. He would now like to plan a full study to examine what causes this “flow.” Does this seem like a reasonable next step? If so, justify why. If not, explain what next step you would suggest.

Fibrinogen has a diffusion coefficient in saline of approximately 2 × 10–7 cm2/s at 25 ˚C. It is a rod-shaped molecule whose length is roughly 10 times its radius. Estimate the length of this molecule.

We want to determine how fast the alveolar CO2 concentration can change in response to changes in blood CO2 concentration. Assume a spherically shaped alveolus (of radius 0.015 cm) with a spatially uniform internal gaseous composition at time t = 0. Calculate the time necessary to achieve 95% equilibration when the CO2 concentration in the alveolar wall is suddenly changed at t = 0. You may neglect any mass-transfer effects of the wall tissue and the liquid film in the alveolus. The diffusion coefficient of CO2 in air is 0.14 cm2/s.

Considering your answer, how important is the diffusion resistance to mass transport through the gas contained within the alveolus?

Fluorescein is a small fluorescent tracer molecule that is used in a wide variety of physiologic studies. In the eye, it is used in a technique known as fluorophotometry to characterize the transport characteristics of aqueous humor, the fluid that fills the anterior chamber behind the cornea.

A drop of fluorescein (50 μl of 0.005%, by mass, fluorescein in saline) is placed onto the cornea and spreads evenly over the corneal surface to create a thin film. It then diffuses through the cornea (D = 1 × 10–6 cm2/s) and enters the aqueous humor on the back side of the cornea. The thickness of the cornea is about 0.05 cm and its radius is 0.5 cm.

This book arose out of a need that frequently faced us, namely coming up with problems to use ashomework in our classes and to use for quizzes. We have found that many otherwise excellenttextbooks in transport phenomena are deficient in providing challenging but basic problems thatteach the students to apply transport principles and learn the crucial engineering skill of problemsolving. A related challenge is to find such problems that are relevant to biomedical engineeringstudents.

The problems included here arise from roughly the last 20–30 years of our collectiveteaching experiences. Several of our problems have an ancestry in a basic set of fluid mechanicsproblems first written by Ascher Shapiro at MIT and later extended by Ain Sonin, also at MIT. RogerKamm at MIT also generously donated some of his problems that are particularly relevant tobiomedical transport phenomena. Thanks are due to Zdravka Cankova and Nirajan Rajkarnikar, whohelped with proof-reading of the text and provided solutions for many of the problems.

Consider the case of mitral regurgitation. If the connection between the ventricle and atrium can be treated as a hole of diameter 0.2 cm, estimate the rate of leakage from the ventricle into the atrium at a time when the ventricular pressure is 100 mm Hg and the atrial pressure is 10 mm Hg. (Note that 760 mm Hg = 1.01 × 105 Pa.) You can neglect unsteadiness. Use a blood density of 1.06 g/cm3 and a viscosity of 0.04 g/cm per s.

One way to measure cardiac output is to inject a tracer into the blood stream and watch how quickly it disperses. This requires a rapid injection of tracer, which is often done by using a tracer-injector device.

One such injector consists of a large syringe that creates a pressure of 300 kPa. The tracer flows from the injector through 4 m of smooth plastic tubing of internal diameter 3 mm, then through a smooth 23-gauge needle (4 cm long, internal diameter 0.455 mm) before entering a peripheral vein where the pressure is 10 mm Hg (1.333 kPa). Neglecting minor losses and the height difference between the injector and the vein, what will the flow rate of tracer be?

In a blood oxygenator, the blood spends approximately 2 s passing through the device. Estimate a minimum value for hm, the mass transfer coefficient characterizing oxygen transport from the plasma to the inside of a red blood cell (RBC), such that the RBCs are completely (i.e. >99%) oxygenated before leaving the oxygenator (100% oxygenated refers to the maximum oxygen loading the cells can achieve in this oxygenator if the RBCs were left in the oxygenator for a very long time). You may assume that the RBCs entering the device are 50% oxygenated. Let the volume of each RBC be 98 µm3 and its surface area be 130 µm2. The diffusion coefficient of oxygen both in plasma and inside of the cell can be taken as 2 × 10−5 cm2/s.

A porous block attached to an air line emits tiny air bubbles (of diameter 0.15 mm, 21% oxygen) at the bottom of a fish tank filled with water at 25 °C. The absolute pressure in the bubbles at the bottom of the tank is 1.07 × 106 g/cm per s2 when they are released. They reach their terminal upward velocity almost instantly, since they enter the tank at nearly this velocity. Find the rate of oxygen transfer from one of these bubbles to the water. You may neglect the concentration of oxygen already in the water. The diffusion coefficient of oxygen in water is 2 × 10−5 cm2/s.

Scope

This chapter describes the stratified pattern observed in gas–liquid flows, for which liquid flows along the bottom of a conduit and gas flows along the top. The gas exerts a shear stress on the surface of the liquid. It is desired to calculate the average height of the liquid layer and the pressure gradient for given liquid and gas flow rates. The flow is considered to be fully developed so that the height of the liquid is not changing in the flow direction and the pressure gradient is the same in the gas and liquid flows.

In order to consider stratified flow in circular pipes, the simplified model of the flow pattern, presented by Govier & Aziz (1972), is exploited. The interface is pictured to be flat. At large gas velocities, some of the liquid can be entrained in the gas. This pattern is considered in Section 12.5 entitled “the pool model” for horizontal annular flow.

Prologue

Horizontal annular flows differ from vertical annular flows in that gravity causes asymmetric distributions of the liquid in the wall layer and of droplets in the gas flow. The understanding of this behavior is a central problem in describing this system. Because of these asymmetries, entrainment can increase much more strongly with increasing gas velocity than is found for vertical flows.

Theoretical analyses of the influence of gravity on the distribution of liquid in the wall film and on the distribution of droplets in the gas phase are reviewed. As with vertical annular flows, entrainment is considered to be a balance between the rate of atomization of the wall film and rate of deposition of droplets. Because of the asymmetric film distribution, the local rate of atomization varies around the pipe circumference. This is treated theoretically by assuming that the local rate is the same as would be observed for vertical annular flow. Gravitational settling contributes directly to deposition so that the rate of deposition is enhanced. Thus, at low gas velocities, entrainment can be much smaller for horizontal annular flows than for vertical annular flows.