# 9 mass transport processes in artificial organs

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• Mass transport processes of membrane artificial lung

ENT 318/3 Artificial Organs

• Gas transport in bloodBy Henrys Law, the concentration of dissolved O2, Cd;Cd = o2 Po2 Where o2 is the O2 solubility coefficient (mol m-3kPa-1) and Po2 is the oxygen partial pressure (kPa).The amount of oxygen combined with haemoglobin is given by fractional saturation, S.If all haemoglobin is combined with oxygen it is said to be fully saturated and S = 1.0.Cb = Bo2 Hb SWhere Hb is the mass concentration of haemoglobin (kg/m3) and Bo2 is the binding capacity of haemoglobin (5.978 x 10-2 mol/kg)

• Fig. Schematic of flow in a tube

P1 is always greater than P2 if:a) the tube is uniform in area and shapeb) there is no acceleration (constant flow with time)c) the tube is horizontalFluid flow in tubes

• There may be Poiseuille flow if:a) flow is steadyb) flow is fully developed (away from entrances, discontinuities, bends)c) the tube is uniformPoiseuille Flow is the pressure loss

is the length of pipe

is thedynamic viscosity

is thevolumetric flow rate

• Fluid particles move in streamlines parallel to the tube walls axisymmetric flow.These streamlines may be visualised by: injection of dye or neutral density particles, or using Doppler techniques (Ultrasound or Laser light)

Fig. Schematic showing the features of Poiseuille Flow

Note that the centre line velocity Uc = 2 U and velocity at the wall Uw = 0 (the no-slip condition)The distribution of fluid velocities across the tube (velocity profile) is parabolic.dUcUrrFeatures of Poiseuille Flow

• a)at the entrance, all the fluid moves at the same velocity (flat profile)b) the tube surface retards the fluid (no-slip condition Ud = 0)viscosity modifies the initial flat velocity profile as more of the fluid is sheared. Fluid at the centre of the tube must then be acceleratedas fluid nearer the tube wall is slowed (Law of conservation of mass)d) far from the entrance, the flow will become parabolicEntrance FlowFig. 15.3 Schematic of how a fluid enters a tube

• In the healthy circulation, the shear stress even at the blood vessel wall is normally less than 10 Nm-2, but can be much higher in various disease states or in artificial organs.Erythrocytes are very deformable and bend very easily.

1 - 10 Nm-2: cells spin less then at lower shear and tend to align with their flay axis parallel to the flow (alignment tends to decrease viscosity)

>10 Nm-2: cells are elongated to cigar shapes

>150 Nm-2: membranes are stretched and become leaky, first losing ions and later haemoglobin, leading to haemolysisEffects of shear on blood components

• Diffusion transport occurs because of the intrinsic motion (Brownian motion) of the molecules in a material. Diffusion and heat transport occur through molecular motion and the equations used to describe them are completely analogous.When molecular motion is rapid (as in a gas) diffusion will occur faster than when the molecules are constrained (as in a solid)Diffusion

• When diffusion in a system is occurring at a constant rate, it is said to be in a steady state and can be expressed using Fick's 1st law:

J = flux (mass/time)D = Diffusion coefficientA = area across which diffusion occursa = activity coefficientx = distance along which diffusion occurs

Activity: = activity coefficientC = concentration (mass/volume). For "ideal" solutions, = 1 and the assumption of ideality is commonly made for aqueous solutions in biology.Steady state diffusion

• In the microcirculation (systemic and pulmonary)

a) the surface area (A) is large

b) distances between the blood and surrounding tissues (x) is small.

Diffusion in the body

• On integrating the Fick equation for small diffusion gradients: D ~ x2/2t

x = typical diffusion distancet = diffusion time

Diffusion times for O2 in tissue1 cm ~ 16 hrs 1 mm ~ 8 min 100 m ~ 5 sec

Diffusion in the body

• Diffusion through complex systems such as the endothelial layer is often difficult to define in terms of the solubilities and the dimensions diffusion distances etc) of each of the components (membranes, cells, fibres, matrix materials etc).

To overcome these problems it is convenient to define a permeability coefficients (P) using the equation:

J = P A C

Where P = permeability coefficient and J is the mass flux in the absence of any fluid movement. For diffusion into or out of cells, C = Cin -Cout

Permeability Coefficients

• Oxygen diffusing into the blood in the lungs or an oxygenator reacts with haemoglobin, with the result that the concentration of free O2 in the blood remains at a very low level. This maintains a high gradient for diffusion.

dC/dt = diffusion flux + reaction rate

Case 1: Rapid reaction: Overall rate determined by transport rate e.g. CO2 uptake rate in lungs is determined by rate at which it is delivered to blood - limited by diffusion across alveolar surface.

Case 2: Slow reaction:Rate of transport of a material is determined by the rate at which it is removed by reaction, e.g. O2 delivery to metabolising tissues.Diffusion with reaction

• This is the layer in which the velocity of fluid is increasing with distance from the wall.That is the region in which viscous retardation is having an important role in determining flow patterns.

In Poiseuille flow the boundary layer thickness () = r (the tube radius).

At the entrance to the tube, the layer is thin: The growth of the boundary layer depends on the balance between the inertia of the fluid and the retarding viscous forces.

Hydrodynamic boundary layer

• When convective and diffusive transport are occurring in the same (or exactly opposite) directions:

Total flux = Jdiffusion + Jconvection

though convection is likely to distort diffusion (concentration) gradients, so there will be interaction.

Interaction of convection with diffusion

• On plotting the position reached by the material, we describe the "mass transport boundary layer (MTBL)

Fig. Schematic of mass boundary layerm is the thickness of the layer at position L along the tube. As the MTBL thickens, the gradient for diffusion between the wall and the fluid decreases, reducing the rate of mass transport into the fluid. In the steady state, the rate of diffusion into the fluid equals the rate of convection downstream. Mass transport boundary layers

• QuestionTwo membrane oxygenators X and Y are operated in series. A total transfer rate of 3.0 x 10-4 mol/s is achieved when a blood flow rate of 1.0 x 10-4 mol/s passes through the series combination. The O2 saturation at inlet of X and outlet of Y 0.65 and 0.95 respectively. Calculate the hemoglobin concentration. Given that BO2 = 0.0598 mol/kg.Answer : 167.22kg/m3

• Analysis for O2 Transfer: Advancing Front Theory

Consideration for tubular geometry, no membrane resistance and laminar flow.

• Assumptions of the model are:Oxygen transport in the outer zone is due to radial diffusion of dissolved oxygen onlyAxial convection of dissolved oxygen is neglected in the outer zoneThe axial decrease in the rate of convective transfer of unoxygenated haemoglobin in the core is equal to the diffusive flux if dissolved oxygen from the outer zone.

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