Only problem 4
Microsoft Word - Bioeng1220_HW4_fall2021.docx Copyright 2021, University of Pittsburgh, All rights reserved. No distribution without written permission. Homework #4 BioE 1220 [Mass transport: Diffusion, Reaction, Diffusion Coefficients] Assigned 11/01/21 – Due 11/16/21 at 5pm Problem #1 – Mass Balance in a Geometry with Changing area vs. a constant area The goal of this problem is to compare concentration profiles and molar flux in a geometry where the area is constant (case I) versus a geometry where the area (case II) is increasing. Consider steady one-dimensional diffusion through a funnel of varying cross section A(x) (see below). Neglect chemical reactions and convection. Boundary conditions: C(x=0) = C0 and C(x=L) = CL, with C0 > CL A) At steady state, write the mass balance in the control volume enclosed with the dashed lines above. The solid walls (black line) are impermeable (no flux). Find an equation for the molar flux N(x) due to diffusion and the varying cross-section area A(x). Derive this equation in the differential form (take limit as Δx->0), so that you can find the concentration profile: N(x) = f(x) . B) Use the differential equation you derived in A), the boundary conditions above, and Fick’s 1st law: ?! = −?! "#" "$ to find an equation that you can solve and find concentration profile as a function of x for Case I: a constant radius ?(?) = ?% Case II: a linearly increasing radius ?(?) = ?% *1 + $ & - versus C) Calculate the flux for the two cases above (a linearly increasing radius versus a constant radius). Is the flux positive or negative? D) Calculate ?! (?) ∙ ?(?) for each case ?(?) = ?% *1 + $ & -and ?(?) = ?%. Is the product of the molar diffusive flux and the area dependent on the x coordinate or not? E) Use your results from D to explain the difference in the concentration profile: why is one profile linear versus the other profile being non-linear? Hint: The integral you ‘ll need to solve is: ∫ '(')*$)# ?? = − ' *#$)* + const Copyright 2021, University of Pittsburgh, All rights reserved. No distribution without written permission. Problem #2 – Diffusion through tissue layers arranged parallel Consider steady one-dimensional diffusion through two tissues (tissue 1 has a diffusion coefficient Di,1 and tissue 2 has a diffusion coefficient Di,2) that are connected in parallel. Each tissue has the same length L. The partition coefficient for tissue 1 is Φ1 and the partition coefficient for tissue 2 is Φ2. You are given the boundary conditions: C=C0 at x=0, and C=CL at x=L a) Find the concentration profile in each tissue b) Develop an expression for the steady state molar flux. c) Calculate the diffusive resistance ? = -. / for each tissue component and the total diffusive resistance for the tissue construct. Compare the diffusive resistance to the example with two tissues connected in series that we did in class. Problem #3 – Diffusion coefficients A virus with a molecular weight of MW=40 million daltons (1Da = g/mol), a density of 1.3 g/ml, and a measured diffusion coefficient in saline of 3 × 10-8 cm 2 /s (at 25 °C) has been discovered. The boltzman constant kB = 1.38 × 10 -16 (cm2 g) / (s2 K), the viscosity of saline μ = 0.01 g / (cm s). a) Assume that the virus is a spherical particle. Use the appropriate form of the Stokes Einstein equation and calculate the equivalent radius of the virus R. b) Find the density of the virus assuming that it is a sphere and using the formula ? = 01 /$ ' 2 , where ?3 = 6.02 ∙ 1045 6789#:89; 678 (????????%?&'() *+,)-./0.234 (Table 6.3 in Truskey) for a cylinder of radius ? = 5.6?? and the length L calculated above and find the diffusion coefficient. How does it compare to the measured diffusion coefficient (D=3 × 10-8 cm 2 /s). What does this tell you about the virus? Can it be approximated as a sphere or as a cylinder? Copyright 2021, University of Pittsburgh, All rights reserved. No distribution without written permission. Problem #4 – Diffusion-Reaction: Determination of anoxic radius The figure below shows the effects of occluding an artery in the cerebellum of the opossum by a spore (arrow). Capillaries in the cerebellum are the elongated structures distributed throughout the tissue, one of which is occluded by the spore. The nerve cells have died in a 25-μm region surrounding the blocked capillaries. The nerve cells around the neighboring functional capillaries are intact. Now we want to model oxygen diffusion surrounding a single capillary that is not occluded and find the cylinder radius RD such that all of the tissue in the cylinder receives an adequate oxygen supply. You can model the capillary as a cylinder with radius R0 (R0 = 5 μm), where oxygen diffuses through the capillary into the surrounding tissue until we reach a radius r=RD where anoxia occurs (oxygen level is zero and there is no more oxygen transfer). Oxygen concentration inside the capillary is uniform C0 = 95 mm Hg (the partial pressure which is equivalent to the concentration of oxygen in the tissue). Let the tissue surrounding this capillary have an uniform oxygen-consumption rate of s = -180 mm Hg/s and let the diffusion coefficient of oxygen through the tissue be D= 1.7 × 10–5 cm2/s. You can assume that there is no convection inside the tissue, and that the problem is steady state. a) Write the conservation equation in differential form for oxygen diffusion in the appropriate coordinate system. What terms can you neglect? b) Integrate the resulting equation and list the unknowns. How many unknowns do you have? c) How many boundary conditions do you need to find the unknowns? What can you assume the boundary conditions to be at the anoxic border? (hint: what is the transport of oxygen there and the actual oxygen concentration?). d) Now, after you apply the boundary conditions you will obtain a non-linear equation for the radius (RD) at which the tissue would become anoxic. You will need to this equation either numerically or plot it using MATLAB. Hint: if you use the MATLAB symbolic solve (e.g. “syms Rd_sol”, where yzero the equation you want to solve), you can use the command “solve (yzero ==0, Rd_sol)” Hint: Make sure that the units work out, when you write 11.11. When tumors grow, they become vascularized so that the tumor cells can get adequate oxygen. We are interested in how large a tumor can become without having its own blood supply. Consider a spherical, avascu- lar tumor of radius R under steady-state conditions. We would like to find how large R can become while still having the tumor receive an adequate oxygen supply. The tumor tissue consumes oxygen at a rate (per unit volume) of 10–5 g/cm3 per s. Let the diffusivity of oxygen in the tumor be 2 × 10–5 cm2/s. The tissue surrounding 50 µm 25 µm 96 Steady diffusion and conduction :DDAC 53 4B 97 B9 5 B7 D7B C :DDAC B9 . 0 / 3 7 8B :DDAC 53 4B 97 B9 5 B7 2 7BC D 8 1 DDC4 B9: , 9 3D C 4 75D D D:7 .3 4B 97 . B7 D7B C 8 C7 3 3 34 7 3D