Rhodamine solution in water

BIOE2397 Homework #6

(Homework #6 is due on Wednesday, 12/5, 11:59pm)

Problem 1 [5 points]: If resting membrane potential of a cardiac cell is -90mV, the concentration of 𝐾𝐾+ outside the cell is 5mM and membrane permeability for all cations (except for 𝐾𝐾+) are zero. What is the concentration of 𝐾𝐾+ inside of the cell?

Problem 2 [10 points]: A cell’s resting membrane potential is -50mV, and the cell’s diameter is about 10ΞΌm. The capacitance of the cell membrane is about 0.5 ΞΌF/cm2, and the internal concentration of 𝐾𝐾+ is 150mM. What is the maximum surface density of 𝐾𝐾+ in the cell? What is the surface density of 𝐾𝐾+ needed to produce the resting membrane potential?

Problem 3 [25 points]: The relative importance of advection versus diffusion is described by a non-dimensional parameter called the Peclet number, 𝑃𝑃𝑃𝑃 = 𝑓𝑓(𝑒𝑒,𝐷𝐷, π‘₯π‘₯), where 𝑒𝑒 is the velocity, 𝐷𝐷 is the diffusion coefficient, and π‘₯π‘₯ is the distance to the point of interest. Using dimensional analysis, find the form of 𝑃𝑃𝑃𝑃 such that 𝑃𝑃𝑃𝑃 β‰ͺ 1 is advection dominated, and 𝑃𝑃𝑃𝑃 ≫ 1 is diffusion dominated. For stream with 𝑒𝑒 = 0.3 m/s and 𝐷𝐷 = 0.05 m2/s, where are diffusion and advection equally important?

Problem 4 [40 points]:

A student injects 5 mL of 20% Rhodamine solution in water (with specific gravity 1.15 g/mL) instantaneously and uniformly over the tube cross-section (𝐴𝐴 = 0.8 cm3) at the point π‘₯π‘₯ = 0 and the time 𝑑𝑑 = 0. The tube is filled with stagnant water. Assume the molecular diffusion coefficient if 𝐷𝐷 = 0.13Β·10-4 cm2/s.

β€’ What is the concentration at π‘₯π‘₯ = 0 at the time 𝑑𝑑 = 0?

β€’ What is the standard deviation of the concentration distribution 𝑑𝑑 = 1 second after injection?

β€’ Plot the maximum concentration in the tube, πΆπΆπ‘šπ‘šπ‘šπ‘šπ‘šπ‘š(𝑑𝑑), as a function of time over the interval 𝑑𝑑 = [0, 24 h].

β€’ How long does it take until the concentration over the region π‘₯π‘₯ = [-1, +1] meters can be treated as uniform? (A uniform distribution is one where the minimum concentration within a region is no less than 95% of the maximum concentration within that same region.)

BIOE2397 Homework #6

SUBMISSION FORMAT:

You will need to include the plots generated by MATLAB code into your homework document, and convert this document into a PDF. Submit your PDF and MATLAB code files in a single ZIP archive file to:

o Ms. Madeleine Lu (mlu5@uh.edu)

o Prof. Sergey Shevkoplyas (sshevkoplyas@uh.edu)

The filename must start with your last name, first name ‐‐ for example, for a student named John Doe, the filename is “Doe, John ‐ Homework 6.zip”