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2. Cylindrical Muscle. A cylindrical muscle of radius b is excised from the animal and is hung in a physiological solution. The partial pressure of
2. Cylindrical Muscle. A cylindrical muscle of radius b is excised from the animal and is hung in a physiological solution. The partial pressure of O2 in the solution is Po. Assume zero-order kinetics for the O2 consumption rate M (a) Using Infinitesimal Differential Element Analysis in cylindrical coordinates, derive a differential equation for mass balance for a radially symmetric problem, concentration C(r), by considering mass balance between two concentric cylinders of radius r an d rdr Hint: the final equation may look like (rj)+R (b) Substitute Fick's Law of Diffusion in cylindrical coordinates jand use Henry's or law CzuP where is the Bunsen's solubility coefficient (it is common to use P for gases like oxygen). Derive the differential equation for pressure (P) in the muscle at steady state. Note that oxygen concentration is conventionally expressed in units of ml O/ml tissue using the conversion 1 mole O2 22,400 ml O2 at STP (standard temperature and pressure) Hint the final expression may look like Dar (e) Find P(r) analytically assuming Pin 0 [hint: there is only one BC but the equation is of the 2nd order, thus you will get two constants of integration. Where is the BC to determine the second constant? The general solution however will have a term that goes to oo when r goes to 0, at the center of the muscle; this term should disappear for the solution to be finite and have physical sense, thus the coefficient in front of this term should equal 0. This effectively replaces the 2nd boundary condition and is typical for cylindrical and spherical coordinates. Same applies to Problem #3] (d) Express Pmin (minimum pressure within the muscle) (e) The muscle metabolic rate Mo is such that Pmin 0 (the center of the muscle becomes hypoxic) whereas P 0 for r>0. Express Mo in this case (f) Determine the value of Mo when Pn 0 (in ml O2/ml tissue/s) for b-0.3 mm, Po-60 mmHg, D-1.6x10-5 cm!2/s, =3x10-5 nlo2l/mm Hg (g) Using MATLAB plot P(r) for the value of Mo determined in (f) [20 points 2. Cylindrical Muscle. A cylindrical muscle of radius b is excised from the animal and is hung in a physiological solution. The partial pressure of O2 in the solution is Po. Assume zero-order kinetics for the O2 consumption rate M (a) Using Infinitesimal Differential Element Analysis in cylindrical coordinates, derive a differential equation for mass balance for a radially symmetric problem, concentration C(r), by considering mass balance between two concentric cylinders of radius r an d rdr Hint: the final equation may look like (rj)+R (b) Substitute Fick's Law of Diffusion in cylindrical coordinates jand use Henry's or law CzuP where is the Bunsen's solubility coefficient (it is common to use P for gases like oxygen). Derive the differential equation for pressure (P) in the muscle at steady state. Note that oxygen concentration is conventionally expressed in units of ml O/ml tissue using the conversion 1 mole O2 22,400 ml O2 at STP (standard temperature and pressure) Hint the final expression may look like Dar (e) Find P(r) analytically assuming Pin 0 [hint: there is only one BC but the equation is of the 2nd order, thus you will get two constants of integration. Where is the BC to determine the second constant? The general solution however will have a term that goes to oo when r goes to 0, at the center of the muscle; this term should disappear for the solution to be finite and have physical sense, thus the coefficient in front of this term should equal 0. This effectively replaces the 2nd boundary condition and is typical for cylindrical and spherical coordinates. Same applies to Problem #3] (d) Express Pmin (minimum pressure within the muscle) (e) The muscle metabolic rate Mo is such that Pmin 0 (the center of the muscle becomes hypoxic) whereas P 0 for r>0. Express Mo in this case (f) Determine the value of Mo when Pn 0 (in ml O2/ml tissue/s) for b-0.3 mm, Po-60 mmHg, D-1.6x10-5 cm!2/s, =3x10-5 nlo2l/mm Hg (g) Using MATLAB plot P(r) for the value of Mo determined in (f) [20 points
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