Download dafoster/ChE 259/2010 ChE 259/Problem Set 1 Solutions

BME/ChE 259 Problem Set #1 (2010)
1. The oxygen diffusion coefficient in tissue is about 1.1x10 -5 cm2/s. The fluid filtration velocity
is typically 1 m/s.
(a) Assuming that convection and diffusion are equal in this system, how large a distance
between capillaries would be needed for convection to influence oxygen transport to
tissues?
Solution
The relative importance of convection and diffusion is evaluated by the Peclet number.
Solve for L, assuming that convection equals diffusion so that Pe=1,
(b) Based upon reported values for the distance between capillaries, do you think that
convection is an important mechanism for oxygen transport in tissues?
Solution
The distance between capillaries is 10-4 m = 0.01 cm.
So convection is negligible compared to diffusion in this problem.
2. The alveolar epithelium, basement membrane and lung capillary endothelium are typically 1
m thick. Under resting conditions, hemoglobin binding with oxygen reaches a steady state
of about 0.33 seconds. Is oxygen diffusion across the alveolus a significant factor in the time
required for the hemoglobin to oxygenate as it transverses the capillary?
Solution
See Table 1.5 in the text for values of
,
With a diffusion time of 0.0005 sec, diffusion is much faster than reaction time of 0.33 s, so
diffusion does not delay the oxygenation process.
3. The permeability of normal rabbit arterial endothelium to LDL is 5 x 10 -9 cm/sec. The
diffusion coefficient of LDL in the rabbit arterial wall is 1 x 10 -10 cm2/sec. The rabbit aorta
thickness is 150 m.
a. Determine the Biot Number
Solution
b. What does this result indicate about the endothelium as a barrier to LDL transport?
Solution
The result indicates that the resistance to LDL transport provided by the endothelium is similar
to that provided by the arterial wall.
4. During exercise, the cardiac output can rise to 25 L/min from a resting level of 5 L/min. The
heart rate of a well-trained athlete might rise from 60 beats/min to 105 beats/min and the
mean arterial pressure can rise from 100 to 130 mm Hg. The heart rate of a sedentary person
can rise from 72 to 125 beats/min and the mean arterial pressure can rise from 100 to 150
mmHg. Determine the volume of blood ejected during each heartbeat (stroke volume) and
the peripheral resistance for an athlete and a sedentary person. Assess the power of the left
side of the heart for the athlete and the sedentary person.
where P is power,
is the mean arterial pressure, V is the ventricular volume, W is work, and f
is the heart rate in beats/sec. Make sure that your units are consistent! (Note: 1 mmHg =
133.3 Pa, and 1Pa =1N/m2 ).
Solution
where CO is the cardiac output (L/min), SV is the stroke volume (L) and HR is the heart rate
(beats/min).
CO is 5 L/min at rest and 25 L/min at exercise.
Athlete
At rest:
At exercise:
Sedentary person:
At rest:
At exercise:
Resting
Heart Rate
(beats/min)
Exercise
Heart Rate
(beats/min)
Resting Mean
Arterial Pressure
(mm Hg)
Exercise
Mean
Arterial
Pressure
(mm Hg)
Stroke
Volume
at Rest
(L)
Stroke
Volume
at
Exercise
(L)
Peripheral
Resistance at Rest
(mm Hg/L /min)
Peripheral
Resistance at
Exercise
(mm Hg/L/ min)
60
72
105
125
100
100
130
150
0.0833
0.0694
0.238
0.2
20
20
5.2
6
Athlete
Sedentary
person
Athlete
At rest:
At exercise:
Sedentary person:
At rest:
At exercise:
To calculate work and power, we can assume that the mean arterial pressure is constant and
where
is the stroke volume.
Athlete
At rest:
At exercise:
Sedentary Person:
At rest:
At exercise:
Athlete
Sedentary person
Work at Rest
(Nm)
1.11
0.925
Work at Exercise
(Nm)
4.12
4.0
Power at Rest
(J/s)
1.11
0.924
Power at Exercise
(J/s)
7.21
8.33