CHE 311 (Fall 2010)
__________________
LAST NAME, FIRST
Second Optional Problem set (Due 11.45 or 1:00 PM on Dec. 3 2010)
1) Reynolds Number in Human Lungs
As is well known, oxygen and carbon dioxide transfers occur, predominately, in the small
alveolar sacs within the lungs of mammals. Figure 1 (from Truskey) is a photograph of one lobe
of a human lung and 1b is a diagram representing the various branches of passageways. During
quiet breathing, a typical adult human breathes at a volumetric flow rate of 0.5 x 10 -3 m3/s while
under vigorous breathing the volumetric flow rate increases to 2.0 x 10-3 m3/s. Table 1 lists the
approximate dimensions of each generation.
Assuming that there are no entrance effects, and that each bifurcation is uniform, calculate the
Reynolds number at each generation listed in Table 1 for quite and vigorous breathing.
Figure 1.
Generation
Trachea
1
2
3
4
Diameter
(cm)
1.80
1.22
0.83
0.56
0.45
Length (cm)
Generation
12.0
4.76
1.9
0.76
1.27
5
10
15
20
Table 1.
Diameter
(cm)
0.35
0.13
0.066
0.045
Length (cm)
1.07
0.46
0.2
0.083
2) Blood Flow in Microvessel
When the blood microvessel dimensions approach the dimensions of the red cell, the red cell is
squeezed through capillaries and the effective viscosity increases. To model this and predict the
apparent viscosity, we will assume that the red cells pass through the capillaries as a continuous
train equal to the capillary length, L. The cells have a radius Rc The plasma flows in a thin gap
between the cell and the vessel wall (See Figure below). The gap thickness is = R – Rc, and
the viscosity is . Because the gap is so thin relative to the vessel wall thickness, assume that
the velocity profile is linear and equals Vcy/ where y = R – r and = R- Rc.
(a) Determine an expression for the shear stress acting on the cells if the velocity in the gap can
be approximated as:
(b) Perform a force balance on the cell to relate the pressure and shear stresses. Use this result to
relate the cell velocity to:
(c) Relate the average velocity with the cell velocity.
(d) Use the results from parts (b) and (c) to derive an expression for the effective viscosity of the
red cell suspension
in a tube.
(e)
0.1 <
by matching the red cell velocity to the mean velocity for Poiseuille flow
Show that the ratio
increases as the ratio
/ R < 0.4 or Rc / R varies from 0.6 to 0.9.
/ R increases for
3) Fluid Shear and Industrial Fermentors
As the “resident Biochemical Engineers” with an industrial pharmaceutical company, you have
been asked to consult on the following problem. The production scale fermentors are used to
produce an antibiotic that employs a mycelia organism. There is evidence in these fermentors
that fluid shear is a critical parameter that influences the antibiotic formation. It has been
proposed to you that design calculations be performed so that the shear at the impeller tip is
doubled form that of the present operating conditions. There are diverse opinions as to how to
increase the shear. You have been asked to elucidate quantitatively the following suggestions.
A. The plant manager who is presently in charge of the fermentors states the following. He
does not want to change the horse power of the electric motor that is presently on the
fermentor. If this constraint must be fulfilled, what would you do to the impeller
diameter and speed?
B. The director of research states that the obvious way to double the shear is to double the
impeller speed. If this were done what would be the consequences of approach to other
parameters in the process?
C. A very astute plant operator suggests that you decrease the impeller diameter to 75% of
the present impeller. If this is done, what must be the changes in other operating
parameters: in particular the impeller speed and power.
Assumptions:
a)
b)
c)
d)
You may assume turbulent flow exists in all conditions above.
Assume physical properties of the fluid are constant.
Neglect the effect of aeration on power absorption.
Please define all symbols used.
4) Scale-up of Mycelial Fermentations
In the scale-up of mycelial fermentations one often finds more than one criteria that must be
satisfied simultaneously. For example, in vitamin B12 fermentations, one must operate the
fermentor at some given mass transfer coefficient but at the same time one cannot exceed a
certain impeller tip speed in order to avoid shear damage to the organism. A pilot scale
fermentor has been used to obtain the necessary data for scale-up. The pilot scale fermentor and
operating conditions are shown in the following table:
Pilot Scale Fermentor:
Liquid Volume
Liquid Height to tank diameter ratio
(HL/Dt)
Impeller speed
Impeller Diameter to tank diameter
ratio (Di/Dt)
1000 liters
2.0
200 RPM
0.5
It has also been shown that scale-up can be performed for mass transfer purposes at equal power
per unit volume. However, the maximum allowable tip speed of the impeller must not exceed
the value shown in the tabulation for the pilot scale fermentor shown above.
As a Biochemical Engineer you have been asked to scale-up the fermentation to 100,000 liters.
The two criteria, equal power per unit volume and equal impeller tip speed must be satisfied
simultaneously. You have been asked to specify in the 100,000 liter fermentor the pertinent
fermentor geometries and impeller dimensions and operating conditions (numerical values of
power per unit volume and impeller tip speed). In order to make you life easier, you may
assume:
1. Neglect aeration effect on power
2. Viscosity and density of broth similar to water
3. The power number is constant above a Reynolds number of 1000 and the Power number is
6.0