In addition, create a Python script that calculates the velocity of a falling particle over time, the particle having…

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In addition, create a Python script that calculates the velocity of a falling particle over time, the particle having started from rest.

The script should produce the following two plots. The second plot is the same as in Example 2.1 on page 42.

6,10-5

5

Velocity (m/s)

0.5

Time (s)

1.5

2

104

v/v₁

0.8

0.6

0.4

0.2

O

2

3

UT LOW REYNOLDS NUMBER PARTICLE DYNAMICS AND STOKES LAW

second term in Eq. 2.14 has the units of inverse time (T). Inverting this group

of parameters yields the particle characteristic time t

T=

C.pd²

18μ

(2.15)

We want to solve Eq. 2.14 for the situation in which we have just released the

particle from rest, and it is allowed to rise or fall. For the initial condition v=0

at t=0, the solution to Eq. 2.14 is

v=Pr- Pl_rg [1-exp(-)]

D=

PP

U = Pr PL T

Pr

Some interesting features of the solution can be seen without much effort.

First, at = 0, Eq. 2.16 predicts that v= 0; hence, our required initial condition

is indeed satisfied. As →∞, the term in square brackets approaches unity

asymptotically; hence, the particle velocity approaches a constant value known

as the terminal velocity:

tg=

41

CAP,-P₁) d²

18μ

บ

P-PL 18

Pr

8

(2.16)

(2.17)

Example 2.1. What relevance does the particle characteristic time r have in

Eq. 2.16? Calculate the particle characteristic time for a 1 um sphere settling

in air. The density of the sphere is 1.1 x 10’kg/m².

SOLUTION. Even though we already know approximately what the function

looks like, it is interesting to plot Eq. 2.16. Using Eq. 2.17, Eq. 2.16 can be

rearranged to

–[1-exp(-)]

Note that in this form, the equation is “nondimensional” the dimension of

the left-hand and the right-hand side is unity. It is often advantageous to

develop functions in this way, since the functions become more “universal”:

If we plot u/u, versus /r as shown above, the plot is the same for any param-

eter values (i.e., P P. d, and u) we might choose.

Note that when t=r (i.e.,t/t-1), the y-axis has the value 1-exp(-1)=0.632.

In other words, when the characteristic time is reached, the particle reaches

63.2% of its terminal velocity. Since Eq. 2.16 implies that the particle takes an

infinite amount of time to reach the final terminal velocity, it is convenient to v/v

1.0

0.8

2

047

02-

LOW-CONCENTRATION PARTICLE SUSPENSIONS AND FLOWS

U/1

Example 2.1

characterize the process with a “characteristic time.” Although the value 63.2%

is not particularly special, the characteristic time derived above does seem to

be a fundamental parameter grouping that derives from our physical and

mathematical analysis. Throughout this book we will keep an eye out for these

kinds of guideposts

To calculate the characteristic time, we need to know the viscosity and

density of air. Taking the ambient temperature to be 20°C, the viscosity of ar

is 1.8 x 10 kg-m¹-s. Also, recall from Table 2.1 that the Cunningham sip

function for a 1-um particle in air at 25°C is 1.168. The characteristic time in

air is then

TH

PC 1.1×10 kg/m³ (1.0×10 m) 1.168

18μ 18(1.8x 10 kg-m-¹)

= 4.0x 10*s

It is apparent that the characteristic time is very small. The practical signif

cance of such a small characteristic time is that in many of our analyses, w

can argue that particles are essentially always sedimenting at their terminal

velocitics.

aw. Therefore, Stokes’ law refers to the low Reynolds number (Re<1) termin
The terminal velocity expression, Eq.2.17, is what most people call Stoke
ttling velocity of a spherical particle in a quiescent fluid. In addition, create a Python script that calculates the velocity of a falling particle over time, the particle having started from rest.
The script should produce the following two plots. The second plot is the same as in Example 2.1 on page 42.
6,10-5
5
Velocity (m/s)
0.5
Time (s)
1.5
2
104
v/v₁
0.8
0.6
0.4
0.2
O
2
3
UT LOW REYNOLDS NUMBER PARTICLE DYNAMICS AND STOKES LAW
second term in Eq. 2.14 has the units of inverse time (T). Inverting this group
of parameters yields the particle characteristic time t
T=
C.pd²
18μ
(2.15)
We want to solve Eq. 2.14 for the situation in which we have just released the
particle from rest, and it is allowed to rise or fall. For the initial condition v=0
at t=0, the solution to Eq. 2.14 is
v=Pr- Pl_rg [1-exp(-)]
D=
PP
U = Pr PL T
Pr
Some interesting features of the solution can be seen without much effort.
First, at = 0, Eq. 2.16 predicts that v= 0; hence, our required initial condition
is indeed satisfied. As →∞, the term in square brackets approaches unity
asymptotically; hence, the particle velocity approaches a constant value known
as the terminal velocity:
tg=
CAP,-P₁) d²
18μ
บ
P-PL 18
Pr
41
8
(2.16)
Example 2.1. What relevance does the particle characteristic time r have in
Eq. 2.16? Calculate the particle characteristic time for a 1 um sphere settling
in air. The density of the sphere is 1.1 x 10'kg/m².
V
- - [1-exp(-:)]
(2.17)
SOLUTION. Even though we already know approximately what the function
looks like, it is interesting to plot Eq. 2.16. Using Eq. 2.17, Eq. 2.16 can be
rearranged to
Note that in this form, the equation is "nondimensional" the dimension of
the left-hand and the right-hand side is unity. It is often advantageous to
develop functions in this way, since the functions become more "universal":
If we plot u/u, versus t/r as shown above, the plot is the same for any param-
eter values (i.e., P. P. d, and u) we might choose.
Note that when t=r (i.e.,t/t-1), the y-axis has the value 1-exp(-1)=0.632.
In other words, when the characteristic time is reached, the particle reaches
63.2% of its terminal velocity. Since Eq. 2.16 implies that the particle takes an
infinite amount of time to reach the final terminal velocity, it is convenient to v/v
1.0
0.8
2
047
02-
LOW-CONCENTRATION PARTICLE SUSPENSIONS AND FLOWS
U/1
Example 2.1
characterize the process with a "characteristic time." Although the value 63.2%
is not particularly special, the characteristic time derived above does seem to
be a fundamental parameter grouping that derives from our physical and
mathematical analysis. Throughout this book we will keep an eye out for these
kinds of guideposts
To calculate the characteristic time, we need to know the viscosity and
density of air. Taking the ambient temperature to be 20°C, the viscosity of ar
is 1.8 x 10 kg-ms. Also, recall from Table 2.1 that the Cunningham sip
function for a 1-um particle in air at 25°C is 1.168. The characteristic time in
air is then
TH
PC 1.1x10 kg/m³ (1.0×10 m) 1.168
18μ 18(1.8x10 kg-m-¹)
= 4.0x10*s
It is apparent that the characteristic time is very small. The practical signif
cance of such a small characteristic time is that in many of our analyses, w
can argue that particles are essentially always sedimenting at their terminal
velocitics.
aw. Therefore, Stokes' law refers to the low Reynolds number (Re<1) termin
The terminal velocity expression, Eq. 2.17, is what most people call Stoke
ttling velocity of a spherical particle in a quiescent fluid.
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