Chemistry Reference
In-Depth Information
Figure 1.8
Taylor vortices.
R
o
and R
i
are the outer and inner cylinder radii of the Couette filled with a fluid
of density r and viscosity Z. Figure 1.8 illustrates the flow pattern of Taylor
vortices that are formed when the Taylor number is exceeded.
1.3.5 The Reynolds Number
The Taylor vortices described above are an example of stable secondary flows.
Now, at even high shear rates the secondary flows become chaotic and
turbulent flow occurs. This happens when the inertial forces exceed the viscous
forces in the liquid. The Reynolds number gives the value of this ratio and in
general is written in terms of the linear liquid velocity, u, the dimension of the
shear gradient direction (the gap in a Couette or the radius of a pipe), the liquid
density and the viscosity. For a Couette we have:
R
e
¼
O
c
ð
R
o
þ
R
i
Þð
R
o
R
i
Þ
r
2Z
ð
1
:
19a
Þ
where R is the radius of the moving cylinder. When we write this in terms of the
shear rate:
g
ð
R
o
R
i
Þ
2
r
Z
R
e
E
ð
1
:
19b
Þ
Another common geometry used for laboratory measurement of viscosity is a
cone and plate with a small included angle, a. Values of a are typically in the
range 1-51. This geometry is used to give a constant shear rate because at any
point on the plate the ratio of the tangential velocity (rO) to the gap is constant.
A suitable expression for the Reynolds number with the cone angle in degrees is:
2
gr
Z
pRa
180
R
e
E
ð
1
:
20
Þ
