Environmental Engineering Reference
In-Depth Information
Note
: When putting variables in dimensionless terms, use quantities that can be measured
or determined (i.e.,
L
,
C
∞
,
u
∞
). Note that
δ
is
not
a good choice.
Select dimensionless variables:
x
∗
=
y
∗
=
/
/
,
x
L
;
y
L
δ
δ
(
note:
L
used here and not
since
is a variable and unknown)
u
∗
=
v
∗
=
v/
/
u
∞
,
u
u
∞
;
v
(
note:
u
∞
used here since there is no constant value of
in the
y
-direction)
C
A
=
C
A
/
C
∞
.
Rearrange the dimensionless variables and substitute into the equation:
x
∗
L
y
∗
L
u
∗
u
∞
,v
=
v
∗
u
∞
,
C
A
C
∞
,
=
,
=
,
=
C
A
=
x
y
u
∂
C
A
∂
∂
C
A
∂
2
C
A
∂
C
L
C
L
D
AB
C
∞
L
2
∂
u
∗
u
∞
x
∗
+
v
∗
u
∞
y
∗
=
y
∗
2
.
Rearrange:
D
AB
ν
∂
C
A
∂
C
A
∂
2
C
A
∂
2
C
A
∂
2
C
A
∂
u
∗
∂
x
∗
+
v
∗
∂
D
AB
Lu
∞
∂
ν
Lu
∞
ScRe
∂
1
y
∗
=
=
=
y
∗
2
.
y
∗
2
y
∗
2
If
ReSc
is large (inertial effects dominate), one can neglect right side of equation.
If
ReSc
is small (diffusional effects dominate), right hand side of equation is significant.
A.4
Summary of the uses of dimensionless numbers
A.4.1
Correlate data
This is one important use of the Buckingham Pi Theorem. When the relationship between
the various quantities is unknown (i.e., there is no equation to relate them), the dimen-
sionless numbers provide a basis for obtaining an equation which fits the data. The exact
coefficients and exponents for the dimensionless numbers are obtained from a best fit of
the experimental data.
Example
626
Re
1
/
2
Sc
1
/
3
Sh
=
0
.
(correlates the
average
value of
k
for flow over a flat plate of length
L
).
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