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o
-D
at
t ' = O
(5.4-7)
Our dimensionless groups in Equation (5.4-3) appeared as dimensionless
variables both in the differential equation and in the boundary condition, and as a
dimensionless coefficient in the differential equation.
Integrating Equation (5.4-6) using Equation (5.4-7) gives
Applying the initial condition gives
C=ln[%] = ln[h',]
which yields
(5.4-8)
(5.4-9)
(5.4- 10)
so it can be seen that all three of the dimensionless groups appear in the
integrated relation.
This illustrates two things which we take without proof
The dimensionless products we determine via the systematic
method have significance in the fundamental differential
equation(s) describing the process and its (their) boundary
conditions.
Another way to find sets of dimensionless products is via de-
dimensionalizing the differential equation and boundary
Chapter 5: Application of Dimensional Analysis 249
conditions (of course, this presupposes our ability to write the
equations and boundary conditions in their dimensional form).
The differential equation describing laminar flow of an incompressible
Newtonian fluid of constant viscosity in the axial direction in a pipe, neglecting
pressure gradients other than in the longitudinal direction, is
(5.4-1 1)
Make this equation dimensionless by using the following dimensionless
variables
Solutions
Noting that
(5.4- 13)
(5.4- 14)
(5.4- 15)
(5.4- 16)
(5.4- 17)
Applying the chain rule and substituting in terms of dimensionless variables
250 Chapter 5: Application of Dimensional Analysis
(5.4-18)
(5.4-19)
(5.4-20)
(5.4-21)
(5,422)
For the same dimensionless boundary conditions, all systems described by
the above differential equation will have the same solution
v; = Vi(X', t') (5.4-23)
Coincidence of the dimensionless velocity profiles indicates (by definition)
dynamic similarity between two systems.
It does not matter whether a liquid or a gas flows in the pipe, whether the
pipe is large or small, etc. The solution will be the same for the same values of
the dimensionless groups. This means, for example, that we can generate (d) in
the laboratory using convenient fluids such as air and water, yet obtain a
solution applicable to corrosive, toxic, or highly viscous fluids, etc. (as long as
they are Newtonian, which is assumed in the original equation).
Ze 5.4-2 One-dimensional e n e r w
The differential equation describing energy transport for a fluid at constant
pressure in one dimension is
Chapter 5: Application of Dimensional Analysis 251
(5.4-24)
Make this equation dimensionless using the following dimensionless variables
Solution
Using the chain rule
(5.4-26)
Using the definitions of the dimensionless variables
which yields
By multiplying and dividing by p this can be written as
(5.4-27)
(5.4-28)
(5 4 2 9 )
where the first bracketed term is the reciprocal of the Prandtl number and the
second bracketed term is the reciprocal of the Reynolds number.
2.52 Chapter 5: Application of Dimensional Analysis
Two systems described by this equation, each system with the same
Reynolds and Prandtl number, each subject to the same dimensionless boundary
conditions, will each be described by the same solution function
T* = T*(x*, to)
Such systems are designated as thermally similar.
(5.4-30)
Example 5.4-3 Mass transport in a binarv mixture
The differential equation describing mass transport in one dimension of
component, A, in a binary mixture of A and B is
(5.4-31)
For this example we have switched to the z-direction rather than the x-direction
for the differential equation to avoid possible confusion between the
concentration XA and the coordinate x.
Make this equation dimensionless using the dimensionless variables
Solution
Using the chain rule
az* a ax, ax:, az* +vz7----;- = I, -- --- ( 1 ax, ax; a Z * ax, ax; at* ax; at* at ax, a Z aZ AB aZ dz* ax; az* az (5.4-33) ---
Using the definitions of the dimensionless variables
Chapter 5: Application of Dimensional Analysis 253
which reduces to
ax; ax; D~~ a2x; - + v z 7 = --
at* (v)D aze2
After multiplying and dividing by (pp) we can write
(5.4-34)
(5.4- 35)
(5.4-36)
where the first bracketed term is the reciprocal of the Schmidt number and the
second bracketed term is the reciprocal of the Reynolds number.
Two systems described by this equation, each system with the same
Reynolds and Schmidt number, each subject to the same dimensionless boundary
conditions, will each be described by the same solution function
x; = x;(z*, t*) (5.4-37)
Examde 5.4-4 ExtraDolatinP model results from one categorr
of momentum, heat. or mass transport to another
The concept of model can be extended even to permit taking data in one
transport situation and applying it to another. Consider the following three
cases, which are treated in detail later in each of the appropriate chapters.
CASE ONE
Consider a flat plate adjacent to a quiescent fluid of infinite extent in the
positive x-direction and the z-direction. Let the plate be instantaneously set in
motion at a constant velocity.
254 Chapter 5: Application of Dimensional Analysis
The governing differential equation
(5.4-38)
has boundary conditions
t = 0: v, = 0, allx (initialcondition)
x = 0: vy = V, all t > 0 (boundarycondition)
x = 00: v, = 0, all t > 0 (boundarycondition)
If we dedimensionalize the dependent variable in the equation using
(5.4-39)
and combine the independent variables in a dimensionless similarity
transformation
q = - X
Jairr (5.4-40)
we reduce the problem to a dimensionless ordinary differential equation where
now, instead of velocity being a function of two independent variables,
v; = v; (x, t) (5.4-41 )
the dimensionless velocity is now a new function of only one independent
variable, q. We therefore use a new symbol for the function, namely, 0
v; = cp(q) (5.4-42)
We have reduced the problem (omitting the intermediate steps which are
covered later) to an ordinary differential equation
cp&quot;+2qcp' = 0 (5.4-43)
with boundary conditions
Chapter 5: Application of Dimensional Analysis
whose solution is
(5.4-44)
(5.4-45)
255
The solution of our problem in terms of the original variables is then
(5.4-46)
CASE TWO
Consider a heat conduction in the x-direction in a semi-infinite (bounded
only by one face) slab initially at a uniform temperature, Ti, whose face suddenly
at time equal to zero is raised to and maintained at Ts.
The goveming differential equation
(5.4-47)
has boundary conditions
t = 0: T = T,, allx (initialcondition)
x = 0: T = T,, all t > 0 (boundarycondition)
x = m: T = T,, all t > 0 (boundarycondition)
If we dedimensionalize tbe dependent variable in the equation using
T-Ti
= 'm' (5.4-48)
and combine the independent variables in a dimensionless similarity
transformation
256 Chapter 5: Application of Dimensional Analysis
(5.4-49)
q = L m
we reduce the problem to a dimensionless ordinary differential equation where
now, instead of temperahm being a function of two independent variables,
T' = T' (x, t) (5.4-50)
the dimensionless temperature is now a new function of only one independent
variable, q. We therefore use a new symbol for the function, namely, Q
T* = (P(q) (5.4-5 1)
We have reduced the problem (omitting the intermediate steps which are
covered later) to an ordinary differential equation
cp&quot;+2qcp' = 0
with boundary conditions
whose solution is
cp = l-erf[q]
= erfc [q]
(5.4-52)
(5.4-53)
(5.4-54)
The solution of our problem in terms of