Fenômentos de Transporte
1047 pág.

Fenômentos de Transporte


DisciplinaFenômenos de Transporte I12.740 materiais111.783 seguidores
Pré-visualização50 páginas
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"+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"+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