Ch 2 Velocity Distribution
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CHAPTER 2. VELOCITY DISTRIBUTIONS IN LAMINAR FLOW In this chapter we show how one may calculate the laminar velocity profiles for some flow systems of simple geometry. These calculations make use of the definition of viscosity and the concept of a momentum balance. Actually, a knowledge of the complete velocity distributions is not usually needed in engineering problems. Rather, we need to know • the maximum velocity, • the average velocity, or • the shear stress at a surface These quantities can be obtained easily once the velocity profiles are known. In the first section we make a few general remarks about differential momentum balances (Shell momentum balances and boundary conditions) Then, in subsequent sections, we work out in detail several classical examples of viscous ftow patterns. These systems are too simple to be of engineering interest; it is certainly true that they represent highly idealized situations, but the results find considerable use in the development of numerous topics in engineering fluid mechanics. 1
Transcript of Ch 2 Velocity Distribution
CHAPTER 2. VELOCITY DISTRIBUTIONS IN LAMINAR FLOW
In this chapter we show how one may calculate the laminar velocity profiles for
some flow systems of simple geometry.
These calculations make use of the definition of viscosity and the concept of a
momentum balance.
Actually, a knowledge of the complete velocity distributions is not usually needed in
engineering problems.
Rather, we need to know
• the maximum velocity,
• the average velocity, or
• the shear stress at a surface
These quantities can be obtained easily once the velocity profiles are known.
In the first section we make a few general remarks about differential momentum
balances (Shell momentum balances and boundary conditions)
Then, in subsequent sections, we work out in detail several classical examples of
viscous ftow patterns.
These systems are too simple to be of engineering interest; it is certainly true that they
represent highly idealized situations, but the results find considerable use in the
development of numerous topics in engineering fluid mechanics.
The methods and problems given in this chapter apply to steady-state flow only. By
‘steady-state’ means that the conditions at each point in the stream do not change
with time.
We will discuss velocity distribution in LAMINAR FLOW.
Here we’ll see how may calculated the laminar velocity profiles for some flow systems of
simple geometry, which are:
1. Flow of a falling film
2. Flow through a circular tube, and
3. Flow through an annulus
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SHELL MOMENTUM BALANCE: BOUNDARY CONDITIONS
Problems in this section are approached setting up over a thin ‘shell’ of fluid.
For the steady state flow, momentum balance. For steady flow, the momentum balance
is:
-
+
-
+
= 0
Momentum may enter and leave the system according to
1. Newtonian (or non-newtonian) expression (molecular)
2. With overall fluid motion (convective)
Forces acting on the system are:
PRESSURE FORCES (acting on surface)
GRAVITY FORCES (acting on the whole volume)
BOUNDARY CONDITIONS COMMONLY USED:
1. At solid fluid interfaces fluid velocity equals to the surface velocity
If A is solid at x=x; Vx=VA
2. At liquid-gas interfaces the momentum flux (velocity gradient) in the liquid phase is
almost zero.
tin liquid = 0 or dx/dy (in liq.)= 0
3. All liquid-liquid interfaces the momentum flux perpendicular to the inteface and the
velocity continuous across the interface.
A
B
interface
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The following procedure for solving velocity distribution for laminar flow
problems will be used:
1. First we identify the velocity component
2. Then we write a momentum balance on shell of finite thickness
3. We let this thickness approach zero and use the mathematical defination of the first
derivative to obtain differential equation describing the momentum flux distribution
4. Then we insert the Newtonian expression for the momentum flux to obtain a differential
equation for the velocity distribution
5. These differential equations will be integrated using appropiate boundary conditions.
6. Final information on velocity profile then be used to calculate other quantities, such as;
AVERAGE VELOCITY
MAXIMUM VELOCITY
VOLUME RATE OF FLOW
PRESSURE DROP
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FLOW OF FALLING FILM
At first example, we consider the flow of a fluid along and inclined flat surface.
Consider we have liquid reservuar and liquid flows downhill on a solid surface.
Such films have been studied in connection with wetted-wall towers, evaporation, Gas
experiments, and application of coatings to paper rolls.
We will study on a L region which is enough far from enterence and ends. In this Region
velocitiy of z component Vz does not depend on z direction.
Control volume element
{x}
{z}W
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We begin by setting up a z-momentum balance
over a system thickness X,
on a plate along Z=0 to Z= L, and extending a
distance W in the y direction (VOLUME
ELEMENT).
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2. average velocity <Vz> over a cross section of the film
3. volumetric flow rate Q is obtained from the average flow velocity or by integration
of the velocity distribution
4. The film thickness is given in terms of average velocity, the volume rate of flow,
or the mass rate of flow per unit width of wall ( ┌ )
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FLOW THROUGH A CIRCULAR TUBE
The flow of fluids in circular tubes is encountered frequently in physics, chemistry, biology,
and engineering. The laminar flow of fluids in circular tubes may be analyzed by means of the
momentum balance described in before.
The only new feature introduced here is the use of cylindrical coordinates ,which are the
natural coordinates to describe positions in a circular pipe.
We consider then the steady laminar flow of a fluid of constant density ‘in a "very long" tube of length L and radius R; we specify that the tube be “very long" because we want to assume that there are no "end effects"; that is we ignore the faet that at the tube entrance and exit the flow will not necessarly parallel everywhere to the tube surface.
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FLOW THROUGH AN ANNULUS
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