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the entity balance to momentum is that we
will generate a vector equation rather than a scalar equation.
We will, therefore, utilize balance equations, not all of which are
conservation equations, and conservation equations, all of which are
balance equations. Conservation equations are a sub-set of balance equations.
Another special case of the entity balance which is often of interest is for a
steady-state process. By a steady-state process we mean one which does not
change in time - if we look at the process initially, then look at it again after
some time has elapsed, the process still looks the same. This means that we are
concerned with the entity balance in the form of rates as expressed in Equation
(1.2-3). Note that this does not imply that all rates are zero. It does imply that
accumulation rate is zero, because the very nature of accumulation means that
content of the system is changing in time. Input and output rates, however, can
be different from zero so long as they are constant in time, as can generation
rates - it is only necessary that these rates balance one another so that there is no
accumulation.
Chapter 1: Essentials 17
For a steady-state process, therefore, the entity balance is written as
Input rate + Generation rate = Output rate
or
[ Generation rate = Output rate - Input rate1
(1.2.2-1)
1.3 The Continuum Assumption
We will treat only models in which each fluid property (for example, the
density, the temperature, the velocity, etc.) is assumed to have a unique value
at a given point at a given time. Further, we assume these values to be
continuous in space and time. Even though we know that fluids are composed
of molecules, and therefore, within volumes approaching molecular size, density
fluctuates (and temperature is hard even to define), in order to use the concepts
of calculus and differential equations we model such properties as continuous.
For example, we define the density at a point as the mass per unit volume;
but, since a point has no volume, it is not clear what we mean. By our
definition we might mean
(1.3-1)
where p is the density, and Am is the mass in an elemental volume AV.
Consider applying this definition to a gas at uniform temperature and
pressure. Assume for the moment that we can observe Am and AV
instantaneously, that is, that we can &quot;freeze&quot; the molecular motion as we make
any one observation. For a range of values of AV suppose we make many
observations of Am/AV at various times but at the Same point.
At large values of AV, because the total number of molecules in our
volume would fluctuate very little, the density from successive observations
would be constant for all practical purposes. As we approach smaller volumes,
however, our observed value would no longer be unique, because a gas is not a
continuum, but is made up of small regions of large density (the molecules)
connected by regions of zero density (empty space). In fact, we would see a
situation somewhat as depicted in Figure 1.3-1, where, as our volume shrinks,
we begin to see fluctuation in the observed density even in a gas at constant
temperature and pressure.
18 Chapter 1: Essentials
0.001 0.01 0.1 1 10 100 1000
relative volume
Figure 1.3-1 Breakdown of continuum assumption
For example, if our AV were of the same order of volume as an atomic
nucleus, the observed density would vary from zero to the tremendous density of
the nucleus itself, depending on whether or not a nucleus (or, perhaps, some
extra-nuclear particle) were actually present in AV at the time of the observation.
To circumvent such problems we make the arbitrary restriction that we will treat
only problems where we may assume that the material behavior we wish to
predict can be described adequately by a model based on a continuum assumption.
In practice, this usually means that we do not attempt to model problems where
distances of the order of the mean free paths of molecules are important.
The continuum assumption does fail under some circumstances of interest.
For example, the size of the interconnected pores of a prous medium may be of
the order of the mean free patb of gas molecules flowing through the pores. In
this situation, &quot;slip flow&quot; (where the velocity at the gas-solid interface is not
zero) can result. Another example is in the upper atmosphere where the mean
free path of the molecules is very large - in fact, it may become of the order of a
hundred miles. In this situation, slip flow can occur at the surface of vehicles
traversing the space.
Chapter I : Essentials 19
1.4 Fluid Behavior
The reader is encouraged to make use of the resources available in flow
visualization, e.g., Japan, S , 0. M. E., Ed. (1988). Visualized flow: Fluid
motion in basic and engineering situations revealed by flow visualization.
Thermodynamics and Fluid Mechanics. New York, NY, Pergamon, Japan, T. V.
S. o., Ed. (1992). Atlas of Visualization. Progress in Visualization. New York,
NY, Pergamon Press, and Van Dyke, M. (1982). An Album of Fluid Motion.
Stanford, CA, The Parabolic Press, which contain numerous photos, both color
and black and white. There are also many films and video tapes available. There
is nothing quite so convincing as seeing the actual phenomena that the basic
models in fluid mechanics attempt to describe - for example, that the fluid really
does stick to the wall.
1.4.1 Laminar and turbulent flow
We normally classify continuous fluid motion in several different ways: for
example, as inviscid vs. viscous; compressible vs. incompressible.
Viscosity is that property of a fluid that makes honey and molasses &quot;thicken&quot;
with a decrease in temperature. Inviscid flow is modeled by assuming that there
is no viscous effect. What we sometimes call ideal fluids or ideal fluid flow is
modeled by not including in the equations of motion the property called
viscosity. Viscous flow is modeled including these viscous effects.
The viscosity of a fluid can range over many orders of magnitude. For
example, in units of pPa s, air has a viscosity of about 10-5; water, about 10-3;
honey, about 10; and molten polymers, perhaps 104; a ratio of largest to
smallest of 1,OOO,OOO,OOO. (This is by no means the total range of viscous
behavior, merely a sample of commonly encountered values in engineering
problems .)
In viscous flow we can have two further sub-classifications. The first is
luminur flow, a flow dominated by viscous forces. In steady laminar flow, the
fluid &quot;particles&quot; march along smoothly in files as if on a railroad track. If one
releases dye in the middle of a pipe in which fluid is flowing at steady-state, as
shown in Figure 1.4.1-1, at low velocities the dye will trace out a straight line.
At higher velocities the fluid moves chaotically as shown and quickly mixes
across the entire pipe cross-section. The low velocity behavior is Zaminar and
the higher velocity is turbulent.
20 Chapter I : Essentials
Figure 1.4.1-1 Injection of dye in pipe flow1* laminar (top) to
turbulent (bottom)
Laminar flow is dominated by viscous forces and turbulent flow is
dominated by inertial forces, In a plot of point velocity versus time we would
see a constant velocity in steady laminar flow, but in &quot;steady&quot; turbulent flow we
would see instead a velocity varying chaotically about the time-average velocity
at the point.
Later in the text we will discuss dimensionless numbers and their meaning.
At this point we merely state that value of the Reynolds number, a
dimensionless group, lets us differentiate between laminar and turbulent flow.
The Reynolds number is composed of the density and the viscosity of the fluid
flowing in the pipe, the pipe diameter, and the area-averaged velocity, and
represents the