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2001, W. E. Haisler Chapter 2: Conservation of Mass 1

Chapter 2

CONSERVATION of MASSFOR A CONTINUUM

MASS going in - MASS going out= change in MASS

(during some time period)

(for the system/control volume as a function of space and time)


The definition of the continuum system/control volume depends on your need. It could be the

universe,this building,

a living organism, ora differential volume of an object.

The Coordinate System is another choice.


Consider a tank with fluid flowing into and out of the tank:

Figure 1.1: Fluid Flow Into and Out of a Tank


For the macro-view, the system might be:

Figure 1.2: Entire Tank Taken as the System


Or, the system boundary might be a smaller micro-view:

Figure 1.3: Possible “Systems” That Could be Chosen


Must consider flow into and out of the smaller "system":

Figure 1.4: Smaller System Chosen Within the Tank


Choose a Cartesian coordinate system for convenience to define the system boundary and the mass flow directions:

Figure 1.5: Rectangular System (Possibly Differential in Size)


What is a continuum and at what length scale is a material a continuum? Is it possible to have too small a volume element?

A continuum is defined to be a material system for which all quantities of interest (variables) can be defined at every point as functions of space (r) and time (t). Consider density:

Continuum Hypothesis :

The above definition implies that mass density can be defined as a function of position and time and therefore at every point in a continuum, provided that the limit shown above exists as the volume . The mass density, , is in general a function of position and time ( = (r, t)).


Length scales which define a continuum:

Building – Brick – Microphotographlength scale ~ 10 m length scale ~ 0.1 m of a Composite -

length scale ~ 10-6 m

Crystallographic lattice for a BCC crystal

water molecules

Water

Steel plate Steel

Water pipe


The length scale must be chosen such that variables of interest (for example, density) are observed as being constant with respect to the volume element chosen (at some position and time within the material).


Consider Conservation of Mass for a differential volume element. Consider a Cartesian coordinate system for a volume element located at position x,y,z and with dimensions x, y, z:


In conservation of mass, if we consider mass going in (or out); this implies the following:

1. there is mass or mass/volume ().2. there must be a velocity of the fluid (

).

3. the mass must be flowing into something (a volume) and through some surface area (the “in and out” area).

4. For a differential volume x y z, the mass flow rate in the x direction through some area yz is given by

and the total mass flow for a time increment t is


So where did the last result come from? How do we determine the appropriate terms to put into the conservation equation? We need the amount of mass entering the system during a certain period. The mass is entering through a certain area at a certain velocity. We can think of it this way. Define the mass flux to be mass flow rate per unit area, or,

Then the mass entering for a certain time period could be written as(mass flux) x (area of entrance) x (time period of observation) or

2001, W. E. Haisler Chapter 2: Conservation of Mass 14So what is the mass flux in the x direction? Must be connected to density and the velocity in x direction. So we write . This is the amount of mass per unit time that flows in the x direction though an area whose normal points in the x direction. Only the component of flow normal to the area enters the area.

In terms of mass flux, we could conveniently write conservation of mass in the following way:

(mass flux in) (area of entrance) (time period of observation)

- (mass flux out) (area of exit) (time period of observation)

= change in mass within system during time period of observation.


A mass flow example: Consider each “ball” to be a mass particle weighing 1 g and 1 mm in diameter, and that the balls are one layer deep (plane of the paper). The mass particles are flowing uniformly to the right with a velocity such that it takes each ball 1 second to pass through the opening.

Note that 6 balls (6 g) will pass through the opening each second. Hence the mass flow rate is 6 g/sec. In 3 seconds, 18 g has passed through the opening.

2001, W. E. Haisler Chapter 2: Conservation of Mass 16Suppose now that the mass particles are flowing to the right with a velocity that is inclined from the horizontal.

Note that in 3 sec, a row of 3 balls will have moved up an imaginary plane oriented at 60 (balls are moving at 1 ball per sec through the plane but their velocity is oriented at 60). Note that there is room for only 3 rows of balls to get through the projected area of the opening. Hence in 3 seconds, 9 balls will pass through the opening, i.e., a mass flow rate of 3 g/sec, or 1/2 of the previous case. Note that the perpendicular component of the velocity is . Hence, .

2001, W. E. Haisler Chapter 2: Conservation of Mass 17You can consider the mass flow rate through an area in two different ways:

1. Only the perpendicular velocity of the balls carries mass through the opening so that

2. The mass can flow through the projected area that it “sees,” i.e., the area perpendicular to the velocity vector, so that .

Either way, you get exactly the same result.

2001, W. E. Haisler Chapter 2: Conservation of Mass 18Another mass flow example: Consider a faucet. If water flows from a faucet with velocity , and the cross-sectional area of the faucet exit is A, what is the mass flow rate?

Consider a time . In this time, a cylindrical volume of water, V, will be discharged of length L and area A: V= A L

If the water velocity is , then in time a water particle will have traveled: s= . Hence, over time , the length of cylindrical volume of water discharged is L=

. Hence, the volume of water discharged in is: V=A . The volumetric flow rate is the volume of water per unit time:

The mass flow rate is mass per unit time, or the volumetric flow rate times the density of the water:

L vA


Consider mass flow in the x direction with density and velocity flowing into the volume xyz through the surface yz during a time increment of t.

The mass entering the left boundary is: and exiting the right boundary is: .


For the entire volume, mass in - mass out:

x direction:

y direction: +

z direction: +

For the volume, change in mass (for time t) :

=


Conservation of mass requires that change in mass = mass in - mass out (for a given time interval). Hence we have:

= + +

Divide by xyzt to obtain


Take the limit of each term; x, y, z, t 0,


CONSERVATION OF MASS(The Continuity Equation)

valid for any point x,y,z in a continuum and any time t.


Recall that the vector operator is called the divergence and is defined by

Hence, conservation of mass (continuity equation) can be written as

In vector notation, the conservation equation is valid for any coordinate system.


Conservation of Mass (cylindrical coordinate system)


For the solution of most engineering physics problems, we must have the appropriate1. Governing equations (conservation of mass, momentum, energy, etc.)2. Boundary conditions (what is known or assumed on the boundary of the system). This could be known displacements, pressures, temperature, heat flux, mass flow rate, velocity, etc.3. Initial conditions (values of system variables which are known at some initial time).


Some examples of conservation of mass (continuity):

1. Steady state : variables are not a function of time.

2. Incompressible: = constant


3. Steady state, incompressible, 1-D flow between two parallel plates (Poiseulle flow)

Boundary Conditions:

Continuity gives:

for plane motion, independent of z


At this point, conservation of mass (continuity) tells us that the velocity in the x direction is some function of y as shown below:

What else can we determine about the velocity distribution, or other variables like pressure in the fluid?

Nothing! … Until we consider conservation of liner momentum and define more information about the fluid (constitutive equations) and kinematics for the fluid.


4. Laminar, steady state flow through a cylindrical tube:

Boundary Conditions: .

Thus: which implies .Assume symmetry with respect to so that .


Lets take a look at what a velocity field looks like and what conservation of mass implies. Suppose over a square planar region, the velocity in Cartesian coordinates is given by:

You should verify that and satisfy COM. Lets use Maple to plot the velocity field and over the square region.


Notice that on the top and bottom boundaries, fluid is flowing into the square region. On the left and right boundaries, fluid is flowing out of the region. Velocity is greatest near the lower right corner and almost zero near the center.


Here are two localized plots (zoomed in). Note that at some points, the velocity becomes zero.

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