Conservation of mass
Henryk Kudela
Contents
1 Principle of Conservation of mass. Transport theorem. 1
1 Principle of Conservation of mass. Transport theorem.
The conservation of mass principle is one of the most fundamental principles in nature. We are all familiar with this principle, and it is not difficult to understand. As the saying goes: you cannot have your cake and eat it too!.
For closed systemΩ(t), which we mean a collection of unchanging contents, so all particles move together inside the regionΩ(t) with boundary S(t). The mass of the system can be express by the density of fluid:
Msys= Z
Ω(t)ρ(x,t) (1)
The conservation of mass can be express as follows:
Definition 1. Let the density ρ(t, x), x ⊂Ω, be a positive, smooth function and let v be a vector field with with map motionΦ(t, x). We sayρ,v satisfy the principle of conservation of mass if
d dt
Z
Ω(t)ρ(t, x) dυ = 0 (2)
The (2) can be rewrite as:
d
dtMsys= 0 (3)
It is worth to emphasize thatΩ(t),ρ and v may change with time, but they must do so in a way that leaves Msys unchanged if the conservation of mass is fulfil.
Theorem 1. The principle of conservation of mass is satisfied by ρ,v if and only if any of the following equivalent condition hold:
1. dtdρ+ρdiv v= 0 2. ∂ρ∂t + div (ρv) = 0
3. ∂∂tRΩ(t= f ixed)ρdυ0= −RS(t= f ixed)ρv dS0
Remark 1. The equation 1) and 2) are called continuity equation. The item 3. in the above theorem related to the CONTROL VOLUMES, (CV) analysis of conservation of mass. Control volumes and their associated control surfaces are defined for convenience on solving any problem.
For a system and a fixed, non-deforming control volume that are coincident at an instant of time, as illustrated in Fig.(1) one have:
time rate of change of the mass of the coincident system =
time rate of change of the mass of the contents of the coincident control volume + net rate of flow of mass through the control surface
d dt
Z
Ω(t)ρdυ= ∂
∂t Z
CVρdυ+ Z
Sρv· n dS (4)
In (4), we express the time rate of change of the system mass as the sum of two control volume
Figure 1: System and control volume at three different instances of time. (a) System and control volume at time t−δt(b) System and control volume at time t, coincident condition. (c) System and control volume at time t+δt.
quantities, the time rate of change of the mass of the contents of the control volume,
∂
∂t Z
CVρdυ (5)
and the net rate of mass flow through the control surface, Z
Sρv· n dS (6)
The integrand v· n, in the mass flow rate integral represents the product of the component of velocity, v, perpendicular to the small portion of control surface and the differential area, dA.
Thus, v· n is the elemental volume flow rate, dqV, through dA andρv· n is the elemental mass flow rate, dqm, through dA. Furthermore, the sign of the dot product v· n is ” + ” for flow out of the control volume and ”− ” for flow into the control volume since n is considered positive when it points out of the control volume. When all of the differential quantities, are summed over the entire control surface,ρv· n as indicated by the integral the result is the net mass flow rate through
the control surface. Z
CVρv· n = ˙mout− ˙min (7)
Proof. The proof of the theorem based on the Reynolds transport theorem
Theorem 2. Let v(t, x) be smooth vector field onΩparallel to∂Ω, with map flowΦ(t, x), and let f(t, x) be a function onΩ. Then
d dt
Z
Ω(t)f(t, x) dυ = Z
Ω(t)(d f
dt + f div v) dυ (8)
where v= (v1,v2,v3),operator div (v) =∂∂vx1
1+∂∂vx2
2 +∂∂vx3
3
Putting in equation f ≡ρwe will at once obtain the(1), and after some manipulation it is not difficult to obtain(2). (3) stems from (4).
Remark 2. One -dimensional version of transport theorem one can obtain from Leibniz theorem which allows for differentiation an integral whose limits of integration are functions of the variable with which one need to differentiate:
d dt
Z x=b(t)
x=a(t) f(x,t)dt = Z b
a
∂f
∂t +db
dt f(b,t) −da
dt f(a,t) (9)
It is of interest to consider the qualitative implications of this formula. Leibnitz’s formula implies that the time-rate of change of a quantity f in a region (interval in this case) [a(t); b(t)] that is changing in time is the rate of change of f within the region plus the net amount of f crossing the boundary of the region due to movement of the boundary f· u. In particular, note that d a(t)dt = u(a) and d b(t)dt = u(b) are the speeds with which the respective boundary points are moving.
Example 1. We now apply continuity equation of theorem 1 to simple situation. We assume that control volume ΩS is fixed in space, and that density does not depend on time. We assume also that entrance and exit are also fix3ed in both space and time( see fig)
It is useful to divide the control surface area into two distinct parts:
1. Se(t), the area of entrance and exits through which fluid may enter or live the control volume 2. Sf(t) the area of solid fixed surface
Due to fact thatΩandρare no longer function of time we can write continuity equation in integral
form as follows: Z
Se
ρu· n dA = 0 (10)
Figure 2: Control volume corresponding to flow through an expanding pipe
Observe that in this case Se consist of two pieces area: A1 - the entrance and A2 - the exit.If we assume, that flow properties are uniform across each cross section (i.e. we assume averaged one-dimensional flow) we have
Z
Se
ρu· n dA = −ρ1U1A1+ρ2U2A2= 0 (11) or
ρ1U1A1=ρ2U2A2 (12) If we now assume that flow to be incompressible soρ1=ρ2we see that
U1= U2
A2
A1
(13) and for the situation as in fig.(2), it follows that U1>U2. Thus, a steady incompressible flow must speed up when going through a constriction and slow down in an expansion, simply due to conservation of mass. It is easily checked that the units in (11) correspond to mass per unit time, and we call this the mass ow rate, denoted qm . In the present case uniform flow in each cross section was assumed, so we have
qm=ρUA (14)
and in general we define the mass flow rate qmas qm=
Z
Se
ρu· n dA (15)
for any given surface Se. Finally, we should note that in the steady-state condition represented by (11) we can express this simply as
(qm)in= (qm)out (16)
A useful related concept is the volume flow rate, sometimes called the ”discharge,” usually denoted by qV and defined in our simple 1-D case by
qV = UA (17)
As with the mass ow rate, there is a more general definition, qV =
Z
Se
u· n dA (18)
which reduces to (18) if U is constant across the given cross section corresponding to Se. This formula shows that if the volume ow rate through a given area is known, we can immediately calculate the average velocity as
U=Q A
. The preceding example has shown that in a steady flow the mass flow rates must be constant from point to point, and ifρis constant the volume flow rates must also be constant.
Example 2. Incompressible, laminar water flow develops in a straight pipe having radius R as indicated in Fig. At section (1), the velocity profile is uniform; the velocity is equal to a con- stant value U and is parallel to the pipe axis everywhere. At section (2), the velocity profile is axisymmetric and parabolic, with zero velocity at the pipe wall and a maximum value of at the centerline. How are U and umaxrelated? How are the average velocity at section(2), V2, and umax
related? SOLUTION. An appropriate control volume is sketched (dashed lines) in Figure:3. The
Figure 3:
application of (10) to the contents of this control volume yields
−ρA1U+ Z
Sρv· n dS = 0 (19)
or, since the component of velocity, V , perpendicular to the area at section (2) is u2, and the element cross-sectional area dA2, is equal to 2πr Eq. (19) becomes
−ρA1U+ Z R
0 ρu22πr= 0 (20)
Since the flow is considered incompressible,ρ1=ρ2, The parabolic velocity relationship for flow through section (2) is used in Eq. (20) to yield
−ρA1U+ 2πZ R
0
(1 − r R
2)r = 0 (21)
Integrating, we get from Eq. (21)
−ρπR2U+ 2πumax(r2 2 − r4
4R2)|R0 = 0 (22)
or
umax= 2U (23)
Thus, the average velocity at section(2), is one-half the maximum velocity,V2= umax2
During this course I will be used the following books:
References
[1] F. M. White, 1999. Fluid Mechanics, McGraw-Hill.
[2] B. R. Munson, D.F Young and T. H. Okiisshi, 1998. Fundamentals of Fluid Mechanics, John Wiley and Sons, Inc. .
