fluid_mechanics

4.4 The Reynolds Transport Theorem 169
If B is an extensive parameter of the system, then the value of it for the system at time
t is
B sys 1t2 B cv 1t2
since the system and the fluid within the control volume coincide at this time. Its value at time
t dt is
B sys 1t dt2 B cv 1t dt2 B I 1t dt2 B II 1t dt2
Thus, the change in the amount of B in the system in the time interval dt divided by this time
interval is given by
dB sys
dt
B sys1t dt2 B sys 1t2
dt
B cv1t dt2 B I 1t dt2 B II 1t dt2 B sys 1t2
dt
By using the fact that at the initial time t we have B sys 1t2 B cv 1t2, this ungainly expression may
be rearranged as follows.
The time rate of
change of a system
property is a Lagrangian
concept.
dB sys
dt
B cv1t dt2 B cv 1t2
dt
B I1t dt2
dt
B II1t dt2
dt
(4.10)
In the limit dt S 0, the left-hand side of Eq. 4.10 is equal to the time rate of change of B for the
system and is denoted as DB sysDt. We use the material derivative notation, D1 2Dt, to denote this
time rate of change to emphasize the Lagrangian character of this term. 1Recall from Section 4.2.1
that the material derivative, DPDt, of any quantity P represents the time rate of change of that
quantity associated with a given fluid particle as it moves along.2 Similarly, the quantity DB sysDt
represents the time rate of change of property B associated with a system 1a given portion of fluid2
as it moves along.
In the limit dt S 0, the first term on the right-hand side of Eq. 4.10 is seen to be the time
rate of change of the amount of B within the control volume
V 2
B
lim cv 1t dt2 B cv 1t2
0B 0 a cv
rb dVb
cv
dtS0 dt
0t 0t
(4.11)
(2)
t = 0
The third term on the right-hand side of Eq. 4.10 represents the rate at which the extensive parameter
B flows from the control volume, across the control surface. As indicated by the figure in the
margin, during the time interval from t 0 to t dt the volume of fluid that flows across section
122 is given by dV II A 2 d/ 2 A 2 1V 2 dt2. Thus, the amount of B within region II, the outflow
region, is its amount per unit volume, rb, times the volume
B II 1t dt2 1r 2 b 2 21dV II 2 r 2 b 2 A 2 V 2 dt
where b 2 and r 2 are the constant values of b and across section 122. Thus, the rate at which this
property flows from the control volume, B # r
out, is given by
δV II
B # B
out lim II 1t dt2
r
dtS0 dt
2 A 2 V 2 b 2
(4.12)
V 2
δt
Similarly, the inflow of B into the control volume across section 112 during the time interval
dt corresponds to that in region I and is given by the amount per unit volume times the volume,
dV I A 1 d/ 1 A 1 1V 1 dt2. Hence,
B I 1t dt2 1r 1 b 1 21dV 1 2 r 1 b 1 A 1 V 1 dt
where b 1 and r 1 are the constant values of b and across section 112. Thus, the rate of inflow of
the property B into the control volume, B # r
in, is given by
(2)
t = δt
B # B
in lim I 1t dt2
r
dtS0 dt
1 A 1 V 1 b 1
(4.13)