Modern Engineering Thermodynamics

50 CHAPTER 2: Thermodynamic Concepts
EXAMPLE 2.4 (Continued )
Solution
From Eq. (2.19), we compute the muzzle velocity as
V projectile = 1 + m
pendulum
½2gRð1 − cos⁡yÞŠ 1/2 = 1 + 5:0kg
½29:81 ð m/s 2 Þð1:5mÞð1 − cos 15° ÞŠ 1/2
m projectile
0:01 kg
= 5:0 × 10 2 m/s
Exercises
6. To bring down large game requires at least 2500. ft·lbf of impact energy (the impact energy of a high-speed projectile is
its kinetic energy on impact). Determine the necessary impact velocity for the following projectiles (recall that there are
7000 grains in 1 lbm): (a) a 200. grain bullet, (b) 300. grain bullet, and (c) a 500. grain bullet. Answers: (a) 2370 ft/s,
(b) 1940. ft/s, (c) 1500 ft/s.
7. Determine the displacement angle produced when a 0.3125 lbm baseball traveling at 90.0 miles per hour is “caught” by
a ballistic pendulum having a 3.00 ft support cable and a mass of 180. lbm. Answer: 1.33°.
2.12 CONSERVATION OF MASS
An important application of the balance equation is to one of the basic conserved physical quantities, mass.
Since mass is conserved in all nonnuclear reactions, its net production in any system is zero. Therefore,
Eq. (2.10) tells us that the mass balance equation has the form
Net gain in mass by the Net mass transported into
= (2.20)
system during time dt the system during time dt
This statement can be cast in mathematical form via Eq. (2.11) as m G = m T (since m P = 0),andEq.(2.12)
provides the rate form of this balance equation as _m G = _m T (since _m P = 0). In more precise mathematical
language, the mass balance (MB) measured over some time interval δt can be written as
Mass balance ðMBÞ over time δt = ðδmÞ system
= ∑m in −∑m out (2.21)
and the mass rate balance (MRB) becomes
Mass rate balance ðMRBÞ = dm
dt
system
= ∑ _m in −∑ _m out (2.22)
One of the most common uses of the conservation of mass balance equation is in chemistry. Chemical reaction
equations are simply mass balances. Though reaction equations are not usually written as equalities, the left-hand
side is the total mass of reactants used and the right-hand side is the total mass of products produced by the reaction.
Since mass is conserved in chemical reactions, these two masses must be equal. For example, the reaction
indicated by the equation A + B = C + D means that the mass of A plus the mass of B is the same as the mass of C
plus the mass of D. These equations are valid either for individual molecules or groups of molecules that bear the
same reaction relation as the individual molecules. These molecular groups, called moles, form the macroscopic
basis for chemistry and chemical engineering. This concept is illustrated in the following example.
EXAMPLE 2.5
The total mass of a system is conserved in a chemical reaction, but the mass of any particular chemical species is not necessarily
conserved. Show that the following chemical reaction is just a closed system mass balance for the chemical species
involved:
n a A + n b B ! n c C + n d D
where A, B, C, and D are the chemical species, and n a , n b , n c , and n d are their molar amounts.
Solution
When a chemical reaction occurs in a closed system, the mass transport term vanishes and then a mass balance for a chemical
species X becomes “the mass of X gained in the reaction must equal the mass of X produced by the reaction and this must equal the