Modern Engineering Thermodynamics

46 CHAPTER 2: Thermodynamic Concepts
EXAMPLE 2.2 (Continued )
Since this is a simple first-order ordinary differential equation, we can separate the variables to obtain
dN
N = αdt
Then defining N 0 as the population at time t = 0, this equation can be integrated as
Z N N0
t Z =t
dN
N = αdt
t =0
to give
ln ⁡ N
= αt
N 0
And inverting the logarithm gives
e ln ðN/N0Þ = N N 0
= e at
or
N = N 0 e αt :
Thus the population increases or decreases exponentially depending on the sign of α.
Exercises
1. Develop a balance equation for the number of hamburgers in your room. Answer: Net hamburgers in the room = Net
hamburgers brought into the room + Net hamburgers made inside the room.
2. The growth rate discussed in Example 2.2 is often called a geometric growth rate. Malthus argued that the food supply of
a population often grew only at a constant, or arithmetic, rate, dF/dt = β, where F is the size of the food supply at time t
and β is a constant. Write a rate balance equation for the food supply F using this growth rate and solve it for F as a
function of time t. Answer:
dF
= _F G = β = _F T + _F P
dt system
and solving for F gives F = βt + F 0 , where F 0 is the size of the food supply at time t = 0.
3. In Example 2.2, the constant α is called the growth rate when it is greater than zero. Determine a general expression for
the time t D required for a population to double. Answer: t D = [ln(2)]/α.
2.11 THE CONSERVATION CONCEPT
In classical physics, a quantity is said to be conserved if it can be neither created nor destroyed. The basic laws of
physics would not produce unique balance equations if it were not for this concept. Whereas a balance equation
can be written for any conceivable quantity, conserved quantities can be discovered only by human research and
observation. The outstanding characteristic of conserved quantities is that their net production is always zero,
and therefore their balance equations reduce to these simpler forms:
When X is conserved, X Production = _X Production = 0, and
X Gain = X Transport
(2.13)
_X Gain = _X Transport
(2.14)
This may not seem like much of a reduction at first, but it is a very significant simplification of the general
balance equations. It means that we need not worry about property production or destruction mechanisms and
how to calculate their effects. Equations (2.13) and (2.14) turn out to be very effective working equations for
engineering design and analysis purposes.
Thus far, scientists have empirically discovered four major entities that are conserved: mass (in nonnuclear
reactions), momentum (both linear and angular), energy (total), and electrical charge. These yield the four basic
laws of physics: the conservation of mass, the conservation of momentum, the conservation of energy, and the
conservation of charge. The conservation of energy is also called the first law of thermodynamics.
If we let E be the total energy of a system, then its conservation is written as
E Production = E P = 0 (2.15)