Modern Engineering Thermodynamics

136 CHAPTER 4: The First Law of Thermodynamics and Energy Transport Mechanisms
Now Let us Write a Thermodynamics Problem
Step A. We limit it to a closed system and use the energy balance as our primary equation:
1Q 2 − 1 W 2 = m ðu 2 − u 1 Þ + V2 2 − V2 1
+ gðZ
2 − Z 1 Þ
2g c
g c
Choose the material. Let the system contain an ideal gas. Then auxiliary equations pv = RT and u 2 − u 1 =
c v (T 2 − T 1 ) can be used.
Choose the unknowns. With two independent equations, we can have two unknowns. Let us choose
1Q 2 and p 2 as the unknowns. We can solve for 1 Q 2 from the energy balance and solve for p 2 = RT 2 /v 2 .
If we put the system in a rigid container, then 1 W 2 = m∫pdv = 0, because for a sealed rigid container,
v = constant, then dv = 0. Let us also add the condition that the process must be isothermal, then
T 2 = T 1 and thus u 2 − u 1 = c v (T 2 − T 1 ) = 0. Further, let us also require that V 2 = V 1 , then the energy
balance reduces to
1Q 2 − 0 = m 0 + 0 + gðZ
2 − Z 1 Þ
g c
system
Now all we need to do is specify m, Z 1 , and Z 2 and we can compute 1 Q 2 .
Step B. The next step is to write a scenario, or a short story, that uses these processes and values to create a
thermodynamic problem. Let us try this:
There are 5.00 kg of hydrogen gas (an ideal gas) at 20.0°C and 0.300 MPa sealed inside a wooden barrel
(a rigid container) at the top of Niagara Falls. The barrel is not insulated and is maintained at a constant
temperature (i.e., isothermal) as it travels over the falls in contact with the water. Determine
a. The heat transfer from the barrel as it travels 50.0 m vertically between the top and bottom of the falls.
b. The final pressure inside the barrel at the bottom of the falls.
Note that the problem scenario does not have to be deadly serious, you can write problem statements
around anything your imagination can conceive.
Step C. Now we must work the problem in the forward direction to see if all the necessary information has
been provided, so let us try it.
Solution
The problem solving technique requires that we start by reading the problem statement carefully.
Step 1 ask us to draw a sketch of the system (the barrel going over the Niagara Falls, see Figure 4.22) and identify
the material in the system, it is the hydrogen in the barrel.
Step 2 asks us to identify the unknowns. Even though we just wrote the problem statement, it is important to
read it again, carefully, to check for errors and completeness. The problem statement should contain clarifying
statements so that the reader need not make any unreasonable assumptions. For example, in our problem statement,
we identified the hydrogen as an ideal gas, because it is not obvious to a beginning thermodynamics student
which materials behave like an ideal gas and
which do not. Also, while it may be obvious to you
when you wrote the problem statement that a barrel
is to be modeled as a sealed, rigid container, itisadvisable
to tell the reader this in clear terms, since the
purpose of the problem should be to test the problem
solving skills of the reader, not his or her ability
to read your mind about how to interpret unfamiliar
things. The unknowns here are clearly specified in
items (a) and (b) at the end of the problem statement.
They are find 1 Q 2 and p 2 .
Step 3 asks us to identify the system’s typeandits
states. The system here is closed (because the barrel is
sealed). We should be able to identify the system states
from the information given in the problem statement:
State 1 State 2
T 1 = 20:0°C T 2 = T 1 = 20:0°C
p 1 = 0:300 Mpa ?
system
5.00 kg of hydrogen gas
at 20.0°C and 0.300 MPa
Niagara falls
50.0 m
State 1 State 2
FIGURE 4.22
A barrel going over Niagara Falls.