MEC2101 Thermodynamics
Module 4: Energy Analysis of
Closed Systems
Slides based on Chapter 4 of Y. A. Cengel and M. A. Boles,
“Thermodynamics, An Engineering Approach”, 5th Edition,
McGraw-Hill, 2006.
D. R. Buttsworth, March, 2016.
1
Moving Boundary Work
• Work = Force x distance
𝛿𝑊 = 𝐹. 𝑑𝑠
𝛿𝑊 = 𝑃. 𝐴 . 𝑑𝑠 = 𝑃. (𝐴. 𝑑𝑠)
𝛿𝑊 = 𝑃. 𝑑𝑉
𝑊 = ∫ 𝑃. 𝑑𝑉
• Will be +ve for an increase in
volume (i.e. magnitude
corresponds to Wout)
• Will be –ve for a decrease in
volume (ie magnitude
corresponds to Win)
2
Moving Boundary Work –
P-V Diagram
• Area under the process curve
on a 𝑃 − 𝑉 diagram is the
magnitude of work (out or in)
associated with a closed
system.
• Work will be +ve (i.e. work done
by system, hence magnitude is
𝑊𝑜𝑢𝑡 ) if process arrow is in
direction of increasing 𝑉.
• Work will be –ve (i.e. work done
on system, hence magnitude is
𝑊𝑖𝑛 ) if process arrow is in
direction of decreasing 𝑉.
3
Moving Boundary Work – Idealized
Processes
• Isochoric/isometric, 𝑊 = 0
• Isobaric, 𝑊 = 𝑃(𝑉2 − 𝑉1 )
• Isothermal 𝑊 =
𝑉2
𝑃𝑉 log 𝑒
𝑉1
note: if isothermal, 𝑃𝑉 = 𝑃1 𝑉1 = 𝑃2 𝑉2
• Polytropic, (𝑃𝑉 𝑛 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡)
𝑊=
𝑃2 𝑉2 −𝑃1 𝑉1
1−𝑛
note: 𝑛 ≠ 1
• Remember that if you have an ideal gas,
𝑃𝑉 = 𝑚𝑅𝑇
can be applied in conjunction with any of the above
expressions.
4
Energy Balance for Closed Systems
• Combining previous results:
Δ𝑈 = 𝑄𝑖𝑛 − 𝑄𝑜𝑢𝑡 + (𝑊𝑖𝑛 − 𝑊𝑜𝑢𝑡 )
• or
Δ𝑈 = 𝑄𝑛𝑒𝑡,𝑖𝑛 − 𝑊𝑛𝑒𝑡,𝑜𝑢𝑡
• where
𝑄𝑛𝑒𝑡,𝑖𝑛 = 𝑄𝑖𝑛 − Q out
𝑊𝑛𝑒𝑡,𝑜𝑢𝑡 = 𝑊𝑜𝑢𝑡 − 𝑊𝑖𝑛
• For a cycle, 𝑈 = 0 so,
𝑊𝑛𝑒𝑡,𝑜𝑢𝑡 = 𝑄𝑛𝑒𝑡,𝑖𝑛
5
Moving Boundary Work for a Cycle
• Cycle – if a system returns to
its initial state at the end of
the process
• Net work is the difference
between the output work of
the system (path A) and the
input work of the system
(path B).
• 𝑊𝑛𝑒𝑡 = 𝑊𝑜𝑢𝑡 – 𝑊𝑖𝑛
• Accordingly, a positive value
represents a net work output,
and a negative value
represents a net work input.
6
Specific Heats
• “the energy required to raise the temperature of a unit
mass of a substance by one degree” – depends on the
process.
Constant Volume Specific Heat, Cv (J/kgK)
• If volume of material remains constant during energy
transfer, and
• for small temperature changes (T = T2 – T1)
𝑢
𝐶𝑣 ≈
or 𝑢 ≈ 𝐶𝑣𝑇
𝑇
Constant Pressure Specific Heat, Cp (J/kgK)
• If pressure within material remains constant during
energy transfer, and
• for small temperature changes (𝑇 = 𝑇2 – 𝑇1)
ℎ
𝐶𝑝 ≈
or ℎ ≈ 𝐶𝑝𝑇
𝑇
Which is likely to be larger, 𝐶𝑣 or 𝐶𝑝? Why?
7
Specific Heats for Ideal Gases
Definition of enthalpy, ℎ = 𝑢 + 𝑃𝑣
If ideal gas, 𝑃𝑣 = 𝑅𝑇
Thus ideal gas enthalpy, ℎ = 𝑢 + 𝑅𝑇
Or in terms of a change of state
ℎ = 𝑢 + 𝑅 𝑇
• Hence, 𝐶𝑝 = 𝐶𝑣 + 𝑅
• Generally, 𝐶𝑝 and 𝐶𝑣 vary with temperature.
• However, the specific heat ratio,
•
•
•
•
𝑘 =
𝐶𝑝
𝐶𝑣
varies only slightly with temperature for an ideal gas.
• If variation of 𝐶𝑝, 𝐶𝑣 or 𝑘 is significant, evaluate
properties at the mean temperature (if suitable
tables are available).
8
Specific Heats for Solids and Liquids
• 𝐶𝑣 and 𝐶𝑝 are identical for incompressible
substances
• 𝐶𝑣 = 𝐶𝑝 = 𝐶
• Generally, the specific heat still varies with
temperature.
• Again, a typical approach would be to calculate
energy (heat) transfer using a single (average) value
over a small range of temperature.
9