MAE 204 — THERMODYNAMICS
PART 2: ENERGY TRANSFER, WORK & THE
FIRST LAW
Clean, re-typeset study notes based on the supplied textbook material (pp. 61–70).
TOPIC
WHAT YOU SHOULD KNOW
Heat transfer
Meaning, direction, adiabatic processes, total heat, heat rate, and specific heat transfer.
Work transfer
How work differs from heat; sign convention; path functions.
Electrical work
Electrical work and power relations.
Mechanical work
Force–displacement, shaft, spring, elastic bar, surface tension, lifting, and acceleration.
Nonmechanical work
Generalized force/displacement and examples.
First Law
Energy conservation and how heat/work change system energy.
Big picture: Heat and work are not properties stored inside a system. They are energy-transfer
mechanisms that occur across the system boundary. The First Law then accounts for how these
transfers change the energy of the system.
1. Heat Transfer
Heat is energy transferred between a system and its surroundings because of a temperature difference.
Heat naturally transfers from a higher-temperature region toward a lower-temperature region.
1.1 Adiabatic process
• An adiabatic process is one in which there is no heat transfer between the system and surroundings.
Q=0
• A process can be adiabatic because the system is well insulated, or because the system and surroundings
are at the same temperature so there is no driving force for heat transfer.
• Adiabatic ≠ isothermal. Adiabatic means Q = 0; it does not necessarily mean temperature is constant.
• A system can still change temperature during an adiabatic process because energy can cross the
boundary as work.
Remember: “No heat transfer” does not mean “no energy transfer.” In an adiabatic process, work can
still transfer energy.
1.2 Total heat, specific heat transfer, and heat rate
• Total heat transferred during a process is denoted by Q. Common units: kJ or Btu.
• Heat transfer per unit mass is called specific heat transfer.
q = Q / m [kJ/kg]
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• The heat-transfer rate is the amount of heat transferred per unit time and is denoted by Q̇.
Q̇ = dQ/dt [kJ/s = kW]
• If Q̇ varies with time, integrate it over the process:
Q = ∫₁² Q̇ dt
• If Q̇ is constant:
Q = Q̇ Δt
1.3 Heat-transfer mechanisms
Mechanism
Definition
Key idea
Conduction
Energy transfer through interaction of particles in a
substance.
More energetic particles transfer energy to less
energetic neighboring particles.
Convection
Energy transfer between a surface and an adjacent
moving fluid.
Combines conduction near the surface with
fluid motion.
Radiation
Energy transfer by electromagnetic waves (photons).
Does not require direct physical contact.
1.4 Historical idea of heat
• The older caloric theory treated heat as a massless fluid that could flow from hot bodies to cold bodies.
• Experiments involving friction showed that heat could be generated continuously, contradicting the idea
that heat was a conserved substance.
• Joule's experiments helped establish that heat is an energy transfer rather than a material fluid.
• Modern thermodynamics treats heat as energy in transit, not a substance contained in a system.
2. Energy Transfer by Work
Work, like heat, is an energy interaction between a system and its surroundings. If energy crosses the
boundary and the mechanism is not caused by a temperature difference, it is classified as work.
2.1 What is work?
• Work is energy transfer associated with a force acting through a distance.
• Examples include a rising piston, rotating shaft, or electrical charges crossing a system boundary.
• Work is measured in kJ (or Btu).
w = W / m [kJ/kg]
• Work done per unit time is power, Ẇ, with units kJ/s = kW.
2.2 Heat vs. work
Heat
Work
Energy transfer caused by a temperature difference.
Energy transfer not caused by a temperature difference.
Occurs across the boundary.
Occurs across the boundary.
Not a property stored in the system.
Not a property stored in the system.
Path function.
Path function.
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2.3 Sign convention
• Formal convention: heat transfer to the system is positive; heat transfer from the system is negative.
• Work done by the system is positive; work done on the system is negative.
• Alternatively, use subscripts in and out to explicitly state the direction.
• If the direction is unknown, assume a direction. A negative calculated result means the actual direction is
opposite the assumed direction.
Exam habit: Always draw the system boundary and label Qin, Qout, Win, and Wout before writing the
energy equation.
2.4 Heat and work are path functions
• Heat and work depend on the process path, not just the initial and final states.
• They are therefore called path functions and are represented using inexact differentials, δQ and δW.
• Properties such as volume and internal energy depend only on the state and are called point functions;
they use exact differentials, dV and dU.
∫₁² dV = V₂ − V₁ = ΔV
∫₁² δW = W₁→₂ (not ΔW)
Critical distinction: A system can have a state with a definite volume or internal energy, but it does not
“contain” heat or work. Heat and work exist only as transfers during a process.
3. Electrical Work
When electrical charge crosses the system boundary through an electrical potential difference, energy is
transferred as electrical work.
Wₑ = V N
• V = electrical potential difference (voltage).
• N = amount of electrical charge transferred.
Ẇₑ = V I [W]
• I is the electric current, i.e., charge flowing per unit time.
• If V and I vary with time:
Wₑ = ∫₁² V I dt
• If V and I are constant:
Wₑ = V I Δt
3.1 Useful electrical power forms
Ẇₑ = V I = I²R = V²/R
• Use these equivalent forms when a resistance R and either V or I are given.
Unit check: V·A = W. For an energy result in kJ, remember that W·s = J and 1000 J = 1 kJ.
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4. Mechanical Forms of Work
4.1 Force–displacement work
• For a constant force F acting in the direction of displacement s:
W=Fs
• If the force changes with position:
W = ∫₁² F ds
• The sign depends on the direction of the force relative to the motion.
4.2 Requirements for a work interaction
• There must be a force acting on the boundary.
• The boundary must move.
• A force with zero displacement produces no mechanical work.
• Boundary displacement without a force also produces no work transfer.
4.3 Shaft work
A rotating shaft transfers energy through torque. For constant torque T and n revolutions:
T=Fr
s = 2π r n
Wsh = 2π n T
• Shaft power is work per unit time:
Ẇsh = 2π ṅ T
• Here ṅ is the number of revolutions per unit time. For rpm, convert minutes to seconds when calculating
power in watts or kW.
4.4 Shaft-power example
• Given T = 200 N·m and 4000 rpm:
Ẇsh = 2π(4000 rev/min)(200 N·m)(1 min/60 s)
Ẇsh ≈ 83.8 kW
• This illustrates that shaft power increases with both torque and rotational speed.
5. Spring, Elastic, and Surface-Tension Work
5.1 Spring work
• For a spring, a differential displacement dx under force F gives:
δWspring = F dx
• For a linear spring:
F = kx
• x is measured from the undeformed (unstretched/uncompressed) position.
Wspring = ½ k (x₂² − x₁²)
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• k is the spring constant, typically in N/m or kN/m.
Common mistake: Do not automatically use W = FΔx when the spring force changes with x. For a
linear spring, integrate F = kx; the result is the ½k(x₂² − x₁²) expression.
5.2 Work done on elastic solid bars
• A solid bar in its elastic range can be modeled similarly to a spring.
• Normal stress is:
σ=F/A
• Therefore:
W
elastic
= ∫₁² F dx = ∫₁² σ A dx
• This applies while the material remains in the elastic range and returns toward its original dimensions
when the load is removed.
5.3 Surface-tension work
• Stretching a liquid film requires work because surface-tension forces act along the film boundary.
Wsurface = ∫₁² σ dA
• σ is surface tension, with units N/m.
• For a rectangular film with a movable wire of length b, the film has two surfaces, so a displacement dx
changes area by:
dA = 2b dx
• The factor 2 comes from the two liquid–air surfaces.
5.4 Raising or accelerating a body
• Raising a body increases its gravitational potential energy.
ΔPE = mg(z₂ − z₁)
• Accelerating a body increases its kinetic energy.
ΔKE = ½m(V₂² − V₁²)
• If friction and other losses are neglected, the required work equals the corresponding increase in potential
or kinetic energy.
• In real machines, additional work/power is required to overcome friction, rolling resistance, and other
losses.
6. Worked Examples You Should Know
Example A Candle burning in an insulated room
• System: the room, including the air and candle.
• Because the room is well insulated, no heat crosses the boundary:
Q=0
• Chemical energy in the candle is converted into other forms of internal energy within the system.
ΔU = 0 (for the process described)
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Lesson: A process can involve energy conversion and temperature changes internally while still having
Q = 0 if the system is insulated.
Example B Potato heated in an oven
• System: the potato.
• The oven is hotter than the potato, creating a temperature difference.
• Energy crosses the potato boundary because of that temperature difference, so the transfer is heat.
Example C Electric oven as the system
• If the entire oven, including its heating element, is the system, electrons cross the system boundary and
transfer electrical energy.
• That energy transfer is classified as electrical work, not heat.
• If the system is instead only the air inside the oven, the heating element is inside the system boundary
and energy reaches the air because of a temperature difference. That transfer is heat.
Very important: The same physical energy transfer can be classified as heat or work depending on
where you draw the system boundary.
Example D Car climbing a hill
• For a man + cart system moving up a ramp at angle θ, the vertical rise is L sinθ. The work needed against
gravity is:
W = mgL sinθ
• For m = 200 kg, g = 9.8 m/s², L = 100 m, θ = 20°:
W ≈ 67.0 kJ
• If only the 100-kg cart is the system, the required work is approximately 33.5 kJ.
Example E Car acceleration
• For a 900-kg car accelerating from rest to 80 km/h in 20 s on a level road:
W = ΔKE = ½m(V₂² − V₁²)
• Convert 80 km/h to m/s before substitution:
80 km/h = 22.22 m/s
W ≈ 222 kJ
Ẇavg = W/Δt ≈ 11.1 kW
• This is the ideal acceleration power increase only; actual vehicle power must also overcome resistance
and losses.
7. Nonmechanical Forms of Work
Not every work interaction is mechanical. A useful general framework is to identify a generalized force and
a corresponding generalized displacement.
δW = F dx
• Electrical work: generalized force → voltage; generalized displacement → electrical charge.
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• Magnetic work: generalized force → magnetic field strength; generalized displacement → total magnetic
dipole moment.
• Electrical polarization work: generalized force → electric field strength; generalized displacement →
polarization of the medium.
General idea: Work is a mode of energy transfer. The exact mathematical form depends on the type of
generalized force and displacement.
8. The First Law of Thermodynamics
The First Law is the conservation of energy principle. Energy can neither be created nor destroyed during a
process; it can only change forms.
8.1 Energy accounting
• A system may exchange energy with its surroundings through heat and work.
• The system may also change its stored energy, such as internal, kinetic, or potential energy.
• Every joule of energy entering, leaving, or changing form must be accounted for.
8.2 Common energy forms
Energy form
Expression / idea
Internal energy, U
Microscopic energy associated with the system.
Kinetic energy, KE
Macroscopic motion: KE = ½mV²
Potential energy, PE
Position in a gravitational field: PE = mgz
Heat, Q
Energy crossing the boundary due to temperature difference.
Work, W
Energy crossing the boundary by a mechanism other than temperature difference.
8.3 Mechanical energy conversion
• A falling body demonstrates conservation of energy.
• As height decreases, potential energy decreases.
• With negligible air resistance, the decrease in potential energy becomes an increase in kinetic energy.
−ΔPE = ΔKE
• Equivalently, total mechanical energy remains constant when no losses or external transfers are present.
8.4 Adiabatic processes and the First Law
• In an adiabatic process, Q = 0, but work interactions may still occur.
• Experiments show that for all adiabatic processes between the same two states, the net work is the same,
regardless of the particular process path.
• This observation provides the foundation for defining energy as a state quantity and for the First Law.
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9. PART 2 — Formula Sheet
Quantity
Formula
Units / Notes
Specific heat transfer
q = Q/m
kJ/kg
Heat rate
Q̇ = dQ/dt
kW
Heat, variable rate
Q = ∫ Q̇ dt
kJ
Heat, constant rate
Q = Q̇Δt
kJ
Specific work
w = W/m
kJ/kg
Electrical work
Wₑ = VN
J or kJ
Electrical power
Ẇₑ = VI
W
Electrical power
Ẇₑ = I²R = V²/R
W
Force work
W = Fs
kJ
Variable-force work
W = ∫ F ds
kJ
Shaft work
Wsh = 2πnT
kJ (with consistent units)
Shaft power
Ẇsh = 2πṅT
W or kW
Spring force
F = kx
N
Spring work
Wspring = ½k(x₂² − x₁²)
J or kJ
Elastic bar work
Welastic = ∫σA dx
J or kJ
Surface work
Wsurface = ∫σ dA
J or kJ
Potential energy
PE = mgz
J
Kinetic energy
KE = ½mV²
J
10. Exam Checklist
• 1. Define the system before deciding whether energy transfer is heat or work.
• 2. Draw the system boundary and mark energy entering/leaving.
• 3. Ask: is the transfer caused by a temperature difference? If yes → heat.
• 4. If not, determine the work mode: electrical, shaft, spring, force–displacement, etc.
• 5. Check whether the process is adiabatic: if yes, Q = 0.
• 6. Remember: heat and work are path functions; U, V, PE, and KE are state-related quantities.
• 7. Convert units before substituting: km/h → m/s, rpm → rev/s when needed, J ↔ kJ.
• 8. For variable force, use integration rather than W = FΔx.
• 9. For a spring, use W = ½k(x₂² − x₁²).
• 10. For acceleration, use ΔKE; for lifting, use ΔPE.
• 11. Always check the sign and physical direction of the final answer.
ONE-SENTENCE SUMMARY: Heat and work are ways energy crosses a system boundary; the First Law
ensures that all such energy transfers and changes in stored energy balance exactly.
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