Week 2 – Chapter 3: Energy, Energy Transfer and General Energy Analysis (Summary Notes)

Forms of Energy and the Energy Framework

• Energy forms can be broadly categorized as macroscopic and microscopic contributions to the total energy of a system.

• Total energy, denoted as E, is the sum of all forms of energy a system possesses.

• Thermodynamics concerns changes in the total energy, not its absolute value.

• Macroscopic forms of energy (with respect to an outside reference frame): kinetic energy and potential energy.

• Microscopic forms of energy (related to molecular structure and activity): internal energy.

• Internal energy, U, is the sum of all microscopic forms of energy.

• The macroscopic energy of an object changes with velocity and elevation, while microscopic energies reside in the molecular structure.

• Central distinction:

  • Total energy E = sum of macroscopic and microscopic forms.
  • Thermodynamics tracks changes ΔE, not E itself.

Internal Energy and Its Components

• Internal energy, U, is the sum of all microscopic energies in a system.
• Microscopic energy forms that contribute to sensible energy include:
  • Sensible energy: portion of internal energy associated with the kinetic energies (motion) of molecules.
  • Latent energy: energy associated with phase changes (e.g., melting, vaporization).
  • Chemical energy: energy associated with atomic bonds within molecules.
  • Nuclear energy: energy associated with strong nuclear bonds inside the nucleus.
• Relationship:
  • \text{Internal energy} \ U = \text{Sensible} + \text{Latent} + \text{Chemical} + \text{Nuclear}
  • Thermal energy is the portion of internal energy that is sensible plus latent: \text{ Thermal} = \text{Sensible} + \text{Latent}

Energy and Energy per Mass

• Total energy of a system can be stored (static forms) or can be transferred (dynamic forms) when interacting with surroundings.

• Dynamic energy interactions are recognized at the system boundary as they cross it and represent energy gained or lost during a process.

• The two energy interactions for a closed system are:

  • Heat transfer (Q)
  • Work (W)

• The criterion that distinguishes heat from work:

  • Heat transfer occurs due to a temperature difference between system and surroundings.
  • If there is no driving temperature difference, the energy interaction is work.

• Energy forms per unit mass (often used in flow problems):

  • Kinetic energy per unit mass: \(e_{k} = \frac{V^2}{2}\
  • Potential energy per unit mass: \(e_{p} = g z\
  • Total energy per unit mass: \(e{t} = e{k} + e_{p} + u) where u is the specific internal energy.

Mechanical Energy

• Mechanical energy is the form of energy that can be converted to mechanical work by an ideal device (e.g., an ideal turbine).

• Common macroscopic mechanical energies:

  • Kinetic energy
  • Potential energy

• For a flowing fluid, define:

  • Mechanical energy per unit mass
  • Rate of mechanical energy for a flowing fluid
  • Mechanical energy change of a fluid during incompressible flow per unit mass
  • Rate of mechanical energy change during incompressible flow

• Example concept (illustrative): mechanical energy transfer in an ideal hydraulic turbine coupled to an ideal generator shows how potential energy change or pressure drop drives work output.

The First Law of Thermodynamics

• The First Law expresses energy conservation: energy can neither be created nor destroyed; it only changes forms.
• For all adiabatic processes between two specified states of a closed system, the net work done is the same regardless of process details (energy balance principle for adiabatic paths).
• In an adiabatic process, heat transfer Q = 0, and the net work done equals the change in total energy: \(W = \Delta E\
• General statement: energy cannot be created or destroyed; it can only change forms.

Energy Balance: Net Change and Components

• Energy change of a system during a process, \(\Delta E)\, is the net difference between energy entering and leaving the system:

  • \(\Delta E = E{\text{in}} - E{\text{out}}\

• The total energy change can be decomposed into contributions from internal, kinetic, and potential energies:

  • \Delta E = \Delta U + \Delta KE + \Delta PE\

• Where each term is:

  • \(\Delta U = m (u{2} - u{1})\
  • \(\Delta KE = \frac{1}{2} m (V{2}^{2} - V{1}^{2})\
  • \(\Delta PE = m g (z{2} - z{1})\

• For stationary systems (no motion or elevation change between states):

  • \(\Delta KE = 0\
  • \(\Delta PE = 0\
  • Therefore, \Delta E = \Delta U (AE = AU)\

• Internal energy change is the primary contributor to energy change for stationary systems: \[\Delta AU = m (u{2} - u{1})\

• In many practical energy balances, the energy balance equation is written as:

  • \(\Delta E = E{\text{in}} - E{\text{out}} = Q - W + \dot{m}{\text{in}} h{\text{in}} - \dot{m}{\text{out}} h{\text{out}}\
  • For a closed system (no mass flow), this reduces to: \(\Delta E = Q - W\

Mechanisms of Energy Transfer (Ein and Eout)

• The energy content of a control volume can be changed by:
  • Heat transfer (Q)
  • Work transfer (W)
  • Mass flow (through inlet and outlet boundaries)
• For a closed mass (no mass flow): only heat transfer and work interactions apply.
• For a cycle, the energy change is zero: \(\Delta E = 0 \Rightarrow Q = W\

Sign Convention and Path vs State Functions

• Formal sign convention:
  • Heat transfer to a system is positive; heat transfer from a system is negative.
  • Work done by a system is positive; work done on a system is negative.
• Alternative convention used in many texts: indicate directions with subscripts in and out (in/out notation).
• Distinctions:
  • Heat and work are boundary phenomena (only defined as they cross system boundaries).
  • Systems possess energy, but not heat or work as state properties.
  • Heat and work are process (path) functions, not state functions; i.e., their magnitudes depend on the path taken between two states. Properties are point (state) functions with exact differentials, while heat/work have inexact differentials.

Electrical Work

• Electrical work and electrical power concepts:
  • Electrical work occurs when potential difference and current change with time.
  • If potential difference and current remain constant, electrical work is proportional to time.
• Electrical power concept: \text{Power} = \text{Voltage} \times \text{Current} \ (P = VI)
• Sign conventions mirror general heat/work conventions as applicable to electrical interactions.

Mechanical Forms of Work

• Conditions for a work interaction to exist between a system and its surroundings:
  • There must be a force acting on the boundary, and
  • The boundary must move.
• Work is computed as
  • \text{Work} = \text{Force} \times \text{Displacement},
  • For non-constant force, integrate along the boundary path.
• Examples of mechanical work forms include shaft work, spring work, and other boundary-force interactions.

Shaft Work

• Torque- and shaft-based work:
  • A force F acting through a moment arm r generates a torque T.
  • The force acts through a distance s, delivering shaft work.
  • Power transmitted through a shaft is shaft work per unit time.

Spring Work

• Work done by a spring due to elongation under a load:
  • For a linear elastic spring, displacement x is proportional to applied force F with spring constant k (kN/m).
  • Incremental work for a differential displacement dx is dW = F dx, leading to the familiar elastic work relation after integration.
  • The displacement doubles when the force doubles for a linear spring, illustrating linearity (Hooke's law).

Electrical and Mechanical Integration in Energy Systems

• Real systems involve combinations of heat, work (mechanical and electrical), and mass flow in energy transfer and conversion.
• The energy interactions at the system boundary include:
  • Heat transfer (Q)
  • Work transfer (W), including shaft work and electrical work
  • Mass flow (ṁ) carrying energy into or out of the control volume

The First Law: Worked Example Scenarios (Adiabatic and Electrical)

• Adiabatic energy balance examples illustrate the relationship between energy change and work when Q = 0:
  • The energy increase of an adiabatic system equals the work done on the system (W_in) and can take forms such as electrical work or shaft work.
• A battery example shows that electrical work done on an adiabatic system equals the increase in the system’s energy.
• In adiabatic shaft or turbine scenarios, the work transfer is equal to the energy increase: W = ΔE.

Energy Change of a System (AE) and Specific Flows

• Energy change of a system (AE) is the difference between final and initial total energy:
  • \Delta E = E{\text{final}} - E{\text{initial}}\
• If the system undergoes a process with heat transfer, work transfer, and possible mass flow, the energy balance accounts for all these terms.

Energy Balance and Practical Expressions

• For a general system:
  • \Delta E = Q - W + \sum{i}(\dot{m}{i} h{i,out} - \dot{m}{i} h_{i,in})\
• For a closed system (no mass flow): \Delta E = Q - W\

Energy Transfer by Heat, Work, and Mass Flow

• Heat transfer: energy transfer due to temperature difference across boundaries.
• Work transfer: energy transfer due to boundary forces and displacements.
• Mass flow: energy transfer carried by mass entering or leaving the control volume (includes enthalpy terms in many applications).
• In a closed cycle (steady operation returning to the same state), \Delta E = 0 ⇒ Q = W (net heat transfer equals net work transfer for a cycle).

Energy Conversion Efficiencies

• Efficiency is a measure of how effectively a process converts or transfers energy.

• Example: Efficiency of a water heater is the ratio of energy delivered to the house by hot water to the energy supplied to the water heater.

• Heating values of fuels:

  • Heating value of a fuel is the amount of heat released when a unit amount of fuel at room temperature is completely burned and the combustion products are cooled to room temperature.
  • Lower heating value (LHV): when the water leaves as vapor (latent heat of vaporization not recovered).
  • Higher heating value (HHV): when the water in combustion products is condensed and the heat of vaporization is recovered.

• Building and appliance performance metrics:

  • AFUE (Annual Fuel Utilization Efficiency): accounts for combustion efficiency and other losses such as heat losses to unheated areas and start-up/cooldown losses.

Generators, Power Plants, and Efficiencies

• Generator: device that converts mechanical energy to electrical energy.
• Generator efficiency: ratio of electrical power output to mechanical power input.
• Thermal efficiency of a power plant: ratio of net electrical power output to rate of fuel energy input.
• Lighting efficacy: lumens per watt (lm/W).
• Overall efficiency of a power plant: product of component efficiencies (e.g., turbine and generator).
• Example: The overall efficiency of a turbine-generator is the product of turbine and generator efficiencies:
  • \eta{overall} = \eta{turbine} \times \eta_{generator}\
  • If \eta{turbine} = 0.75 and \eta{generator} = 0.97, then \eta_{overall} = 0.75 \times 0.97 = 0.7275 \approx 0.73\

Efficiencies of Mechanical and Electrical Devices

• Mechanical efficiency (pump/motor or turbine): ratio of useful mechanical energy output to mechanical energy input.
• Electrical efficiency (generator or motor): ratio of electrical energy output to electrical energy input.
• Overall efficiency combines mechanical and electrical aspects along with any thermal losses to give the net useful energy transfer.
• In energy system diagrams, overall efficiency reflects the fraction of input energy that ends up as useful output energy (e.g., electrical power delivered).

Practical Implications and Environmental Considerations

• Using energy-efficient appliances reduces overall energy demand and environmental pollutants.
• Combustion of fuel produces:
  • Carbon dioxide (CO₂): contributes to global warming.
  • Nitrogen oxides (NOₓ) and hydrocarbons: contribute to smog.
  • Carbon monoxide (CO): toxic.
  • Sulfur dioxide (SO₂): contributes to acid rain.

Summary of Key Concepts (Chapter 3)

• Forms of energy:
  • Macroscopic: kinetic + potential energy.
  • Microscopic: internal energy (sensible + latent + chemical + nuclear).
• Energy transfer mechanisms:
  • Heat transfer, work transfer, and mass flow.
• First Law of Thermodynamics:
  • Energy can change forms but not be created/destroyed.
  • For adiabatic processes, the net work depends only on initial and final states, with Q = 0.
• Energy balance for a system:
  • AE = \Delta E = \Delta U + \Delta KE + \Delta PE.
  • For stationary systems, AE = AU.
  • General energy balance: \Delta E = Q - W + sum(\dot{m}h) differences across boundaries.
• Sign conventions and path vs state properties:
  • Heat and work are boundary/path functions; energy and state variables are path-independent state functions (with exact differentials).
• Efficiency and energy valuation:
  • Fuel heating values (LHV vs HHV) and AFUE for buildings.
  • Component efficiencies (turbine, generator, pump) determine overall plant efficiency.
  • Thermal efficiency = Net work output / Rate of fuel energy input.
• Real-world relevance:
  • The design and operation of energy systems rely on balancing energy inputs and outputs while optimizing efficiencies to minimize fuel use and environmental impact.

Key Equations (LaTeX)

• Internal energy decomposition:
  U = ext{Sensible} + ext{Latent} + ext{Chemical} + ext{Nuclear}
• Thermal energy:
  ext{Thermal energy} = ext{Sensible} + ext{Latent}
• Total energy change:
  \Delta E = \Delta U + \Delta KE + \Delta PE
• Internal energy change (specific):
  \Delta U = m\,(u2 - u1)
• Kinetic energy change (specific):
  \Delta KE = \tfrac{1}{2}\,m\,(V2^2 - V1^2)
• Potential energy change (specific):
  \Delta PE = m\,g\,(z2 - z1)
• Energy entering vs leaving:
  \Delta E = E{\text{in}} - E{\text{out}}
• Adiabatic relation (Q = 0):
  Q = 0 \Rightarrow W = \Delta E
• First law for adiabatic processes (illustrative):
  W_{\text{in}} = \Delta E = \Delta U + \Delta KE + \Delta PE
• Turbine-generator overall efficiency:
  \eta{overall} = \eta{turbine} \times \eta_{generator}
• Example values:
  \eta{turbine} = 0.75, \quad \eta{generator} = 0.97 \Rightarrow \eta_{overall} = 0.75 \times 0.97 \approx 0.73
• Thermal efficiency of a power plant:
  \eta_{thermal} = \frac{\text{Net electrical energy output}}{\text{Rate of fuel energy input}}
• Sign convention (textual): heat to system and work by system are positive; heat from system and work on system are negative.
• Electrical power:
  P = V I
• Work (boundary interaction, general):
  \text{Work} = \text{Force} \times \text{Displacement}
• Mass flow energy balance (control volume):
  \Delta E = Q - W + \sumi ( \dot{m}i h{i,out} - \dot{m}i h_{i,in} )
• Heat transfer per unit mass vs rate (conceptual):
  Q = \text{heat transfer}, \quad \dot{Q} = \text{heat transfer rate}
• For a closed cycle: \Q = W\