MEC2405 Week 3

Overview

  • Topic: Ideal cycles for energy systems (ESs) and TESTA calculations in MEC2405, Week 3 (S2 2024).
  • Core idea: Use idealized (quasistatic) cycles composed of simple q-processes to analyze energy systems. These idealizations provide upper-bound efficiency estimates; real processes are typically worse due to irreversibilities and finite rates.
  • Key ideas
    • State changes are assumed quasistatic, so intermediate states are equilibrium states and obey the State Postulate.
    • Any ES cycle is modeled as a sequence of simple q-processes, where only one intensive property remains constant at a time. Important simple q-processes: isobaric, isochoric, isothermal, isentropic, isenthalpic.
    • Cycles can be represented in multiple 2D state diagrams (P–v, T–s, etc.). A closed loop in one diagram corresponds to a closed loop in all others, with the same cycle name (e.g., Otto Cycle).
  • Applications: Petrol engines (Otto Cycle), Diesel engines, Stirling engines, Brayton cycles (gas power plants), Rankine cycles (steam/vapor-liquid plants), Vapour-Liquid cycles (refrigeration/heat pumps).
  • Tool: TESTA app for property calculations and state diagrams; used via a PC PC Model (VL systems) or PG Model (Ideal-Gas systems).

Fundamental concepts and notation

  • State Postulate (two independent intensive properties determine the state of a simple compressible system).
  • Quasistatic process: infinitesimally slow so the system remains in equilibrium at all times.
  • Internal energy change in ideal-gas processes is tied to temperature, and in isentropic processes, entropy is constant.
  • Adiabatic process: no heat transfer between system and environment (Q = 0). Not restricted to quasistatic; an adiabatic process in a cycle can be non-quasistatic, but in ideal q-process analysis we model some steps as quasistatic adiabatic (isentropic).

Idealization and simple q-processes (definitions)

  • Isobaric (constant pressure): P = const.
  • Isochoric (constant volume): V = const.
  • Isothermal (constant temperature): T = const.
  • Isentropic (constant entropy): S = const. For an ideal gas, this implies p v^γ = const and T v^{γ−1} = const, where γ = cp/cv.
  • Isenthalpic (constant enthalpy): H = const. Ideal throttles are modeled as isenthalpic (no heat, no work, no KE/PE changes).h<em>2=h</em>1h<em>2=h</em>1

Otto Cycle: closed-cycle ideal gas engine (example)

  • State sequence (as described in the transcript):
    • 0 → 1: isobaric process (P = const)
    • 1 → 2: isentropic compression (s = const) → rapid compression with negligible heat transfer
    • 2 → 3: isochoric pressurization (V = const) → heating at constant volume
    • 3 → 4: isentropic expansion (s = const)
    • 4 → 1: isochoric depressurization (V = const)
  • Naming: This idealized cycle is named Otto Cycle (after Nicolaus Otto, spark-ignition engine).
  • Observations about representations:
    • A cycle drawn on any state diagram (P–v, T–s, etc.) represents the same physical cycle; the loop closes when returning to the initial state.
    • In the Otto cycle, the two isobaric segments appear as horizontal in the P–v diagram but are often neglected in the T–s or other diagrams for simplified first-cut analyses since they contribute less to the cycle’s efficiency in the idealized model.
    • If a more accurate upper-bound estimate is needed, more segments can be added (more simple q-processes or a smooth curve), increasing computational complexity (spreadsheets or MATLAB/Python).
  • Isentropic processes in an ideal gas: compression raises temperature; expansion lowers temperature, with no heat transfer (Q ≈ 0 for the isentropic steps). See details below in the isentropes section.
  • Practical implications: The cycle illustrates the relation between heat addition, work output, and the roles of pressure-volume and temperature-entropy relationships in idealized engines.

Isentropes for ideal gases (conceptual derivation, key results)

  • Relationship basics for an ideal gas (IG):

    • The entropy S for an ideal gas is a function of T and P (or T and v).
    • For an isentropic process (S = const), the following commonly used relationships hold (γ = cp/cv):
    • pvextγ=extconstantp v^{ ext{γ}} = ext{constant}
    • Tvextγ−1=extconstantT v^{ ext{γ}-1} = ext{constant}
    • p1−extγTextγ=extconstantp^{1- ext{γ}} T^{ ext{γ}} = ext{constant}

  • Implications of the isentrope shapes in P–v and T–s diagrams:

    • Along a single isentrope in a P–v diagram, p and v are constrained by the above relation, so the curve is not linear and generally steeper than isotherms.
    • In a T–s diagram, an isentrope is a vertical line (s = const) by definition; higher entropy shifts isentropes diagonally in P–v terms, but on T–s the s coordinate is fixed for the isentrope.

  • Temperature changes during isentropic steps (IG):

    • Isentropic compression → temperature increases (volume decreases and pressure increases, following p v^{γ} = constant).
    • Isentropic expansion → temperature decreases (volume increases and pressure decreases).

  • Why these happen (qualitative):

    • During compression, work is done on the gas. In an adiabatic (no heat transfer) and isentropic process, all that input energy goes into increasing the internal energy, hence temperature rises.
    • During expansion, gas does work on surroundings; with no heat transfer, internal energy decreases, lowering the temperature.

  • Isentropic vs isothermal contrast:

    • Isentropic: work changes temperature via energy transfer, with entropy fixed.
    • Isothermal: temperature fixed; internal energy for an IG is a function of T only, so ΔU = 0, and heat transfer balances the work: if compressed (Win > 0), heat must flow out (Q < 0) to keep T constant; if expanded (Wout < 0), heat flows in (Q > 0).

  • Practical visual: represent a cyclic process with isothermal and isentropic segments (e.g., an idealized “rectangular” cycle on a T–s diagram: alternating isothermal (horizontal in T–s) and isentropic (vertical in T–s) legs).

Isobaric and isochoric features in state diagrams

  • In an ideal gas IMAGINE a cycle with isobars and isochores:
    • Isobaric segments in P–v diagram are horizontal lines (since P = const, v changes).
    • In the corresponding P–v diagram, as isobars rise (higher P), the curve sits higher in the diagram.
    • Isochores (V = const) are vertical lines in the P–v diagram.
  • In a T–s diagram:
    • Isothermal segments are horizontal (T = const).
    • Isentropic segments are vertical (s = const).
  • For VL (vapour–liquid) systems, state diagrams (P–v, T–s) show a VL dome; above the VL dome, the state is superheated gas; within the two-phase region, temperature remains constant during phase change along an isobar.
  • Compressed-liquid approximation (VL):
    • The properties of a compressed liquid can be approximated by the saturated liquid properties at the same temperature. This is a common simplification near the compressed-liquid region.

Some important ideal cycles (non-exhaustive)

  • Otto Cycle (gas IG working fluid; spark-ignition engine) – described above.
  • Diesel Cycle (compression-ignition engines):
    • In the Diesel cycle, fuel ignition is modeled as an isobaric process (heat addition at constant pressure) instead of isochoric heat addition in the Otto cycle.
  • Stirling Engine Cycle:
    • Transfers heat between hot and cold sides through external heat source; gas undergoes two isochoric processes (de- and re-pressurization) when moving between hot and cold sides; regenerator (not shown in some animations) stores heat between the two sides and is central to efficiency.
  • Brayton Cycle (Gas Power Plants):
    • Open-flow cycle for air (or other gas) through compressor, combustion chamber, and turbine.
    • Idealization: isentropic compression, isobaric heating (heat addition), isentropic expansion, and isobaric cooling (heat rejection).
    • State diagrams: p–v and T–s for air treated as an ideal gas; real plant includes continuous flow and heat exchange with the environment.
  • Rankine Cycle (Vapour-Liquid Power Plants):
    • Working fluid is VL; isentropic pumping (liquid); isobaric heating in boiler to become steam; isentropic expansion in turbine; isobaric condensation back to liquid; pump and turbine are vertical lines in the P–v diagram; boiler and condenser lie along isobars.
  • Vapour–Liquid Compression Cycle (Refrigeration/Heat Pumps):
    • Reverse of Rankine: isentropic compressor (gas) increases pressure; high-temperature isobaric condenser rejects heat; low-temperature isobaric evaporator absorbs heat; instead of a turbine, an isenthalpic throttle valve reduces pressure (state 3 → 4), entering a two-phase mixture.
    • This throttle's isenthalpic line is visible in the P–v diagram as the two-phase drop between states; h stays constant across the throttle: h<em>3=h</em>4h<em>3 = h</em>4
  • Open vs Closed cycles:
    • Closed cycles: working fluid returns to initial state after one cycle (e.g., Otto).
    • Open cycles: working fluid flows through devices (e.g., Brayton, Rankine) with mass exchange; not all state variables return to a single initial state within a single device loop.
  • The role of the regenerator (Stirling):
    • A regenerator stores heat between the hot and cold sides of the engine to improve efficiency; conceptually, it reduces external heat transfer requirements and increases overall efficiency.

Idealized components and devices in ES cycles

  • Ideal Isentropic Turbines, Compressors, and Pumps
    • Quasistatic state change for each unit mass.
    • Device is perfectly adiabatic; hence each kg of fluid changes state isentropically (S = const).
    • KE and PE changes are neglected.
    • Isentropic compression is used for compressors; isentropic expansion for turbines; pumps compress liquids isentropically.
    • In this unit, only isentropic turbines are encountered; hydroelectric turbines exist but are not typical in ESs involving heat-work conversions.
  • Ideal Isobaric Heat Exchangers (Boilers/Condenser):
    • Two fluids exchange energy as heat with zero pressure drop; no mechanical work; KE/PE changes neglected.
    • Each kg of fluid undergoes an isobaric q-process as it passes through the exchanger.
  • Ideal Isenthalpic Throttle Valves (Throttle):
    • No heat transfer, no work; KE/PE changes neglected.
    • Enthalpy remains constant for each kg of fluid flowing through the throttle: h<em>2=h</em>1h<em>2 = h</em>1

TESTA: Using the TESTA app for property calculations and state diagrams

  • Accessing TESTA:
    • TESTA can be accessed in the Study Area of the online textbook by Bhattacharjee, via the Pearson portal. Register with the Pearson portal if required.
    • Steps: Open TESTA → TESTapps tab → choose the model (PC Model for VL systems or PG Model for Ideal-Gas systems).
  • Two example problem types described in the transcript
    • Example 1: Find properties of water at 3.5 MPa saturated vapor, then superheat by 5 °C.
    • Steps:
      • In TESTA, select PC Model and PC TESTapp (HTML5).
      • Choose SI units; fluid = H2O.
      • Check p1 and x1 to enter P = 3.5 MPa and x1 = 1.00 (saturated vapor).
      • Read saturation temperature Tsat at 3.5 MPa (from the app): approximately 515.8 °C.
      • To add superheat: uncheck x1 and check T1; input T1 = 515.8 + 5 = 520.8 °C; the app updates to the desired properties.
      • View graphics: select p–v state diagram to visualize the state.
    • Example 2: Find properties of air at 1 MPa superheated by 303 K.
    • Steps:
      • In TESTA, select PG Model; PG TESTapp (HTML5).
      • SI units; fluid = Air.
      • Check p1 and T1 checkboxes; input P = 1 MPa, T = 303 K; app outputs properties.
  • Using TESTA in practice
    • The app emphasizes that a state is specified by two intensive properties and that you should choose the appropriate fluid model (PC for VL systems, PG for IG systems).
    • The PDF accompanying TESTA shows additional examples of simple two-state, single-step q-processes; videos and examples reinforce hand calculations and the use of the app for verification.
  • Practical notes
    • The TESTA interface includes a System States panel with various models; for MEC2405, PC Model (VL) and PG Model (IG) cover the needed cases.
    • The State Panel allows selection of units (SI) and the desired working fluid (e.g., H2O for VL; Air for IG).
    • The Graphics Panel can plot the p–v diagram (and other diagrams) to visualize the cycle states.

Connections, implications, and reflections

  • Relationship between work, heat, and entropy:
    • In isentropic segments, heat transfer is negligible; work changes lead to temperature changes in IGs; entropy remains constant.
    • In isothermal segments, temperature is fixed; any heat transfer is balanced by equal and opposite work, keeping ΔU = 0 for ideal gases; heat transfer is directly tied to entropy change.
  • Practical relevance:
    • Real ES devices deviate from the idealized cycles due to irreversibilities, heat losses, finite-rate processes, friction, and non-ideal fluid behavior; ideal cycles provide upper bounds and a baseline for comparison.
    • The choice of state diagrams (P–v, T–s, etc.) helps visualize how the same thermodynamic process looks in different representations and why certain segments contribute more to efficiency.
  • Ethical, philosophical, or practical implications:
    • Understanding ideal cycles informs design choices for energy efficiency and emissions reductions in engines and power plants; this has real-world consequences on energy policy, environmental impact, and engineering ethics.

Summary of key formulas and concepts (compact reference)

  • Simple q-processes definitions (IG and VL):
    • Isobaric: P = const
    • Isochoric: V = const
    • Isothermal: T = const
    • Isentropic: S = const → for IG: pvextγ=extconstant,extandTvextγ−1=extconstantp v^{ ext{γ}} = ext{constant}, ext{ and } T v^{ ext{γ}-1} = ext{constant}
    • Isenthalpic: H = const → for throttles: h<em>2=h</em>1h<em>2 = h</em>1
  • Ideal-gas isentropic relations (γ = cp/cv):
    • pvextγ=extconstantp v^{ ext{γ}} = ext{constant}
    • Tvextγ−1=extconstantT v^{ ext{γ}-1} = ext{constant}
    • p1−extγTextγ=extconstantp^{1- ext{γ}} T^{ ext{γ}} = ext{constant}
    • Temperature-pressure-volume changes for an IG on an isentrope: rac{T2}{T1} = igg( rac{p2}{p1}igg)^{ rac{ ext{γ}-1}{ ext{γ}}}, \ rac{v2}{v1} = igg( rac{p1}{p2}igg)^{ rac{1}{ ext{γ}}}
  • Enthalpy in throttles: h<em>2=h</em>1h<em>2 = h</em>1
  • Energy/heat relations in isothermal IG processes: for an ideal gas, ΔU=nCVΔT;extifT=extconst,ΔU=0extandQ=WΔU = n C_V ΔT; ext{ if } T= ext{const}, ΔU=0 ext{ and } Q=W in an idealized isothermal step.
  • Internal energy-temperature relationship for IG: U ∝ T (for an ideal gas, U = f(T))

Note on figures and diagrams mentioned

  • P–v diagrams: illustrate pressure vs. specific volume; isentropic segments are not generally vertical; isochoric segments are vertical; isobaric segments are horizontal.
  • T–s diagrams: isentropic steps are vertical lines (S = const); isothermal steps are horizontal lines (T = const).
  • VL diagrams: the VL dome delineates compressed-liquid, saturated-liquid, two-phase, saturated-vapor, and superheated-vapor regions; the compressed-liquid approximation ties VL properties to saturated-liquid values at the same temperature.
  • Isentropes for IGs appear as p–v curves following p v^γ = const; higher entropy moves isentropes outward in the P–v plane.

Appendix: Quick actions in TESTA (practical checklist)

  • Access TESTA via the Pearson portal and register if needed.
  • In TESTA: open TESTapps → choose PC Model (VL) or PG Model (IG).
  • For a VL problem (PC Model):
    • State selection: System States → PC TESTapp (HTML5).
    • SI units; select fluid (e.g., H2O).
    • Use p1 and x1 checkboxes to specify P and quality x1 for saturated state; read Tsat.
    • For a superheated state: switch to T1 input; keep P fixed; enter T1 = Tsat + ΔT.
    • View p–v diagram in Graphics Panel to visualize the state.
  • For an IG problem (PG Model):
    • State selection: System States → PG TESTapp (HTML5).
    • SI units; select fluid (e.g., Air).
    • Input P and T (using p1 and T1 checkboxes) to obtain properties.
  • Use the PDF resources and in-app videos to reinforce understanding of two-state, single-step q-processes and how to perform hand calculations and verify with TESTA.

Endnotes

  • The Week 3 material emphasizes the connection between state diagrams and cycle representations, the rationale for using simple q-processes, and the role of idealizations in building intuition before tackling more detailed, real-world systems.
  • The TESTA app is a practical tool to visualize and verify state properties and to generate isobaric, isochoic, isentropic, and isothermal segments for both VL and IG systems.