Thermodynamics Lecture – Power Cycles, Refrigeration, Transient Analysis & Intro to 2nd Law

Course Logistics / Administrative Remarks

  • Instructor is currently grading the mid-term; solutions will be reviewed in an upcoming lecture once marking is complete.
  • Grades will be posted on Brightspace; arrangements will be made for students to view their marked copies (possible on-campus office hours).
  • Upcoming tutorials will centre on steady-state, steady-flow (SSSF) device analysis (Chapter 4 problems).
  • No quiz next week; lectures continue with Chapter 5 and beyond.

Quick Recap – SSSF Devices (from previous lecture)

  • Devices treated as open systems operating at steady state–steady flow:
    • Heat exchangers (boiler, condenser, recuperators, etc.)
    • Turbines & compressors
    • Pumps (for liquids) / throttling valves
    • Nozzles & diffusers
  • Key idea:
    For each device, if you know its purpose you may safely neglect certain terms in \dot Q,\;\dot W,\;\Delta KE,\;\Delta PE when writing the 1st-law rate form
    \dot Q{cv}+\sum \dot min h{in} = \dot W{cv}+\sum \dot m{out} h{out} \;\Bigl( + \text{KE} + \text{PE terms}\Bigr).
  • Pumps/Compressors: work into device (usually small heat losses, (\Delta KE\approx\Delta PE\approx0)).
  • Turbines: work out of device (adiabatic, (\Delta KE\approx\Delta PE\approx0)).
  • Heat exchangers: large \dot Q, negligible \dot W, often operate at approximately constant pressure.
  • Throttles: h{in}!\approx!h{out} (isenthalpic), large pressure drop, no work.

Basic Rankine (Steam) Power-Plant Cycle

  • Purpose: Convert a positive heat input Q{B} (boiler) into net work output via turbine WT.
  • Working fluid: liquid water / steam.
  • Four fundamental devices composing the closed loop
    1. Pump: 1 → 2 – increases pressure of sub-cooled liquid at almost constant specific volume.
    2. Boiler: 2 → 3 – constant-pressure heat addition; liquid is vaporised to super-heated steam.
    3. Turbine: 3 → 4 – adiabatic expansion; high-pressure steam produces shaft work.
    4. Condenser: 4 → 1 – constant-pressure heat rejection; condenses exhaust to saturated/sub-cooled liquid.
  • Two operating pressures only: low (condenser) and high (boiler/turbine inlet).
  • T–v diagram characteristics
    • Pump stroke ≈ vertical line (tiny \Delta v).
    • Boiler: horizontal rightward path (constant (P_{high})).
    • Turbine: downward/rightward curve to low pressure inside or just outside vapour dome.
    • Condenser: horizontal leftward to compressed-liquid region.
  • Rule of thumb for turbine exhaust: stay near saturated-vapour line (quality x \gtrsim 0.9) to limit blade erosion but avoid excessive condenser load.

Detailed Worked Example – Basic Steam Power Plant

Given

  • \dot m = 10\;\mathrm{kg\,s^{-1}} (water)
  • Pump power input W_P = 200\;\text{kW}
  • Pressures: P1=P4=15\;\text{kPa},\;P2=P3=2\;\text{MPa}
  • Temperatures / qualities:
    T1 = 45^{\circ}\text{C}\; (\text{compressed liquid}),\quad T3 = 300^{\circ}\text{C}\; (\text{super-heated}),\quad x_4 = 0.90

(A) Turbine power
Assumptions: adiabatic, SSSF, negligible KE/PE.
\dot WT = \dot m\,(h3-h4) Values (steam tables): h3=3023.5\;\text{kJ kg}^{-1}, \; h4 = 2361.7\;\text{kJ kg}^{-1} Result \boxed{\dot WT = 6.62\;\text{MW}}

(B) Pump exit temperature
First-law (pump): -\dot WP = \dot m\,(h1-h2) Compressed-liquid approximation h\approx hf(T)
h1 = 188.42\;\text{kJ kg}^{-1}\;@\;45^{\circ}\text{C}\Rightarrow h2 = 208.42\;\text{kJ kg}^{-1}
Interpolate hf(T) ⇒ T2 \approx 49.8^{\circ}\text{C}

(C) Boiler heat input
Heat-exchanger, SSSF, \dot W=0, \Delta KE,\Delta PE\approx0
\dot QB = \dot m\,(h3-h_2) = 10\,(3023.5-208.4) \approx 28.15\;\text{MW}

(D) Observations
Thermal efficiency (ideal) \eta = \dfrac{\dot W{net}}{\dot QB} \approx \tfrac{6.62\;\text{MW}}{28.15\;\text{MW}} \approx 0.235 (≈ 23.5 %).
Real plants add feed-water heaters, reheat stages, etc., to raise (\eta).

Refrigerator (Vapour-Compression) Cycle

  • Goal: Remove heat Q{in} from a low-temperature region (refrigerated space) and reject Q{out} to ambient while consuming work W_{comp}.
  • Key components (ideal single-stage cycle)
    1. Evaporator (low-pressure): liquid–vapour mixture absorbs Q_{in} → becomes super-heated vapour.
    2. Compressor: raises pressure & T of vapour (work input).
    3. Condenser (high-pressure): vapour rejects Q_{out} → saturated liquid.
    4. Throttle / Expansion Valve: isenthalpic drop back to low pressure, producing 2-phase mixture.
  • T–v diagram: two pressure levels; horizontal constant-p heat transfer in evaporator & condenser; vertical-ish compression; throttling follows constant enthalpy line inside dome.
  • Performance metric: Coefficient of performance \text{COP}R = \dfrac{Q{in}}{W_{comp}}.
  • Must select refrigerant with low boiling point at evaporator pressure (e.g., R-134a, ammonia).

Transient-Flow (Unsteady) Analysis (Chapter 4, Topic 4)

When SSSF assumptions break down (e.g., tanks being filled/emptied):

  • Control volume properties change with time but are assumed uniform at any instant (lumped parameter).
  • Inlet/outlet streams: constant (\dot m) and constant state during the process.

Mass conservation over finite interval \Delta t:
m2-m1 = m{in}-m{out}

General 1st-law form (no a priori simplifications):
Q{1\rightarrow2}+m{in}h{in}-m{out}h{out}=\bigl(m2u2-m1u1\bigr)+\bigl(KE+PE\bigr)+W{1\rightarrow2}

Worked Example – Filling a Rigid Ammonia Tank

Data

  • Rigid tank volume V=0.35\;\text{m}^3 initially contains NH(3) at T1=-20^{\circ}\text{C},\;x_1=0.5.
  • Supply line (inlet): T{in}=80^{\circ}\text{C},\;P{in}=800\;\text{kPa} (super-heated).
  • Valve opened until tank reaches P2=600\;\text{kPa} with final quality x2=1 (sat. vapour); tank still rigid (no work).
    Find (a) T2, (b) m{in}, (c) Q_{1\rightarrow2}.

Results

  • (a) Interpolating NH(3) saturation tables: T2=9.2^{\circ}\text{C}.
  • (b) Masses using m=V/v:
    • m1 = 1.12\;\text{kg},\quad m2 = 1.658\;\text{kg}
      ⇒ \boxed{m_{in}=0.537\;\text{kg}}.
  • (c) Energy balance (no shaft work, no outlet stream):
    Q{1\to2}=m2u2-m1u1-m{in}h_{in} = \boxed{548.5\;\text{kJ (heat in)}}

Motivation for the Second Law (Start of Chapter 5)

  • First Law is an energy balance; does not predict direction of processes.
  • Everyday observation: hot coffee cools to room temperature; the reverse (room air spontaneously heating coffee) never happens even though 1st Law would allow it energy-wise.
  • Example tank-and-fan “cycle” from earlier lecture operates only in one direction; reversing violates observed reality.
  • Therefore, a second restriction on processes is required → 2nd Law.

Heat Engines – Definition & Conceptual Example

Definition

  • A heat engine is a closed cycle that
    1. Receives net positive heat Q_H>0 from a high-temperature reservoir (source),
    2. Rejects heat Q_L<0 to a low-temperature reservoir (sink),
    3. Delivers net positive work W_{net}>0.

Thought-experiment engine (single piston, removable mass)

  • Stage 1 (1→2): Add heat Q_H from hot source → gas pressure rises, piston lifts heavy mass ⇒ positive work.
  • Stage 2 (2→1): Remove mass, cool gas against cold sink (heat Q_L out) until piston returns to stops ⇒ small negative work.
  • Net cycle work W{net}=W{1\to2}+W_{2\to1}>0 (area enclosed on P–V diagram).
  • Energy flows from hot to cold; part converted to useful work.
  • Previews later 2nd-Law results: efficiency bound, irreversibility concept, Carnot limits.

Miscellaneous Study Tips & Connections

  • Always start with mass conservation then 1st Law; add 2nd Law constraints where applicable (Ch 5-7).
  • Know how to read property tables (saturated, super-heated, compressed-liquid) and perform linear interpolation.
  • Memorise sign conventions:
    Q{cv}>0 (heat into system), W{cv}>0 (work by system).
  • Compressed-liquid approximation: {u,h,v}{\text{CL}}\approx{uf,hf,vf}_{T} when high-pressure data absent.
  • For ideal gases: u=u(T)=\int cv\,dT, h=h(T)=\int cp\,dT independent of pressure.
  • Typical exam pitfalls: forgetting to convert kJ↔J, dropping sign, neglecting kinetic/potential energy assumptions when invalid.

Suggested Further Resources

  • Brightspace PDF of curated YouTube links for visualisations (turbines, compressors, refrigeration animations, etc.).
  • Re-work tutorial/lecture examples with tables open to gain property-lookup speed.
  • Preview Chapter 5 sections on entropy, reversibility, Carnot cycle before next lecture.