GATE Marine Engineering Thermodynamics and Marine Engineering Flashcards

24 sample cards from the Thermodynamics and Marine Engineering deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. Are the Kelvin-Planck and Clausius statements equivalent?

   Yes. They are two equivalent expressions of the Second Law: a violation of one implies a violation of the other. This is proved by showing that a hypothetical device violating one statement could be coupled to produce a violation of the other.

2. Define a reversible process and list conditions for reversibility.

   A reversible process can be reversed so that both the system and surroundings return to their initial states with no net change. Conditions: quasi-static (infinitesimally slow), no friction, no unrestrained expansion, no heat transfer across a finite temperature difference, no mixing/dissipative effects.

3. Define an irreversible process and give common causes of irreversibility.

   An irreversible process cannot be reversed without leaving permanent changes in the surroundings. Causes include friction, unrestrained (free) expansion, heat transfer through a finite temperature difference, mixing, throttling, inelastic deformation, and chemical reactions.

4. State the Carnot Theorem (a corollary of the Second Law).

   No heat engine operating between two given thermal reservoirs can be more efficient than a reversible (Carnot) engine operating between the same two reservoirs. Hence $\eta_{irrev} < \eta_{rev}$.

5. State the corollary of Carnot's theorem regarding all reversible engines.

   All reversible engines operating between the same two thermal reservoirs have the same efficiency, independent of the working substance. Their efficiency depends only on the reservoir temperatures.

6. How is the thermodynamic (absolute) temperature scale defined using a reversible engine?

   For a reversible engine the ratio of heat transfers equals the ratio of absolute temperatures: $$\frac{Q_1}{Q_2} = \frac{T_1}{T_2}$$ This defines the Kelvin (absolute) temperature scale, independent of any thermometric substance.

7. List the four reversible processes that make up the Carnot cycle.

   (1) Reversible isothermal heat addition at $T_H$, (2) reversible adiabatic (isentropic) expansion, (3) reversible isothermal heat rejection at $T_L$, and (4) reversible adiabatic (isentropic) compression back to the start.

8. Write the thermal efficiency of a Carnot heat engine.

   $$\eta_{Carnot} = 1 - \frac{Q_L}{Q_H} = 1 - \frac{T_L}{T_H}$$ where temperatures are absolute (Kelvin). It is the maximum possible efficiency between reservoirs at $T_H$ and $T_L$.

9. Write the coefficient of performance (COP) of a Carnot refrigerator.

   $$\text{COP}_{ref} = \frac{Q_L}{W_{net}} = \frac{Q_L}{Q_H - Q_L} = \frac{T_L}{T_H - T_L}$$ where $Q_L$ is heat removed from the cold space.

10. Write the coefficient of performance (COP) of a Carnot heat pump.

    $$\text{COP}_{hp} = \frac{Q_H}{W_{net}} = \frac{Q_H}{Q_H - Q_L} = \frac{T_H}{T_H - T_L}$$ Also, $\text{COP}_{hp} = \text{COP}_{ref} + 1$.

11. State the Clausius Inequality.

    For any cyclic process, $$\oint \frac{\delta Q}{T} \leq 0$$ Equality holds for a reversible cycle; the inequality ($<0$) holds for an irreversible cycle. $T$ is the absolute temperature of the boundary where $\delta Q$ crosses.

12. How is entropy defined from the Clausius inequality?

    Since $\oint (\delta Q/T)_{rev} = 0$, the quantity $(\delta Q/T)_{rev}$ is an exact differential of a property called entropy: $$dS = \left(\frac{\delta Q}{T}\right)_{rev}$$ Entropy is a point function with units $\mathrm{J/K}$.

13. What is the relation between entropy change and heat for reversible vs irreversible processes?

    $$dS \geq \frac{\delta Q}{T}$$ Equality ($dS = \delta Q/T$) for a reversible process; inequality ($dS > \delta Q/T$) for an irreversible process. The difference is entropy generated by irreversibility.

14. State the principle of increase of entropy for an isolated system.

    For an isolated system (no heat or work crossing the boundary) entropy can never decrease: $$dS_{iso} \geq 0$$ It increases for irreversible processes and stays constant for reversible ones, defining the direction of spontaneous change.

15. Write the entropy change of an ideal gas in terms of temperature and volume.

    $$\Delta S = m\left( c_v \ln\frac{T_2}{T_1} + R \ln\frac{V_2}{V_1} \right)$$

16. Write the entropy change of an ideal gas in terms of temperature and pressure.

    $$\Delta S = m\left( c_p \ln\frac{T_2}{T_1} - R \ln\frac{p_2}{p_1} \right)$$

17. Write the entropy change of an ideal gas in terms of pressure and volume.

    $$\Delta S = m\left( c_v \ln\frac{p_2}{p_1} + c_p \ln\frac{V_2}{V_1} \right)$$

18. What is the entropy change of an ideal gas during a reversible isothermal process?

    With $T_2 = T_1$: $$\Delta S = mR \ln\frac{V_2}{V_1} = mR \ln\frac{p_1}{p_2} = \frac{Q}{T}$$

19. What is the entropy change in a reversible adiabatic process, and what is such a process called?

    Since $\delta Q = 0$ and the process is reversible, $\Delta S = 0$. The process is isentropic (constant entropy).

20. Write the two Tds (Gibbs) equations relating entropy to other properties.

    $$T\,dS = dU + p\,dV \quad\text{(first Tds equation)}$$ $$T\,dS = dH - V\,dp \quad\text{(second Tds equation)}$$ These hold for any simple compressible substance.

21. Compare heat and work added in isothermal vs adiabatic expansion of an ideal gas between the same volume ratio.

    Isothermal: $T$ constant, $\Delta U=0$, all heat added becomes work, $Q=W=mRT\ln(V_2/V_1)$. Adiabatic: $Q=0$, work is done at the expense of internal energy so temperature falls, $W = -\Delta U = mc_v(T_1-T_2)$. For the same volume ratio, isothermal work exceeds adiabatic work.

22. Relate the polytropic index $n$ to whether heat is added or rejected during gas expansion (with $1<n<\gamma$).

    For expansion with $1<n<\gamma$, the factor $\dfrac{\gamma-n}{\gamma-1}>0$, so $Q$ and $W$ have the same sign: heat is added to the gas while it does positive work. The specific heat $c_n = c_v\dfrac{\gamma-n}{1-n}$ is negative in this range.

23. What is a perpetual motion machine of the second kind (PMM2)?

    A PMM2 is a hypothetical device that produces work continuously while exchanging heat with only a single reservoir, i.e. with 100% efficiency. It violates the Kelvin-Planck statement of the Second Law and is impossible to build.

24. Define a thermal reservoir and distinguish a source from a sink.

    A thermal reservoir is a body with very large heat capacity that can supply or absorb finite heat without any change in its temperature. A source supplies heat (e.g., furnace at $T_H$); a sink absorbs heat (e.g., atmosphere at $T_L$).

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