Chapter 10

Question 1

What does the First Law of Thermodynamics state for a closed system?

Answer 1

The change in internal energy equals the work done on the system plus the heat added to the system (ΔE = ΔW + ΔQ).

Question 2

What is the infinitesimal work done by a gas during expansion?

Answer 2

dW = -p dV.

Question 3

What is the per-unit-mass form of the First Law of Thermodynamics?

Answer 3

de = -p d(1/ρ) + δq.

Question 4

What is the ideal gas law in terms of density?

Answer 4

p = ρRT / MW.

Question 5

What does MW represent in the ideal gas law?

Answer 5

The molecular weight of the gas.

Question 6

What is the formula for internal energy of a calorically perfect gas?

Answer 6

e = c_v T.

Question 7

What is c_v?

Answer 7

Specific heat at constant volume.

Question 8

How is enthalpy defined for a system?

Answer 8

H = E + pV.

Question 9

How is specific enthalpy defined per unit mass?

Answer 9

h = e + p/ρ.

Question 10

What is the enthalpy of an ideal gas in terms of temperature?

Answer 10

h = c_p T.

Question 11

What is the relation between c_p and c_v?

Answer 11

c_p = c_v + R/MW.

Question 12

What is the definition of the specific heat ratio γ?

Answer 12

γ = c_p / c_v.

Question 13

What is the approximate value of γ for air?

Answer 13

γ = 1.4.

Question 14

What is an adiabatic process?

Answer 14

A process in which no heat is transferred (ΔQ = 0).

Question 15

What is a reversible process?

Answer 15

A process that occurs slowly enough to maintain thermodynamic equilibrium and can be reversed.

Question 16

What is the temperature-density relation for an isentropic ideal gas process?

Answer 16

T/T₀ = (ρ/ρ₀)^(γ−1).

Question 17

What is the temperature-pressure relation for an isentropic ideal gas process?

Answer 17

T/T₀ = (p/p₀)^((γ−1)/γ).

Question 18

What makes shock waves irreversible?

Answer 18

They generate entropy and destroy isentropic behavior.

Question 19

When do isentropic relations fail?

Answer 19

In the presence of shock waves.

Question 20

What is the entropy change for reversible heat transfer?

Answer 20

dS = δQ/T.

Question 21

What is the per-unit-mass entropy relation?

Answer 21

ds = δq/T.

Question 22

What is the fundamental thermodynamic identity involving entropy?

Answer 22

T ds = de + p d(1/ρ).

Question 23

What is an isentropic process?

Answer 23

A reversible adiabatic process with constant entropy.

Question 24

What is the total energy per unit mass of a moving fluid?

Answer 24

e + u²/2.

Question 25

What is stagnation enthalpy?

Answer 25

h₀ = h + u²/2.

Question 26

When is stagnation enthalpy conserved?

Answer 26

Along streamlines in steady

Question 27

What does Bernoulli’s law fail in compressible flow?

Answer 27

Because density is not constant.

Question 28

What replaces Bernoulli’s law in compressible flow?

Answer 28

Conservation of stagnation enthalpy.

Question 29

What is the mass conservation equation for compressible flow?

Answer 29

Dρ/Dt + ρ(∇·u) = 0.

Question 30

What is the momentum equation for compressible flow?

Answer 30

Du/Dt + (∇p)/ρ = g.

Question 31

When is entropy conserved in a flow?

Answer 31

Outside boundary layers and in the absence of shocks.

Question 32

What is the condition for entropy conservation in steady flow?

Answer 32

Entropy is constant along streamlines.

Question 33

What are the perturbed flow variables in sound wave analysis?

Answer 33

ρ = ρ₀ + ρ'

Question 34

What assumption makes the sound wave equations linear?

Answer 34

Primed quantities are much smaller than base quantities (small perturbations).

Question 35

What is the thermodynamic relation between pressure and density perturbations?

Answer 35

p' = (∂p/∂ρ)_s ρ'.

Question 36

What is the definition of the square of the sound speed?

Answer 36

c² = (∂p/∂ρ)_s.

Question 37

What is the sound speed in a perfect gas?

Answer 37

c = √(γRT/MW).

Question 38

What physical phenomenon does the sound speed represent?

Answer 38

The speed of propagation of small pressure and density disturbances.

Question 39

What kind of equation governs sound waves in gases?

Answer 39

The wave equation.

Question 40

What is the wave equation for density perturbations?

Answer 40

∂²ρ'/∂t² − c₀²∇²ρ' = 0.

Question 41

Exercise 1: Calculate c_v and c_p for air with γ = 1.4.

Answer 41

Let R = R_specific. Then c_v = R/(γ−1) = R/0.4 and c_p = γR/(γ−1) = 1.4R/0.4 = 3.5R.

Question 42

Exercise 2: Give an example of a reversible process.

Answer 42

Slow compression or expansion of a gas using a frictionless piston with nearly equal pressure on both sides.

Question 43

Exercise 2: Give an example of an irreversible process.

Answer 43

Free expansion of a gas or expansion with shock waves and turbulence.

Question 44

Exercise 3: What is the functional form of entropy for a perfect gas?

Answer 44

s(T,ρ) = c_v ln(T) − R ln(ρ) + constant

Question 45

Exercise 4: Compute the sound speed of air at 293 K (γ = 1.4).

Answer 45

c ≈ √(γRT) ≈ √(1.4 × 287 × 293) ≈ 343 m/s.