Notes

1 Energy

• The ability to do work
• Characterizes the state of a system
• Potential energy is the energy in storage
• Kinetic energy is the energy is use
• Renewable energy can be harvested again in a short period of time
• Non-renewable energy can be obtained only once in our lifetime

1.1 Forms of Energy

• Mechanical energy
• Elastic energy
• Sound energy
• Thermal energy
• Electric and magnetic energy
• Chemical energy
• Radiation energy
• Nuclear energy

1.2 Units

• $\text{J}$, $\text{kWh}$
• $\text{cal}$, $\text{Cal/kcal}$
• $\text{bbl}$ (barrel of oil)
• $\text{BTU}$ (British Thermal Unit)
• $\text{toe}$ (tons of oil equivalent), $\text{mt}$, $\text{bt}$

$1\text{ kWh}$	$3.6\times 10^6\text{ J}$
$1\text{ toe}$	$42\times 10^9\text{ J}$
$1\text{ Cal}$	$4184\text{ J}$
$1\text{ BTU}$	$1055\text{ J}$
$1\text{ bbl}$	$42\text{ US gallons}\approx 159\text{ L}\approx 1700\text{ kWh heat}\approx 680\text{ kWh electricity}$

$\text{pu}1Cal$ is the same as the energy required to increase $\text{pu}1L$ of water by $\text{pu}{1}^{\circ }C$

1.3 Power

$$P=\frac{E}{t}$$

• $\text{J/s, W, kW, MW, GW, TW}$
• $\text{hp}$ (horse power)

$1\text{ hp}$	$746\text{ W}$

1.4 Chemical Reaction

$$\text{Net Energy Release}=\text{Energy in Bonds Formation}-\text{Energy in Bond Breaking}$$

2 Temperature

• The ability of a substance to release heat to other substances

2.1 Zeroth Law of Thermodynamics

If A and C are in thermal equilibrium with B, then A and $C$ are in thermal equilibrium

2.2 Temperature Scales

$$\begin{array}{rl}{T}_F& =\frac{9}{5}{T}_c+32\\ T_K& =T_C+273.15\end{array}$$

2.3 Ideal Gas Law

$$pV=nRT=N{k}_{b}T$$

$n$	Number of mole
$R$	Gas constant ($\text{pu}8.31Jmol-1K-1$)
$N$	Number of particles
$k_b$	Boltzman’s constant ($\text{pu}1.38E-23JK-1$)

$p\to 0,\frac{pV}{nT}\to R$

Ideal gas is defined as a gas of point-like gas particles with negligible inter-particle force. Since there are no interactions, there are no potential energy.

2.4 Kinetic Gas Theory

$$pV=2N\left(\frac{1}{2}m⟨v_x^2⟩\right)=2N{E}_{kin}$$ $$\begin{array}{rl}pV=\cancel{N}{k}_{b}T& =2\cancel{N}\left(\frac{1}{2}m⟨v_2^2⟩\right)\\ \frac{1}{2}m⟨v_x^2⟩& =\frac{1}{2}{k}_{B}T\end{array}$$ $$\begin{array}{rl}& ⟨v^2⟩=⟨v_x^2⟩+⟨v_y^2⟩+⟨v_z^2⟩=3⟨v_x^2⟩\\ & ⟨E_{kin}⟩=\frac{1}{2}m⟨v^2⟩=\frac{3}{2}{k}_{B}T\end{array}$$

2.4.1 Degree of Freedom

The kinds of kinetic or rotational energy

• Monoatomic gas
  • 3 translations
  • $j=3$
• Diatomic gas
  • 3 translations + 2 rotations
  • $j=5$

2.4.2 Internal Energy

$$U=\frac{j}{2}N{k}_{B}T$$

2.5 Specific Heat Capacity

$$Q=mc\Delta T$$

Gas with same degree of freedom has similar heat capacity

2.6 Latent Heat

$$Q=mL$$

Changes in temperature and phase are accompanied by a change in internal energy, i.e. $\Delta U=Q$

Thermodynamics

Internal Energy

$$\begin{array}{rl}& U=\frac{j}{2}nRT\\ & \Delta U=\frac{j}{2}nR\Delta T\end{array}$$

Isothermal

$$W_{\text{on}}=-nRT\ln \left(\frac{V_b}{V_a}\right)$$

• Expansion: $V_b>V_a$ and $W_{\text{on}}<0$
• Compression: $V_b<V_a$ and $W_{\text{on}}>0$
• $Q=-W$
  • Since $\Delta U=Q+W$ and $\Delta T=0$ implies $\Delta U=0$

Isochoric

Isobaric

Entropy

$$\Delta S=\Delta S_1+\Delta S_2=\Delta Q\left(\frac{1}{T_2}-\frac{1}{T_1}\right)$$

• $\Delta S=0$ for a reversible process
• $\Delta S>0$ for an irreversible process

Carnot Engine

1. Isothermal expansion
2. Adiabatic expansion
3. Isothermal compression
4. Adiabatic compression

Engine Efficiency

$$e_{\text{any engine}}\le 1-\frac{T_c}{T_h}$$

Heat Pump

$$\begin{array}{rl}& {\text{COP}}_{\text{cool}}=\frac{Q_c}{W}=\frac{Q_c}{Q_h-Q_c}\le \frac{T_c}{T_h-T_c}\\ & {\text{COP}}_{\text{heat}}=\frac{Q_h}{W}=\frac{Q_h}{Q_h-Q_c}\le \frac{T_h}{T_h-T_c}\end{array}$$

All $Q$ are magnitude
$\text{COP}$ is larger if $\Delta T$ is smaller, i.e. refrigerators work best in cold environment, heat pumps work best in hot environment

Heat Pump Cycle (Cooling)

1. Isochoric cooling (outside, release heat to surround)
2. Adiabatic expansion (valve, temperature decrease)
3. Isochoric heating (indoor, absorb heat from indoor)
4. Adiabatic compression (compressor, temperature increase)

Hydro-Power

Run-of-the-River System

$$\begin{array}{rl}P=\frac{\Delta K}{t}& =\frac{\frac{1}{2}M{v}^2}{t}\\ & =\frac{1}{2}\frac{M}{t}{v}^2\\ & =\frac{1}{2}\left(\frac{pV}{t}\right)v^2\\ & =\frac{1}{2}(\rho Av)v^2\\ & =\frac{1}{2}\rho A{v}^3\end{array}$$

Pumped Sgtorage Stations

• Almost no start time required (fossil fuel power statins require 0.5h)
• Larger than 70% efficiency

Load Factor

$$\frac{\text{Output}}{\text{Possible output}}$$

Three Gorges Dam

• Yangtze river
• FLood control
• Electricity generation
• 32 main water tubines, each with capacity of $\text{pu}700MW$

Tide Pool

$$\frac{(2\rho phA)gh}{A\cdot \text{pu}6.2hours}=\frac{2\rho g{h}^2}{\text{pu}6.2hours}$$

Consider center of mass

Wind Energy

Wind is created by uneven surface heating

$$\begin{array}{rl}{E}_{\text{kin}}& =\frac{1}{2}m{v}^2\\ & =\frac{1}{2}(\rho Avt)v^2\\ & =\frac{1}{2}\rho At{v}^3\\ & \\ P& =\frac{\frac{1}{2}m{v}^2}{t}\\ & =\frac{1}{2}\rho A{v}^3\end{array}$$

Betz Limit

Max efficiency

$$V_{\text{out}}=\frac{1}{3}{V}_{\text{in}}$$