Thermal Machines: Comprehensive Summary of Work and Heat

Energy Exchanges in Thermal Machines

• Operating Principle: The operating principle of thermal machines is fundamentally based on energy exchanges between a working fluid and two parts of the external environment that are held at different temperatures.

• Modes of Exchange: Energy is exchanged in two primary forms:

  • Work ($W$)

  • Heat ($Q$)

• Energy Content Modification: These exchanges directly modify the energy content of the working fluid, defined by the total energy equation:

  • $E_{tot} = E_{macro} + U$

  • The change in energy is expressed as: $\Delta E_{tot} = \Delta E_{macro} + \Delta U = W + Q$

  • For elementary (infinitesimal) exchanges and variations: $dE_{tot} = dE_{macro} + dU = \delta W + \delta Q$

Work Done by a Force and Pressure

• Elementary Work of a Force: The elementary work ($\delta W$) done by a force (\vec{F}) is the energy received by the system during an elementary displacement ($d\vec{l}$):

  • $\delta W(\vec{F}) = \vec{F} \cdot d\vec{l}$

• Pressure Force Equivalence: Any force (\vec{F}_i) applied to a surface area ($A$) can be considered equivalent to a pressure ($P_i$):

  • $\vec{F}_i = \int P_i d\vec{A}$

  • The elementary surface ($dA$) is oriented perpendicular to the surface itself.

Work Done by Pressure Forces in Mechanical Systems

• System Configuration: Consider a fluid inside a cylinder closed by a piston with cross-sectional area ($A$). The system is subject to the pressure of the external environment ($P_e$) and any other external force ($\vec{F}$) applied to the piston.

• Total External Force and Pressure:

  • The intensity of the total external force ($F_{ext}$) is: $F_{ext} = P_e A + F$

  • This is equivalent to a total external pressure: $P_{ext} = \frac{F_{ext}}{A} = P_e + \frac{F}{A}$

• Elementary Work Formula:

  • The external force causes an elementary displacement ($dl$) of the piston in the direction of the force.

  • The work received by the fluid is: $\delta W = F_{ext} dl = P_{ext} A dl$

  • Since $A dl$ represents the change in fluid volume ($dV$), and because the work received and the change in volume have opposite signs, the formula is: $\delta W = -P_{ext} dV$

• Sign Convention for Work:

  • Compression: The fluid receives work (\delta W > 0) while its volume decreases (dV < 0).

  • Expansion: The fluid releases work (\delta W < 0) while its volume increases (dV > 0).

• Total Work Calculation: The work received by the system during a process from state $A$ to state $B$ is the integral of the elementary work:

  • $W_{AB} = \int_{A}^{B} \delta W = -\int_{V_A}^{V_B} P_{ext} dV$

Work in Specific Processes

• Constant External Pressure Process: If $P_{ext} = \text{const.}$, the work is calculated as:

  • $W_{AB} = -P_{ext} \int_{V_A}^{V_B} dV = -P_{ext}(V_B - V_A) = -P_{ext} \Delta V_{AB} = P_{ext}(V_A - V_B)$

• Isochoric Process: During an isochoric process, the volume of the system remains constant:

  • $dV = 0$ and $\Delta V_{AB} = 0$

  • The work exchanged is zero: $\delta W = 0$ and $W_{AB} = 0$

• Reversible Process:

  • Mechanical Equilibrium: Occurs when the internal pressure ($P$) of the system is equal to the external pressure ($P_{ext}$): $P = P_{ext}$.

  • Quasi-static Manner: If the process occurs slowly enough, equilibrium is valid throughout the process. This allows for reversible processes that can lead back to the initial state via the opposite direction.

  • Work Formula for Reversible Process: $\delta W = -P dV$ and $W_{AB} = -\int_{V_A}^{V_B} P dV$, where pressure is a state function $P(V, n, T)$.

• Geometric Interpretation of Work: In a $P(V)$ diagram, the work received by the system during a reversible process corresponds to the opposite of the area under the curve $P(V)$.

Work of Reversible Processes for Ideal Gases

• Reversible Isobaric Process ($P = \text{const.}$):

  • $W_{AB} = -P(V_B - V_A) = P(V_A - V_B)$

  • Using the ideal gas equation of state: $W_{AB} = n_A R T_A - n_B R T_B$

  • For a closed system: $W_{AB} = nR(T_A - T_B) = -nR \Delta T_{AB}$

• Reversible Isothermal Process of a Closed System ($T = \text{const.}$):

  • The elementary work is: $\delta W = -\frac{nRT}{V} dV$

  • Integrating from state $A$ to $B$: $W_{AB} = -nRT \int_{V_A}^{V_B} \frac{dV}{V} = -nRT \ln\left(\frac{V_B}{V_A}\right) = nRT \ln\left(\frac{V_A}{V_B}\right)$

  • Alternative expressions using the equation of state:

    • $W_{AB} = P_A V_A \ln\left(\frac{V_A}{V_B}\right) = P_B V_B \ln\left(\frac{V_A}{V_B}\right)$

    • Since $\frac{V_A}{V_B} = \frac{P_B}{P_A}$ for an isothermal process, then: $W_{AB} = nRT \ln\left(\frac{P_B}{P_A}\right) = P_A V_A \ln\left(\frac{P_B}{P_A}\right) = P_B V_B \ln\left(\frac{P_B}{P_A}\right)$

Nature of Work and Heat as Process-Dependent Quantities

• Path Dependence: Work is not a state function. Transitioning between state $A$ and state $B$ via different processes (e.g., isothermal vs. isobaric followed by isochoric) results in different areas under the curves in a $P(V)$ diagram.

• Mathematical Notation: Elementary work is designated as a differential form $\delta W$, not a total differential $dW$.

Definition and Characteristics of Heat

• Formal Definition: The elementary heat exchanged ($\delta Q$) is defined in equivalence to the work of a pressure force:

  • $\delta Q = T_{ext} \delta S_e$

  • Where $T_{ext}$ is the temperature of the external environment and $\delta S_e$ is the elementary entropy exchanged.

• Total Heat Calculation: For a process from $A$ to $B$:

  • $Q_{AB} = \int_{A}^{B} \delta Q = \int_{A}^{B} T_{ext} \delta S_e$

• Constant External Temperature Process ($T_{ext} = \text{const.}$):

  • $Q_{AB} = T_{ext} \int_{A}^{B} \delta S_e = T_{ext} S_{e, AB}$

• Adiabatic Process: A process with no entropy exchange ($\delta S_e = 0$), which implies no heat exchange: $\delta Q = 0$ and $Q_{AB} = 0$.

Heat in Reversible and Isentropic Processes

• Reversible Process and Thermal Equilibrium: The system temperature ($T$) equals the external temperature ($T_{ext}$).

• Entropy Equation: For a reversible process, no entropy is created ($\delta S_c = 0$). The change in entropy ($dS$) is solely due to the entropy exchanged ($\delta S_e$):

  • $dS = \delta S_c + \delta S_e = \delta S_e$

• Reversible Heat Formula: $\delta Q = T dS$ and $Q_{AB} = \int_{S_A}^{S_B} T dS$.

• Isentropic Process: A process where the entropy of the system remains constant ($dS = 0$, $\Delta S_{AB} = 0$).

  • The heat exchanged is zero ($\delta Q = 0$, $Q_{AB} = 0$).

  • An isentropic process is equivalent to a reversible adiabatic process.

• Geometric Interpretation of Heat: In an entropic diagram ($T(S)$), the heat received during a reversible process corresponds to the area under the curve $T(S)$. Like work, heat depends on the nature of the process and is not a state function.

• Reversible Isothermal Heat ($T = \text{const.}$):

  • $Q_{AB} = T \int_{S_A}^{S_B} dS = T(S_B - S_A) = T \Delta S_{AB}$

  • This confirms that isothermal does not mean adiabatic; energy can still be exchanged as heat.

Heat Capacities and Temperature Change

• Definition: Heat capacity represents the energy required as heat to increase the system's temperature by a specific amount during an isochoric or isobaric process.

• Isochoric Process ($V = \text{const.}$):

  • $\delta Q = C_V dT$

  • $Q_{AB} = \int_{T_A}^{T_B} C_V dT$

  • For an ideal gas where $C_V$ is constant: $Q_{AB} = C_V(T_B - T_A) = C_V \Delta T_{AB}$

• Isobaric Process ($P = \text{const.}$):

  • $\delta Q = C_P dT$

  • $Q_{AB} = \int_{T_A}^{T_B} C_P dT$

  • For an ideal gas where $C_P$ is constant: $Q_{AB} = C_P(T_B - T_A) = C_P \Delta T_{AB}$

• Heat Capacity Relationships:

  • C_P > C_V because at constant pressure, additional energy must be supplied for the expansion of the system to achieve the same temperature increase (\Delta Q_{isobaric} > \Delta Q_{isochoric} for the same $\Delta T$).

  • Heat Capacity Ratio (Gamma): \gamma = \frac{C_P}{C_V} > 1

Scientific Measurements of Heat Capacity

• SI Unit: The units for system heat capacities are $J \cdot K^{-1}$.

• Specific Heat Capacity ($c_V, c_P$): Related to mass ($m$):

  • $C_V = m c_V$ and $C_P = m c_P$

  • Unit: $J \cdot K^{-1} \cdot kg^{-1}$

• Molar Heat Capacity ($c_{V,m}, c_{P,m}$): Related to the amount of substance ($n$):

  • $C_V = n c_{V,m}$ and $C_P = n c_{P,m}$

  • Unit: $J \cdot K^{-1} \cdot mol^{-1}$

Heat Flux (Power)

• Definition: Heat flux or current ($\phi$) is the heat received per unit of time ($dt$), representing power:

  • $\phi = \frac{\delta Q}{dt}$

  • SI Unit: $J \cdot s^{-1} = W$

  • Analogous to electric current: $I = \frac{dq}{dt}$.

Mechanisms of Heat Exchange

• Conduction: Heat transfer through a material.

  • $\phi = \lambda \frac{A}{e} (T_{ext} - T)$

  • Where $\lambda$ is thermal conductivity ($W \cdot m^{-1} \cdot K^{-1}$) and $e$ is thickness.

• Convection: Heat transfer involving fluid movement.

  • $\phi = h A (T_{ext} - T)$

  • Where $h$ is the convection coefficient ($W \cdot m^{-2} \cdot K^{-1}$).

• Radiation: Energy transfer via electromagnetic waves.

  • $\phi = \sigma A [\epsilon T_{ext}^4 - (1 - \alpha) T^4]$

  • Stefan-Boltzmann Constant: $\sigma = 5.67 \times 10^{-8} W \cdot m^{-2} \cdot K^{-4}$

  • Parameters:

    • $\epsilon$: Emissivity of the external environment ($0 \leq \epsilon \leq 1$).

    • $\alpha$: Reflectance of the system ($0 \leq \alpha \leq 1$).