Energy Analysis of Closed Systems

Learning Objectives and Preamble

• Moving Boundary Work: Examine boundary work (also known as $\rho dV$ work) in the context of applications such as automotive engines and compressors.
• First Law of Thermodynamics: Identify the first law for closed systems, characterized by a fixed mass.
• Energy Balance: Develop a general energy balance specific to closed systems.
• Specific Heat Definitions: Define specific heat at constant volume ($c_v$) and specific heat at constant pressure ($c_p$).
• Ideal Gas Relationships: Relate specific heats to the calculation of changes in internal energy ($u$) and enthalpy ($h$) for ideal gases.
• Incompressible Substances: Describe substances that are incompressible (solids and liquids) and determine changes in their internal energy and enthalpy.
• Problem Solving: Solve energy balance problems for closed (fixed mass) systems.

Moving Boundary Work ($PdV$ Work)

• Definition: Moving boundary work, or $PdV$ work, refers to the expansion and compression work observed in a piston-cylinder device.
• Differential Work: The work associated with a moving boundary is called boundary work. The gas performs a differential amount of work denoted as $\delta W_b$.
• Mathematical Derivation:     * $\delta W_b = F \, d s = P A \, d s = P \, d v$     * The total boundary work is the integral: $W_b = \int_{1}^{2} P \, dV \, [kJ]$
• Sign Convention for Work:     * Expansion: W_b > 0     * Compression: W_b < 0
• P-V Diagram Representation:     * The area under the process curve on a Pressure-Volume ($P-V$) diagram is equal in magnitude to the work done during a quasi-equilibrium expansion or compression process of a closed system.     * Boundary work is path-dependent: The work done depends on the specific path followed between the initial and final states, not just the states themselves.
• Net Work in a Cycle: The net work done during a thermodynamic cycle is the difference between the work done by the system and the work done on the system, represented by the area enclosed by the cycle curve on a $P-V$ diagram.

Boundary Work for Specific Processes

• Constant Volume (Isochoric) Process:     * In a constant volume process, $dV = 0$.     * $W_b = \int_{1}^{2} P \times 0 \, dV = 0$
• Constant Pressure (Isobaric) Process:     * In a constant pressure process, the pressure $P$ is constant and can be moved outside the integral.     * $W_b = \int_{1}^{2} P \, dV = P(V_2 - V_1)$
• Isothermal Compression of an Ideal Gas:     * At a constant temperature, the ideal gas law is $PV = mRT = C$, where $C$ is a constant.     * Therefore, $P = \frac{C}{V}$.     * $W_b = \int_{1}^{2} P \, dV = \int_{1}^{2} \frac{C}{V} \, dV = C \ln\left(\frac{V_2}{V_1}\right)$     * Applying state 1 values: $W_b = P_1 V_1 \ln\left(\frac{V_2}{V_1}\right)$

Polytropic Processes

• Process Definition: During actual expansion and compression processes of gases, the relationship between pressure and volume is often described by the equation $P = CV^{-n}$, where $C$ and $n$ are constants.
• Integration for Work:     * $W_b = \int_{1}^{2} P \, dV = \int_{1}^{2} CV^{-n} \, dV = \frac{P_2 V_2 - P_1 V_1}{1 - n}$
• Polytropic Process in Ideal Gases ($PV = mRT$):     * The work expression for an ideal gas becomes $W_b = \frac{mR(T_2 - T_1)}{1 - n}$, provided that $n \neq 1$.
• Special Case of $n = 1$:     * When $n = 1$, the process is isothermal.     * $W_b = \int_{1}^{2} P \, dV = \int_{1}^{2} CV^{-1} \, dV = PV \ln\left(\frac{V_2}{V_1}\right)$

Energy Balance for Closed Systems

• General Energy Balance Principle: For any system undergoing any process, the energy balance is stated as:     * $E_{in} - E_{out} = \Delta E_{system} \, [kJ]$     * $E_{in} - E_{out}$ represents the net energy transfer by heat, work, and mass.     * $\Delta E_{system}$ represents the change in internal, kinetic, potential, and other forms of energy.
• Rate Form of Energy Balance:     * $\dot{E}<em>{in} - \dot{E}</em>{out} = \frac{dE_{system}}{dt} \, [kW]$     * $\dot{E}<em>{in} - \dot{E}</em>{out}$ is the rate of net energy transfer.     * $\frac{dE_{system}}{dt}$ is the rate of change in total energy.
• Relation Between Total and Rate Quantities:     * $Q = \dot{Q} \Delta t$     * $W = \dot{W} \Delta t$     * $\Delta E = \left(\frac{dE}{dt}\right) \Delta t \, [kJ]$
• Sign Convention and First Law Formula:     * Positive (+) for heat input ($Q_{in}$) and work output ($W_{out}$).     * Negative (-) for heat output ($Q_{out}$) and work input ($W_{in}$).     * Standard equation: $Q_{net,in} - W_{net,out} = \Delta E_{system}$     * Simplified: $Q - W = \Delta E$     * Where $Q = Q_{net,in} = Q_{in} - Q_{out}$     * Where $W = W_{net,out} = W_{out} - Win$
• Empirical Nature of the First Law: The first law of thermodynamics cannot be proven mathematically. However, no known process in nature has ever violated it, which serves as sufficient proof of its validity.

Constant-Pressure Process Example Analysis

• General Analysis: For a closed system undergoing a quasi-equilibrium constant-pressure process, assuming heat $Q$ is into the system and work $W$ is from the system:     * $Q - W = \Delta U + \Delta KE + \Delta PE$     * Assume $\Delta KE = 0$ and $\Delta PE = 0$.     * The total work is split into other work and boundary work: $W = W_{other} + W_b$.     * $Q - W_{other} - W_b = U_2 - U_1$     * Substituting boundary work $W_b = P_0(V_2 - V_1)$, where $P_0 = P_1 = P_2$:     * $Q - W_{other} - P_0(V2 - V1) = U_2 - U_1$     * $Q - W_{other} = (U_2 + P_2 V_2) - (U_1 + P_1 V_1)$
• Introduction of Enthalpy ($H$):     * Recognizing that $H = U + PV$:     * $Q - W_{other} = H_2 - H_1 \, [kJ]$
• Key Conclusion: For a constant-pressure expansion or compression process, the relationship is $\Delta U + W_b = \Delta H$.

Specific Heats

• General Definition: Specific heat is the energy required to raise the temperature of a unit mass of a substance by one degree.
• Specific Heat at Constant Volume ($c_v$): The energy required to raise the temperature of a unit mass of a substance by one degree while maintaining a constant volume.     * Definition: $c_v = \left(\frac{\partial u}{\partial T}\right)_v$     * This represents the change in internal energy with temperature at constant volume.
• Specific Heat at Constant Pressure ($c_p$): The energy required to raise the temperature of a unit mass of a substance by one degree while maintaining a constant pressure.     * Definition: $c_p = \left(\frac{\partial h}{\partial T}\right)_p$     * This represents the change in enthalpy with temperature at constant pressure.
• Comparison: $c_p$ is always greater than $c_v$. This is because at constant pressure, the system is allowed to expand, requiring additional energy to perform expansion work.
• Properties and Units:     * $c_v$ and $c_p$ are thermodynamic properties.     * Common units are $kJ/kg \cdot ^{\circ}C$ or $kJ/kg \cdot K$.     * $c_v$ is related to internal energy changes, while $c_p$ is related to enthalpy changes.

Ideal Gases

• Ideal Gas Definition: A gas whose temperature, pressure, and specific volume are related by the equation of state $PV = RT$.
• Internal Energy dependency: Both analytically and experimentally, it is demonstrated that for an ideal gas, internal energy is a function of temperature only: $u = u(T)$.
• Enthalpy dependency: Using the definition $h = u + PV$ and substituting $PV = RT$, we get $h = u + RT$. This implies enthalpy is also a function of temperature only: $h = h(T)$.
• Temperature Effects: For ideal gases, $u, h, c_v,$ and $c_p$ vary with temperature only:     * $u = u(T); \, h = h(T); \, c_v = c_v(T); \, c_p = c_p(T)$
• Differential Changes:     * $du = c_v(T) \, dT$     * $dh = c_p(T) \, dT$
• Process Changes (State 1 to State 2):     * $\Delta u = u_2 - u_1 = \int_{1}^{2} c_v(T) \, dT \, [kJ/kg]$     * $\Delta h = h_2 - h_1 = \int_{1}^{2} c_p(T) \, dT \, [kJ/kg]$
• Ideal-Gas Specific Heats: Real gases at low pressures approach ideal-gas behavior. Their specific heats depend only on temperature and are called zero-pressure specific heats ($c_{p0}, c_{v0}$).

Calculation Methods for Ideal Gas Properties

There are three primary ways to calculate $\Delta u$ and $\Delta h$:

1. Tabulated Data: Using indices of $u$ and $h$ data (e.g., Table A-2c). This is the easiest and most accurate method.     * $\Delta u = u_2 - u_1$     * $\Delta h = h_2 - h_1$
2. Specific Heat Functions: Integrating $c_v$ or $c_p$ as a function of temperature. This is very accurate.     * $\Delta u = \int_{1}^{2} c_v(T) \, dT$     * $\Delta h = \int_{1}^{2} c_p(T) \, dT$
3. Average Specific Heats: Using constant specific heats at an average temperature value. This is reasonably accurate if the temperature interval ($\Delta T$) is small.     * $\Delta u \cong c_{v,avg}(T_2 - T_1) \, [kJ/kg]$     * $\Delta h \cong c_{p,avg}(T_2 - T_1) \, [kJ/kg]$

Specific Heat Relations for Ideal Gases

• Relationship between $c_p$ and $c_v$:     * Starting with $h = u + RT$ and differentiating: $dh = du + R \, dT$     * Substituting definitions: $c_p \, dT = c_v \, dT + R \, dT$     * $c_p = c_v + R \, [kJ/kg \cdot K]$
• Molar Specific Heats:     * $\bar{c}_p = \bar{c}_v + R_u \, [kJ/kmol \cdot K]$     * Where $R$ is the gas constant and $R_u$ is the universal gas constant.
• Specific Heat Ratio ($k$):     * Defined as $k = \frac{c_p}{c_v}$.     * For air at room temperature, $k = 1.4$.

Solids and Liquids (Incompressible Substances)

• Definition: An incompressible substance is one whose specific volume (or density) remains constant during a process.
• Approximation: Solids and liquids can be treated as incompressible without significant sacrifice in accuracy.
• Specific Heat Characteristics:     * For incompressible substances, the constant-volume and constant-pressure specific heats are identical ($c_p = c_v = c$).     * Specific heats depend on temperature only: $du = c_v \, dT = c(T) \, dT$.
• Internal Energy Changes:     * $\Delta u = u_2 - u_1 = \int_{1}^{2} c(T) \, dT \, [kJ/kg]$     * For small temperature intervals: $\Delta u \cong c_{avg}(T_2 - T_1) \, [kJ/kg]$.
• Enthalpy Changes:     * Starting with $h = u + PV$ and differentiating with constant specific volume ($v = const, dv = 0$):     * $dh = du + d(PV) = du + V \, dP + P \, dv$     * $dh = du + V \, dP$     * Integrating: $\Delta h = \Delta u + V \Delta P \cong c_{avg} \Delta T + V \Delta P \, [kJ/kg]$.

Applications for Incompressible Substances

• Solids: The term $V \Delta P$ is insignificant. Therefore, $\Delta h = \Delta u \cong c_{avg} \Delta T$.
• Liquids Cases:     1. Constant Pressure Processes: (e.g., heaters where $\Delta P = 0$). Here, $\Delta h = \Delta u \cong c_{avg} \Delta T$.     2. Constant Temperature Processes: (e.g., pumps where $\Delta T = 0$). Here, $\Delta h = V \Delta P$.

References

• Y. A. Çengel, J. M. Cimbala, R. H. Turner. Fundamentals of Thermal-Fluid Sciences. 5th ed. McGraw-Hill, 2017.
• ASHRAE Handbook of Fundamentals SI version. Atlanta, GA: American Society of Heating, Refrigerating, and Air-Conditioning Engineers, Inc., 1993.
• ASHRAE Handbook of Fundamentals SI version. Atlanta, GA: American Society of Heating, Refrigerating, and Air-Conditioning Engineers, Inc., 1994.