KZ

Closed Systems

Study Notes on Energy Analysis of Closed Systems

Moving Boundary Work

  • Moving boundary work (also known as boundary work or PdV work) is associated with the expansion or compression of a gas in devices like automotive engines and compressors. [1, 2]

  • During this process, the inner face of the piston moves, hence the name "moving boundary work." [2]

  • While thermodynamics can't precisely calculate the moving boundary work in real-world engines and compressors due to the high piston speeds and non-equilibrium conditions, it can be analysed for quasi-equilibrium processes, where the system remains near equilibrium throughout. [3, 4]

  • Quasi-equilibrium processes are good approximations for real engines with slow piston speeds, and they provide the theoretical maximum work output for engines and minimum work input for compressors. [4]

  • In a quasi-equilibrium process, the differential work (δWb) done as the piston moves a distance ds is given by: δWb = F ds = PA ds = P dV [5] where: F is the force on the piston A is the cross-sectional area of the piston P is the absolute pressure dV is the differential change in volume

  • The total boundary work (Wb) is obtained by integrating this differential work over the volume change from state 1 to state 2: Wb = ∫1 2 P dV [6]

  • This integral requires the relationship between P and V during the process, which is the equation of the process path on a P-V diagram. [6]

  • The magnitude of the work done during a quasi-equilibrium expansion or compression process equals the area under the process curve on a P-V diagram. [7]

  • The net work done during a cycle is the difference between the work done during expansion and the work done during compression. [8]

  • If the relationship between P and V is given as experimental data, the work can be determined graphically by calculating the area under the P-V curve. [9]

Energy Balance for Closed Systems

  • The energy balance for any closed system can be expressed as: Ein - Eout = ΔEsystem [10] where:

  • Ein represents energy transfer into the system

  • Eout represents energy transfer out of the system

  • ΔEsystem represents the change in the system's energy

  • For a closed system undergoing a cycle, the initial and final states are identical, meaning ΔEsystem = 0, simplifying the energy balance to: Ein = Eout [11]

  • As closed systems have no mass flow across their boundaries, their energy balance can be expressed in terms of heat (Q) and work (W) interactions as: Wnet,out = Qnet,in [12] This means the net work output during a cycle equals the net heat input.

Specific Heats

  • Specific heat is the energy needed to raise the temperature of a unit mass of a substance by one degree. [13]

  • There are two main types of specific heat:

  • Specific heat at constant volume (cv): The energy required to raise the temperature while keeping the volume constant. [13]

  • Specific heat at constant pressure (cp): The energy required to raise the temperature while keeping the pressure constant. [13]

  • cp is always greater than cv for a given substance, as at constant pressure, the system expands and requires energy for expansion work. [13]

Internal Energy, Enthalpy, and Specific Heats of Ideal Gases

  • The internal energy (u) of an ideal gas depends solely on its temperature: u = u(T). [14]

  • Similarly, the enthalpy (h) of an ideal gas is also a function of temperature only: h = h(T). [15]

  • As a consequence, the specific heats cv and cp of an ideal gas are also functions of temperature only. [15]

  • This means that at any given temperature, an ideal gas will have specific values for u, h, cv, and cp, regardless of its specific volume or pressure. [15]

  • The changes in internal energy (Δu) and enthalpy (Δh) of an ideal gas can be calculated as: Δu = u2 - u1 = ∫T1 T2 cv(T) dT (kJ/kg) [16] Δh = h2 - h1 = ∫T1 T2 cp(T) dT (kJ/kg) [17] where T1 and T2 are the initial and final temperatures, respectively.

  • Ideal-gas specific heats, also known as zero-pressure specific heats (cp0 and cv0), are specific heats of real gases at low pressures, where they closely approximate ideal gas behaviour. These are usually provided as third-degree polynomials in tables. [17, 18]

  • Over small temperature ranges, specific heat can be approximated as constant, leading to the simplified equations: Δu = cv,avg (T2 - T1) (kJ/kg) [19] Δh = cp,avg (T2 - T1) (kJ/kg) [20] where cv,avg and cp,avg are the average specific heats over the temperature interval.

Specific Heat Relations of Ideal Gases

  • For ideal gases, cp and cv are related by the following equation: cp = cv + R (kJ/kg·K) [21] where R is the gas constant.

  • When specific heats are given on a molar basis, the universal gas constant Ru is used instead of R: cp = cv + Ru (kJ/kmol·K) [22]

  • The specific heat ratio (k) is defined as: k = cp/cv [23]

Internal Energy, Enthalpy, and Specific Heats of Solids and Liquids

  • Incompressible substances have a constant specific volume (or density). [24]

  • Solids and liquids can be approximated as incompressible substances with high accuracy. [24]

  • For incompressible substances: cp = cv = c [25] Therefore, only one specific heat value (c) is needed.

Please note: This is not an exhaustive set of study notes on the topic. Always refer to your textbook and other learning materials for a complete understanding.