Engineering Thermodynamics: Systems, Processes, and First Law Analysis

Law of Conservation of Energy

• Fundamental Definition: The law of conservation of energy states that energy can neither be created nor destroyed; it can only be transformed or converted from one form to another. The total energy content of an isolated system remains constant.

• Modes of Energy Conversion:

  • Electrical energy to mechanical energy (e.g., electric motor).

  • Heat energy to mechanical energy (e.g., internal combustion engine or steam turbine).

  • Chemical energy to heat energy (e.g., combustion of fuel).

Thermodynamic System, Boundary, and Surroundings

• Thermodynamic System: Defined as any specified region in space, quantity of matter, or group of matter within a prescribed boundary that is selected for thermodynamic analysis.

• Surroundings: All matter and space outside the prescribed boundary of the thermodynamic system.

• Universe: The combination of the thermodynamic system and its surroundings taken together:   $\text{Universe} = \text{System} + \text{Surroundings}$

• Boundary: The real or imaginary envelope or surface that encloses the system and separates it from the surroundings. A boundary may be fixed or movable.

• Concrete Example (Gas Cylinder / Tank):

  • System: The gas contained inside the tank.

  • Boundary: The physical surface of the tank container.

  • Surroundings / Universe: The environment and space external to the tank.

Classification of Thermodynamic Systems

• Closed System (Control Mass):

  • Mass Transfer: Strictly zero ($m = \text{constant}$).

  • Energy Transfer: Energy in the form of heat ($Q$) and work ($W$) can cross the boundary into or out of the system.

  • Example: A fixed mass of gas enclosed inside a cylinder fitted with a movable piston. When heat is added from an external source, the gas temperature increases and causes the gas to expand, pushing the piston outward. Work is performed by the expanding gas as heat and work cross the boundary, while gas mass remains constant.

• Open System (Control Volume):

  • Mass Transfer: Mass flows continuously across the system boundary into and out of the system.

  • Energy Transfer: Energy in the form of heat and work flows across the boundary.

  • Control Volume: A fixed region in space defined for thermodynamic analysis of open systems.

  • Example: A centrifugal compressor, steam turbine, or pump where working fluid enters and exits the system alongside energy flow.

• Isolated System:

  • Mass Transfer: No mass transfer across the boundary ($\Delta m = 0$ ).

  • Energy Transfer: No heat or work transfer across the boundary ($Q = 0$, $W = 0$).

  • Example: Liquid stored inside a perfectly insulated thermos flask (thermo flask).

Thermodynamic Properties

• Definition: Any measurable physical quantity in defined units that characterizes the physical state or condition of a system (e.g., pressure, temperature, volume, density).

• Intensive Properties:

  • Definition: Physical properties that are independent of the mass or size of the system.

  • Examples: Temperature ($T$), pressure ($p$), density ($\rho$).

  • Specific Extensive Properties: When any extensive property is divided by mass ($m$), it becomes a specific property and behaves as an intensive property:

  • Specific volume: $v = \frac{V}{m}$

  • Specific internal energy: $u = \frac{U}{m}$

• Extensive Properties:

  • Definition: Physical properties that depend directly on the mass or size of the system.

  • Examples: Total volume ($V$), total internal energy ($U$), total enthalpy ($H$), total kinetic energy.

  • Example Scenario: In a tank containing a mixture of gas A and gas B, the total volume $V$ depends on the combined mass of the gases present.

State, Process, Path, Phase, and Cycle

• State: The complete condition of a system at any given moment, defined uniquely by a set of measured thermodynamic properties (e.g., $(p_1, v_1)$).

• Change of State: The operation in which a system undergoes a change in one or more of its property values.

• Thermodynamic Process: The transformation of a system from an initial equilibrium state to a final equilibrium state along a specified path.

• Path: The continuous sequence of intermediate equilibrium states passed through by a system during a change of state.

• Phase: A quantity of matter that is uniform throughout in chemical composition and physical state (solid, liquid, or gas).

  • Homogeneous System: A system that consists of a single uniform phase.

  • Heterogeneous System: A system that consists of two or more distinct phases.

• Thermodynamic Cycle: A sequence of thermodynamic processes carried out such that the final state coincides exactly with the initial state, restoring all initial property values.

Thermodynamic Equilibrium

• Definition: A system is in a state of thermodynamic equilibrium when no net change occurs in any of its macroscopic properties over time. Achieving complete thermodynamic equilibrium requires three simultaneous conditions:

• Mechanical Equilibrium: Absence of any unbalanced forces within the system and between the system and its surroundings ($\sum F = 0$).

• Chemical Equilibrium: Absence of chemical reactions within the system and absence of mass transfer or concentration gradients across regions.

• Thermal Equilibrium: Absence of temperature gradients within the system and between the system and surroundings when connected via a diathermic wall.

Zeroth Law of Thermodynamics

• Statement: When system A is in thermal equilibrium with system B, and system B is independently in thermal equilibrium with system C, then system A and system C are also in thermal equilibrium with each other.

• Practical Significance: Serves as the fundamental baseline for temperature measurement and thermometer calibration.

Reversible and Quasi-Static Process

• Quasi-Static Process: A process carried out so infinitely slowly that the system remains arbitrarily close to a state of thermodynamic equilibrium at all times.

• Reversible Process: A process that can be reversed along the exact same path without leaving any permanent change or net effect on either the system or the surroundings.

Work Transfer

• Thermodynamic Definition: Work is done by a system if the sole effect on the surroundings could be reduced to the raising of a weight.

• Sign Convention for Work:

  • Work done by the system on the surroundings = Positive ($+W$) (e.g., expanding gas).

  • Work done on the system by the surroundings = Negative ($-W$) (e.g., gas compression).

• Work is a Path Function:

  • Work transfer depends on the specific path followed during a change of state and not merely on the end states.

  • Mechanical work done during a reversible non-flow expansion/compression process corresponds to the area under the process curve on a pressure-volume ($p\text{-}V$) diagram:     $W_{1-2} = \int_{V_1}^{V_2} p\,dV$

  • For a gas changing state from A ($p_1, V_1$) to B ($p_2, V_2$) along different paths (A-1-B, A-2-B, or A-3-B), the area under each curve is different, demonstrating that work is a path function.

Heat Transfer

• Thermodynamic Definition: Heat is energy transfer across a system boundary solely due to a temperature difference between the system and its surroundings.

• Sign Convention for Heat:

  • Heat added into the system = Positive ($+Q$).

  • Heat rejected or leaving out of the system = Negative ($-Q$).

• Heat is a Path Function: Heat transfer depends on the process path traversed between the initial and final states.

First Law of Thermodynamics

• Basis: Founded on the Law of Conservation of Energy.

• Statement for a Cyclic Process: When a thermodynamic system undergoes a complete cyclic process, the cyclic integral of heat transfer is equal to the cyclic integral of work transfer:   $\oint \delta Q = \oint \delta W$

• Statement for a Process (Change of State): When a system undergoes a change of state during a process, part of the heat supplied is converted into work output, and the remainder is stored as internal energy ($U$) within the system:   $Q = W + \Delta U$   $\Delta U = Q - W$   where $\Delta U = U_2 - U_1$ is the change in internal energy, assuming kinetic and potential energy changes are negligible.

• Perpetual Motion Machine of the First Kind (PMM-I):

  • Definition: A hypothetical device that continuously delivers mechanical work output without receiving any heat energy input from an external source.

  • Violation: Violates the First Law of Thermodynamics and is physically impossible to construct ($W \neq 0$ when $Q = 0$).

Classification of Thermal Processes

• Non-Flow Process (Closed System):

  • System mass remains fixed ($m = \text{constant}$).

  • Divided into Reversible Non-Flow Processes and Irreversible Non-Flow Processes.

• Flow Process (Open System):

  • Mass continuously crosses control volume boundaries.

  • Steady Flow Process: Fluid properties and energy flow rate across control volume boundaries do not change with time.

  • Unsteady Flow Process: Fluid properties and energy flow rate vary with time.

Reversible Non-Flow Processes

• Constant Volume (Isochoric) Process:

  • Condition: $V_1 = V_2 = V$ ($dV = 0$).

  • $p\text{-}V\text{-}T$ Relationship: $\frac{p}{T} = \text{Constant} \implies \frac{p_1}{T_1} = \frac{p_2}{T_2}$.

  • Work Transfer: $W_{1-2} = \int_{V_1}^{V_2} p\,dV = 0$

  • Change in Internal Energy: $\Delta U = m C_v (T_2 - T_1)$

  • Heat Transfer: $Q_{1-2} = W_{1-2} + \Delta U = m C_v (T_2 - T_1)$

  • Change in Enthalpy: $\Delta H = m C_p (T_2 - T_1)$

  • Thermal Characteristics: Heating increases pressure and temperature; cooling decreases pressure and temperature.

• Constant Pressure (Isobaric) Process:

  • Condition: $p_1 = p_2 = p$

  • $p\text{-}V\text{-}T$ Relationship: $\frac{V}{T} = \text{Constant} \implies \frac{V_1}{T_1} = \frac{V_2}{T_2}$.

  • Work Transfer: $W_{1-2} = p(V_2 - V_1) = m R (T_2 - T_1)$

  • Change in Internal Energy: $\Delta U = m C_v (T_2 - T_1)$

  • Heat Transfer: $Q_{1-2} = W_{1-2} + \Delta U = m R (T_2 - T_1) + m C_v (T_2 - T_1) = m C_p (T_2 - T_1)$

  • Change in Enthalpy: $\Delta H = m C_p (T_2 - T_1) = Q_{1-2}$

  • Key Property: Heat transfer during a constant pressure process equals the change in enthalpy.

• Constant Temperature (Isothermal) Process:

  • Condition: $T_1 = T_2 = T$

  • $p\text{-}V\text{-}T$ Relationship: $p V = \text{Constant} \implies p_1 V_1 = p_2 V_2$

  • Work Transfer: $W_{1-2} = \int_{V_1}^{V_2} p\,dV = p_1 V_1 \ln\left(\frac{V_2}{V_1}\right) = 2.303 p_1 V_1 \log_{10}\left(\frac{V_2}{V_1}\right)$

  • Change in Internal Energy: $\Delta U = m C_v (T_2 - T_1) = 0$

  • Heat Transfer: $Q_{1-2} = W_{1-2} + \Delta U = W_{1-2} = p_1 V_1 \ln\left(\frac{V_2}{V_1}\right)$

  • Change in Enthalpy: $\Delta H = m C_p (T_2 - T_1) = 0$

• Hyperbolic Process:

  • Condition: $p V = \text{Constant}$

  • Characteristics: For an ideal gas, the hyperbolic process matches the isothermal process.

  • Work Transfer: $W_{1-2} = 2.303 p_1 V_1 \log_{10}\left(\frac{V_2}{V_1}\right)$

  • Change in Internal Energy: $\Delta U = m C_v (T_2 - T_1)$

  • Heat Transfer: $Q_{1-2} = 2.303 p_1 V_1 \log_{10}\left(\frac{V_2}{V_1}\right)$

  • Change in Enthalpy: $\Delta H = m C_p (T_2 - T_1)$

• Reversible Adiabatic (Isentropic) Process:

  • Condition: $Q_{1-2} = 0$ (perfect insulation against heat transfer).

  • $p\text{-}V\text{-}T$ Relationship: $p V^\gamma = \text{Constant} \implies p_1 V_1^\gamma = p_2 V_2^\gamma$, where $\gamma = \frac{C_p}{C_v}$.

  • Work Transfer: $W_{1-2} = \frac{p_1 V_1 - p_2 V_2}{\gamma - 1} = \frac{m R (T_1 - T_2)}{\gamma - 1}$

  • Change in Internal Energy: $\Delta U = m C_v (T_2 - T_1) = -W_{1-2}$

  • Heat Transfer: $Q_{1-2} = 0$

  • Change in Enthalpy: $\Delta H = m C_p (T_2 - T_1)$

• Polytropic Process:

  • Condition: $p V^n = \text{Constant} \implies p_1 V_1^n = p_2 V_2^n$, where $n$ is the polytropic index.

  • Work Transfer: $W_{1-2} = \frac{p_1 V_1 - p_2 V_2}{n - 1} = \frac{m R (T_1 - T_2)}{n - 1}$

  • Change in Internal Energy: $\Delta U = m C_v (T_2 - T_1)$

  • Heat Transfer: $Q_{1-2} = W_{1-2} + \Delta U = \left(\frac{\gamma - n}{\gamma - 1}\right) W_{1-2}$

  • Change in Enthalpy: $\Delta H = m C_p (T_2 - T_1)$

Summary Table of Non-Flow Reversible Processes

Steady Flow Energy Equation (SFEE)

• Continuity (Mass Conservation) Equation:   $\dot{m} = \frac{A_1 V_1}{v_1} = \frac{A_2 V_2}{v_2}$   where $A$ is cross-sectional area, $V$ is fluid velocity, and $v$ is specific volume.

• General SFEE per Unit Mass:   $h_1 + \frac{V_1^2}{2} + g Z_1 + q = h_2 + \frac{V_2^2}{2} + g Z_2 + w$   where $h = u + p v$ is enthalpy, $V$ is fluid velocity, $g Z$ is potential energy per unit mass, $q = \frac{Q}{\dot{m}}$ is heat transfer per unit mass, and $w = \frac{W}{\dot{m}}$ is work transfer per unit mass.

• Work Transfer in a Steady Flow Process:   $dW = -V\,dp \implies W_{1-2} = -\int_{p_1}^{p_2} V\,dp$

Engineering Applications of SFEE

• Steam Boiler (Heat Absorbing System):

  • Function: Generates steam by transferring heat to water at constant pressure.

  • Assumptions: Work transfer $W = 0$, changes in kinetic energy $\Delta KE \approx 0$, and potential energy \Delta PE \approx 0$.\n * **SFEE Simplification**: Q = h_2 - h_1\n\n\n\n* **Evaporator (Heat Absorbing System)**:\n * **Function**: Vaporizes liquid refrigerant by absorbing heat from a surrounding space.\n * **Assumptions**: W = 0$,$\Delta KE \approx 0$,$\Delta PE \approx 0$.

  • SFEE Simplification: $Q = h_2 - h_1$

• Steam Condenser (Heat Rejection System):

  • Function: Condenses exhaust steam to water by rejecting heat to cooling water.

  • Assumptions: $W = 0$, $\Delta KE \approx 0$, $\Delta PE \approx 0$.

  • SFEE Simplification: $-Q = h_2 - h_1 \implies Q = h_1 - h_2$

• Steam Turbine (Work Developing System):

  • Function: Expands high-pressure steam across rotor blades to generate mechanical work output.

  • Assumptions: Adiabatic operation ($Q = 0$) and \Delta PE \approx 0$.\n * **SFEE Simplification**: W = (h_1 - h_2) + \frac{V_1^2 - V_2^2}{2}\n * **When velocity change is negligible**: W = h_1 - h_2\n\n\n\n* **Rotary Compressor (Work Absorbing System)**:\n * **Function**: Increases gas pressure via continuous mechanical work input.\n * **Assumptions**: Work input -W$, adiabatic operation ($Q = 0$),$\Delta PE \approx 0$,$\Delta KE \approx 0$.

  • SFEE Simplification: $-W = h_2 - h_1 \implies W = h_2 - h_1$

• Reciprocating Compressor (Work Absorbing System):

  • Function: Compresses air in a piston cylinder while heat is lost to cylinder walls.

  • SFEE Simplification: $-W = (h_2 - h_1) - Q \implies W = (h_2 - h_1) + Q_{\text{loss}}$

• Steam Nozzle (Velocity Increasing Device):

  • Function: Converts fluid enthalpy drop into kinetic energy to produce a high-velocity steam jet.

  • Assumptions: Adiabatic casing ($Q = 0$), no mechanical work done ($W = 0$), \Delta PE \approx 0$.\n * **SFEE Simplification**: h_1 + \frac{V_1^2}{2} = h_2 + \frac{V_2^2}{2} \implies V_2 = \sqrt{V_1^2 + 2(h_1 - h_2)}\n * **When inlet velocity V_1$is negligible ($V_1 \approx 0)**:\n    V_2 = \sqrt{2(h_1 - h_2)}$$