2.2 First Law of Thermodynamics

2.2 First Law of Thermodynamics (AMeE0202)

1. Conservation of Mass and Energy

Conservation of Mass (Mass Balance)

• Principle: Mass is neither created nor destroyed in a system.

• General Form: $\frac{dm_{CV}}{dt} = \sum \dot{m}_{in} - \sum \dot{m}_{out}$ where $m_{CV}$ is the mass within the control volume.

• For Steady-Flow Process (mass in control volume constant): $\sum \dot{m}_{in} = \sum \dot{m}_{out}$

Conservation of Energy (First Law of Thermodynamics)

• Principle: Energy can neither be created nor destroyed; it can only change forms or be transferred.

2. Internal Energy and Enthalpy

Internal Energy ($U$)

• The total energy stored within a system's molecules.

• Sum of microscopic kinetic and potential energies of the molecules.

• An extensive property (kJ).

• Specific internal energy: $u = U/m$ (kJ/kg).

Enthalpy ($H$)

• A composite property defined for convenience, especially in flow processes.

• $H = U + PV$

• An extensive property (kJ).

• Specific enthalpy: $h = u + Pv$ (kJ/kg).

• For ideal gases: $\Delta h = c_p \Delta T$ and $\Delta u = c_v \Delta T$.

3. Specific Heat

Definition

• The amount of heat required to raise the temperature of a unit mass of a substance by one degree.

Types

1. Specific Heat at Constant Volume ($c_v$):

   • For a constant-volume process.

   • Relates to internal energy change: $\Delta u = \int_{1}^{2} c_v dT \quad \text{or for constant } c_v: \quad \Delta u = c_v \Delta T$

   • $c_v = \left( \frac{\partial u}{\partial T} \right)_v$

2. Specific Heat at Constant Pressure ($c_p$):

   • For a constant-pressure process.

   • Relates to enthalpy change: $\Delta h = \int_{1}^{2} c_p dT \quad \text{or for constant } c_p: \quad \Delta h = c_p \Delta T$

   • $c_p = \left( \frac{\partial h}{\partial T} \right)_P$

Relationship for Ideal Gases

$c_p - c_v = R$

$k = \frac{c_p}{c_v}$ where $k$ is the specific heat ratio (isentropic index).

4. Heat and Work Transfer

Heat Transfer ($Q$)

• Energy transfer due to a temperature difference between the system and its surroundings.

• Sign Convention: Heat transfer into the system is positive ($Q > 0$). Heat transfer out of the system is negative ($Q < 0$).

• Path function (depends on process).

Work Transfer ($W$)

• Energy transfer due to a force acting through a distance.

• Sign Convention: Work done by the system on the surroundings is positive ($W > 0$). Work done on the system is negative ($W < 0$).

• Path function (depends on process).

Boundary Work (Moving Boundary Work)

• Work associated with the expansion or compression of a gas (or fluid) in a piston-cylinder device.

• $\delta W_{boundary} = P dV$ $W_{boundary} = \int_{1}^{2} P dV$

• For a constant-pressure process: $W = P (V_2 - V_1)$

• For a polytropic process ($PV^n = \text{constant}$): $W = \frac{P_2 V_2 - P_1 V_1}{1 - n} \quad (n \ne 1)$

5. First Law for Closed Systems

• General Form (Energy Balance): $\Delta E_{system} = Q_{in} - W_{out}$ where $\Delta E_{system} = \Delta U + \Delta KE + \Delta PE$.

• For stationary systems (no changes in kinetic or potential energy): $Q - W = \Delta U$ or $Q_{net,in} - W_{net,out} = U_2 - U_1$ where $Q_{net,in} = Q_{in} - Q_{out}$ and $W_{net,out} = W_{out} - W_{in}$.

6. Basic Thermodynamic Processes

Isothermal Process (Constant Temperature, $T = \text{constant}$)

• For an ideal gas: $PV = \text{constant}$.

• $\Delta U = 0$ and $\Delta H = 0$ for ideal gases (since $\Delta T = 0$).

• $Q = W$ (from first law, since $\Delta U = 0$).

• Boundary work for ideal gas: $W = P_1 V_1 \ln \left( \frac{V_2}{V_1} \right) = mRT \ln \left( \frac{V_2}{V_1} \right)$

Isobaric Process (Constant Pressure, $P = \text{constant}$)

• $W = P (V_2 - V_1)$

• $\Delta H = Q$ (from first law for a closed system with $P = \text{constant}$).

• For ideal gases: $\Delta H = m c_p \Delta T$.

Isochoric Process (Constant Volume, $V = \text{constant}$)

• $W = 0$ (no boundary work).

• $Q = \Delta U$ (from first law).

• For ideal gases: $\Delta U = m c_v \Delta T$.

Adiabatic Process (No Heat Transfer, $Q = 0$)

• For a reversible adiabatic process (isentropic process):

  • $PV^k = \text{constant}$ (for ideal gas with constant $c_p, c_v$).

  • $TV^{k-1} = \text{constant}$.

  • $\frac{T_2}{T_1} = \left( \frac{P_2}{P_1} \right)^{(k-1)/k}$

• From first law: $-\Delta U = W$ (work is done at the expense of internal energy).

Polytropic Process

• A generalized process following $PV^n = \text{constant}$.

• $n$ is the polytropic index.

  • $n = 0$: Isobaric process.

  • $n = 1$: Isothermal process (ideal gas).

  • $n = k$: Isentropic (reversible adiabatic) process.

  • $n = \infty$: Isochoric process.

• Boundary work (for $n \ne 1$): $W = \frac{P_2 V_2 - P_1 V_1}{1 - n}$

• Heat transfer can be found from first law: $Q = \Delta U + W$.

7. First Law for Open Systems (Control Volume Analysis)

Steady-Flow Energy Equation (SFEE)

• For a steady-state, steady-flow process (properties at any point within the CV are constant with time, and mass flow rates are constant): $\dot{Q}_{in} - \dot{W}_{out} = \sum_{out} \dot{m} \left( h + \frac{V^2}{2} + gz \right) - \sum_{in} \dot{m} \left( h + \frac{V^2}{2} + gz \right)$

• For a single-stream, steady-flow device (one inlet, one outlet): $\dot{Q}_{in} - \dot{W}_{out} = \dot{m} \left[ (h_2 - h_1) + \frac{V_2^2 - V_1^2}{2} + g(z_2 - z_1) \right]$

• Often, changes in kinetic and potential energy are negligible: $\dot{Q}_{in} - \dot{W}_{out} = \dot{m} (h_2 - h_1)$

Applications of SFEE

• Nozzles/Diffusers: $W=0$, often $Q \approx 0$, $h_1 + \frac{V_1^2}{2} = h_2 + \frac{V_2^2}{2}$.

• Turbines/Compressors/Pumps: Often $Q \approx 0$, $\dot{W}_{out} = \dot{m} (h_1 - h_2)$ (turbine), $\dot{W}_{in} = \dot{m} (h_2 - h_1)$ (compressor/pump).

• Throttling Valves: $W=0$, $Q \approx 0$, $\Delta KE \approx 0$, $\Delta PE \approx 0$. Thus, $h_1 = h_2$ (isenthalpic process).

• Heat Exchangers: $W=0$, often $\Delta KE \approx 0$, $\Delta PE \approx 0$. Energy balance: $\dot{m}_h (h_{h,in} - h_{h,out}) = \dot{m}_c (h_{c,out} - h_{c,in})$.

8. Unsteady (Transient) Flow Applications

• For processes where properties within the control volume change with time (e.g., filling or emptying a tank).

• General Energy Balance for a Control Volume: $\frac{dE_{CV}}{dt} = \dot{Q}_{in} - \dot{W}_{out} + \sum_{in} \dot{m}_i \left( h_i + \frac{V_i^2}{2} + gz_i \right) - \sum_{out} \dot{m}_e \left( h_e + \frac{V_e^2}{2} + gz_e \right)$ where $E_{CV} = U_{CV} + KE_{CV} + PE_{CV}$.

• For a uniform-flow process (properties inside the CV are uniform at any instant): $(m_2 u_2 - m_1 u_1)_{CV} = Q - W + \sum m_i h_i - \sum m_e h_e$ (neglecting kinetic and potential energy changes of the CV contents and flow streams).