Application of the First Law of Thermodynamics to Variable

In engineering thermodynamics, real-world gas compression and expansion processes rarely conform perfectly to idealized

In engineering thermodynamics, real-world gas compression and expansion processes rarely conform perfectly to idealized extremes. They are seldom completely adiabatic, nor are they truly isothermal. To bridge the gap between theoretical models and actual physical behavior, engineers rely on the concept of a polytropic process.

A polytropic process is defined by a specific power-law relationship between the absolute pressure $P$ and the volume $V$ of a system:

$$PV^n = C$$

where $C$ is a constant, and $n$ is the polytropic index. The value of $n$ is dictated by the specific working fluid and the nature of heat transfer occurring during the process. By adjusting this index, the polytropic framework elegantly encompasses several standard thermodynamic processes:

  • When $n = 0$, the process is isobaric ($P = C$).
  • When $n = 1$, the process is isothermal ($PV = C$).
  • When $n = k$ (where $k$ is the specific heat ratio), the process is isentropic or adiabatic ($PV^k = C$).
  • As $n \to \infty$, the process becomes isochoric ($V = C$).
    According to the First Law of Thermodynamics, the boundary work $W$ done by a closed system during a quasi-static process is defined by the integral of pressure with respect to volume:

$$W = \int_{V_1}^{V_2} P , dV$$

For a system obeying the polytropic relation $PV^n = C$, we can express pressure as $P = C V^{-n}$. Substituting this into the integral yields two distinct scenarios:

  1. For $n \neq 1$:
    $$W = \int_{V_1}^{V_2} C V^{-n} , dV = C \left[ \frac{V^{1-n}}{1-n} \right]_{V_1}^{V_2} = \frac{P_2 V_2 - P_1 V_1}{1-n}$$
    To align with standard engineering sign conventions, this is typically rearranged as:
    $$W = \frac{P_1 V_1 - P_2 V_2}{n-1}$$

  2. For $n = 1$ (Isothermal Process):
    $$W = \int_{V_1}^{V_2} \frac{C}{V} , dV = P_1 V_1 \ln \frac{V_2}{V_1}$$

In practical engineering calculations, if the initial state $(P_1, V_1)$ and the compression ratio $\varepsilon = V_1/V_2$ are known, these work expressions can be significantly streamlined for rapid evaluation.

Applying the First Law of Thermodynamics

The general mathematical expression of the First Law for a closed system is:

$$Q = \Delta U + W$$

where $Q$ represents the heat transferred into the system, $\Delta U$ is the change in internal energy, and $W$ is the boundary work done by the system.

1. Change in Internal Energy ($\Delta U$)

For an ideal gas, internal energy is a function solely of temperature. Regardless of the path taken, the change in internal energy depends only on the temperature difference between the initial and final states:

$$\Delta U = m c_v (T_2 - T_1)$$

where $m$ is the mass of the gas and $c_v$ is the specific heat at constant volume.

2. Heat Transfer ($Q$)

By substituting the polytropic work $W$ and the internal energy change $\Delta U$ into the First Law equation, we obtain the expression for heat transfer during the process:

$$Q = m c_v (T_2 - T_1) + \frac{P_1 V_1 - P_2 V_2}{n-1}$$

Utilizing the ideal gas law ($PV = mRT$), this equation can be rewritten entirely in terms of temperature:

$$Q = m c_v (T_2 - T_1) + \frac{mR(T_1 - T_2)}{n-1}$$

By factoring out common terms and applying the thermodynamic relationships $R = c_p - c_v$ and $k = c_p/c_v$, we can derive a highly simplified formula for heat transfer in a polytropic process:

$$Q = m \frac{k-n}{k-1} c_v (T_2 - T_1)$$

3. Physical Significance

The derived equation $Q = m \frac{k-n}{k-1} c_v (T_2 - T_1)$ provides profound insight into the relationship between the polytropic index $n$ and heat exchange:

  • If $n = k$: $Q = 0$, confirming the process is adiabatic with no heat transfer.
  • **If $1 < n < k$**: During a compression process (where $T_2 > T_1$), $Q$ becomes negative, indicating that the system rejects heat to its surroundings. This scenario is highly prevalent in actual reciprocating compressors, where cooling jackets or intercoolers are used to extract heat and lower the discharge temperature.
  • If $n = 1$: The process is isothermal. During isothermal compression, all the mechanical work added to the system is immediately dissipated as heat to maintain a constant temperature.

Comprehensive Calculation Example

Scenario:
1 kg of air (modeled as an ideal gas with $k=1.4$, $c_v=0.718 \text{ kJ/(kg·K)}$, and $R=0.287 \text{ kJ/(kg·K)}$) undergoes a polytropic compression. The initial state is $P_1 = 100 \text{ kPa}$ and $T_1 = 300 \text{ K}$. The compression ratio is $\varepsilon = V_1/V_2 = 8$, and the polytropic index is $n = 1.3$.

Calculation Steps:

  1. Determine the final temperature $T_2$:
    Using the polytropic temperature-volume relation $\frac{T_2}{T_1} = \left( \frac{V_1}{V_2} \right)^{n-1}$:
    $$T_2 = 300 \times (8)^{1.3-1} = 300 \times 8^{0.3} \approx 300 \times 1.866 = 559.8 \text{ K}$$

  2. Calculate the boundary work $W$:
    $$W = \frac{mR(T_1 - T_2)}{n-1} = \frac{1 \times 0.287 \times (300 - 559.8)}{1.3 - 1} \approx \frac{-74.56}{0.3} = -248.5 \text{ kJ}$$
    (The negative sign indicates that work is done on the system during compression.)

  3. Calculate the change in internal energy $\Delta U$:
    $$\Delta U = m c_v (T_2 - T_1) = 1 \times 0.718 \times (559.8 - 300) \approx 186.6 \text{ kJ}$$

  4. Calculate the heat transfer $Q$:
    Applying the First Law ($Q = \Delta U + W$):
    $$Q = 186.6 + (-248.5) = -61.9 \text{ kJ}$$
    The negative result signifies that the system rejects $61.9 \text{ kJ}$ of heat to the surroundings during compression.

Conclusion

The polytropic process serves as a vital bridge in engineering thermodynamics, connecting idealized theoretical models with actual physical operations. By introducing the polytropic index $n$, engineers can flexibly characterize gas behavior across varying pressure and volume conditions while accounting for simultaneous heat exchange.

When applying the First Law of Thermodynamics to analyze these processes, a logical methodology should be followed:

  1. Identify the polytropic index $n$ and the known initial and final state parameters.
  2. Utilize the relation $PV^n = C$ or corresponding temperature-volume relations to solve for any missing state variables.
  3. Calculate the boundary work $W$ and the internal energy change $\Delta U$ independently.
  4. Apply $Q = \Delta U + W$ to determine the heat transfer, and verify the physical validity of the result by examining the relationship between $n$ and $k$.