2.3 Applications of the First Law

Energy Analysis using the First Law

Energy balances in systems

The First Law of Thermodynamics says energy is conserved: it can change forms, but it can't be created or destroyed. For any system, the change in internal energy ($\Delta U$) equals the net energy added through heat and work.

For a closed system (no mass crosses the boundary), the energy balance is:

$\Delta U = Q - W$

Here, $Q$ is heat added to the system and $W$ is work done by the system. A piston-cylinder device is the classic example. Pay attention to the sign convention: heat into the system is positive, and work done by the system is positive. Some textbooks flip the work sign, so check which convention your course uses.

For an open system (mass flows in and out), you need to account for the energy carried by that mass. The balance becomes:

$\Delta U = Q - W + \sum m_i h_i - \sum m_e h_e$

where $m_i$ and $m_e$ are the masses entering and exiting, and $h_i$ and $h_e$ are their specific enthalpies. Turbines, compressors, and nozzles are all open systems. The enthalpy terms capture both the internal energy and the flow work ($Pv$) that the fluid carries with it.

For steady-state, steady-flow devices (where conditions don't change over time and $\Delta U = 0$), the energy balance simplifies to:

$Q - W = \sum m_e h_e - \sum m_i h_i$

This simplified form is what you'll use most often for turbines, compressors, and heat exchangers on homework problems.

Heat engines and refrigerators

Heat engines convert thermal energy into mechanical work. They absorb heat $Q_H$ from a hot reservoir, convert part of it to net work $W_{net}$, and reject the remaining heat $Q_C$ to a cold reservoir. Think of internal combustion engines or steam turbines.

Thermal efficiency measures how much of the input heat gets converted to useful work:

$\eta = \frac{W_{net}}{Q_H} = 1 - \frac{Q_C}{Q_H}$

An efficiency of 1 (100%) would mean all the heat becomes work, which never happens in practice.

Refrigerators and heat pumps do the opposite: they use work input to move heat from a cold space to a hot space. The difference between them is what you care about:

• A refrigerator is designed to remove heat from the cold space. Its performance is measured by:

$COP_R = \frac{Q_C}{W_{net}}$

• A heat pump is designed to deliver heat to the warm space. Its performance is measured by:

$COP_{HP} = \frac{Q_H}{W_{net}}$

Notice that $COP_{HP} = COP_R + 1$. This follows directly from the First Law ($Q_H = Q_C + W_{net}$), and it's a useful check on your calculations. COP values are often greater than 1, which is why we don't call them "efficiencies."

Thermodynamic Cycles and Efficiency

Efficiency of thermodynamic cycles

The Carnot cycle is the theoretical best-case scenario: a fully reversible cycle operating between two thermal reservoirs. It consists of four processes:

1. Isothermal expansion (heat absorbed from hot reservoir)
2. Adiabatic expansion (temperature drops to $T_C$)
3. Isothermal compression (heat rejected to cold reservoir)
4. Adiabatic compression (temperature rises back to $T_H$)

Its efficiency depends only on the reservoir temperatures (which must be in absolute units, i.e., Kelvin):

$\eta_{Carnot} = 1 - \frac{T_C}{T_H}$

No real engine operating between the same two temperatures can exceed this efficiency. That's what makes Carnot so important: it sets the upper bound.

The Rankine cycle is the practical vapor power cycle used in most steam power plants. Its four processes are:

1. Isentropic compression in a pump (liquid pressurized)
2. Constant-pressure heat addition in a boiler (liquid becomes steam)
3. Isentropic expansion in a turbine (steam produces work)
4. Constant-pressure heat rejection in a condenser (steam returns to liquid)

Rankine efficiency is:

$\eta_{Rankine} = \frac{W_{net}}{Q_{in}} = \frac{W_t - W_p}{Q_{in}}$

where $W_t$ is turbine work output, $W_p$ is pump work input, and $Q_{in}$ is heat added in the boiler. In practice, the pump work is usually very small compared to the turbine work, so $W_{net} \approx W_t$.

Performance of real-world systems

Real systems always perform worse than their ideal counterparts because of irreversibilities, which include:

• Friction in bearings, seals, and moving parts
• Heat transfer across finite temperature differences in heat exchangers (the larger the temperature gap, the greater the irreversibility)
• Unrestrained expansion, such as flow through throttling valves where pressure drops without producing work

To analyze a real system, follow these steps:

1. Identify the sources of irreversibility and energy loss in the system
2. Apply the First Law energy balance, accounting for those losses
3. Calculate the actual efficiency or COP and compare it to the ideal value

Strategies to improve real-world efficiency:

• Minimize irreversibilities: reduce friction through better lubrication and design, optimize heat exchanger sizing to reduce temperature differences
• Recover waste heat: cogeneration uses rejected heat for heating or industrial processes; regeneration preheats the working fluid using exhaust energy
• Use high-performance materials: advanced alloys and ceramics can withstand higher temperatures, allowing operation closer to Carnot limits

The gap between a system's actual efficiency and its Carnot efficiency tells you how much room there is for improvement. A large gap suggests significant irreversibilities that could potentially be reduced through better engineering.