4.5 Energy balance for reactive systems

Energy balance for reactive systems

Energy balances for reactive systems account for the energy released or absorbed when chemical bonds break and form during reactions. Without this, you can't accurately predict temperatures, heat transfer needs, or cooling requirements in a reactor. This topic builds on the non-reactive energy balances you've already seen by adding a reaction enthalpy term.

Formulating energy balance equations

The general energy balance for a reactive system adds a heat of reaction term to the familiar terms for heat transfer ($Q$) and work ($W$). This term captures the energy difference between products and reactants due to the chemical transformation itself.

The reaction enthalpy ($\Delta H_{rxn}$) quantifies that energy. You calculate it from standard enthalpies of formation:

$\Delta H_{rxn} = \sum \Delta H_f^\circ (\text{products}) - \sum \Delta H_f^\circ (\text{reactants})$

• If $\Delta H_{rxn} < 0$, the reaction is exothermic (releases heat). This acts as an energy source inside the system.
• If $\Delta H_{rxn} > 0$, the reaction is endothermic (absorbs heat). This acts as an energy sink.

How the reaction term enters the balance depends on the reactor type:

• Batch reactors: The heat of reaction appears as a source/sink term alongside $Q$ and $W$. You're tracking the total energy change of a fixed mass over time.
• Continuous flow reactors: The reaction enthalpy is embedded in the enthalpy difference between outlet and inlet streams. You compare $\dot{H}_{out}$ to $\dot{H}_{in}$, and the reaction's contribution shows up in the changed composition and temperature of the outlet.

Incorporating phase changes

Reactions sometimes cause or coincide with phase changes (e.g., water produced as vapor in combustion, or a solid reactant melting). These phase changes carry their own energy cost or contribution through latent heat, and you must include them in the balance.

• Vaporization (liquid → gas) requires energy input (endothermic). The latent heat of vaporization ($\Delta H_{vap}$) gets added as an energy sink.
• Condensation (gas → liquid) releases energy (exothermic), acting as a source.
• Melting/freezing follow the same logic with the latent heat of fusion ($\Delta H_{fus}$).

If you skip a phase change that actually occurs, your energy balance will be significantly off. For example, if a combustion reaction produces water and you assume it exits as liquid when it actually exits as vapor, you'll overestimate the useful heat output by roughly 44 kJ per mole of water (the latent heat of vaporization at 25°C).

Solving energy balance problems

Problem-solving approach

A systematic method keeps you from missing terms or making sign errors. Follow these steps:

1. Define the system boundaries. Decide what's inside your control volume (the reactor, a section of pipe, etc.) and what crosses the boundary (streams, heat, work).
2. Specify initial and final states. Pin down the temperature, pressure, composition, and phase of every stream or batch at the start and end.
3. Gather thermodynamic data. Look up specific heat capacities ($C_p$), latent heats, and standard enthalpies of formation ($\Delta H_f^\circ$) from thermodynamic tables or references.
4. Write the energy balance equation. For a steady-state flow system, this typically looks like: $Q - W = \dot{H}_{out} - \dot{H}_{in}$ where the enthalpy terms include sensible heat changes, latent heat contributions, and the heat of reaction.
5. Substitute known values and solve for the unknown (temperature, $Q$, $W$, etc.). Keep careful track of signs: heat added to the system is positive $Q$; heat removed is negative.

Example problems

These are the types of problems you should be comfortable setting up:

• Batch reactor temperature rise: Given an exothermic $\Delta H_{rxn}$ and the system's total heat capacity ($nC_p$), find the adiabatic temperature change: $\Delta T = \frac{-\xi \Delta H_{rxn}}{nC_p}$, where $\xi$ is the extent of reaction.
• Cooling requirement for a flow reactor: For an exothermic reaction at constant temperature, the heat removal rate $\dot{Q}$ must equal the rate of heat generation from the reaction.
• Combustion efficiency: Compare the actual useful energy output to the fuel's heating value to get efficiency. For instance, if methane (heating value ≈ 890 kJ/mol) is burned and only 750 kJ/mol of useful heat is recovered, the efficiency is about 84%.
• Distillation column energy balance: Account for latent heat of vaporization in the reboiler, sensible heat changes across stages, and condenser duty. The reaction term may be zero here, but the phase-change terms dominate.

Energy requirements of reactions

Combustion processes

Combustion is the most common exothermic reaction in engineering. A fuel (hydrocarbon, hydrogen, biomass) reacts with oxygen, releasing heat. The heating value of a fuel tells you how much energy you get per unit mass or mole under standard conditions.

There are two heating values to know:

• Higher heating value (HHV): Assumes water in the products condenses to liquid, so you recover the latent heat. For methane, HHV ≈ 890 kJ/mol.
• Lower heating value (LHV): Assumes water stays as vapor. For methane, LHV ≈ 802 kJ/mol.

Which one you use depends on whether your system actually condenses the water.

Efficiency depends on several factors:

• Air-to-fuel ratio: Too little air causes incomplete combustion (wasted fuel, CO formation). Too much air means you're heating excess nitrogen, which carries energy out the exhaust.
• Heat recovery: Capturing waste heat from exhaust gases (e.g., through economizers or preheaters) improves overall efficiency.

Chemical synthesis reactions

Many industrial synthesis reactions are endothermic and need energy input. The Haber-Bosch process for ammonia ($N_2 + 3H_2 \rightarrow 2NH_3$) is actually exothermic ($\Delta H_{rxn} \approx -92$ kJ/mol), but it requires high temperatures (400–500°C) and pressures (150–300 atm) to achieve reasonable reaction rates, so the overall process still consumes significant energy.

To evaluate energy efficiency of a synthesis process:

• Compare the total energy input (heating, compression work, separation) to the energy content or economic value of the product.
• Thermodynamic efficiency = useful energy output ÷ total energy input. This gives you an upper bound on how well the process can perform.

Pinch analysis is a practical tool for improving efficiency in processes with multiple hot and cold streams. It identifies the minimum heating and cooling utilities needed by finding the "pinch point" where hot and cold composite curves are closest. This helps you design heat exchanger networks that reuse energy internally rather than importing it all from external sources.

Feasibility of reactive systems

Energy balance considerations

Before building a reactor or process, you need to confirm that the energy balance actually closes in a practical way.

• Can available energy sources (fuel, electricity, waste heat from other units) meet the process energy demand?
• How do operating conditions affect the balance? Raising temperature may speed up a reaction but increase heat losses. Raising pressure may shift equilibrium favorably but costs compression work.
• Product yield and energy efficiency often trade off against each other. A higher conversion might require disproportionately more energy, so there's usually an economic optimum.

Optimization techniques

• Sensitivity analysis: Vary one parameter at a time (inlet temperature, flow rate, excess air ratio) to see which ones most affect $Q$, efficiency, or product yield. Focus your optimization on those high-impact variables.
• Process integration: Heat integration pairs hot outlet streams with cold inlet streams to reduce external heating and cooling. Mass integration does the same for material streams, recycling where possible.
• Modeling and simulation: Tools like Aspen Plus or HYSYS let you test different configurations and operating conditions computationally before committing to hardware.
• Renewable energy sources: Solar thermal or biomass can supply process heat in some cases, reducing fossil fuel dependence. Whether this is feasible depends on the temperature level required and the local availability of the resource.