Intro to Chemical Engineering Unit 4 Review: Energy Balances

Key Concepts and Definitions

• Energy the capacity to do work or transfer heat
• Thermodynamics the study of energy and its transformations
• System a specific region or quantity of matter under study
• Surroundings everything external to the system
  • Can exchange energy and/or mass with the system
• State variables properties that describe the state of a system (temperature, pressure, volume)
• Process a change in the state of a system due to energy transfer or work
• Equilibrium a state where no changes occur in the system properties over time
  • Thermal equilibrium: no temperature gradient
  • Mechanical equilibrium: no pressure gradient

Energy Balance Fundamentals

• Conservation of energy principle energy cannot be created or destroyed, only converted from one form to another
• Energy balance an accounting of all energy entering, leaving, and accumulating within a system
  • Applies the conservation of energy principle
• Reference states chosen states for assigning a value of zero to a specific property (enthalpy, entropy)
• Heat transfer energy transfer due to a temperature difference between the system and surroundings
  • Occurs through conduction, convection, or radiation
• Work energy transfer due to a force acting over a distance
  • Includes expansion/compression work, shaft work, and electrical work
• Internal energy a state function representing the total kinetic and potential energy of a system
• Enthalpy a state function equal to the sum of internal energy and the product of pressure and volume ($H = U + PV$)

Types of Energy in Chemical Systems

• Kinetic energy energy associated with the motion of an object ($KE = \frac{1}{2}mv^2$)
• Potential energy energy stored due to an object's position or configuration
  • Gravitational potential energy: $PE = mgh$
  • Chemical potential energy: energy stored in chemical bonds
• Thermal energy (heat) energy associated with the random motion of particles in a substance
• Mechanical energy the sum of kinetic and potential energy in a system
• Chemical energy energy stored in chemical bonds and released or absorbed during chemical reactions
• Electrical energy energy associated with the flow of electric charges
• Nuclear energy energy released during nuclear reactions (fission or fusion)

First Law of Thermodynamics

• States that the change in internal energy of a system equals the heat added minus the work done by the system ($\Delta U = Q - W$)
• Applies the conservation of energy principle to thermodynamic systems
• Heat ($Q$) is positive when added to the system and negative when removed from the system
• Work ($W$) is positive when done by the system on the surroundings and negative when done on the system by the surroundings
• For a cyclic process, the change in internal energy is zero ($\Delta U = 0$)
  • Heat added equals work done ($Q = W$)
• Enthalpy change ($\Delta H$) is often used in place of internal energy change for processes at constant pressure

Closed vs. Open Systems

• Closed system (control mass) a fixed amount of mass with no exchange of matter with the surroundings
  • May exchange energy (heat and work) with the surroundings
• Open system (control volume) a region in space with mass flowing in and out
  • Exchanges both energy and mass with the surroundings
• Isolated system does not exchange energy or mass with the surroundings
• Adiabatic system does not exchange heat with the surroundings but may exchange work
• Steady-state system has no change in properties over time (mass flow rates, temperatures, pressures remain constant)
  • Applicable to many open systems (heat exchangers, turbines, pumps)

Energy Balance Equations and Calculations

• General energy balance equation: $\Delta U = Q - W + \sum m_i(h_i + \frac{v_i^2}{2} + gz_i) - \sum m_e(h_e + \frac{v_e^2}{2} + gz_e)$
  • $\Delta U$: change in internal energy
  • $Q$: heat added to the system
  • $W$: work done by the system
  • $m_i$, $m_e$: mass flow rates in and out
  • $h_i$, $h_e$: specific enthalpies in and out
  • $v_i$, $v_e$: velocities in and out
  • $z_i$, $z_e$: elevations in and out
• Simplified energy balance for a closed system: $\Delta U = Q - W$
• Steady-state, steady-flow energy balance: $0 = \dot{Q} - \dot{W} + \sum \dot{m}_i(h_i + \frac{v_i^2}{2} + gz_i) - \sum \dot{m}_e(h_e + \frac{v_e^2}{2} + gz_e)$
  • $\dot{Q}$, $\dot{W}$, $\dot{m}$: rates of heat, work, and mass flow
• Enthalpy balance for a steady-state, steady-flow process: $0 = \dot{Q} - \dot{W}_s + \sum \dot{m}_ih_i - \sum \dot{m}_eh_e$
  • $\dot{W}_s$: rate of shaft work

Real-World Applications

• Power plants use energy balances to optimize efficiency and minimize waste heat
  • Boilers, turbines, condensers, and pumps are analyzed as open systems
• Refrigeration cycles (air conditioners, refrigerators) rely on energy balances to calculate heat removal and work input
  • Evaporators, compressors, condensers, and expansion valves are treated as open systems
• Heat exchangers (shell-and-tube, plate) are designed using energy balances to determine heat transfer rates and outlet temperatures
• Chemical reactors (batch, continuous stirred-tank, plug-flow) use energy balances to account for heat of reaction and temperature changes
• Distillation columns employ energy balances to calculate reboiler and condenser duties, as well as stage temperatures and compositions
• Fuel cells and batteries are analyzed using energy balances to determine efficiency and power output

Common Pitfalls and Tips

• Ensure consistent units throughout calculations (SI or English)
• Pay attention to sign conventions for heat and work (positive or negative)
• Clearly define the system and surroundings for each problem
• Identify the type of system (closed, open, steady-state) to apply the appropriate energy balance equation
• Account for all forms of energy entering and leaving the system (heat, work, mass flow)
• Use reference states consistently when calculating changes in properties (enthalpy, entropy)
• Double-check that the energy balance equation is satisfied (inputs = outputs + accumulation)
• Simplify the energy balance equation when appropriate (neglect kinetic/potential energy, assume steady-state)