chemical process balances unit 13 study guides

Key Concepts

• Simultaneous material and energy balances involve analyzing systems where both mass and energy are conserved
• Requires understanding the interactions between material flows and energy transfers within a system
• Involves setting up and solving a system of equations that describe the conservation of mass and energy
  • These equations are often linear and can be solved using matrix algebra or other mathematical techniques
• Assumes steady-state conditions where the accumulation of mass and energy within the system is zero
• Applies to a wide range of chemical processes, including reactors, heat exchangers, and distillation columns
• Helps in designing, optimizing, and troubleshooting chemical processes by predicting the behavior of the system
• Key variables include flow rates, compositions, temperatures, pressures, and enthalpies of the streams entering and leaving the system

Fundamental Principles

• Conservation of mass: The total mass entering a system must equal the total mass leaving the system plus any accumulation within the system
  • For steady-state processes, the accumulation term is zero, simplifying the mass balance equation
• Conservation of energy: The total energy entering a system must equal the total energy leaving the system plus any accumulation within the system
  • Energy can be in the form of heat, work, or changes in the enthalpy of the streams
• Steady-state assumption: The properties of the system do not change with time, implying that the accumulation terms in the mass and energy balance equations are zero
• Ideal gas behavior: Many simultaneous balances involve gas streams, which are often assumed to behave as ideal gases to simplify the equations
• Thermodynamic properties: Enthalpy, heat capacity, and other thermodynamic properties are essential for setting up the energy balance equations
• Stoichiometry: Chemical reactions within the system must satisfy stoichiometric relationships, which are used in the mass balance equations
• Phase equilibrium: When multiple phases are present, equilibrium relationships (e.g., Raoult's law, Henry's law) are used to relate the compositions of the phases

Types of Simultaneous Balances

• Non-reactive systems: Simultaneous balances for systems without chemical reactions, such as heat exchangers, mixing vessels, and separators
  • Focus on the conservation of mass and energy, with no generation or consumption of species due to reactions
• Reactive systems: Simultaneous balances for systems with chemical reactions, such as reactors and reactive distillation columns
  • Involve the generation and consumption of species according to stoichiometric relationships
  • Require additional equations to describe the reaction kinetics or equilibrium
• Open systems: Systems that exchange mass and energy with their surroundings, such as flow reactors and continuous distillation columns
• Closed systems: Systems that do not exchange mass with their surroundings but may exchange energy, such as batch reactors and closed-loop heating/cooling systems
• Adiabatic systems: Systems that do not exchange heat with their surroundings, simplifying the energy balance equation
• Isothermal systems: Systems that maintain a constant temperature, often achieved by exchanging heat with the surroundings

Setting Up Equations

• Identify the system boundaries and the streams entering and leaving the system
• Define the variables for the flow rates, compositions, temperatures, pressures, and enthalpies of each stream
• Write the mass balance equations for each component in the system
  • For non-reactive systems, the mass balance equations equate the inlet and outlet flow rates of each component
  • For reactive systems, include the generation or consumption of species due to reactions using stoichiometric relationships
• Write the energy balance equation for the system
  • Include the enthalpy changes of the streams, heat transferred, and work done by or on the system
  • Use thermodynamic properties (e.g., heat capacities, heats of reaction) to express the enthalpy changes in terms of the system variables
• Introduce additional equations as needed, such as equilibrium relationships, reaction kinetics, or constraints on the system variables
• Simplify the equations by making appropriate assumptions (e.g., ideal gas behavior, constant properties) and substituting known values

Solving Techniques

• Algebraic manipulation: Rearrange the equations to solve for the unknown variables explicitly
  • Suitable for simple systems with a small number of equations and variables
• Substitution: Express one variable in terms of others and substitute it into the remaining equations
  • Reduces the number of equations and variables, making the problem easier to solve
• Matrix methods: Express the equations in matrix form and use matrix algebra to solve for the unknown variables
  • Efficient for systems with a large number of equations and variables
  • Can be solved using Gaussian elimination, Cramer's rule, or matrix inversion
• Iterative methods: Use initial guesses for the unknown variables and iteratively refine the solution until convergence is achieved
  • Useful for systems with nonlinear equations or complex dependencies between variables
  • Examples include Newton-Raphson method, successive substitution, and Wegstein's method
• Computer-aided solutions: Use software packages (e.g., MATLAB, Python libraries) to solve the system of equations numerically
  • Advantageous for complex systems with a large number of equations and variables or nonlinear behavior

Common Applications

• Heat exchangers: Determine the outlet temperatures and heat duty for co-current or counter-current heat exchangers
  • Involves simultaneous mass and energy balances, with no chemical reactions
• Reactors: Predict the outlet composition, temperature, and conversion for various reactor types (e.g., batch, CSTR, PFR)
  • Requires mass balances with reaction stoichiometry and an energy balance considering the heat of reaction
• Distillation columns: Determine the composition and flow rates of the distillate and bottoms streams, as well as the heat duties of the condenser and reboiler
  • Involves simultaneous mass and energy balances around the entire column and individual stages
• Absorption and stripping: Predict the composition and flow rates of the gas and liquid streams leaving the column
  • Requires mass balances for each component and an energy balance considering the heat of absorption or desorption
• Combustion processes: Analyze the flue gas composition, adiabatic flame temperature, and heat released in combustion systems
  • Involves mass balances with combustion stoichiometry and an energy balance considering the heat of combustion
• Evaporators and crystallizers: Determine the composition and flow rates of the product streams and the heat duty required
  • Requires mass balances for each component and an energy balance considering the latent heat of vaporization or crystallization

Troubleshooting & Tips

• Double-check the system boundaries and ensure all streams and components are accounted for in the mass and energy balances
• Verify the consistency of units throughout the equations and convert if necessary
• Start with simple assumptions (e.g., ideal gas behavior, constant properties) and gradually relax them if needed
• Use degrees of freedom analysis to ensure the system of equations is solvable
  • The number of independent equations must equal the number of unknown variables
• Identify and handle any redundant equations that may arise due to the conservation of mass and energy
• Simplify the equations by identifying and eliminating variables that are not of interest or can be easily calculated later
• Validate the results by checking if they make physical sense and satisfy the conservation laws
  • Perform a reality check on the magnitudes and signs of the calculated values
• Conduct sensitivity analyses to understand the impact of uncertainties in the input parameters on the solution
• Break down complex systems into smaller, more manageable subsystems and solve them sequentially or iteratively

Practice Problems

1. A heat exchanger is used to cool a hot oil stream from 150°C to 100°C using cooling water. The cooling water enters at 25°C and leaves at 50°C. If the flow rate of the oil is 5 kg/s and its specific heat capacity is 2 kJ/kg·K, determine the required flow rate of the cooling water, assuming its specific heat capacity is 4.18 kJ/kg·K.

2. A steady-state CSTR is used to carry out the gas-phase reaction A + B → C. The reactor is fed with a mixture of 60% A and 40% B at a total flow rate of 100 kmol/h. The reaction has a conversion of 80% for component A. Determine the composition and flow rate of the reactor effluent.

3. An adiabatic combustion chamber burns methane (CH4) with 20% excess air. The methane and air enter the chamber at 25°C and 1 atm. Assuming complete combustion and neglecting any changes in kinetic and potential energy, determine the adiabatic flame temperature of the combustion products. The standard heats of formation for CH4, CO2, and H2O are -74.8, -393.5, and -241.8 kJ/mol, respectively.

4. A distillation column is used to separate a binary mixture of benzene and toluene. The feed contains 50 mol% benzene and enters the column at a rate of 100 kmol/h. The distillate contains 95 mol% benzene, and the bottoms contain 5 mol% benzene. Determine the flow rates of the distillate and bottoms streams.

5. An evaporator is used to concentrate a salt solution from 10 wt% to 50 wt% by evaporating water. The feed enters the evaporator at a rate of 1000 kg/h. Determine the flow rates of the product solution and the evaporated water, assuming no salt is lost in the vapor stream.