Cheatsheet: Steady-State Systems

1. Steady-State Fundamentals

1.1 Definition

1.2 Key Characteristics

• ∂/∂t = 0 for all system variables (temperature, pressure, composition, flow rates)
• Input rates = Output rates for mass and energy
• No time-dependent changes in system inventory
• Does not imply equilibrium (reactions may still occur at constant rates)

2. Mass Balances

2.1 General Mass Balance Equation

2.2 Component Balances

• Total mass balance: Σṁ_in = Σṁ_out
• Component balance: Σ(ṁ_i)_in + R_i = Σ(ṁ_i)_out
• R_i = net generation rate of component i (positive for production, negative for consumption)
• For non-reacting species: R_i = 0

2.3 Degree of Freedom Analysis

2.4 Basis Selection

• Choose basis as amount of feed, product, or time (e.g., 100 kg feed, 1 hour of operation)
• Use convenient values (100 kg, 1 kmol, 1 m³) to simplify calculations
• For composition problems, use 100 kg or 100 kmol basis
• Scale results to actual process rates after solving

3. Multiple Unit Systems

3.1 Solution Strategies

• Sequential modular: Solve units one at a time in process flow order
• Simultaneous solution: Solve all equations for all units together
• Recycle streams require iterative solution or simultaneous equations
• Tear streams: Break recycle loops to enable sequential solution

3.2 Mixing and Splitting

3.3 Recycle Systems

• Overall balance: Input from outside = Output to outside + Accumulation
• Fresh feed + Recycle = Total feed to unit
• Recycle ratio = ṁ_recycle / ṁ_{fresh feed}
• At steady-state, recycle composition and flow rate are constant

4. Reactive Systems

4.1 Extent of Reaction

• ξ has units of moles
• ν_i = stoichiometric coefficient (negative for reactants, positive for products)
• Independent reactions = number of species - rank of stoichiometric matrix

4.2 Conversion and Yield

4.3 Limiting and Excess Reactants

• Limiting reactant: Reactant that would be completely consumed first based on stoichiometry
• Excess reactant: Reactant present in greater than stoichiometric amount
• Percent excess = [(n_actual - n_stoich) / n_stoich] × 100%
• Theoretical yield: Maximum product based on limiting reactant

4.4 Reactor Mass Balances

• Write balances on atomic species (conserved) or molecular species with reaction terms
• Atomic balances: Σ(a_ijn_j)_in = Σ(a_ijn_j)_out
• Molecular balances: n_i,out = n_i,in + ν_iξ
• Use atomic balances when reaction stoichiometry is unknown

5. Energy Balances

5.1 General Energy Balance

• Q = heat added to system (positive when added)
• W = work done by system (positive when done by system); W_s = shaft work
• ΔH = H_out - H_in = enthalpy change

5.2 Enthalpy Calculations

5.3 Reference States and Heat of Formation

• Choose reference state: specific T and P (often 25°C, 1 atm)
• ΔH_f° = standard heat of formation at 25°C, 1 atm
• ΔH_f° = 0 for elements in standard state
• Path independence: ΔH depends only on initial and final states

5.4 Energy Balance with Reaction

• Heat of reaction method: Calculate sensible heat changes relative to T_ref, add reaction heat
• Heat of formation method: Calculate absolute enthalpies using heats of formation
• Both methods give identical results when applied correctly

5.5 Adiabatic Reaction Temperature

• Adiabatic: Q = 0, so ΔH_total = 0
• Exothermic reactions increase temperature; endothermic reactions decrease temperature
• Solve iteratively: Guess T_out, calculate ΔH, adjust until ΔH = 0
• Maximum adiabatic temperature: Complete conversion with no heat loss

6. Phase Equilibrium

6.1 Vapor-Liquid Equilibrium

• Ideal solution: Raoult's Law applies; α_ij = P_i^sat / P_j^sat
• More volatile component: Higher K-value, lower boiling point
• Bubble point: Temperature where first bubble forms (Σy_i = 1)
• Dew point: Temperature where first drop forms (Σx_i = 1)

6.2 Flash Calculations

• β = V/F = vapor fraction
• Solve Rachford-Rice equation iteratively for β
• Calculate compositions: x_i = z_i / [1 + β(K_i - 1)]; y_i = K_ix_i

6.3 Distillation

6.3.1 Binary Distillation

• Overhead (distillate) enriched in more volatile component
• Bottoms enriched in less volatile component
• Reflux ratio: R = L/D (liquid returned / distillate withdrawn)
• Minimum reflux: R_min corresponds to infinite stages

6.3.2 McCabe-Thiele Method

• Assumptions: Constant molal overflow (equimolal vaporization/condensation)
• Rectifying section: y_n+1 = (R/(R+1))x_n + x_D/(R+1)
• Stripping section: y_m+1 = (L'/V')x_m - (L'/V' - 1)x_B
• Step off stages between operating lines and equilibrium curve

7. Separation Processes

7.1 Absorption and Stripping

• Operating line for absorption: Y = (L/G)(X - X_in) + Y_in
• Minimum liquid rate: (L/G)_min at pinch point (operating line touches equilibrium)
• Actual liquid rate: L/G = (1.2 to 2.0)(L/G)_min

7.2 Extraction

• Solute transfers from feed phase to solvent phase
• Distribution coefficient: K_D = concentration in extract / concentration in raffinate
• Single-stage extraction: Material balance and equilibrium relationship
• Countercurrent extraction: Analogous to absorption calculations

7.3 Crystallization

• Mass balance: Solute in feed = Solute in crystals + Solute in mother liquor
• Mother liquor at saturation (equilibrium) at final temperature

8. Psychrometry and Humidification

8.1 Definitions

8.2 Key Relationships

8.3 Psychrometric Processes

• Adiabatic saturation: Humid gas contacted with water; exits saturated at adiabatic saturation T
• Wet-bulb temperature: Approximates adiabatic saturation temperature
• Dew point: Temperature at which H₂O begins to condense (H = H_s at T_dp)
• Heating: H constant, T increases, H_r decreases
• Cooling: H constant until dew point, then H_s decreases along saturation curve

9. Solution Techniques

9.1 Sequential Modular Approach

• Number all streams; label known and unknown variables
• Perform degree-of-freedom analysis on each unit
• Solve units in sequence where DOF = 0
• For recycles, guess tear stream variables, solve forward, iterate until convergence

9.2 Simultaneous Solution

• Write all balance equations for all units simultaneously
• Solve system of linear or nonlinear equations
• Use matrix methods for linear systems
• Use Newton-Raphson or other methods for nonlinear systems

9.3 Iterative Calculations

• Convergence criterion: |x_new - x_old| <>
• Typical tolerance: 10⁻⁴ to 10⁻⁶ depending on accuracy requirements

9.4 Common Pitfalls

• Inconsistent units (convert all to common basis)
• Incorrect basis (ensure all streams referenced to same basis)
• Missing streams (check all inlets and outlets)
• Incorrect stoichiometry in reactive systems
• Forgetting recycle contributions in overall balances
• Using wrong reference state for enthalpy calculations

10. Unit Operations Summary

10.1 Heat Exchangers

• Energy balance: Q = ṁ_hotC_p,hot(T_hot,in - T_hot,out) = ṁ_coldC_p,cold(T_cold,out - T_cold,in)
• LMTD method: Q = UAΔT_lm
• ΔT_lm = (ΔT₁ - ΔT₂) / ln(ΔT₁/ΔT₂)
• Countercurrent: More efficient than cocurrent

10.2 Pumps and Compressors

• η = efficiency (0 < η="">< 1);="">_s = shaft work (positive = work input)
• Isentropic: Reversible adiabatic process (ideal)

10.3 Reactors

• At steady-state, CSTR and PFR have constant outlet composition
• -r_A = reaction rate (moles reacted per volume per time)