GATE CH cheat sheet

Material balance on a steady-state unit: Σ(inputs) = Σ(outputs) + accumulation; for a non-reactive system with C components, write C−1 independent balances plus one total balance.

Recycle with fractional conversion per pass X, recycle ratio R and fresh feed F: overall conversion depends on R; the recycle stream composition is set by the separator/recycle split, not by the reactor inlet composition.

Energy balance (open, steady): Q − W_s = Σ n_out h_out − Σ n_in h_in; use reference-state enthalpies consistently and include heats of reaction via formation enthalpies or a standard heat of reaction.

Heat of reaction at temperature T: ΔH_r(T) = ΔH_r(T_ref) + ∫Σν_i C_p,i dT; Kirchhoff's law, valid when no phase change occurs between T_ref and T.

Adiabatic reactor temperature rise: ΔT = (−ΔH_r)·X / (Σ Θ_i C_p,i), where Θ_i is moles of species i per mole of key reactant in the feed.

Raoult's law for ideal VLE: y_i P = x_i P_i^sat(T); relative volatility α_ij = P_i^sat/P_j^sat, and α > 1 means component i is lighter and enriches in the vapor.

Log-mean driving force for heat transfer: ΔT_lm = (ΔT_1 − ΔT_2)/ln(ΔT_1/ΔT_2); Q = U A ΔT_lm, with U based on the same area as the film coefficients.

Friction factor: laminar (Re < 2100) f = 16/Re with ΔP = 32μLu/D²; turbulent use Colebrook or Blasius f = 0.079 Re^(−1/4) for smooth pipes, 4000 < Re < 10^5.

CSTR vs PFR for first-order reaction: CSTR V = v·X/(k(1−X)); PFR V = (v/k)·ln(1/(1−X)); for the same X and k, the CSTR volume is always larger.

Dimensionless groups to recognize: Re = ρDu/μ (inertia/viscous), Pr = C_p μ/k, Nu = hD/k, Sc = μ/(ρD_AB), Sh = k_c D/D_AB; heat–mass transfer analogy: Nu/Pr^(1/3) = Sh/Sc^(1/3).

Mass transfer with chemical reaction: for a first-order reaction in a film, the enhancement factor is Ha/tanh(Ha), where Hatta number Ha = √(k D_AB)/k_c; Ha >> 1 means reaction-limited, Ha << 1 means diffusion-limited.

Drying and humidification: use humidity Y = 0.622 p_w/(P − p_w); wet-bulb equals dry-bulb only at saturation; adiabatic saturation lines coincide with constant wet-bulb lines for air–water.

Distillation shortcut: Fenske for minimum stages N_min = ln[(x_D/(1−x_D))·((1−x_B)/x_B)]/ln α_avg; Underwood for minimum reflux; Gilliland correlation links N, N_min, R, R_min.

Leaching and extraction: Kremser equation for countercurrent stages, (Y_N − Y_0)/(Y_1 − Y_0) = (S^(N+1) − S)/(S^(N+1) − 1), with absorption factor S = L/(mV).

Reaction orders from data: use the differential method (plot ln(−dC_A/dt) vs ln C_A) or the integral method by testing 1/(1−X) vs t for second order and −ln(1−X) vs t for first order.

Non-isothermal reactor stability: for a CSTR, plot heat generated (S-shaped, Arrhenius) against heat removed (straight line); intersections give steady states - up to three, with the middle one unstable.

Packed bed pressure drop uses the Ergun equation: ΔP/L = 150μu(1−ε)²/(ε³d_p²) + 1.75ρu²(1−ε)/(ε³d_p); the two terms are viscous and inertial contributions.

Thermodynamic consistency checks: at the azeotrope y_i = x_i and α = 1; Gibbs–Duhem must hold for activity coefficients, and an azeotrope cannot be separated by ordinary distillation alone.

First-order plus dead-time process: G(s) = K_p e^(−θs)/(τs + 1); the Ziegler–Nichols reaction-curve settings use K_c = 1.2τ/(K_p θ), τ_I = 2θ, τ_D = 0.5θ.

Ultimate-gain tuning: find K_cu at which the loop oscillates continuously under proportional-only control; then K_c = K_cu/1.7, τ_I = P_u/2, τ_D = P_u/8 for PID.

Second-order underdamped response: transfer function ω_n²/(s² + 2ζω_n s + ω_n²); overshoot depends only on ζ, and decay ratio for ζ = 0.216 gives about quarter-amplitude damping.

Closed-loop stability: Routh–Hurwitz on the characteristic equation 1 + G_c G_p = 0; a sign change in the first column means a right-half-plane pole, i.e. instability.

Control valve sizing: liquid flow Q = C_v √(ΔP/G); cavitation occurs when the vena-contracta pressure falls below vapor pressure - check the cavitation index and use choked-flow limits.

Relief system sizing: the API orifice area for vapor service scales as A ∝ W/(K_d P_1 √(M/(T Z)))·√(k/(k+1) terms); always compare required area with the next standard orifice letter.

HAZOP guide words: NO/NOT, MORE, LESS, AS WELL AS, PART OF, REVERSE, OTHER THAN - applied to nodes and parameters (flow, pressure, temperature, level) to generate deviations.

Flammability: the lower flammability limit sets the minimum fuel concentration for ignition; inerting keeps the oxygen concentration below the limiting oxygen concentration (LOC), typically a few percent below LFL conditions.

Cost estimation: the six-tenths rule C_2 = C_1 (S_2/S_1)^0.6; the Lang factor multiplies delivered equipment cost by roughly 3–5 depending on whether the plant is solid, fluid, or mixed processing.

Equation of state: van der Waals (P + a/V_m²)(V_m − b) = RT with a = 27R²T_c²/(64P_c), b = RT_c/(8P_c); critical compressibility Z_c = 3/8, which overpredicts for real fluids.

Departure functions: for a cubic EOS, residual enthalpy and entropy come from integrating (∂P/∂T)_V and (∂P/∂V)_T; at the ideal-gas limit both residuals vanish, so always state the reference.

Fugacity coefficient from the EOS: ln φ_i = (1/RT)∫_V^∞ [(∂P/∂n_i) − RT/V] dV − ln Z; for an ideal gas φ = 1 and fugacity equals partial pressure.

Activity coefficient models: Margules and van Laar for binary VLE; Wilson cannot predict liquid–liquid split, while NRTL and UNIQUAC can, which is why they are preferred for partially miscible systems.

Maxwell–Stefan diffusion: driving force = Σ (x_i x_j/(D_ij))·(u_i − u_j); for a binary mixture it reduces to Fick's law with D_AB, and the multicomponent D matrix is not simply pairwise.

Conduction: steady 1-D slab q = kA ΔT/L; cylindrical shell q = 2πkL ΔT/ln(r_2/r_1); lumped capacitance is valid when Bi = hL/k < 0.1.

Convection correlations: Dittus–Boelter Nu = 0.023 Re^0.8 Pr^n (n = 0.4 heating, 0.3 cooling); for laminar fully developed tube flow Nu = 3.66 (constant wall temperature) or 4.36 (constant heat flux).

Radiation: blackbody emissive power E_b = σT⁴ with σ = 5.67×10⁻⁸ W/(m²·K⁴); the Stefan–Boltzmann law for gray surfaces includes view factors, and view factors obey reciprocity A_i F_ij = A_j F_ji.

Chemical equilibrium: K_p = Π (p_i)^ν_i and ΔG° = −RT ln K; K depends only on temperature, and van 't Hoff gives d(ln K)/dT = ΔH°/(RT²).