Unit operation equations

Bioreactors stirred_tank_bioreactor fed_batch_bioreactor continuous_bioreactor airlift_bioreactor

Grows cells on a sugar and nitrogen feed and turns the substrate into biomass and product, run as a batch, fed-batch, continuous or air-lift culture.

V_work = Q t_cycle;  V_vessel = 1.25 V_work
V_work = 1,000 L/h x 30 h = 30,000 L;  V_vessel = 1.25 x V_work = 37,500 L

V_work = Q t_cycle;  V_vessel = 1.25 V_work
V_work = 1,000 L/h x 45.3 h = 45,300 L;  V_vessel = 1.25 x V_work = 56,600 L

bioreactor_separation, _run_fermentation_kinetics · fermentation_kinetics.simulate_fermentation, size_production_vessel · gases in fermenter_gas_balance

Yields

order        user value → molecule database → stoichiometric estimate → host Y_XS (biomass) → host y_ps
degree of reduction   γ = (4C + H − 2O − 3N)/C
stoichiometric        Y_Cmol = min(1, γ_S/γ_P);   Y_theo = Y_Cmol·(MW_P/C_P)/(MW_S/C_S);   Y = 0.65·Y_theo
biomass ceiling       D_G = 200 + 18(6 − C)^1.8 + exp{[(3.8 − γ)²]^0.16·(3.6 + 0.4C)}      (Heijnen & van Dijken 1992)
                      −ΔG_cat = 118·γ aerobic, 40 anaerobic
                      Y_X,max = 1/(1 + D_G/ΔG_cat);   Y_XS clamped to it
element check         (Y_XS, Y_i) projected onto a feasible C/H/O/N/electron balance; unlocked yields only move down

Oxygen supply

u_g      = vvm·V/60 / (πD²/4),   D = (4V/(π·H/D))^(1/3)
kLa      stirred, coalescing       3600 · 0.026 · (P/V)^0.40 · u_g^0.50      (van 't Riet 1979)
         stirred, non-coalescing   3600 · 0.002 · (P/V)^0.70 · u_g^0.20
         bubble column             3600 · 0.32 · u_g^0.7;   airlift × 0.7
         × viscosity penalty (1 + k_visc·X)^n_visc
C*       water solubility at T and P_abs × 10^−(0.14·I + 0.014·c_org)   factor [0.3, 1]
OUR_max  = 0.90 · (vvm·60/24.5) · y_O2 · 32   g/(L·h)                    gas feed ceiling

Kinetics (adaptive Euler, dt 0.0005–0.25 h)

temperature   m_G = 4.5·exp[−(69000/8.314)(1/T − 1/298)];   m_s, m_O2 × m_G(T)/m_G(T_ref)
pH            μmax × r(pH)/r(pH_opt),   r(x) = 1/(1 + 10^(pH_low − x) + 10^(x − pH_high))
substrate     Monod S/(Ks + S);  Haldane S/(Ks + S + S²/Ki);  Contois S/(Ksx·X + S)
dissolved O₂  kLa·(C* − C) = [μmax·f_S·C/(C + K_O2)/Y_XO2 + m_O2]·X          quadratic in C
product       f_P = max(0, 1 − P/Pmax)^n                                        Levenspiel
growth        μ = μmax · f_S · f_O2 · f_P;   if OUR > OUR_max:  μ ≤ (OUR_max − m_O2·X)·Y_XO2/X
uptake        q_s = μ/Y_XS + m_s + β·f_P/Y_PS                                   Herbert–Pirt
balances      dX/dt   = (μ − k_d − D)·X
              dP/dt   = (α·μ + β·f_P)·X − D·P,   α = Y_PS/Y_XS               Luedeking–Piret
              dS/dt   = −q_s·X + D·(S_f − S)
              intracellular product ≤ 0.30 g/g DCW (0.35 inclusion bodies)
DO control    below 0.5·setpoint, stirrer speed N/N₀ rises (≤ 3);  P/V = P₀·(N/N₀)³
FED-BATCH     V₀ = 0.45·V_max
              on demand    F = min(0.02·V_max, [q_s·X·V + (0.05·Ki − S)·V] / S_f)
              exponential  plan = (μ_set/Y_XS + m_s)·X·V,  μ_set = 0.25·μmax
CONTINUOUS    washout if D + k_d > μmax·f_S(S_feed);  steady state after 5/D at < 0.2 %/h change
STOP          harvest titer → productivity peak (P/(t + t_turnaround) < 0.95 × max) → washout/steady
              → vessel full → substrate spent → stall (μ ≤ 0.02·μmax) → time limit (200 h)
HEAT          metabolic 14.4 kJ/g O₂ × OUR (anaerobic 0.55 kJ/g substrate) + agitation − evaporation

Mass balance of the broth

SPECIFIED TITER
  S_X = X/Y_XS,   S_P = P/Y_PS,   S_growth = the binding one
  S_required = S_growth / (1 − f_m)                 f_m = maintenance fraction ≤ 0.5
  fed-batch feed  F = (S_req + R − S₀) / (1 − (S_req + R)/c_f),   R = 3 g/L residual
  shortfall       X, P × S_available/S_required  (a locked titer is refused instead)
  nitrogen        X ≤ N_available / f_N,   f_N = 28.014/246.26  (biomass C10H18O5N2)
PREDICTED / STOICHIOMETRIC
  S_consumed = X·S_in;   X = Y_XS·S_consumed;   P = Y_PS·S_consumed    (or from the integrator)
  titer ceiling   per product min(solubility/5, top of typical range), class wall when a titer is declared
GASES (element balance, g per L broth)
  CO₂ = C_substrate − C_biomass − C_products
  NH₃ = max(0, N_out − N_substrate);   H₂O = (H_in + H_NH3 − H_out)/2
  O₂  = (O_out + O_in,CO2/H2O − O_in)/2        anaerobic: routed to an electron sink (H₂)
  water stripped = 1.20 · air per L · [W(T, 1) − W(20 °C, 0.5)] · 1000,   ≤ 5 % of liquid
VOLUME
  Q_out = (m_in + Q_fed)·1000 / (1000 − g_net),   g_net = O₂ + CO₂,fixed − CO₂ − H₂O,stripped

Sizing (sets capex)

cycle       fill = drain = 0.5·(V_unit/1000)^0.35 h;  sterilise 1.5;  inoculate 0.5;  clean 2.0 h
batch       V_work = Q · t_cycle   (fixed point with n vessels);   V_vessel = V_work · 1.25
continuous  V_work = Q / D;         V_vessel = V_work · 1.25
n           = ⌈V_vessel / V_max⌉      V_max microbial 500 000, mammalian/insect 25 000, plant cell 100 000 L
seed train  stages N = ⌈ln(V_p/V₀)/ln(1/r)⌉ ≤ 5,  r = 0.10 microbial / 0.20 mammalian,  V₀ 500 / 800 L
            each stage priced on its own volume × n_trains^0.90
capex       C = n^0.90 · C_ref · ((V_vessel/n) / 48 000 L)^0.65,   floor $150k
anaerobic   commodity tank anchor $1.388M at 3 785 000 L  (NREL F-300)

• Refused: aeration on an anaerobic step, no usable nitrogen source, a declared titer the substrate or kinetics cannot reach, no growth from the pitch, CO₂ fixation beyond the CO₂ fed.
• The perfusion bioreactor was removed on 2026-09-23. A flowsheet that still names it gets a named refusal pointing to continuous_bioreactor.

Conversion reactor conversion_reactor enzymatic_hydrolysis

Converts a feed component into a product in a stirred reactor, by a chemical reaction or by enzymes breaking down a polymer such as cellulose.

V_liquid = Q tau;  V_vessel = 1.25 V_liquid
V_liquid = 1,000 L/h x 24 h = 24,000 L;  V_vessel = 1.25 x V_liquid = 30,000 L

V_liquid = Q tau;  V_vessel = 1.25 V_liquid
V_liquid = 1,000 L/h x 84 h = 84,000 L;  V_vessel = 1.25 x V_liquid = 105,000 L

conversion_reactor_separation, _conversion_reactor_single_row, resolve_conversion_reaction

MODES
  stoichiometric   X and τ as set
  equilibrium      K(T) = K_ref·exp[−(ΔH·1000/8.314)(1/T − 1/T_ref)];   X ≤ K/(1 + K)
  kinetic          k = k_ref·exp[−(Ea·1000/8.314)(1/T − 1/T_ref)]        k_ref 0.035 /h at 50 °C
                   k_app = k/(1 + P/K_I);   x′ = X/X_max
                   batch/PFR  τ = −ln(1 − x′)/k;     CSTR  τ = x′/(k(1 − x′))
                   enzyme decay  τ = −(1/k_d)·ln[1 + (k_d/k)·ln(1 − x′)],  k_d = ln2/t½;   τ ≤ 168 h
BALANCE (per reaction row)
  converted   = X·m_S,in
  product_i  += converted · Y_i
  substrate   = (1 − X)·m_S,in
  Δwater      = −X·(ΣY − 1)·m_S,in           X scaled down if the water runs out
DUTY
  sensible    (T_rx − T_in)·ṁ·4.186/3600;   reaction  ΔH·(converted/MW_repeat)/3600
SIZING
  V_liquid = Q · max(τ, 0.1 h);   V_vessel = V_liquid · 1.25   → capex on VOLUME
  capex  n^0.90 · C_ref · ((V/n)/V_ref)^0.70
AGITATION (one table, both tiers, on LIQUID volume;  Walas 1990 Ch. 10)
  duty                        kW/m³   band
  blending                    0.07    0.04–0.10
  homogeneous reaction        0.20    0.10–0.30
  reaction with heat transfer 0.65    0.30–1.00
  liquid–liquid               1.00    0.70–1.30
  gas–liquid                  1.50    1.00–2.00
  slurry                      2.00    1.50–2.50
  unagitated pump-around      0.02    NREL T-310
  default duty: insoluble solids > 10 wt% (whole stream, water by volume) → slurry;
  otherwise reaction with heat transfer if the step has a jacket duty, else homogeneous.
  agitation_kw_per_m3 on the step wins.   power = P/V · V_liquid + declared ancillary kWh/m³ · Q

Dilute-acid pretreatment pretreatment_reactor

Cooks lignocellulosic biomass in hot dilute acid so the hemicellulose dissolves and the cellulose opens up to enzymes.

log R0 = log10(tau exp((T - 100)/14.75));  CS = log R0 - pH
log R0 = log10(5 min x exp((158 - 100) / 14.75)) = 2.41;  CS = 2.41 - 1.35 = 1.06

pretreatment_reactor_separation (NREL TP-5100-47764 Area 200; Saeman 1945)

SEVERITY
  pH      = clamp(−log10 M_acid, 0.5, 3)
  log R₀  = log10(τ · exp((T − 100)/14.75));    CS = log R₀ − pH
  f_furf ≥ 0.02 + 0.055·max(0, CS − 0.8)
REACTIONS (g/h)
  xylose   = 1.1363·f_x·m_xylan        furfural = 0.7273·f_f·m_xylan
  glucose  = 1.1111·f_g·m_glucan       HMF      = 0.7778·f_h·m_glucan
STEAM AND FLASH
  Cp       = x_s·1.4 + (1 − x_s)·4.18
  steam    = m·Cp·(T − T_feed)/2100  kg/h
  flash    = Cp·(T − 100)/2257;   30 % of the furfural leaves with the vapour

Anaerobic digester anaerobic_digester

Lets microbes break organic waste down to biogas, a mix of methane and carbon dioxide, without oxygen.

anaerobic_digestion_separation (Metcalf & Eddy 5th ed., ch. 13)

destroyed_i  = x_type · m_i     sugar 0.90, organic acid 0.95, protein 0.75, lipid 0.78, cells 0.50 …   ≤ 75 % of VS
COD removed  = Σ COD_factor · destroyed
Y_CH4        = (22.414/64) · (1 − 1.42·Y_x)   Nm³/kg COD
CH4          = Y_CH4 · COD_removed;   CO2 = C_destroyed − CH4 − C in new biomass
biomass      = Y_x · COD_removed
VOLUME       V_liquid = max(Q·24·HRT,  VS_in·24/OLR)
             V_vessel = V_liquid · 1.15  (declared headroom, band 1.10–1.25)   → capex on VOLUME
power        0.006 kW/m³ · V_liquid + declared ancillary 0.6 kWh/m³ feed (pumps, blower)   same in both tiers
heating      ṁ·4.0·ΔT·(1 − recovery)·1.18;   biogas heat V_CH4·35.8 MJ/Nm³

Biomass digestion (PHA release) biomass_digestion

Digests the cell mass around PHA granules with hypochlorite, optionally with SDS, so the polymer granules are freed for recovery.

f_dig = f_max (1 - exp(-t / tau)) f_dose   (hypochlorite f_max 0.98, tau 0.30 h)
f_dig = 0.98 x (1 - exp(-2 h / 0.30 h)) x 1 = 0.979

biomass_digestion_separation (Berger 1989; Jacquel 2008)

dilution    V_req = (m_cells + m_PHA) / DCW_set(30 g/L);   V_susp = max(Q, V_req)
extent      f_dig = f_max · (1 − e^(−t/τ)) · f_dose
hypochlorite  NaOCl demand = 1.2·m_cells (× 0.5 with SDS);   f_dose = min(1, NaOCl/demand)
              MW loss = clamp(0.20 + 0.10(t − 1)⁺ + 0.30(pct − 5)⁺/25, 0.20, 0.50)
masses      digested = m_cells·f_dig;   PHA freed = m_PHA·(1 − 0.05)
energy      heat = Q_out·4180·ΔT/3.6e6;   steam = heat/0.85

Disc-stack centrifuge disc_centrifugation

Spins a dilute broth through a stack of conical discs so cells and debris settle out, giving a clear liquid and a concentrated sludge.

centrifugation_separation → _size_disc_stack_capacity_and_capture → disc_stack_design.design_disc_stack (catalogue unit centrifugation_disc)

SIGMA (designed geometry)
  Σ = (2π/3) · N_ch · ω²/g · (r_o³ − r_i³)/tanθ · f_corr         ω = 2π·rpm/60
  r_o = 0.88·R_bowl,  r_i = 0.33·r_o,  θ = 40°,  gap 1 mm,  RCF = ω²R/g ≤ 15 000
  search R 0.068–0.25 m, 30–200 channels
CAPTURE
  G(d) = min(1, v(d)·η·Σ / Q_machine)          integrated over a log-normal size distribution
PER-MACHINE CAPACITY  (uptime u = 60·t_int / (60·t_int + 8 s))
  separation   Σ / (Σ per L/h at the capture target) · u
  hydraulic    55 m³/m²/h · πR² · 1000 · u
  sludge       V_sl = π(R² − r_o²)·0.65R·0.55;   wet V_sl/(t_int/60)·u;   dry V_sl·0.35/(t_int/60)·u
  machines     smallest N meeting all four limits, smallest bowl envelope
FREE OIL (above 0.5 g/L light liquid)
  oil recovery 0.97 demulsified (0.95–0.99), 0.55 emulsion;  water in oil 0.3 / 5 wt %
CAPEX (geometry, no repeat-unit discount)
  purchased = N × $650k × (πR²H / (π·0.17²·0.4))^0.62

• Refused when solids carry no particle size and none is declared: capture would be a guess.
• Refused when more than the machine limit is needed, or when the wet discharge exceeds the feed.

Decanter centrifuge decanter_centrifugation

Separates a solids-heavy slurry in a horizontal spinning bowl, with a scroll that pushes the settled solids out continuously.

v50 = Q_m / (2 Sigma eta);  d50 = sqrt(18 mu v50 / (drho g))
v50 = 2.78e-04 m3/s / (2 x 8,000 m2 x 0.2);  d50 = sqrt(18 x 1.17 mPa s x v50 / (100 kg/m3 x 9.81 m/s2)) = 1.36 um

centrifugation_separation → _evaluate_non_disc_centrifuge_capacity_and_capture → _model_centrifuge_sigma

GRADE (Ambler)   v_50 = Q_m / (2·Σ·η);   d50 = √(18μ·v_50 / (Δρ·g))
                 G(d) = min(1, 0.5·(d/d50)²)         G(d50) = 0.5;  integrated over the log-normal
ONE FLEET        Q_σ  = per-machine flow at which the grade integral reaches separation_efficiency
                 N    = ⌈max(Q/Q_hydraulic, Q/Q_σ)⌉
                 capture evaluated at Q/N per machine (≥ target);   capex bills the same N
                 N > 8  → refused: decanter_parallel_limit_exceeded
INSTALLED        declared N → capture at that N, never resized;  Q/N above the hydraulic rating → refused
NO SOLIDS        nothing to capture → hydraulic sizing, capture 1.0

Basket centrifuge basket_centrifugation

Spins a slurry against a filter cloth in a perforated basket, holding back crystals or coarse particles as a cake.

W = V_wash / V_liq;  R = 0.01 + 0.99 x exp(-2.0 W)
W = 58.7 / 112 L/h = 0.523;  R = 0.01 + 0.99 x exp(-2.0 x 0.523) = 0.358

centrifugation_separation (filtering centrifuge: cake wash applies)

CAPTURE (cloth sieving, not settling)
  G(d)    = 0.98 above the cloth cut, 0.70 between 0.5× and 1× the cut, 0 below   (filtration ladder)
  capture = ∫ G(d)·f_m(d) dd over the log-normal size basis;   cloth cut default 10 µm [1–100]
  refused when the median particle is below the cut;  no declared size → stated capture, flagged
CAKE WASH (displacement)
  V_liq   = V_heavy − Σ m_solid/ρ
  V_wash  = ratio · m_cake / 1000;   W = V_wash / V_liq
  residual R = 0.01 + 0.99·e^(−2.0·W)
  kept    = max(R·dissolved_in_cake, saturated-liquor floor)
capacity  hydraulic wall 2 000 L/h;   N = ⌈Q / 2000⌉

Three-phase decanter three_phase_decanter

Splits a feed into a light oil phase, a water phase and solids in one spinning bowl.

unrecovered = residual% x m_heavy;  recovery = 1 - unrecovered / F_oil
recovery = 1 - (0.501 % x 944 kg/h) / 60 kg/h = 0.921

three_phase_decanter_separation

unrecovered oil = min(F_oil·(1 − rec), residual% · (m_feed − F_oil·rec))
recovery        = 1 − unrecovered / F_oil
light phase     oil + water x_w = m_oil·w/(1 − w)
heavy phase     aqueous + all solids

• Refused when the feed has no free light liquid, when the oil is not demulsified, or when Δρ < 0.05 g/mL.

Depth filtration depth_filtration

Traps cells and fine debris inside a thick porous filter medium, the usual step to clarify broth before sterile filtration.

A = V / loading   (single-use cartridges, 400 L per m2)
A = 1,000 L/h / 400 L/m2 = 2.5 m2 of cartridge per hour; 20,000 m2/yr over 8,000 h at $25/m2 = $500,000/yr of media

filtration_separation

balance      shared filter balance, no stated capture
area ratio   = clamp(100 LMH / film-theory average flux, 1, 20),  CF = Q/V_r     capped at 4 for capex
thorough     A = (permeate/100)·ratio;  N = ⌈A/200 m²⌉;  price × ratio^0.85
cartridges   Q·hours / 400 L/m²

Filter press filter_press_filtration

Pumps a slurry into cloth-lined chambers between plates, building a solid cake while the filtrate drains away.

t = (mu alpha c / 2 dP) (V/A)^2 + mu R_m (V/A) / dP;  A = Q t_cycle / (V/A)
2 h = (0.001 Pa s x 1.00e+12 m/kg x 25 kg/m3 / (2 x 12 bar)) (V/A)^2 + 0.001 x 1e11 /m x (V/A) / 12 bar  ->  V/A = 0.827 m3/m2;  A = 1 m3/h x 2 h / 0.827 m3/m2 = 2.42 m2

filtration_separation · cake area in rate_based_models

stated capture 0.98;  ΔP 12 bar;  2 h cycle;  cake DM 350 g/L
capex on cake area A (A_ref 2.65 m², A_max 400 m² per press)

Rotary vacuum drum filter rotary_vacuum_filtration

Pulls liquid through a cloth on a rotating drum under vacuum, leaving a cake that is scraped off on every turn.

t = (mu alpha c / 2 dP) (V/A)^2 + mu R_m (V/A) / dP;  A = Q t_cycle / (V/A)
0.03 h = (0.001 Pa s x 1.00e+12 m/kg x 25 kg/m3 / (2 x 0.7 bar)) (V/A)^2 + 0.001 x 1e11 /m x (V/A) / 0.7 bar  ->  V/A = 0.0209 m3/m2;  A = 1 m3/h x 0.03 h / 0.0209 m3/m2 = 1.43 m2

filtration_separation

stated capture 0.95;  ΔP 0.7 bar;  cycle 0.03 h;  cake DM 250 g/L
capex on cake area A (A_ref 1.55 m², A_max 150 m² per drum)

Vacuum belt filter belt_filtration

Draws the liquid out of a slurry through a moving filter cloth under vacuum, washing the cake before it is discharged.

t = (mu alpha c / 2 dP) (V/A)^2 + mu R_m (V/A) / dP;  A = Q t_cycle / (V/A)
0.05 h = (0.001 Pa s x 3.00e+11 m/kg x 25 kg/m3 / (2 x 0.45 bar)) (V/A)^2 + 0.001 x 1e11 /m x (V/A) / 0.45 bar  ->  V/A = 0.035 m3/m2;  A = 1 m3/h x 0.05 h / 0.035 m3/m2 = 1.43 m2

filtration_separation

stated capture 0.95;  under 3 kg polymer/t DS:  capture = 0.50 + (cap − 0.50)·(dose/3)
ΔP 0.45 bar;  cycle 0.05 h;  cake DM 200 g/L
capex on cake area A (A_ref 1.53 m², A_max 200 m²)
polymer $/yr = V_m³/yr · 0.030 t/m³ · dose · $4/kg

Nutsche filter nutsche_filtration

Filters a batch of slurry through a flat filter plate in a closed vessel, where the cake can be washed in place.

filtration_separation

no stated capture;  ΔP 1.0 bar;  2 h cycle;  2 diavolumes wash;  DM floor 400 g/L
capex on flow, × the membrane area ratio (nominal 100 LMH) in the classic tier

Flocculation flocculation

Adds a flocculant so fine particles clump into larger flocs that settle or filter more easily.

theta = D_s / D_sat;  f_agg = 0.95 x 4 theta (1 - theta);  capture = f_agg G(d_floc) + (1 - f_agg) G(d_primary)
theta = 5 / 10 kg/t DS = 0.5;  f_agg = 0.95 x 4 x 0.5 x (1 - 0.5) = 0.95;  capture = 0.95 x 0.937 + (1 - 0.95) x 1.71e-04 = 0.89

flocculation_separation

DOSE (per tonne of suspended dry solids, La Mer bridging)
  D_s     = dose / X_ss;   default dose = D_opt by class
            pam_sludge 5 kg/t DS [1–10] (WEF MOP-8);  cationic_harvest 50 kg/t DS [20–150] (McNerney 2015)
  θ       = min(1, D_s / D_sat),  D_opt = 0.5·D_sat
  E(θ)    = 4θ(1 − θ);   f_agg = 0.95·E(θ)          θ > 0.8 → overdose restabilisation warning
CAPTURE   f_agg·G(d_floc) + (1 − f_agg)·G(d_primary)      G = the Hazen integral above
          d_floc 200 µm [100–500];  floc Δρ 15 kg/m³ (flocs are mostly water)
SLUDGE    V_sludge = captured dry solids / 40 g/L;  dissolved species split by liquor volume
REAGENT   polymer mass enters the stream and leaves with the sludge;  billed kg/h × price
          (cationic harvest polymers need a declared price)
sizing    G·T = 6e4, G = 60 s⁻¹;  V = Q·t·1.2;  P = μG²V

Gravity thickener gravity_thickening

Lets solids settle under gravity in a large tank, drawing off a thickened underflow and a clear overflow.

A = max(solids load / solids flux, overflow / q_o,design)
A = max(3,600 kg/d / 50 kg/m2/d, 31.9 m2) = 72 m2

gravity_thickener_separation (runs the centrifuge balance with capture and underflow DM)

solids load   suspended (above-solubility) solids only
area          A = max(solids load kg/d ÷ solids flux,  overflow m³/d ÷ q_o,design 20 m³/m²/d)
CAPTURE (Hazen ideal settler, the Σ integral with Σ = A and g = 1)
  q_o      = Q_overflow / A;   v_cut = q_o / η_h,   η_h 0.6 [0.4–0.8] short-circuiting
  G(d)     = min(1, v_t(d)·η_h / q_o)
  capture  = min(declared solids_capture, ∫ G f_m)
underflow     one shared settler balance (thickener and flocculation);  feed already denser than the
              target → whole feed to underflow, "capped … conserved";  below 20 g/L → dilute warning
capex on flow

Hydrocyclone hydrocyclone

Swirls a pressurised feed inside a cone so dense particles are thrown to the wall and leave at the bottom, with no moving parts.

G(d) = 1 / (1 + (d50 / d)^m)
G(100 um) = 1 / (1 + (19.7 / 100)^2.5) = 0.983

hydrocyclone_separation (Plitt 1976, Rietema 1961)

q_unit = 7.5 m³/h · (D/50)² · √(ΔP/1.5);   n = ⌈Q / q_unit⌉
u      = q_unit / (πD²/4);   Eu = ΔP / (½ρu²);   Stk50 = 0.0611 / Eu
d50    = √(Stk50 · 18μD / (Δρ·u))
G(d)   = 1 / (1 + (d50/d)^m)
split  particulates R = R_f + (1 − R_f)·G;   solutes and water R_f
pump   kW = Q/3600 · ΔP·1e5 / 0.65 / 1000

• Cells above 0.1 g/L are refused by the expert rules: they are too small and too light to cyclone.

Microfiltration microfiltration

Holds back cells and particles on a porous membrane while water and dissolved molecules pass through.

J_gel = k ln(C_gel / C_bulk) x 3.6e6;  J = min(J_osm, J_gel, J_crit) x ff
J_gel = 2.24e-05 m/s x ln(240 / 30 g/L) x 3.6e6 = 167 L/m2/h;  J = min(J_osm, J_gel, J_crit) = 167 L/m2/h;  x ff 0.45 = 75.3 L/m2/h (gel-limited on the cell layer)

filtration_separation (absolute membrane)

V_ret      = V_feed · (1 − filtrate_fraction)
relief     V_ret ≥ Σ(m·σ)/DM_max        ≤ 0.95·V_feed, then packing relief (8 iterations)
wash       V_wash = V_ret·N,   V_filtrate = V_feed − V_ret + V_wash
V_cp       = V_feed − V_ret
solid      m_s · [V_ret/(V_ret + (1 − σ)V_cp)] · e^(−N(1 − σ))
liquor     m_l · [V_ret/(V_ret + V_cp)] · e^(−N)
σ          solids 1.0; dissolved 0;  assumed sizes cells 1.0 µm, colloid 0.5, crystal 5, oil 5
Sizing     CF = inlet/retentate → area ratio;  J_nom 100 LMH;  J_crit 220 × 0.45 = 99 LMH

• TMP ≤ 0 refuses: retentate = feed.

Ultrafiltration ultrafiltration_10k ultrafiltration_30k ultrafiltration_100k

Holds back proteins and other large molecules on a membrane with a set molecular-weight cut-off, to concentrate the product or exchange its buffer.

J_gel = k ln(C_gel / C_bulk) x 3.6e6;  J = min(J_osm, J_gel, J_crit) x ff
J_gel = 1.32e-05 m/s x ln(313 / 5 g/L) x 3.6e6 = 196 L/m2/h;  J = min(J_osm, J_gel, J_crit) = 160 L/m2/h;  x ff 0.55 = 88 L/m2/h (the retained protein layer)

J_gel = k ln(C_gel / C_bulk) x 3.6e6;  J = min(J_osm, J_gel, J_crit) x ff
J_gel = 1.26e-05 m/s x ln(324 / 6 g/L) x 3.6e6 = 180 L/m2/h;  J = min(J_osm, J_gel, J_crit) = 160 L/m2/h;  x ff 0.55 = 88 L/m2/h (the retained protein layer)

J_gel = k ln(C_gel / C_bulk) x 3.6e6;  J = min(J_osm, J_gel, J_crit) x ff
J_gel = 5.54e-06 m/s x ln(325 / 1 g/L) x 3.6e6 = 115 L/m2/h;  J = min(J_osm, J_gel, J_crit) = 115 L/m2/h;  x ff 0.55 = 63.4 L/m2/h (the retained protein layer)

ultrafiltration_separation, membrane_operating_point (batch mode)

1. CF clipped to the binding ceiling (≥ 1.0001)
2. V_ret = V_feed / CF
3. dry-matter relief, then packing relief (each ≤ 0.5·V_feed)
4. driving-force derate on V_ret and N
5. species: CF^−(1 − σ) × washout × stream split
6. V_perm = V_feed − V_ret + V_ret·N
Sizing: area_ratio from the flux integral;  J_nom 50 LMH;  J_crit 160 LMH

• Infeasible when a CF above 1 is asked for but the derate is about 0 or the delivered CF is at most 1.0002.
• The three grades differ only in MWCO (10 000, 30 000, 100 000 Da); the panel shows each grade's own catalogue default.

Nanofiltration nanofiltration

Passes water and small monovalent ions through a tight membrane while holding back sugars, divalent ions and small organic molecules.

ultrafiltration_separation (single-stage mode, dense rejection branch)

m_ret/m_in = V_ret / (V_ret + (1 − σ)·V_perm)      σ from the dense branch (charge- and pH-weighted)
Lp 5 LMH/bar,  ff 0.75,  J_nom 25 LMH

Reverse osmosis reverse_osmosis

Pushes water through a dense membrane against the osmotic pressure, leaving almost all dissolved solutes behind.

reverse_osmosis_separation, _ro_feasible_water_recovery

V_ret  = V_feed·(1 − WR),   V_perm = V_feed·WR
solute m_ret = m · V_ret / (V_ret + (1 − σ)·V_perm)
water  m_perm = m_w·WR
π      van 't Hoff (+ virial) on the retentate;   NDP = P − π
WR ceiling (bisection): infeasible if NDP ≤ 0, NDP < design margin, or C_i > solubility(T)
Sizing  once-through;  J_nom 15 LMH,  Lp 1.5,  ff 0.85;  pump Q·ΔP/0.65;  elements $100/m²

• If even WR = 0 is infeasible: permeate 0, retentate = feed (fatal).

Membrane distillation membrane_distillation

Evaporates water through a hydrophobic membrane, driven by a temperature difference, to concentrate a feed without boiling it.

V_conc = V_feed / CF
V_conc = 1,000 L/h / 5 = 200 L/h

membrane_distillation_separation, md_area_penalty_factor

             TPC    B_m (kg/m²/h/Pa)
  DCMD       0.65   1.8e-3
  AGMD       0.80   3.6e-4
  SGMD       0.75   4.0e-4
  VMD        0.70   1.1e-3

T_fm = T_f − ½(1 − TPC)ΔT,    T_pm = T_p + ½(1 − TPC)ΔT
p_sat = 10^(8.07131 − 1730.63/(233.426 + T)) · 133.322  Pa
J     = B_m·(p_sat(T_fm)·a_w − p_sat(T_pm)) · GOR^−0.9          a_w by Raoult, salts count ions
derate = min(1, 20·J_raw/1.64);   removal = (1 − 1/CF)·derate ≤ 0.999
m_w,evap = m_w,in − (V_feed/CF − V_solutes,ret)·ρ_aq             fixed point
volatiles α_MD = min(50, (p_i/p_w)·γ∞·f_neutral·τ),  τ 0.30 (DCMD/AGMD) or 0.70;  carry = 1 − w^α
Area   A = (m_w,evap/1000) / J_avg;   ratio = clamp(1.64/J_avg, 1, 20)
Heat   Q = V_distillate · h_fg(T)/GOR   kWh/m³;   cooling = Q

• Ceiling at packing, or water activity a_w < 0.72. Warnings for ΔT < 5 K and wetting by surfactants or alcohols.

Electrodialysis electrodialysis

Moves ions out of a stream through charged membranes under an electric field, to desalt it or recover an organic acid.

b = removal x CE  (fraction of the salt sent to the brine)
b = 0.9 x 0.85 = 0.765

electrodialysis_separation

V_dil = V_feed·r/(1 + r),   V_conc = V_feed/(1 + r)
b = removal · CE                       (0.765 at defaults)
fraction to brine:  salt b;  small acids (MW < 500) 0.7·b;  other charged 0.5·b;
                    protein, sugar, cells 0.02;  other uncharged 0.05;  water by r

• A fixed-fraction split. There is no stack, current or area model, so capex scales on feed flow alone.

Affinity chromatography affinity

Captures the product on a resin carrying a ligand that binds it specifically, then elutes it as a concentrated, purified pool.

_affinity_capture in _chromatography_separation

θ        = C / (K_d + C)
retained = clip(θ · ligand_util · elution_eff, 0, 1)
then the shared engine: bind–elute, 3 % carryover, base cut 0.18, bed-priced

Ion exchange ion_exchange_cation ion_exchange_anion

Binds charged molecules to a charged resin and releases them with a change in salt or pH, separating them by charge.

B = min(V_f / LBV, max(D / (q util), V_f / 100));  bed = B x t_cycle
B = min(1,000 / 10, max(4,800 g/h / 80 g/L, 1,000 / 100)) L/h;  bed = B x 1.5 h = 90 L

B = min(V_f / LBV, max(D / (q util), V_f / 100));  bed = B x t_cycle
B = min(10,000 / 10, max(549,000 g/h / 60 g/L, 10,000 / 100)) L/h;  bed = B x 1.5 h = 1,500 L

_ion_exchange_capture, _iex_binding (steric mass action, Brooks & Cramer 1992)

CHARGE
  protein    q = −0.12 · MW_kDa · (pH − pI),   |q| ≤ 0.2 · MW_kDa
  small      q = n_basic − Σ 1/(1 + 10^(pKa_i − pH))
  binds      cation: q > 0.05;   anion: q < −0.05
SMA BINDING
  ν          = (0.65 if protein else 1) · |q|
  C_salt     = max(0.005, I·1000/58.44, conductivity · 0.0085)   M
  log10 Keq  = log10 K0 + ν · log10(Λ / C_salt)        Λ = 0.30 M;  K0 protein 0.3, small ion 2.0
  C_elute    = Λ · (K0 · R)^(1/ν)
CAPTURE
  k′         = Keq · R
  capture    = k′/(1 + k′) · min(1, (1 + k′)/max(1, LBV))
RECOVERY TO PRODUCT CUT
  target     capture · 0.93 + (1 − capture) · carryover
  impurity   capture · 0.85 · overlap + (1 − capture) · carryover
  overlap    = 1 / (1 + ((r − 1)/0.10)²),   r = ratio of elution salts

FLOW-THROUGH POLISH MODE (chromatography_mode = flow_through)
  kd         = 5.0 · clip(|q|, 0.05, 2.5) / (1 + 1.2·I);   bound f = kd·R/(1 + kd·R)
  salts      90 % removed;  product loss 2 %
  bed        = max(V_f / 10 BV/h,  captured · 4 / (0.8 · capacity))
  t_run      = clip(0.8 · capacity · bed / captured, 4, 168) h
  regen      8 · bed / t_run  L/h;   regenerant 3.0 kg/kg captured at $0.55/kg

Hydrophobic interaction hydrophobic_interaction

Binds proteins by their hydrophobic surface patches at high salt and releases them as the salt drops.

_hydrophobicity_index in _chromatography_separation

h          starts 0.35; salt −0.25, sugar −0.18, organic acid +0.10, metabolite +0.08,
           protein +0.22 + 0.18·exp(−|pH − pI|/1.2);  −0.06·min(3, |charge|);  clamp [0.02, 0.98]
log k      = −5.48 + 10.0·h + 1.86·M          k bounded 10^±4
retained   = k/(1 + k) · elution_eff

Reverse phase reverse_phase

Separates molecules by hydrophobicity on a non-polar resin, eluting them with an organic solvent.

log k = log k0 - S phi;  bed = (V_f / LBV) x t_cycle
loaded at phi = 0.05 organic;  bed = (200 L/h / 10 BV) x 1.5 h = 30 L

_reverse_phase_capture

log k0     = −0.5 + 3.5·h + 0.2                h = hydrophobicity index (as HIC)
log k      = log k0 − S·φ                        bounded ±4
retained   = k/(1 + k) · elution_eff
solvent    $1 000/day × flow/100 L/h

Size exclusion size_exclusion

Separates molecules by size as they pass through a porous gel, largest first.

B = V_f / load fraction;  bed = B x 3.0 h
B = 5 L/h / 0.02 CV;  bed = B x 3 h = 750 L

_size_exclusion_captures, _sec_cycle_cv

radius     r = 1.25 · 0.066 · MW^(1/3) nm      (or size/2)
Kav        = (1 − clip(r/r_pore, 0, 1))²
captured   = exp(−(Kav − Kav_target)² / (2σ²)) · scale
B          = V_f / load fraction
dilution   = 1.3 + 0.7 · (0.04 − LBV)/(0.04 − 0.003)
V_product  = V_f · dilution;   V_buffer = V_product − 0.03·V_f
bed        = B · 3.0 h    (150 × V_f at the 0.02 default)

• Refused outside 0.3–4 % CV load, or for desalting duty (a desalting column is a different design).

Membrane chromatography membrane_chromatography

Runs flow-through or bind-and-elute chromatography on stacked adsorptive membranes in disposable capsules instead of a packed column.

V_mem = L_target / (loading x 1000)
V_mem = 4,500 g/h / (7.5 kg/L x 1000) = 0.6 L

Ion-exchange physics in the shared engine; sized as capsules

V_product  = V_f · 0.95;   product = ret · 0.05 + ft · 1.0
V_mem      = L_target / (loading · 1000)          flow-through
           = L_target / (DBC · 0.8)               bind–elute
capsules   L/yr = annual kg / (loading · cycles)
capex      flow basis (no resin family)

Activated carbon activated_carbon

Takes colour, odour and trace organics out of a liquid by adsorbing them onto activated carbon.

dose = max(dose to reach the removal target, polish floor);  carbon = dose x V_f / 1000
dose = max(1.96, 5) g/L;  carbon = dose x 1,000 L/h / 1000 = 5 kg/h

activated_carbon_separation, Langmuir helpers

q_eff      = q_max (from molecule DB);  acids × (1 − 1/(1 + 10^(pKa1 − pH)))
b          = 100 · (q_max/0.10)²     clamp [0.05, 1000] L/g
a          = dose · 0.75 · q_eff      inventory per L feed
outlet x   solves  b·x² + (a·b − C_in·b + 1)·x − C_in = 0
dose       = max(bisection to reach the removal target, polish floor)
product    = (m_in − m_ads) · (1 − hold)
carbon     kg/h = dose · V_f / 1000
BED (capex, GAC anchor)
  V_bed    = Q · EBCT/60 · stages;   velocity = depth/(EBCT/60) in 2–20 m/h
opex       single use: dose · ($3.0 + $0.35 disposal)/kg;  regenerated: dose · (1.5 + 0.08·3.0)

• Refused when solids exceed 0.1 g/L, the target is a cell or colloid, log P exceeds 3, or more than 5% of the target would adsorb.

Adsorption–elution adsorption_elution

Loads the product onto a resin or carbon bed, then strips it off with a smaller volume of eluent, capturing and concentrating it in one step.

R_cap = C_t (1 - f_bt) / (f_dyn q_t);  bed = V_f R / rho_bed
R_cap = 3 x (1 - 0.05) / (0.7 x 0.198) = 20.5 g resin/L;  bed = 1,000 L/h x 20.5 / 650 g/L = 31.6 L/h

adsorption_elution_separation (macroporous resin or carbon)

isotherm   q_i = q_eff,i · b_i · C_i / (1 + Σ b_j·C_j)          competitive Langmuir
R_cap      = max_t C_t·(1 − f_bt) / (f_dyn · q_t)                g bed per L feed
R          = max(R_cap, ρ_bed / max_loading_BV)                   ρ_bed resin 650, carbon 480 g/L
bed        = V_f · R / ρ_bed  L/h
bound      = min(C_i·(1 − f_bt), R·f_dyn·q_i) · V_f
washed     weak species × (1 − exp(−wash_BV))
eluted     = (bound − washed) · elution_recovery
eluate     = eluted + elution·φ·ρ_eluent + (elution·(1 − φ) + void)·1000
bill       eluent make-up = organic · (1 − 0.9);  resin bed cycled / life at $35/kg

Molecular sieve dehydration molecular_sieve_dehydration

Removes the last water from a solvent such as ethanol by adsorbing it into a zeolite bed, getting past the azeotrope that distillation cannot.

molecular_sieve_dehydration_separation, _sieve_solve_recycle

WATER BALANCE with recycle (≤ 200 iterations)
  F = m + S;   w_F = (m·w_F0 + S·w_s)/F
  R/D = (w_F − w_D)/(w_R − w_F);   D = F/(1 + R/D);   R = F − D
  S = η_s · R · (1 − w_R)/(1 − w_s)
BED
  W       = (F·w_F − D·w_D)/1000  kg/h water removed
  m_cap   = W · t_h / (q_work · util)
  ρ_v     = p·1e5·(MW/1000) / (8.314·(T + 273.15))
  A       = Q_v / u_s;   D_bed = √(4A/π)
  L       = clip(m_cap/(700·A), max(1.8, D_bed), L/D · D_bed)     ≤ 8 m
  ΔP/L    = 150μ(1 − ε)²u/(ε³d²) + 1.75ρ_v(1 − ε)u²/(ε³d)          Ergun, ε 0.37, d 3 mm
ENERGY
  regeneration 2 900 kJ/kg · W;  superheat 1.8 · ΔT;  stripper h_vap · 3

• Refused above 15 wt% feed water or 0.5 g/L non-volatiles, when dew point + 28 K exceeds 200 °C, or when the recycle does not converge.

Distillation distillation

Separates liquids by boiling point in a column, taking the volatile components overhead and the heavier ones from the bottom.

distillation_separation · energy _energy_distillation

Mass balance

T_op   = water boiling point at column pressure
α_i    = alpha_vs_water(i) at the feed pseudo-binary mole fraction
Key    = volatile target with highest α; light if α_p > 1, else heavy

Light product:   d_p = key_recovery · F_p
                 d_water = d_p/spec − d_p − Σ(non-keys in D)       clamped [0, water in]
Heavy product:   b_p = key_recovery · F_p
                 b_water = b_p/spec − b_p − Σ(non-keys in B)

Azeotrope cap on product mass fraction (water heavy key only):
   override → van Laar crossing where α = 1 → static record (ethanol 0.956, IPA 0.874, …)

Fenske          N_min = ln[(xd/(1−xd))·((1−xb)/xb)] / ln α          α = geometric mean at xb, xf, xd, ≥ 1.01
Underwood       R_min = [xd/xf − α(1−xd)/(1−xf)] / (α − 1)
Non-key split   d_i/b_i = (d_k/b_k) · α_ik^N_min
Gilliland       Y = 0.75·(1 − X^0.5668),  X = (R − R_min)/(R + 1)
Available       N_act = n_stages · E_M
Reflux needed   Y = (N_act − N_min)/(N_act + 1) → X → R = (R_min + X)/(1 − X)
Design reflux   R = max(reflux_ratio, R_needed, 0.1)
Stage-limited   (N_act ≤ N_min,spec): key logits scaled by k = N_ach/N_min,spec
Reported        N_theo = (N_min + Y)/(1 − Y),  stages_required = ceil(N_theo / E_M)
Volumes         V_D = V_feed · m_D/m_total

Energy & sizing

Q_reboiler  = D·(R + 1)·λ_top + max(0, m_feed·cp·(T_bottom − T_in))
Q_condenser = D·(R + 1)·λ_top
Classic tier: 90 kW thermal per m³/h feed

• Refused if N_theo > 150 or α_avg < 1.05 (Kister 1992; Perry's Sec. 13).
• Refused if the target decomposes: T_b(P) > T_decomp − 20 K; the message names the vacuum needed.
• With no volatile target: 30% of the water overhead, others by Kremser φ = α^N/(1 + α^N).

Evaporation evaporation

Boils off water to concentrate a solution, often in several effects that reuse the heat in the vapour.

V_conc = V_feed / CF
V_conc = 1,000 L/h / 5 = 200 L/h

evaporation_separation, _evaporator_water_evaporated · energy _energy_evaporation

Mass balance

V_conc      = V_feed / CF
m_w,kept    = max(0, (V_conc − Σ m_solute/ρ_solute) · 1000)
m_evap      = min(m_w,in − m_w,kept, 0.92 · m_w,in)
w           = 1 − m_evap / m_w,in
T_boil      = Tsat(P) + boiling-point elevation (Raoult bubble point of the concentrate)
f_i         = 1 − w^α_i        overhead fraction of each non-water species

Energy

Multi-effect  Q_heat = (sensible + λ·m_vap / E(N)) / η
MVR           Q_heat = sensible / η,   electricity = 15 kWh/t water evaporated
Classic tier  Q = [m·4.18·(60 − 25) + m_w·λ(60 °C)/E] / 0.92
Capex         flow basis × N^0.70  (× 1.8 for MVR)

• Warns when the CF needs more than the removal ceiling, and when MVR runs below 2 t/h of water.

Evaporative crystalliser evaporative_crystallizer

Boils off solvent until the product exceeds its solubility and crystallises out.

magma = sum m_crystallised / (m_feed - W)
magma = 168 / (1,150 - 590) kg/h = 0.3

evaporative_crystallization_separation · energy in thorough_flowsheet

Mass balance

T_op = Tsat(P) + BPE                    BPE iterated 3× on fresh feed, ≥ 0
Evaporative mode:
  bisect W ∈ [0, 0.95·m_w,in] (24 iterations) until
  magma = Σ m_crystallised / (m_feed − W) = target
  each trial runs the cooling-crystalliser equilibrium at T_op on the concentrated feed
Vacuum-cooling mode (adiabatic flash):
  W·λ(T_op) = m·cp·(T_feed − T_op) + Q_cryst
Outlets:  magma (heavy)  +  condensate W as pure water
Yield       = crystallised target / target in
Selectivity = (crys_t/crys_imp)/(in_t/in_imp)   clamped [0.01, 100]

Energy & sizing

net_latent = max(0, λ(T_boil)·W − Q_cryst)        Q_cryst: ΔH_sol → ΔH_fus → class default (sugar 60 … 120 kJ/kg)
Steam       Q_heat = (sensible + net_latent/E(N)) / 0.92
MVR         Q_heat = sensible/0.92,  +15 kWh/t
Vessel      V = (magma kg/h ÷ 1.2 kg/L) · τ(2 h) · 1.25     reported, not priced
Capex       flow basis × effects multiplier (as evaporation)

• Magma held at ≤ 0.35 solids (Myerson Ch. 6); inherits every cooling-crystalliser refusal.

Cooling crystallisation crystallization

Cools a solution so the solubility of the product falls and it comes out as crystals.

eq = 1 - C_sat V / m;  frac = max(eta eq, 1 - 1.05 (1 - eq))
eq = 1 - 41.4 g/L x 930 L/h / 150 kg/h = 0.743;  frac = max(0.7 x 0.743, 1 - 1.05 x (1 - 0.743)) = 0.731

crystallization_separation, _cryst_partition, _crystallization_with_mother_liquor_recycle

Mass balance

C_sat(T, pH)  van 't Hoff from the 25 °C solubility, then Henderson–Hasselbalch
              factor clamped [0.1, 10], T clamped 0–100 °C, C_sat ≤ 1500 g/L
V_liquor      = m_water/1000 + Σ m_dissolved/ρ_solid          ρ_solid default 1350 g/L
m_solid       = η · max(0, m − C_sat·V_liquor)                 fixed point, 12 iterations
eq            = 1 − C_sat·V/m
frac          = clamp(max(η·eq, 1 − 1.05·(1 − eq)), 0, eq)
Magma cap     (crystals + carried)/m_total ≤ 0.35  → scaled pro rata
Occlusion     m_occl = crystals · f/(1 − f)    ≤ 0.5 × liquor
Recycle       loop feed = fresh + r·liquor;  V_loop = P_total / C_fresh
              purge limit r = (K − 1)/(K·φ − φ_p),  K = 0.95·C_sat/c_0

Energy & sizing

Q_cool  = max(0, m·cp·(T_in − T_final) + Q_cryst),   chiller electricity = Q/3
Vessel  V = Q_loop · τ(4 h) · 1.25                     reported, not priced

• Refused when the target is water-miscible, liquid at T_op, has no solubility record, was never dissolved, or particulates exceed 2 g/L or magma 0.40.

Flash drum flash_drum

Drops the pressure on a liquid so its volatile part vaporises at once, splitting the stream into vapour and liquid.

Q_v = m_v / rho_v;  D = sqrt(4 Q_v / (pi u_max))
Q_v = 74.9 kg/h / 0.699 kg/m3;  D = sqrt(4 Q_v / (pi x 4.05 m/s)) = 0.0968 m

flash_drum_separation

Mass balance

K_i = γ_i(x) · Psat_i(T) / P                van Laar on organic + water pseudo-binary
Rachford–Rice   Σ z_i(K_i − 1) / (1 + ψ(K_i − 1)) = 0      ψ ∈ [0, 1), bisection
Vapour fraction of i = ψK_i / (1 + ψ(K_i − 1))
Non-volatile by class: cells, protein, salt, sugar, amino acid, polymer, MW ≥ 1000

Sizing (reported)

ρ_v    = P·MW_v / (8314.462·T)
u_max  = 0.107 · √((ρ_L − ρ_v)/ρ_v)          Souders–Brown, GPSA
D      = √(4·Q_v / (π·u_max)),   L/D = 4

Solvent recovery solvent_recovery

Distils spent solvent back out of a stream so it can be reused, with the unrecovered part billed as make-up.

solvent_recovery_separation

water_overhead = recovery · water_in
                 (azeotrope: min(water_in, solvent_in·recovery·w/(100 − w)))
unrecovered    = solvent_in · (1 − recovery)
residual       = min(unrecovered, spec·base/(1 − spec))     base = non-solvent + product water
overhead       = solvent_in − residual
make-up        = solvent_in · (1 − recovery)                  billed
residual ppm checked against ICH Q3C

Reactive distillation (esterification) esterification_reactive_distillation

Runs an esterification inside a distillation column, taking the products away as they form so the reaction keeps going.

X_ceil = 1 - (1 - X_eq)^n;  X = min(X_req, X_ceil)
X_ceil = 1 - (1 - 0.571)^8 = 0.999;  X = min(0.98, 0.999) = 0.98

esterification_reactive_distillation_separation, esterification_equilibrium_conversion

Equilibrium   K = (X + e0)(w0 + X) / ((1 − X)(r − X))
Column        X_ceil = 1 − (1 − X_eq)^n                      Taylor & Krishna 2000
Conversion    X = min(X_req, X_ceil, 0.999, 0.999·r)
MW_ester      = MW_acid + MW_alcohol − 18.015
Split         ester 0.995 light; free alcohol 0.999 light; acid & non-volatiles heavy (0.002 carryover)
Bottoms water = heavy_nonwater · (1/0.60 − 1)
Catalyst      inventory = acid kg/h ÷ WHSV;  use = inventory / (life × 8000 h)
Energy        Q = λ_overhead·(R + 1) + sensible to T_bottom      reaction thermoneutral

Dryers spray_drying freeze_drying drum_drying fluid_bed_drying rotary_drying vacuum_tray_drying

Removes water from a product to leave a dry powder or solid, by spray, freeze, drum, fluid-bed, rotary or vacuum tray drying.

product water = rm / (1 - rm) x sum retained
product water = 0.03 / (1 - 0.03) x 223 kg/h = 6.9 kg/h

product water = rm / (1 - rm) x sum retained
product water = 0.04 / (1 - 0.04) x 86.4 kg/h = 3.6 kg/h

product water = rm / (1 - rm) x sum retained
product water = 0.02 / (1 - 0.02) x 171 kg/h = 3.49 kg/h

product water = rm / (1 - rm) x sum retained
product water = 0.05 / (1 - 0.05) x 361 kg/h = 19 kg/h

product water = rm / (1 - rm) x sum retained
product water = 0.05 / (1 - 0.05) x 11.2 kg/h = 0.587 kg/h

drying_separation (shared mass balance) · energy _energy_drying

Mass balance

w              = [rm/(1 − rm) · DM · yield] / m_w,in
f_co,i         = 1 − w^α_i                            co-evaporation (Rayleigh)
retained_i     = m_i · (1 − f_co,i) · yield
product water  = rm/(1 − rm) · Σ retained              ≤ water fed
vapour water   = water in − product water
powder volume  = (DM/600) · (1 + rm)/(1 − rm)

Energy by dryer

Spray / fluid bed / rotary (convective, one formula for both tiers)
  η_air = (T_in − T_out)/(T_in − T_amb) · (1 − f_loss)       per-dryer heat loss in data
          fluid bed 70/40 °C → 0.60;  spray 0.58;  rotary 0.55   (≈ 4.2 MJ/kg water for the fluid bed)
  Q     = λ(T_out)·m_evap/η_air + sensible,   Q ≥ 4500 kJ/kg × m_evap   (Baker & McKenzie 2005)
Drum
  Q = latent/0.75 + sensible;   area = m_evap / 20 kg/m²/h (reported)
Vacuum tray
  P 10 mbar, T_evap = Tsat(P);   Q = (latent + sensible)/0.75
Freeze
  sublimation = m·2838/3600 kW;   freezing = sensible to 0 °C + m_w·333.6 + m·2.05·25
  condenser   = sublimation + m·1.996·15;   electricity = refrigeration/1.25 + vacuum

• Rayleigh co-evaporation overstates volatile loss from a drying droplet (Thijssen & Rulkens 1968). The code says so.

Heat sterilisation heat_sterilization

Heats a medium or feed long enough to kill contaminating organisms, then cools it back down.

k_d = (ln10 / D121) exp(-(Ea/R)(1/T - 1/394.25));  t_hold = Del / k_d
k_d = (ln10 / 2.4 min) x exp(-(283,000 / 8.314) x (1/(140 + 273.15) - 1/394.25));  t_hold = 40 / k_d = 48.2 s

heat_sterilization_separation · hold time in sterility_economics

Balance     target_out = target_in · (1 − yield_loss)       rest kept as heat-degraded
Del factor  k_d = (ln10 / D121) · exp(−(Ea/R)(1/T − 1/394.25))    D121 = 2.4 min, Ea = 283 kJ/mol
            t_hold = ∇ / k_d   (∇ target 40 → 48 s at 140 °C, 132 s at 135 °C)   or declared hold → ∇ delivered
            one resolver (sterility_economics) feeds the balance, the rate panel and the TEA
Energy      Q = m·cp·(T_hold − T_in)·(1 − η)   billed as STEAM at the same setpoint in both tiers,
            plus trim cooling water for the same duty

• Delivered Del below 32.2 is fatal ("Under-sterilised medium"), with the hold that would be needed. The old default of 135 °C for 5 s delivered Del 1.5.

Pasteurisation pasteurization

Heats a liquid briefly to kill vegetative microbes, a milder treatment than sterilisation.

D_T = D_ref x 10^((72 - T)/z);  log reduction = t_hold / D_T
D_T = 1.5 s x 10^((72 - 72) / 6);  log reduction = 15 s / D_T = 10

pasteurization_separation

D_T            = D_ref · 10^((72 − T)/z)
log reduction  = t_hold / D_T                        warning below 5 log (US PMO)
spores         D = 120 s · 10^((121.1 − T)/10)
PU             = (t/60) · 10^((T − 60)/7)
Q              = m·4.186·(T − T_in)·(1 − η)          T_in 25 °C

Liquid–liquid extraction extraction

Moves the product from the water phase into an immiscible solvent in which it dissolves better.

makeup = m_extract (1 - recovery) + m_raffinate (1 - stripper)
makeup = 822 x (1 - 0.995) + 80 x (1 - 0.95) kg/h = 8.11 kg/h

liquid_liquid_extraction_separation · make-up in process_solvents.solvent_makeup_kg_per_hr

V_org = R·F,   V_aq = F
solvent lost to raffinate  m_s,raff = min(m_solv, S_sw·F)
water into extract         m_w,ext = min(m_w,in, m_s,ext · w/(1 − w)),   w = water-in-solvent wt fraction ≤ 0.5
log P                      component → database → class fallback (protein −2, salt −3, sugar −2.5, acid 0.5 …)
acids / amino acids        logP_eff = logP + log10(f_neutral),   f_neutral = 1 − 1/(1 + 10^(pKa − pH))
K_D                        = 10^clamp(logP_eff + solvent correction, −5, 5)
Kremser (countercurrent)   E = K_D·R
                           f_org = (E^(N+1) − E)/(E^(N+1) − 1),   E = 1 → N/(N + 1),   ≤ 0.98
make-up                    charge = F·R·ρ;  raffinate = min(charge, S_sw·F/1000)
                           makeup = (charge − raffinate)(1 − recovery) + raffinate(1 − stripper)
thorough sizing            V_stage = F(1 + R)·(10/60)·1.25 per stage, 0.4 kW/m³

• A water-miscible solvent (ethanol, methanol) is refused: it makes one phase, not two.

pH-swing back-extraction ph_swing_back_extraction

Moves an acidic or basic product back out of the organic solvent into a fresh water phase by changing the pH.

E = D R;  f = (E^(N+1) - E) / (E^(N+1) - 1)   (Kremser, forward extraction)
E = 16.2 x 0.25 = 4.06;  f = (4.06^(1.7 + 1) - 4.06) / (4.06^(1.7 + 1) - 1) = 0.929

ph_swing_back_extraction_separation, _kremser_fraction, _ph_swing_distribution_ratio

pH window      acid: fwd = pKa − 0.5, back = pKa + 4.5;   base: fwd = pKa + 1.0, back = pKa − 4.0   (clamp 1.5–12)
D(pH)          = 10^clamp(logP + log10 f_neutral(pKa, pH) + solvent correction, −5, 5)
N_eff          = N·η
forward        f1 = min(0.98, Kremser(D_fwd·R, N_fwd,eff))
degradation    k = k_ref·2.5^((T − 5)/10);   loss = 1 − exp(−k·t_hold)        acid swing only
back           V_back = back_ratio·R·F;   E_b = (V_aq/V_solv)/D_back;   f_b = min(0.98, Kremser(E_b, N_back,eff))
buffer floor   V_back ≥ m_target / (1.5·c_sat(T, pH_back))
back titrant   mol = m_target/MW/eq + 0.02·V_back       (eq 2 for H2SO4, Ca(OH)2)
solvent        makeup = s_raff(1 − stripper) + s_prod + purge·circulating

• Refused for proteins, polymers, cells or MW > 1500, and for a pKa outside acid 2–8 or base 5.5–11.5.

Precipitation precipitation

Adds salt, solvent or acid so the product, or an impurity, becomes insoluble and drops out as a solid.

m_solid = eta max(0, m - C_sat V_liquor)
m_solid = 0.8 x max(0, 5 kg/h - 4.00e-04 g/L x 1,180 L/h) = 4 kg/h

precipitation_separation, _precip_isoelectric_c_sat, shared partition _cryst_partition

PROTEIN SOLUBILITY (isoelectric / salting-out; Cohn)
  q      = 0.12 · MW_kDa · |pH − pI|
  S_pH   = min(S_w, (S_w/30) · 10^(0.25·q))
  S      = S_pH · 10^(−0.75·I)              I = factor·dose/MW  ((NH4)2SO4 3, NaCl 1, CaCl2 3, alum 15)
ANTISOLVENT
  C_sat × 10^(−4.5·φ)                       φ: xanthan IPA 0.60, hyaluronan EtOH 0.70, default EtOH 0.67
  top-up V = (φ − φ_feed)·F/(1 − φ)
METAL SALT  C_sat × 0.02;   THERMAL (T ≥ 60 °C) proteins C_sat × 0.05
PARTITION
  m_solid = η · max(0, m − C_sat·V_liquor)   η: isoelectric 0.75, salting-out 0.80, antisolvent 0.93, metal 0.85, thermal 0.85
  magma ≤ 0.35 solids;  occluded = crystals·f/(1 − f)
SIZING   V = Q · hold/60 · 1.25,  0.3 kW/m³

• Isoelectric needs a protein with a pI and |pH − pI| ≤ 0.5. A magma above 0.40 solids is refused.

pH adjustment ph_adjustment

Adds acid or base to bring a stream to a target pH.

titrant g/h = mol/L x MW x F
NaOH = 0.878 mol/L x 39.997 g/mol x 1,000 L/h = 35.1 kg/h

ph_adjustment_separation, ph_calculations.titrant_demand

charge balance   SID = Kw/h − h + Σ C_i·α_i(pH),   α = 1/(1 + 10^(pKa − pH))
dose D           bisection (60×, 0–12.5 mol/L) until pH_after(SID + z·D) = target;  deadband 0.05
H2SO4            eq = D·(1 + α_HSO4(pH)),  bisulfate pKa 1.99
lime             mol = D/2
NH4OH            mol = D + mol·α_9.25(pH_in)     fixed point
titrant g/h      = mol/L · MW · F
acid dose        forms a salt: g_salt = mol·F·stoich·MW_salt   (e.g. (NH4)2SO4 132.14)

Reagent dose reagent_dose

Adds a set amount of a liquid or solid reagent to a stream.

V_add = F phi / (1 - phi);  m = V_add rho
V_add = 1,000 L/h x 0.67 / (1 - 0.67) = 2,030 L/h;  m = V_add x 0.789 kg/L = 1,600 kg/h

reagent_dose_separation

liquid (ratio = final volume fraction φ, default 0.67)   V_add = F·φ/(1 − φ);   m = V_add·ρ·1000
solid  (ratio = g per L feed)                            m = ratio·F;   V_add = m/(ρ·1000)
V_out = F + V_add;   every feed concentration × F/V_out
above solubility → suspended;  lime bisected to pH 12.45, the rest is slurry

• Refused at φ ≥ 0.9 or ratio ≤ 0.

Mixing vessel mixing_vessel

Blends streams and additions together in a stirred tank.

C_mix = sum C_k V_k / V_out
C_mix = (500 g/L x 200 L/h syrup + 0 g/L x the dilution water) / 1,000 L/h = 100 g/L

mixing_vessel_separation

inlets are mixed upstream by mass:  C_mix,i = Σ C_i,k·V_k / Σ V_k
target C × (1 − conversion)            conversion default 0
optional target_pH → titrant_demand (NaOH / H2SO4);  kg/h = kg_per_m³ · F/1000
yield = 0.99·(1 − 0.05·conversion)
sizing  V = Q · residence_time/60 · 1.25   (15 min),  0.5 kW/m³

Viral inactivation viral_inactivation

Holds the product at low pH for a set time so enveloped viruses are inactivated, then neutralises it.

target_out = target_in (1 - yield_loss)
target_out = 4.5 kg/h x (1 - 0.02) = 4.41 kg/h

viral_inactivation_separation, _apply_product_loss

target_out = target_in · (1 − yield_loss)          lost part kept as "(denatured)"
low pH: two titrant legs pH_in → 3.8 → 7.0, bisulfate carried between them
sizing  V = Q · hold/60 · 1.25;  capex design flow uses the 2 h batch window

Homogeniser & bead mill high_pressure_homogenizer bead_mill

Breaks cells open to release the product inside them, by forcing them through a valve at high pressure or grinding them with beads.

R = 1 - (1 - R1)^(N (P/800)^a)   (yeast R1 0.60, a 2.2)
R = 1 - (1 - 0.60)^(2 x (800 / 800)^2.2) = 0.84

R = 1 - exp(-k E)   (yeast k 0.85)
R = 1 - exp(-0.85 x 2 kWh/kg) = 0.817

cell_disruption_separation, _homogenizer_release_fraction, _bead_mill_release_fraction

HOMOGENISER (Hetherington / Follows, Middelberg 1995)
  R = 1 − (1 − R1)^(N·(P/800)^a)          ≤ 0.995
      yeast (0.60, 2.2)   gram-negative (0.80, 1.4)   gram-positive (0.55, 1.8)   mammalian (0.97, 1.0)
  energy  kWh/m³ = N·P·1e5/3.6e6;   pump kW = kWh/m³ · Q / η(0.85)
  ΔT_ad   = N·P·1e5 / (1000·4180)
BEAD MILL
  R = 1 − exp(−k·E)                        k: yeast 0.85, gram-positive 0.7, gram-negative 2.5, mammalian 5.0
  P_shaft = max(E·X·Q, 0.45·E·100·Q)
  heat-limited capacity  Q_max = 60 kW / (E·X)
BALANCE
  solubles = m_X·R·(g_HCP + g_NPS);   debris = m_X·R·(1 − g_HCP − g_NPS);   intact = m_X·(1 − R)
  intracellular target: released m·R, the rest stays cell-associated

Hammer & ball mill hammer_mill ball_mill

Grinds solid feed such as biomass or grain into smaller particles, with hammers or tumbling balls.

E = 10 Wi (1/sqrt(P80) - 1/sqrt(F80))
E = 10 x 250 kWh/t x (1/sqrt(2,000) - 1/sqrt(20,000)) = 38.2 kWh/t

E = 10 Wi (1/sqrt(P80) - 1/sqrt(F80))
E = 10 x 200 kWh/t x (1/sqrt(300) - 1/sqrt(3,000)) = 79 kWh/t

milling_separation (Bond 1952)

E [kWh/t dry] = 10·Wi·(1/√P80 − 1/√F80)
shaft kW = E · dry t/h;   motor kW = shaft / 0.90
mass unchanged; every solid leaves at size P80

• Refused above 15% moisture. P80 below 100 µm is clamped.

Alkaline lysis alkaline_lysis

Opens bacterial cells with alkali and detergent to release plasmid DNA, then neutralises so debris and genomic DNA precipitate.

m_wet = m_X / 0.25;  V_resusp = max(F, 7.5 m_wet);  V_out = 3 V_resusp
m_wet = 40 kg/h;  V_resusp = max(200, 7.5 x 40) = 300 L/h;  V_out = 3 x 300 = 900 L/h

alkaline_lysis_separation (plasmid DNA)

m_wet = m_X/0.25;   V_resusp = max(F, 7.5·m_wet/1000);   V_out = 3·V_resusp
NaOH g = M·39.997·V;   SDS g = pct·10·V;   KAc g = M·98.142·V
shear loss = min(0.30, max(0, kb − 20)·0.01);   over-lysis = min(0.9, max(0, t − 5)·0.03)
recovered pDNA = m·lysis_eff·pdna_recovery·(1 − shear)(1 − over-lysis)
floc = other + DNA(1 − leak) + RNA(1 − 0.8) + LPS(1 − 0.05) + 0.95·SDS + 0.9·protein
cooling = V_out·4180·ΔT/3.6e6 / COP 3

Inclusion-body refold inclusion_body_refold

Dissolves insoluble protein aggregates in a denaturant, then dilutes them so the protein folds back into its active form.

beta = C / C*;  Y = ln(1 + beta (1 - exp(-k1 tau))) / beta;  refold = Y_max Y
beta = 0.5 / 0.35 = 1.43;  Y = ln(1 + 1.43 (1 - exp(-0.35 x 16))) / 1.43 = 0.62;  refold = 0.85 x 0.62 = 0.527

inclusion_body_refold_separation, _ib_refold_yield_fraction

WASH        dissolved kept (1/(1 + r))^k;  target kept (1 − loss)^k;  debris removal min(0.8, 0.5 + 0.1·M)
SOLUBILISE  f_sol = 0.96 / (1 + exp(−(M_eff − 5.0)/0.45))       M_eff = M × potency (urea 1, GuHCl 2)
REFOLD      V_refold = m_sol / refold_g_l
  class by disulfides   (Y_max, C*, k1):  0 → (0.92, 3.0, 20);  ≤2 → (0.88, 0.80, 0.50);  ≤4 → (0.85, 0.35, 0.35);
                                          ≤8 → (0.70, 0.10, 0.20);  more → (0.50, 0.02, 0.10)
  temperature           C* = C*_ref·exp(50000/8.314·(1/T − 1/283.15)),  k1 with the opposite sign
  β                     = C_protein / C*;   pulses n ≤ 3, β_eff = β/n
  kinetic yield         Y = ln(1 + β(1 − e^(−k1·τ))) / β
  refold yield          = Y_max · (1/n)·Σ Y(β_eff, k1, τ(n − i)/n)
  non-native            × 0.70 → aggregate
VESSEL      V_work = V_pool · τ;   sized volume = V_work / 0.85   → capex on VOLUME

RNA reduction heat shock rna_reduction_heat_shock

Heats harvested cells briefly so their own enzymes break down RNA, lowering the nucleic-acid content of single-cell protein.

m_out = m_b (1 - f);  RNA removed = m_b rna_in - m_out rna_out
m_out = 20 x (1 - 0.3) = 14 kg/h;  RNA removed = 20 x 0.1 - 14 x 0.02 = 1.72 kg/h

rna_reduction_separation

defaults: 68 °C, 30 min, dry-mass loss f 0.30, RNA 0.10 → 0.02, heat regeneration 0.50
m_out        = m_b · (1 − f)
RNA removed  = m_b·rna_in − m_out·rna_out
leachate     = max(m_b·f, RNA removed)  → nucleotides + cell solubles
Q_heat       = ṁ·4.186·(T_hold − T_in)·(1 − regen);   Q_cool = Q_heat

Ester hydrolysis ester_hydrolysis

Splits an ester back into its acid and alcohol with water in a reactive distillation column.

X_ceil = 1 - (1 - X_eq)^n;  X = min(X_req, X_ceil)
X_ceil = 1 - (1 - 0.667)^8 = 1;  X = min(0.98, 1) = 0.98

ester_hydrolysis_separation, hydrolysis_equilibrium_conversion

defaults: water:ester 4.0 mol/mol, K 2.5, 8 reactive stages, target conversion 0.98
water make-up = max(0, r_w·n_ester·18.015 − water in)
equilibrium   (1/K)(1 − X)(r_w − X) = (X + a0)(X + b0)
column        X_ceil = 1 − (1 − X_eq)^n;   X = min(X_req, X_ceil, 0.999)
acid formed   = X·n·MW_acid;   alcohol formed = X·n·MW_alc
heavy (acid)  water = non-water · (1/0.88 − 1);   light (recycle) alcohol × 0.995

Hydrotreater hydrotreater_upgrading

Reacts a bio-oil or lipid with hydrogen over a catalyst to strip out oxygen and leave a hydrocarbon fuel.

M_HC = Y X M_oil
M_HC = 0.8 x 0.98 x 900 kg/h = 706 kg/h

hydrotreating_separation (PNNL-23227)

Y ≤ max(0.05, (CC·0.855 − G·0.5)/0.855)      carbon ceiling, G = gas fraction 0.07
M_H2 = y·M_oil;   M_HC = Y·X·M_oil;   gas = G·M_oil;   X = 0.98
M_H2O = M_oil(1 + y) − M_HC − (1 − X)M_oil − gas     ≤ M_oil·O_frac·18.015/16
catalyst  V = (M_oil/900) / LHSV(0.6);   Q_rxn = 50 MJ/kg H2
beds      n = ⌈ΔT_ad / 55⌉;   compression 1.35 kWh/kg H2

CHP boiler & turbogenerator chp_boiler_turbogenerator

Burns residues or biogas to raise steam, and runs a turbine on it for electricity and process heat.

P_el = eta_el x Q_fuel / 3.6
P_el = 0.15 x 7,360 MJ/h / 3.6 = 307 kW

chp_boiler_separation, _chp_flue_balance (NREL/TP-5100-47764)

defaults: boiler η_b 0.80, net electrical η_el 0.15, back-pressure mode, excess air 0.20
LHV_i     = 0.94·ΔHc, else class (lipid 38, protein 23, sugar 14.6, cell 21 … MJ/kg)
Q_fuel    = m_comb·LHV − m_water·2.44                       MJ/h
P_el      = η_el · Q_fuel / 3.6                             kW
Q_steam   = max(0, η_b·Q_fuel − 3.6·P_el)
flue      O2 = C + H/4 + S − O/2;  air = O2·(1 + EA)/0.2095
          CO2 = C, H2O = H/2 + moisture, N2 = 0.7905·air + N/2
credit    steam and power credited up to the plant's own demand

• Refused below 35% solids: the fuel does not burn self-supporting.

Wastewater treatment wastewater_treatment

Treats the waste streams of the plant to remove their organic load before discharge.

wastewater_treatment_separation · plant cost in wastewater_cost_lines

COD            = Σ c_i · ThOD_i           ThOD from formula, else class (sugar 1.07, lipid 2.9, alcohol 2.1 …)
effluent COD   = COD · (1 − removal)      physicochemical 0.60, aerobic 0.95, anaerobic–aerobic 0.98
PLANT COST (on every waste stream, OSBL)
  hydraulic    = 70 000 · (V_waste/1000)^0.75
  COD capital  = hydraulic · (m − 1)
  installed    × 3.0
  opex /m³     = 2.5 kWh · price + $1.50 chemicals + $0.80 labour + surcharge

  COD tier       ceiling mg/L   m      surcharge $/m³
  low            500            1.0    0
  medium         2 000          1.5    1.5
  high           10 000         2.5    4.0
  very high      50 000         4.0    8.0          (log-linear between tiers)