Model, equations & assumptions (SI absolute internally; shown in psig and °C)
Display convention: pressures in psig, temperatures in °C. The model works in Pa absolute and K; conversion at the display only: psig = (P_abs − 101 325 Pa) / 6 894.76 Pa/psi, °C = K − 273.15.
Flowsheet
Feed gas at supply pressure Psup passes a fixed restriction (with a check valve) into the suction header. Stage 1 compresses into the interstage volume (intercooled back to T), stage 2 into the discharge volume (aftercooled). A self-acting back-pressure regulator passes gas from discharge to a product header, from which the consumer valve (product demand) draws to atmosphere. PIC-100 holds suction pressure by recycling gas back to suction: low suction pressure opens RV-1 (interstage → suction) first, then RV-2 (discharge → suction). A relief valve PSV-102 on discharge vents to atmosphere if the product side cannot take the throughput.
Volumes (lumped, isothermal, ideal gas; moles and mass tracked separately so MW is transported)
dn_j/dt = Σ ṅ_in − Σ ṅ_out dm_j/dt = Σ ṁ_in − Σ ṁ_out ṅ = ṁ / M_source
P_j = n_j·R·T / V_j M_j = m_j / n_j ρ_j = m_j / V_j = P_j·M_j / (R·T)
(for a single gas this is the familiar dP/dt = R·T/(V·M) · Σ ṁ)
Diaphragm stages (positive displacement)
ṁ_stage = N · V_swept · η_v · ρ_in
η_v = 1 − c·[(P_out/P_in)^(1/k) − 1], clamped to 0 … 1 (clearance re-expansion; polytropic n taken = k)
T_dis = T·[1 + ((P_out/P_in)^((k−1)/k) − 1) / η_is]
Power = ṁ · c_p · T·((P_out/P_in)^((k−1)/k) − 1) / η_is, c_p = k/(k−1) · R/M_in [J/(kg·K)]
Discharge temperature and power are readouts only; gas leaving each stage is cooled back to T before entering the next volume. *Total shaft power excludes mechanical/hydraulic losses of the diaphragm drive.
Valves and restrictions (compressible, ISA-style)
x = (P_up − P_dn)/P_up, x_c = (k/1.4)·x_T, x_e = min(x, x_c) (choked when x ≥ x_c)
Y = 1 − x_e / (3·x_c)
ṁ = A_eff · Y · √(2 · ρ_up · P_up · x_e) (= A_eff·Y·√(2·ρ_up·ΔP) when unchoked)
A_eff = A_max · (position/100) (linear trim); below x = 0.002 the √ is linearised to avoid the zero-ΔP singularity
τ_v · dv/dt = c − v (recycle valves: first-order stroke lag)
Back-pressure regulator (self-acting, proportional)
lift_cmd = clamp((P_dis − P_set) / band · 100, 0, 100) %, τ_B · d(lift)/dt = lift_cmd − lift
BPR flow: discharge → product header, no reverse flow
A purely proportional regulator has droop: discharge pressure sits above the set pressure by (lift needed / 100)·band, e.g. ~2.3 psi at the default throughput.
Split-range PIC-100
e = SP − P_suction (reverse acting: low suction ⇒ more output ⇒ more recycle)
u = Kp·e + I, I ← I + Kp·Ts/Ti·e + (Ts/Ti)·(u_clamped − u_unclamped) (positional PI, back-calculation anti-windup)
RV-1 command = clamp(u / split · 100, 0, 100)
RV-2 command = clamp((u − split)/(100 − split) · 100, 0, 100)
Kp in % output per psi of error; Ti in s (0 = P only). Changing Kp re-initialises I so the output does not bump. AUTO/MANUAL is bumpless (I tracks in MANUAL). Default tuning Kp = 10 %/psi, Ti = 15 s, Ts = 0.2 s, split = 50 %: a +3 psi SP step settles in about 75 s with about 4 % overshoot.
Default plant (generic, illustrative numbers)
| Item | Value | Unit |
|---|
| Gas temperature T (all volumes) | 27 (300 K) | °C |
| Volumes: suction / interstage / discharge / product hdr | 0.50 / 0.10 / 0.30 / 0.20 | m³ |
| Speed N (fixed) | 400 | rpm |
| Swept volume per rev, stage 1 / 2 | 1.6 / 0.30 | L |
| Clearance c, isentropic eff. η_is | 0.05, 0.75 | – |
| Feed MW (slider 2–200), k (auto from MW, or slider) | 28, 1.39 | g/mol, – |
| Feed supply P_sup, restriction A | 30 psig, 17.5 | mm² |
| RV-1 / RV-2 full-open area | 12 / 7 | mm² |
| BPR area, set, band, lag | 9 mm², 250 psig, 7.5 psi, 0.5 s | |
| Consumer valve area (100 %), default demand | 9 mm², 50 % | |
| PSV-102 set, full lift at, area | tracks BPR: max(BPR + 100, 1.10·BPR) psig, full lift +10 % of set; 350 / 385 psig at the 250 psig default, 1100 / 1210 psig at BPR 1000; 6 mm² | |
| x_T (choke factor), x linearisation | 0.70, 0.002 | – |
Default steady state (MW 28, k 1.39, PIC SP 14.5 psig): suction 14.5, interstage 71.8, discharge 252.3, product header 138.7 psig; PIC out ≈ 27 % (RV-1 ≈ 54 %, RV-2 shut); feed = product ≈ 45 kg/h, stage 1 ≈ 82 kg/h, recycle 1 ≈ 36 kg/h; ratios 2.96 / 3.09 (absolute); T_dis ≈ 171 / 177 °C; power ≈ 3.4 + 2.0 kW.
Things to try
Feed supply drop to 20 psig: RV-1 runs to 100 % and RV-2 picks up past the split point; suction stays on SP. At or below the 14.5 psig suction pressure there is no feed at all (check valve) and the machine runs on full recycle. MW step: lighter gas raises the volumetric flow through the feed restriction, so less recycle is needed and stage 1 runs at a higher ratio; heavier gas does the opposite and pushes the loop toward RV-2; at MW 200 the PIC sits near 95 % (both recycles almost wide open), so there is little margin left for a feed upset with very heavy gas. Below about MW 10 at the default supply pressure (e.g. hydrogen, MW 2), the feed restriction passes more gas than stage 1 can swallow even with both recycles shut, so suction rises above SP (PIC at 0 %): this is a capacity limit, not a bug. Demand step down: a PD compressor will not back off, so with the product side restricted the discharge climbs past the BPR set until stage-2 ratio, discharge temperature, and finally PSV-102 take up the difference.
High back pressure (BPR set up to 1000 psig)
The BPR set goes to 1000 psig. PSV-102 is not a fixed cap: its set tracks the BPR at max(BPR + 100 psi, 1.10 × BPR) psig, with full lift at +10 % of that set, so it is 350 / 385 psig at the default 250 psig and 1100 / 1210 psig at 1000 psig. Because the stages are positive displacement, raising the BPR set barely changes throughput; it just raises the stage-2 ratio. At 1000 psig with MW 28 the interstage rises from 72 to 88 psig, the ratios go from 2.96 / 3.09 to 3.51 / 9.90, η_v falls from 0.94 / 0.94 to 0.93 / 0.79, stage-2 discharge reaches about 392 °C (TAHH, shown on purpose) and power rises from 3.4 + 2.0 to 4.0 + 4.8 kW; feed = product stays 45 kg/h. Filling the discharge volume from 250 to 1000 psig takes about 23 min at MW 28, and the product header is starved while that happens. A real diaphragm machine would be limited to roughly 150–200 °C discharge and would need a third stage, or a bigger interstage pressure, for a ratio of 10. Volumetric efficiency reaches zero only at r = (1 + 1/c)^k, which is about 69 at the default c = 0.05, so it is not hit at 1000 psig. If it were (for example, a very large clearance), the stage would stop pumping, pressures would stall below the BPR set, and the η_v ≈ 0 alarm would show. With demand at 0 % and BPR 1000 psig, PSV-102 relieves the whole throughput at about 1111 psig.
Heavy gas (MW above ~60–80) and k
k_auto(MW) = 1.04 + 0.37 / (1 + (MW/52)^5) → 1.41 (MW 2), 1.39 (28), 1.30 (44), 1.16 (60), 1.05 (100), 1.04 (200)
By default k is not a free input: each volume uses k from this simple correlation at its own local MW (so k changes as a new gas works through the plant). It is a smooth curve through typical values (light diatomics ≈ 1.4, CO₂/propane-range ≈ 1.13–1.3, heavy hydrocarbons → ~1.05) and is illustrative only, since real k depends on composition, not MW alone (CO₂ and propane are both MW 44 but have k 1.29 and 1.13). Moving the k slider unticks k follows MW and uses the slider value everywhere. Above about MW 60–80, ideal gas with a constant k is a poor model at these pressures: heavy gases are near or below their dew point (they would condense in the interstage and discharge, and at 250 psig most C5+ vapours cannot exist as gas at 27 °C), compressibility Z falls well below 1, and k drops toward ~1.05. The sim still runs cleanly to MW 200 (no condensation is modelled), and the low k makes discharge temperatures fall to about 50 °C, but treat MW 100–200 results as qualitative.
Assumptions & simplifications
Ideal gas; all volumes isothermal at T (perfect inter/after-cooling, no heat of compression stored); k from the MW correlation above or a user override (real gas k depends on composition: ~1.4 diatomic, ~1.3 CO₂/CH₄, 1.67 monatomic); no condensation, Z = 1; fixed speed; volumetric efficiency from clearance only (no valve, leakage or diaphragm hydraulic effects); stage flow responds instantly (no pulsation); feed composition mixes perfectly in each volume; BPR is proportional only (has droop); valves have linear trim; the feed has a check valve and the BPR passes no reverse flow; consumer and PSV vent to atmosphere (0 psig). Hobby/learning model, not a design tool.