High-Vacuum & Mass Spectrometry Engineering, Kruger-Shapiro Turbomolecular Pump Physics
Turbomolecular Pump Compression Ratio & H2 Back-Streaming
Calculate Rotor Blade Linear Speed vb (m/s), Maxwell Most-Probable Molecular Speed vm (m/s), Blade-to-Molecular Speed Ratio, and Log10 Zero-Flow Compression Ratio log10(K0).
Why a turbomolecular pump with 100,000,000:1 compression on Nitrogen drops to ~160:1 on Hydrogen—letting H2 back-stream out of your roughing foreline!
**Turbomolecular Vacuum Pumps (`TMPs`)** spin multi-stage slanted titanium/aluminum rotor blades (`Rotational Speed N` RPM, `Mean Blade Radius R` m -> **Linear Blade Velocity `v_b = 2 * PI() * R * (N / 60)`** m/s, typically `380–500` m/s!) in free molecular flow (`Kn > 1`) to bat gas molecules preferentially toward the backing foreline! By **Kruger & Shapiro (`1961`) / Gaede Molecular-Drag Theory**, **Zero-Throughput Maximum Compression Ratio `K_0 = P_{fore} / P_{UHV} = EXP(G * (v_b / v_m))`** (`log_{10}(K_0) = (G * (v_b / v_m)) / LN(10)` across Multi-Stage Geometric Constant `G`) depends exponentially on the ratio of blade speed `v_b` to **Maxwell Most-Probable Thermal Speed `v_m = SQRT((2 * R_{gas} * T) / (M / 1,000))`** (`m/s`, where `R_{gas} = 8.3144626` J/(mol·K) and Molar Mass `M` is in `g/mol`)! Because **`v_m` scales as `1 / SQRT(M)`**, light **Hydrogen (`M = 2.016` g/mol, `v_m = 1,560` m/s)** zips `3.73x` faster than **Nitrogen (`M = 28.01` g/mol, `v_m = 418.5` m/s)**—collapsing `K_0` by **six orders of magnitude** and making `H_2` the dominant residual gas in every UHV system!
Verify on a `60,000` RPM magnetically levitated Turbomolecular Pump (`R = 0.068` m) pumping `295` K Nitrogen (`M = 28.01` g/mol) with Multi-Stage Geometric Factor `G` = 18.5 (B6): Blade Speed `v_b` reaches `427.3` m/s (E2), Nitrogen Molecular Speed `v_m` equals `418.5` m/s (E3), Speed Ratio `v_b / v_m` equals `1.021` (E4), and **`log_{10}(K_0) = 8.20`** (`K_0 = 1.58 x 10⁸` compression ratio! E5)!
| Kruger-Shapiro TMP Compression KPI (60,000 RPM, R=0.068 m, T=295 K, G=18.5) | Formula | Verified Output |
|---|---|---|
| Rotor Blade Linear Speed vb (m/s) | 2 * PI() * Mean_Radius_m * (RPM / 60) | 427.3 m/s |
| Most-Probable Gas Speed vm (m/s) | SQRT((2 * 8.3144626 * T_K) / (MW_g_mol / 1000)) | 418.5 m/s N2 (1,559.9 m/s H2!) |
| Blade-to-Gas Speed Ratio (vb / vm) | Blade_Speed_vb / Molecular_Speed_vm | 1.021 N2 (0.274 H2 supersonic gas!) |
| Log10 Max Compression log10(K0) | (Geom_G * (vb / vm)) / LN(10) | 8.20 -> 1.58x10⁸ N2 (2.20 -> 158x H2!) |
Kruger-Shapiro Exponential Molecular Compression & Molar-Mass Scaling Formula
Validate `Rotor_RPM` in `[5,000, 120,000]`, `Mean_Radius R` in `[0.02, 0.30]` m, safe sub-burst tip speed `v_b <= 620` m/s, `Molar_Mass M` in `[2.0, 350.0]` g/mol, `Gas_Temperature T` in `[77, 500]` K, and `Stage_Factor G` in `[3, 35]`!
=LET(nr_p, B2, rb_m, B3, mw_g, B4, tk_g, B5, gf_s, B6, vb_s, 2 * PI() * rb_m * (nr_p / 60), vm_s, SQRT((2 * 8.3144626 * tk_g) / (mw_g / 1000)), IF(OR(nr_p < 5000, nr_p > 120000, rb_m < 0.02, rb_m > 0.3, vb_s > 620, mw_g < 2, mw_g > 350, tk_g < 77, tk_g > 500, gf_s < 3, gf_s > 35), "Invalid TMP Input", ROUND((gf_s * (vb_s / vm_s)) / LN(10), 2)))Hydrogen (M = 2.016 g/mol) 6-Decade Compression Collapse Check (Set Gas MW B4 = 2.016)
Switch from Nitrogen (`28.01` g/mol) to **Molecular Hydrogen (`B4 = 2.016` g/mol)**: Thermal Speed E3 jumps `3.73x` to `1,559.9` m/s, Speed Ratio E4 drops to `0.274`, and **`log_{10}(K_0)` E5 plunges from `8.20` down to `2.20`** (`K_0 ~= 158:1`!)—proving why UHV chambers need Holweck drag stages or dry two-stage backing pumps to hold foreline `P_{H2} < 10⁻⁷` mbar!
=IF(ISTEXT(E5), E5, ROUND(SQRT((2 * 8.3144626 * B5) / (B4 / 1000)), 1))Try the example
Use the original values or change them to test your understanding.