Common formulas for vacuum calculations

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1. Boyle’s law: Volume V, pressure P; P·V = constant (For a gas of constant mass, when the temperature remains unchanged, the pressure of the gas is inversely proportional to its volume.) That is, P1/P2 = V2/V1). 2. Gay-Lussac’s law: When the pressure P remains constant, for a given mass of gas, its volume V is proportional to the absolute temperature T: (V1/V2 = T1/T2 = constant). When the pressure remains constant, for a given mass of gas, an increase (or decrease) in temperature of 1°C results in a corresponding increase (or decrease) in volume of 1/273. 3. Charles’s Law: When the volume V of a gas remains constant, for a given mass of gas, the pressure P is proportional to its absolute temperature T; that is, P1/P2 = T1/T2. At a constant volume, for a given mass of gas, every increase (or decrease) of 1°C in temperature results in an increase (or decrease) in pressure of 1/273 relative to the original value. 4. Mean free path: λ = (5×10^-3)/P (cm)
5. Pumping speed: S = dv/dt (liters/second), or S = Q/P. Where Q is the flow rate (torr·liters/second), P is the pressure (torr), V is the volume (liters), and t is the time (seconds).
6. Conductance: C = Q/(P2 – P1) (liters/second)
7. Vacuum pumping time: For pumping from atmospheric pressure to 1 torr, the formula for calculating the time is t = 8V/S (an empirical formula). Here, V is the volume and S is the pumping rate; typically, the value of t ranges from 5 to 10 minutes. 8. Selection of the maintenance pump: S維 = S前/10
9. Estimation of the pumping speed of a diffusion pump: S = 3D² (D = diameter in cm)
10. Pumping speed of the pre-stage pump for a Roots pump: S = (0.1–0.2)S罗 (l/s)
11. Leakage rate: Q漏 = V(P2 – P1)/(t2 – t1); Q漏 represents the system’s leakage rate in mmHg·l/s, where V is the system volume in liters, P1 is the pressure in the system when the vacuum pump is stopped in mmHg, P2 is the pressure reached in the vacuum chamber after time t in mmHg, and t is the time it takes for the pressure to rise from P1 to P2 in seconds.
12. Selection of the pumping speed for a roughing pump: S = Q1/P预 (l/s); alternatively, S = 2.3V·lg(Pa/P预)/t. Here, S is the effective pumping speed of the mechanical pump, Q1 is the leakage rate of the vacuum system in torr·liter/second, P预 is the desired pre-vacuum level in torr, V is the volume of the vacuum system in liters, t is the time required to reach P预, and Pa is the atmospheric pressure in torr.
13. Selection of the pumping speed for the pre-stage pump: For transfer pumps such as diffusion pumps, oil-enhanced pumps, Roots pumps, and turbomolecular pumps, whose exhaust pressure is below one atmosphere, a pre-stage pump is needed to keep the pressure before them below a critical value. The pre-stage pump must be capable of removing the maximum amount of gas produced by the main pump. Based on the principle that the flow rate at each section of the pipeline remains constant, we have: PnSg ≥ PgS or Sg ≥ Pgs/Pn. Here, Sg is the effective pumping speed of the pre-stage pump in l/s, Pn is the critical pre-stage pressure of the main pump (the maximum exhaust pressure) in l/s, Pg is the highest operating pressure in the vacuum chamber in torr, and S is the effective pumping speed of the main pump at pressure Pg. (l/s) 14. Formula for calculating the pumping speed of a diffusion pump: S = Q/P = (K·n)/(P·t) (liters per second). Where: S – the pumping speed of the pump under test (l/s); n – the number of divisions the oil column rises by in the dropper (divisions); t – the time required for the oil column to rise by n divisions (seconds); P – the pressure measured near the pump outlet (torr); K – the dropper coefficient (torr·liters per second). K = V0·(L/n)·(Υ0/Υm) + Pa△Vt. V0 – the initial volume of the dropper and vacuum tubing (liters); L – the length of the scaled portion of the dropper (mm); n – the number of divisions on the scaled portion of the dropper (divisions); Υ0 – the specific gravity of the oil (grams/cm³); Υm – the specific gravity of mercury (grams/cm³); Pa – the local atmospheric pressure (torr); △Vt – the volume corresponding to one division on the dropper’s scale (liters/division). 15. Formula for calculating the geometric pumping speed of a rotary vane vacuum pump: S = πZnLKv(D2-d2)/(24×104) (l/s). Where: Z is the number of vanes; n is the rotational speed (revolutions per minute); L is the length of the pump chamber; D is the diameter of the pump chamber; d is the diameter of the rotor (cm); Kv is the volume utilization factor (usually 95%). 16. For an O-shaped rubber groove, the depth B = 0.7D, where D is the diameter of the rubber; the width C = 1.6B. 17. For a square rubber groove, the depth B = 0.8A