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p_{a}\cdot S=p_{a}\cdot S_0-C_{R}\left(p_{v}-p_{a}\right)-S_{R}\cdot p_{v}

\frac{p_{a}}{p_{v}}=K_0=\frac{S_0+C_{R}}{C_{R}+S_{R}}

K_0=\frac{S_0}{C_{R}}

K_0=\frac{S_0}{S_{R}}

S=\frac{S_{V}\cdot(p+\Delta p)}{p}

Physical Sign	Unit	Description
S_v	[m³/h]	Pumping speed foreline pump at pressure p_v
p	[mbar]	Inlet pressure of roots pump
\Delta p	[mbar]	Pressure difference adjusted at bypass valve
S	[m³/h]	Pumping speed at roots inlet flange

S_1=\frac{S_0}{1-\frac{1}{K_0}+\frac{S_0}{K_0\cdot S_{V}}}

S=S_0\cdot\left[1-\frac{1}{K_0}\left(\frac{p_{v}}{p_{a}}-1\right)\right]

S=S_0\cdot\left(1-\frac{p_{v}}{K_0\cdot p_{a}}\right)

t_1=\frac{V}{S}\mbox{ln}\frac{p_0}{p_1}

S=\frac{V}{t_1}\mbox{ln}\frac{p_0}{p_1}

S=\frac{2,000l}{300s}\mbox{ln}\frac{1,000}{50}=20\frac{l}{s}=72\frac{m^3}{h}

K_{\Delta}=\frac{p_{v}+\Delta p_{d}}{p_{v}}

p_{a}=\frac{Q}{S}

P_a / hPa	P_v / hPa	S_v / (m³ / h)	Q / (hPa · m³/ h)	K_Δ	K₀	S₁ / (m³ / h)	S₂ / (m³ / h)	t / h	t / s
1,000.0000	1,053.00	90.00	94,770.00	1.05		94.77		0.00490	17.66
800.0000	853.00	92.00	78,476.00	1.07		98.10		0.00612	22.04
600.0000	653.00	96.00	62,688.00	1.09		104.48		0.00827	29.79
400.0000	453.00	100.00	45,300.00	1.13		113.25		0.01359	48.93
200.0000	253.00	104.00	26,312.00	1.27		131.56		0.00652	23.45
100.0000	153.00	105.00	16,065.00	1.53	7.00	160.65	321.56	0.00394	14.18
50.0000	103.00	105.00	10,815.00	2.06	13.00	216.30	382.20	0.00608	21.87
14.9841	56.00	110.00	6,160.00	18.70	18.00	2,053.33	411.10	0.00822	29.58
2.5595	10.00	115.00	1,150.00		36.00		449.30	0.01064	38.30
0.2300	1.00	105.00	105.00		50.00		456.52	0.00670	24.13
0.0514	0.30	75.00	22.50		46.00		437.39	0.00813	29.27
0.0099	0.10	37.00	3.70		40.00		375.17	0.00673	24.23
0.0033	0.06	15.00	0.90		39.00		270.42	0.00597	21.51
0.0018	0.05	5.00	0.25		37.00		135.29
Pump-down time: 344.94 s

Q=p_{vc}\cdot S_1=\frac{R\cdot T}{t}\cdot\left(\frac{m_{water}}{M_{water}}+\frac{m_{air}}{M_{air}}\right)

Where:

S_2=\frac{Q_{air}}{0.3\cdot p_{air}}

S=S_0\cdot\left[1-\frac{1}{K_0}\left(\frac{p_{v}}{p_{a}}-1\right)\right]

p_{a}=\frac{Q}{S_1}

A_{k}=\frac{Q_{water}\cdot m_{water}}{t\cdot\Delta T_{m}\cdot k}

A_k	Condensation surface area	[m²]
Q_water	Specific enthalpy of evaporation	[Ws kg^-1]
m_water	Water vapor mass	[kg]
ΔT_m	Temperature differential between vapor and condensation surface	[K]
k	Thermal transmission coefficient	[W m^-2 K^-1]

p_{b}=\frac{Q_{l}}{S}+\frac{Q_{des,M}(t_3)}{S}+\frac{Q_{des,K}(t_4)}{S}

p_b	Base pressure	[Pa]
Q_l	Gas flow through leaks and permeation	[Pa m³ s^-1]
Q_des,M	Outgassing from the metal surface	[Pa m³ s^-1]
Q_des,K	Outgassing from the seals	[Pa m³ s^-1]

The vessel has the following data:

V	Vessel volume	0.2 m³
A	Vessel surface	1.88 m²
A_k	ealing surface of the FPM seal	0.0204 m²
Q_l		1.0 ⋅ 10 ^-9Pa m³ s^-1
q_desM	Area-related desorption rate of stainless steel	2.7 ⋅ 10 ^-4Pa m³ s^-1 m^-2
q_desK	Area-related desorption rate of FPM	1.2 ⋅ 10 ^-4Pa m³ s^-1 m^-2