Purging Calculations – Liquid

Introduction

The average flow rate of [gasses/liquids] undergoing steady-state, isothermal purging can be represented using the Weymouth equation: 

Q = 0.0181 \left(\frac{T_0}{P_{0}}\right)D^{2.666} \sqrt{\frac{P_1^2 – P_2^2}{G T Z L}} = 0.0361D^{2.666} \sqrt{\frac{P_1^2 – P_2^2}{L}) }= C_1 \sqrt{P_1^2 – P_2^2}

Q = 0.0181 \left(\frac{T_0}{P_{0}}\right)D^{2.666} \sqrt{\frac{P_1^2 - P_2^2}{G T Z L}} = 0.0361D^{2.666} \sqrt{\frac{P_1^2 - P_2^2}{L}) }= C_1 \sqrt{P_1^2 - P_2^2}

Where: 
𝑄 – Quantity of gas flowing [Mscf/h] 
𝑃₁ − Purge Gas Injection Pressure [psia] 
P₂ – Blowoff pressure (18-20 [psia]) 
L – Length of gas line [miles]  
D – Internal pipe diameter [in] 
T₀ – Base measurement temperature (60 F = 520 R) 
P₀ – Base measurement pressure (14.73 lb/in^2 absolute) 
G – Specific gravity of natural gas (0.6)
T – Mean flow temperature (60 F = 520 R)

C₁ –

\frac{0.0361D^{2.666}}{\sqrt L}, \text{ where } D \text{ is in inch } \&\, L \text{ is in miles}

\frac{0.0361D^{2.666}}{\sqrt L}, \text{ where } D \text{ is in inch } \&\, L \text{ is in miles}

The blowoff pressure P₂ should be kept between 18-20 psig and maintained. Assuming critical velocity at the blowout valve: 

Q=KP_2=KP_1\sqrt{\frac{C_1^2}{C_1^2+K^2}}

Q=KP_2=KP_1\sqrt{\frac{C_1^2}{C_1^2+K^2}}

Where:  
K – Blowoff valve coefficient [Mscf/(h*psia)]  

The average pressure of a pressurized pipeline is given as:  

P_{ave}=P_1+P_2-\frac{P_1P_2}{P_1+P_2}

P_{ave}=P_1+P_2-\frac{P_1P_2}{P_1+P_2}

The minimum time required to purge the pipeline (in hours) is given in Method A as:

t_m = \frac{\pi L D^2}{Q}\frac{ P_{ave}}{P_0} = \frac{(0.177) D^2 L P_{ave}}{C_1 \sqrt{P_1^2 – P_2^2}}

t_m = \frac{\pi L D^2}{Q}\frac{ P_{ave}}{P_0} = \frac{(0.177) D^2 L P_{ave}}{C_1 \sqrt{P_1^2 - P_2^2}}

Which is algebraically equivalent to Method B: 

T_m = (0.078) D^2 L \frac{1+ \frac{\frac{C_1^2}{C_1^2K^2}}{1+\sqrt{ \frac{C_1^2}{C_1^2K^2} }}} {C_1 \sqrt{1-\frac{C_1^2}{C_1^2+K^2}}}

T_m = (0.078) D^2 L \frac{1+ \frac{\frac{C_1^2}{C_1^2K^2}}{1+\sqrt{ \frac{C_1^2}{C_1^2K^2}    }}}         {C_1  \sqrt{1-\frac{C_1^2}{C_1^2+K^2}}} 

Where: 
t_m– minimum purge time [h] 

It is recommended to purge the pipeline for twice the minimum purge time calculated above, for conservation’s sake.    

t_rec=2t_m

The total volume of natural gas lost in the purging (in Mscf) is given by: 

V_{loss} = Q t – V_0 =\bigg( C_1 \sqrt{P_1^2 – \frac{C_1^2 P_1^2}{C_1^2 – K^2}} \times \frac{t}{60} \bigg) – V_0

V_{loss} = Q t - V_0 =\bigg( C_1 \sqrt{P_1^2 - \frac{C_1^2 P_1^2}{C_1^2 - K^2}} \times \frac{t}{60}  \bigg) - V_0

V₀ – The original volume of (assumed) air purged from the pipe [Mscf]
t – Purge time [min]

Method A

Recommended purge time is 2T. The minimum purge time in minute is

T_a = \left( \frac{0.117 (D^2) L P_M \sqrt{L}}{0.0361 D^{2.667} \sqrt{P_1^2 – P_2^2}} \right)

T_a = \left( \frac{0.117 (D^2) L P_M \sqrt{L}}{0.0361 D^{2.667} \sqrt{P_1^2 - P_2^2}} \right)

𝑇_𝑎 − Minimum Purge Time (min)
𝑇_𝑟𝑎 − Recommended Minimum Purge Time (min)
𝐶₁ = 0.0361𝐷^2.667/√𝐿

C_1 = \frac{0.0361 D^{2.667}}{\sqrt{L}}\~\V_a = \bigg(C_1 \sqrt{P_1^2 – \frac{C_1^2 P_1^2 }{K^2 + C_1^2}} \left(\frac{T}{60}\right) \bigg)- V_1 \~\ V_{ra} = \bigg(C_1 \sqrt{P_1^2 – \frac{C_1^2 P_1^2 }{K^2 + C_1^2}} \left(\frac{2T}{60}\right) \bigg)- V_1

C_1 = \frac{0.0361 D^{2.667}}{\sqrt{L}}\\~\\V_a = \bigg(C_1 \sqrt{P_1^2 - \frac{C_1^2 P_1^2 }{K^2 + C_1^2}} \left(\frac{T}{60}\right) \bigg)- V_1 \\~\\ V_{ra} = \bigg(C_1 \sqrt{P_1^2 - \frac{C_1^2 P_1^2 }{K^2 + C_1^2}} \left(\frac{2T}{60}\right) \bigg)- V_1

𝑉_𝑎 − Volume of Gas Lost − (min) time (MSCF)
𝑉₁ − Actual Volume of Pipe Section − Assume filled with air before purge (MCF)
𝑉_𝑟𝑎 − Volume of Gas Lost − Recommended time (MSCF)

Method B

Recommended purge time is 2T. The minimum purge time in minute is

T_b = 0.0078 D^2 L \frac{1+ \frac{\frac{C_1^2}{K^2+C_1^2}}{1+\sqrt{ \frac{C_1^2}{K^2+C_1^2} }}} {C_1 + \sqrt{\frac{C_1^2}{K^2 + C_1^2}}}

T_b = 0.0078 D^2 L \frac{1+ \frac{\frac{C_1^2}{K^2+C_1^2}}{1+\sqrt{ \frac{C_1^2}{K^2+C_1^2}    }}}         {C_1 + \sqrt{\frac{C_1^2}{K^2 + C_1^2}}} 

𝑇_𝑏 − Minimum Purge Time (min)
𝑇_𝑟𝑏 − Recommended Minimum Purge Time (min)
𝐶₁ = 0.0361𝐷^2.667/√𝐿

C_1 = \frac{0.0361 D^{2.667}}{\sqrt{L}}\~\V_b = \bigg(C_1 \sqrt{p_1^2 – \frac{C_1^2 p_1^2 }{K^2 + C_1^2}} \left(\frac{T}{60}\right) \bigg)- V_1 \~\ V_{rb} = \bigg(C_1 \sqrt{p_1^2 – \frac{C_1^2 p_1^2 }{K^2 + C_1^2}} \left(\frac{2T}{60}\right) \bigg)- V_1

C_1 = \frac{0.0361 D^{2.667}}{\sqrt{L}}\\~\\V_b = \bigg(C_1 \sqrt{P_1^2 - \frac{C_1^2 P_1^2 }{K^2 + C_1^2}} \left(\frac{T}{60}\right) \bigg)- V_1 \\~\\ V_{rb} = \bigg(C_1 \sqrt{P_1^2 - \frac{C_1^2 P_1^2 }{K^2 + C_1^2}} \left(\frac{2T}{60}\right) \bigg)- V_1

𝑉_𝑏 − Volume of Gas Lost − (min) time (MSCF)
𝑉₁ − Actual Volume of Pipe Section − Assume filled with air before purge (MCF)
𝑉_𝑟𝑏 − Volume of Gas Lost − Recommended time (MSCF)

Case Guide

Part 1: Create Case

Input Parameters

• Blow-off Pressure, just Upstream of the Blow-off Valve (Recommended 18-20) (psig)
• For standard blow-off size (in) (dropdown)
• Blow-off Coefficient (MCFH-psi)
• Internal Diameter of Pipe Section (in)
• Length of Pipe Section (ft) (dropdown)

Part 2: Outputs/Reports

Results

• Flowrate at Critical Velocity (MSCF)
• Purge Gas Injection Pressure (psig)
• Actual Volume of Pipe Section (Assume filled with air before purging) (MCF)
• Calculation Method A/B
  • Minimum Purge Time (min)
  • Recommended Purge Time (min)
  • Volume of Gas Lost – Minimum Time (MCF)
  • Volume of Gas Lost – Recommended Time (MCF)

References

• Pipeline Design for Hydrocarbons Gases and Liquids, Committee of pipeline planning, American Association of Civil Engineers, 1975
• Engineering Data Book, Volume II, Gas Processor Association, Revised Tenth Edition, 1994
• Pipeline Design & Construction, A Practical Approach, American Society of Mechanical Engineers, 2000

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