Part 3 of 6 in the Technical Series: Emergency Pressure Relief & Effluent Handling Systems

# Blog 3: Sizing Relief Systems for Single-Phase & Flashing Two-Phase Flows

Accurately determining the required relief device flow area ($A_n$) and mass flow rate ($W$) is fundamental to process safety engineering. While single-phase gas or liquid sizing follows classical fluid mechanics, two-phase flashing flow introduces dynamic phase equilibrium changes, rapid volumetric expansion, and choking at much higher critical pressure ratios.

# 1. Fundamental Nozzle Flow Model (SI / MKS Units)

The mass flux ($G_o$) through an ideal, frictionless, isentropic relief valve nozzle is calculated by integrating Bernoulli's differential energy balance:

Where:

• $P_o$ = Stagnation pressure inside vessel ($\text{Pa absolute}$)
• $P_n$ = Pressure at nozzle throat ($\text{Pa absolute}$)
• $\rho_n$ = Fluid density at nozzle throat ($\text{kg/m}^3$)

The actual rated mass flow rate ($W_{rated}$) incorporates the certified discharge coefficient ($K_d$) and ASME Code derating factor ($0.90$):

# 2. Single-Phase Flow Sizing Equations

# A. Incompressible Liquid Flow Sizing

For liquids with constant density ($\rho = \text{const}$):

Where: $K_v$ is the viscosity correction factor for $N_{Re} < 10,000$, $K_w$ is the balanced bellows backpressure correction factor, and $P_b$ is backpressure ($\text{Pa}$).

# B. Compressible Gas / Vapor Flow Sizing (Ideal Gas Choked)

Choked flow occurs when fluid velocity at the nozzle throat reaches sonic velocity ($u_n = c$). The critical pressure ratio $\eta_c = P_c / P_o$ is:

For choked gas flow ($\eta \le \eta_c$):

Where (in SI units):

• $C = 3.948 \sqrt{k \left(\frac{2}{k+1}\right)^{\frac{k+1}{k-1}}}$
• $M_w$ = Molecular weight ($\text{kg/kmol}$)
• $T_o$ = Stagnation temperature ($\text{K}$)
• $Z$ = Compressibility factor

# 3. Two-Phase Flashing Flow Sizing (DIERS Methodology)

When a boiling liquid or two-phase mixture relieves, static pressure drops along the nozzle path. Vapor flashes rapidly, expanding the mixture volume and reducing the speed of sound.

# Two-Phase Speed of Sound Phenomenon

The speed of sound in a homogeneous two-phase mixture ($c$) drops dramatically below that of pure gas or liquid:

Example: For a steam-water mixture at void fraction $\alpha = 0.50$, where gas sound speed $c_G = 500 \text{ m/s}$, the two-phase sound speed drops to just $25 \text{ m/s}$! Consequently, two-phase mixtures choke at much higher backpressure ratios ($\eta_c \approx 0.60 - 0.85$) than pure gases ($\eta_c \approx 0.53$).

graph LR
    Vessel[Vessel Stagnation P_o, T_o] --> Nozzle[Relief Valve Nozzle]
    Nozzle --> Flash[Flashing / Vapor Generation]
    Flash --> Choke[Choked Flow at Throat P_c, G_c]
    Choke --> Exit[Discharge Pipe P_b]

# The Leung Omega ($\omega$) Method

The Omega method linearizes the two-phase density relationship relative to stagnation conditions:

# Calculating $\omega$ for Saturated Flashing Fluids:

Where:

• $\alpha_o$ = Initial gas void fraction
• $v_{fo}$ = Liquid specific volume ($\text{m}^3/\text{kg}$)
• $v_{fgo}$ = Specific volume change on vaporization ($\text{m}^3/\text{kg}$)
• $h_{fgo}$ = Latent heat of vaporization ($\text{J/kg}$)
• $C_{po}$ = Liquid heat capacity ($\text{J}/(\text{kg}\cdot\text{K})$)

# Critical Pressure Ratio ($\eta_c$) Determination:

Simpson explicit approximation (accurate to within $2\%$):

# Choked Two-Phase Mass Flux ($G_c$):

# 4. Worked Numerical Example (SI / MKS Units)

Scenario: Bottom relief of saturated liquid Ethylene at stagnation pressure $P_o = 3.261 \text{ MPa}$ ($32.61 \text{ bar}$) and $T_o = -10.55^\circ \text{C}$ ($262.6 \text{ K}$).

• Stagnation Properties:
  • $\rho_{L0} = 384.23 \text{ kg/m}^3$
  • $v_{f0} = 0.002602 \text{ m}^3/\text{kg}$
  • $v_{fg0} = 0.012424 \text{ m}^3/\text{kg}$
  • $h_{fg0} = 246,217 \text{ J/kg}$
  • $C_{pL0} = 5,146 \text{ J}/(\text{kg}\cdot\text{K})$

1. Calculate Omega ($\omega$):

2. Calculate Critical Pressure Ratio ($\eta_c$):

3. Calculate Choked Mass Flux ($G_c$):

4. Calculate Valve Rated Capacity ($W_{rated}$):
   With an effective orifice area $A_n = 1,186 \text{ mm}^2$ ($1.186 \times 10^{-3} \text{ m}^2$) and $K_d = 0.975$:

# ↔️ Series Navigation

• ⬅️ Previous Article: Blog 2: Selecting Pressure Relief Devices & Managing Installation Constraints
• ➔ Next Article: Blog 4: Runaway Reactions & Calorimetry for Emergency Vent Sizing
• 📌 Series Overview & Roadmap: Mastering Pressure Relief & Effluent Handling

Disclaimer: Two-phase relief sizing requires thermodynamic property verification. Use validated simulation tools like SuperChems or CCFlow for detailed design.