# Emergency Vent & Pressure Relief Sizing (API 520 / 521 & DIERS)

# 1. Objective & Regulatory Standards

The Emergency Vent & Pressure Relief Sizing Suite provides chemical, pharmaceutical, and process safety engineers with rigorous sizing calculations for pressure relief valves (PRV / PSV), emergency vents, and rupture disks in accordance with:

1. API Standard 520 Part I (10th Edition): Sizing, Selection, and Installation of Pressure-Relieving Devices in Refineries & Chemical Plants.
2. API Standard 521 (7th Edition): Pressure-relieving and Depressuring Systems (Fire Exposure Modeling).
3. API Standard 526 (7th Edition): Flanged Steel Pressure-relief Valves (Standard Orifice Designations D through T).
4. AIChE / DIERS (Design Institute for Emergency Relief Systems): Runaway Chemical Reaction Two-Phase Flashing Flow Sizing (Leung $\omega$-Method).

# 2. Overpressure Scenarios & Governing Equations

+-------------------------------------------------------------------------------+
|                       EMERGENCY RELIEF SIZING SUITE                           |
+-------------------+--------------------+------------------+-------------------+
|  1. API 521 Fire  | 2. Liquid Thermal  | 3. Process Vapor | 4. DIERS 2-Phase  |
|  Pool fire boil   | Blocked hydraulics | Steady gas load  | Runaway exotherm  |
|  Q = 43200*F*A^.82|  Q_L = beta*Q/rhoCp| A = W/(C*Kd*P1)  | Leung Omega / HEM |
+-------------------+--------------------+------------------+-------------------+
                                        |
                                        v
                 +---------------------------------------------+
                 |    GOVERNING RELIEF CASE EVALUATION         |
                 |      Max(A_fire, A_liq, A_vap, A_diers)     |
                 +---------------------------------------------+
                                        |
                                        v
                 +---------------------------------------------+
                 |       API 526 STANDARD ORIFICE (D to T)     |
                 +---------------------------------------------+

# 2.1. Scenario 1: External Fire Exposure (API 521 / API 520 Part I)

# A. Wetted Surface Area ($A_w$)

For a vertical cylindrical vessel with liquid fill height $H_{liq}$ (capped at API 521 maximum flame envelope height of $7.6\text{ m}$ / $25\text{ ft}$):

# B. Heat Absorption Rate ($Q$)

Where:

• $F = 1.00$: Bare vessel with no drainage or remote location.
• $F = 0.50$: Bare vessel with good drainage and prompt firefighting.
• $F = 0.30$: Water deluge / spray system.
• $F = \frac{U_{ins} (T_{fire} - T_{rel})}{43,200}$: Insulated vessel with high-temperature insulation ($T_{fire} = 904\text{ K}$).

# C. Vapor Relieving Rate ($W$)

# D. Critical Vapor Sizing (API 520 §5.6)

Where:

• $C = 520 \sqrt{k \left(\frac{2}{k+1}\right)^{\frac{k+1}{k-1}}}$ (Ideal Gas Constant)
• $K_d = 0.975$ (Standard certified vapor discharge coefficient)
• $P_1 = P_{set} \times (1 + \text{Overpressure \%}) + P_{atm}$ (Relieving Absolute Pressure in $\text{psia}$)
• $T = \text{Relieving Temperature in }^\circ\text{R}$ ($T_K \times 1.8$)
• $Z = \text{Vapor compressibility factor}$ ($1.0$ ideal)

# 2.2. Scenario 2: Liquid Thermal Expansion (API 520 Part I §5.9)

Used when trapped liquid in a piping header, heat exchanger shell/tube, or jacketed reactor expands due to external heating without vapor generation:

# A. Volumetric Expansion Rate ($Q_L$)

Where:

• $\beta = \text{Cubical coefficient of thermal expansion } [1/\text{K}]$
• $Q_{in} = \text{Heat absorption duty } [\text{kW}]$
• $\rho_L = \text{Liquid density } [\text{kg/m}^3]$
• $C_p = \text{Specific heat capacity } [\text{J/kg}\cdot\text{K}]$

# B. Liquid Relief Orifice Sizing

Where:

• $K_d = 0.65$ (API certified liquid trim coefficient)
• $G = \text{Specific gravity } (\rho_L / 1000)$
• $\Delta P_{psi} = P_1 - P_{back}$ (Differential relieving pressure in $\text{psi}$)

# 2.3. Scenario 3: Process Vapor / Gas Relief (API 520 §5.6)

For continuous process gas discharges, control valve failure, or steady-state boil-off:

# 2.4. Scenario 4: DIERS Runaway Reaction Two-Phase Flow (Leung $\omega$-Method)

During exothermic runaway reactions, boiling liquid swell and bubbling vapor create homogeneous two-phase flashing flow across the relief nozzle:

# A. Leung Dimensionless $\omega$ Parameter

Where $v_0 = V_{vessel} / m_{batch}$ is the mixture specific volume ($\text{m}^3/\text{kg}$).

# B. HEM Critical Mass Flux ($G_{crit}$)

# C. Required Relief Area ($A$) for Tempered Exotherms

# 3. Standard API 526 Orifice Designations

# 4. Worked Numerical Example & Validation

# Problem Statement:

A $5.0\text{ kL}$ ($5.0\text{ m}^3$) jacketed chemical reactor ($D = 1.6\text{ m}$, $H = 2.2\text{ m}$) containing $3,150\text{ kg}$ of Ethanol ($MW = 46.07\text{ g/mol}$, $\Delta H_{vap} = 846\text{ kJ/kg}$, $C_p = 2.44\text{ kJ/kg}\cdot\text{K}$) is set at $P_{set} = 3.0\text{ bar g}$ ($21\%$ fire overpressure allowance $\implies P_1 = 4.64\text{ bar a}$).

# Step 1: Fire Case Evaluation (API 521)

• Wetted area $A_w = \pi (1.6)(1.76) + 1.084(1.6^2) = 8.84 + 2.78 = 11.62\text{ m}^2$.
• Heat input $Q = 43,200 \times 0.50 \times (11.62)^{0.82} = 162.7\text{ kW}$.
• Relieving rate $W = (162.7 \times 3600) / 846 = 692.4\text{ kg/h}$ ($1,526.4\text{ lb/h}$).
• $C_{gas} = 520 \sqrt{1.13 \times (2/2.13)^{2.13/0.13}} = 330.5$.
• Required Area $A_{fire} = \frac{1526.4}{330.5 \times 0.975 \times 67.3 \times 1.0} \sqrt{\frac{632.4 \times 1.0}{46.07}} = \mathbf{0.258\text{ in}^2}$.
• Selected API Orifice: F ($0.307\text{ in}^2$, $1.5" \times 3"$ Flange).

# Step 2: Two-Phase Runaway Reaction (DIERS at $12^\circ\text{C/min}$)

• Heat generation $q_{rxn} = 3150 \times 2.44 \times (12/60) = 1,537.2\text{ kW}$.
• Leung $\omega = 5.24$, $G_{crit} = 1,180\text{ kg/m}^2\cdot\text{s}$.
• Required Area $A_{diers} = \mathbf{2.04\text{ in}^2}$.
• Selected API Orifice: L ($2.853\text{ in}^2$, $4" \times 6"$ Flange).

# Conclusion:

The DIERS Two-Phase Runaway Reaction is the Governing Case ($A = 2.04\text{ in}^2$), dictating an API 526 Letter 'L' Orifice with $4" \times 6"$ Flanges. Sizing for fire case alone would have severely undersized the safety relief device by $\approx 87\%$!