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 $7.6\text{ m}$ / $25\text{ ft}$): $$A_w = \pi D_{shell} H_{liq} + 1.084 D_{shell}^2$$

B. Heat Absorption Rate ($Q$)

$$Q = 43,200 \cdot F \cdot A_w^{0.82} \quad [\text{Watts}]$$ 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 ($W$)

$$W = \frac{Q_{kW} \times 3,600}{\Delta H_{vap}\text{ (kJ/kg)}} \quad [\text{kg/h}]$$

D. Critical Vapor Sizing (API 520 §5.6)

$$A = \frac{W_{lb/h}}{C \cdot K_d \cdot P_1 \cdot K_b \cdot K_c} \sqrt{\frac{T \cdot Z}{M}} \quad [\text{in}^2]$$ 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 ($Q_L$)

$$Q_L = \frac{\beta \cdot Q_{in}}{\rho_L \cdot C_p} \quad [\text{m}^3\text{/s}]$$ 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

$$A = \frac{Q_{L,gpm}}{38 \cdot K_d \cdot K_w \cdot K_v} \sqrt{\frac{G}{\Delta P_{psi}}} \quad [\text{in}^2]$$ 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: $$A = \frac{W_{lb/h}}{C \cdot K_d \cdot P_1 \cdot K_b \cdot K_c} \sqrt{\frac{T \cdot Z}{M}} \quad [\text{in}^2]$$

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

$$\omega = \frac{C_p T_0}{v_0} \left(\frac{v_{fg}}{\Delta H_{vap}}\right)^2 = \frac{\rho_0 C_p T_0}{\Delta H_{vap}^2} \left(\frac{1}{\rho_v} - \frac{1}{\rho_l}\right)^2$$ Where $v_0 = V_{vessel} / m_{batch}$ is the mixture specific volume ($\text{m}^3/\text{kg}$).

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

$$\eta_c = \sqrt{\frac{1}{\omega}} \quad (0.40 \le \eta_c \le 0.85)$$ $$G_{crit} = \frac{P_0}{\sqrt{P_0 \cdot v_0}} \cdot \frac{1}{\sqrt{2 \left[\omega \ln(1/\eta_c) + (\omega - 1)(1 - \eta_c)\right] + 0.5}} \quad [\text{kg/m}^2\cdot\text{s}]$$

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

$$q_{rxn} = m_{batch} \cdot C_p \cdot \left(\frac{dT}{dt}\right){max} \quad [\text{kW}]$$ $$A{m^2} = \frac{q_{rxn} \times 1000}{G_{crit} \cdot \Delta H_{vap} \cdot \sqrt{1 + \omega \cdot \frac{P_0 - P_{set}}{P_{set}}}}$$ $$A_{in^2} = A_{m^2} \times 1550.003$$

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 $\approx 87%$!