Safety Relief Valve (PSV) Sizing: API 520 / 526 & Choked Orifice Areas

Executive Summary & Engineering Relevance

In chemical plants, oil refineries, and offshore facilities, Pressure Relief Valves (PRVs or PSVs) are the final independent mechanical layer of protection against catastrophic overpressure, vessel rupture, and major accident hazards (MAH).

When instrumentation interlocks (SIS / SIL loops), DCS control valves, and human operator intervention fail, a self-actuated spring-loaded or pilot-operated relief valve must open reliably to exhaust hazardous fluids to flare, scrubber, or atmosphere.

Under-sizing a relief valve allows internal vessel pressure to exceed the ASME Boiler & Pressure Vessel Code (Section VIII Division 1) allowable accumulation limits, risking structural plastic deformation or boiling liquid expanding vapor explosions (BLEVE). Conversely, oversizing a relief valve by more than $200\% - 300\%$ leads to rapid, violent cycling known as valve chatter—where the disc hammers against the nozzle seat at acoustic frequencies, destroying the seating surfaces, bending the stem, and severely restricting discharge capacity.

This engineering guide details the mathematical equations of API Standard 520 (Part I) for gas, liquid, and steam relief, standard API Standard 526 orifice designations (letters D through T), certified discharge coefficients ($K_d$), backpressure corrections ($K_b, K_w$), and a step-by-step worked industrial example for a propane deethanizer overhead accumulator.

1. Overpressure Protection Codes & Relief Scenarios (API 521)

Before calculating relief areas, process engineers must analyze all credible abnormal scenarios under API Standard 521 to determine the governing case with the largest required mass relief rate ($W$ in $kg/h$ or $lb/h$).

Allowable Accumulation Limits (ASME Sec VIII Div 1 / API 520)

Vessel Pressure Terminology:
Pressure [% of MAWP]
 121% +---------------------------------------------------------------+ External Pool Fire Limit
      |                   21% Overpressure Margin                     |
 116% +---------------------------------------------------------------+ Multiple Valves Non-Fire
      |                   16% Overpressure Margin                     |
 110% +---------------------------------------------------------------+ Single Valve Non-Fire Limit
      |                   10% Overpressure Margin                     |
 100% +===============================================================+ MAWP (Max Allowable Working Pressure)
      |                Normal Operating Buffer (10-15%)               | = Set Pressure Pset
  85% +---------------------------------------------------------------+ Max Normal Operating Pressure

2. API 526 Standard Effective Orifice Designations

To standardize flange dimensions, center-to-face dimensions, and orifice sizes, the American Petroleum Institute (API Standard 526) establishes 14 standard orifice letter designations ranging from D ($0.110\text{ in}^2$) to T ($26.000\text{ in}^2$):

3. Gas & Vapor Relief Sizing Equations (API 520 Part I)

3.1 Critical Choked Flow vs. Subcritical Flow

Gas flow through a convergent relief valve nozzle accelerates to sonic velocity ($Ma = 1.0$) at the throat if upstream relieving pressure exceeds the critical flow pressure ($P_{cf}$):

$$\frac{P_{cf}}{P_1} = \left(\frac{2}{k + 1} \right)^{\frac{k}{k - 1}}$$

Where: * $k = C_p / C_v$: Ideal gas specific heat ratio ($1.05 - 1.67$). * For typical diatomic gases ($k = 1.40$), $P_{cf} / P_1 = 0.528$. If absolute backpressure $P_b \le 0.528 P_1$, flow is critical (sonic choked).

3.2 Critical Flow Sizing Equation (API 520 Part I)

$$A = \frac{W}{C \cdot K_d \cdot P_1 \cdot K_b \cdot K_c} \cdot \sqrt{\frac{T \cdot Z}{M}}$$

Where: * $A$: Required effective discharge orifice area ($mm^2$). * $W$: Required mass flow rate ($kg/h$). * $P_1$: Relieving pressure in $kPa$ absolute: $$P_1 = [P_{set} \cdot (1 + \text{Overpressure})] + P_{atm}$$ * $T$: Absolute relieving temperature ($K$). * $Z$: Compressibility factor of the relieving gas at $P_1, T$. * $M$: Molecular weight of the relieving vapor ($kg/kmol$). * $K_d = 0.975$: Effective discharge coefficient for API gas nozzles. * $K_c = 1.0$: Rupture disk derating factor ($0.90$ if rupture disk installed upstream). * $K_b$: Backpressure correction factor for balanced bellows valves ($1.0$ for conventional valves in critical flow). * $C$: Gas expansion coefficient function of $k$:

$$C = 0.03948 \cdot \sqrt{k \cdot \left(\frac{2}{k + 1} \right)^{\frac{k + 1}{k - 1}}}$$

(In Imperial units where $A$ is in $in^2$, $W$ in $lb/h$, $P_1$ in $psia$, the coefficient is $C = 520 \sqrt{k [2/(k+1)]^{(k+1)/(k-1)}}$).

4. Liquid & Steam Sizing Formulations

4.1 Incompressible Liquid Sizing (API 520 Part I)

$$A = \frac{11.78 \cdot Q}{K_d \cdot K_w \cdot K_c \cdot K_v} \cdot \sqrt{\frac{SG}{P_1 - P_b}}$$

Where: * $Q$: Volumetric flow rate ($m^3/h$). * $P_1 - P_b$: Differential pressure across the valve ($bar$). * $SG$: Fluid specific gravity at relieving temperature. * $K_d = 0.65$: Certified API liquid discharge coefficient. * $K_w$: Balanced bellows liquid backpressure correction factor. * $K_v$: Viscosity correction factor (evaluated via Reynolds number; $K_v = 1.0$ for water and light hydrocarbons).

4.2 Steam Service (Napier Equation per ASME Section VIII)

$$A = \frac{W}{51.5 \cdot P_1 \cdot K_d \cdot K_n \cdot K_{sh} \cdot K_b} \quad [in^2]$$

Where $K_n$ is Napier's high-pressure correction factor and $K_{sh}$ is the superheat derating factor ($1.0$ for saturated steam).

5. Valve Construction Styles & Backpressure Limits

Backpressure acting on the outlet flange of a PSV is divided into Superimposed Backpressure (present before the valve opens) and Built-up Backpressure (generated by flow friction in the discharge flare header).

Relief Valve Construction Types & Limits:
+-----------------------------------------------------------------------------+
| Conventional Spring-Loaded Valve:                                           |
|   - Backpressure Limit: Built-up Pb ≤ 10% of Pset.                          |
|   - Over 10% Pb causes severe disc flutter, set pressure shift, & chatter.   |
+-----------------------------------------------------------------------------+
| Balanced Bellows Valve:                                                     |
|   - Isolates bonnet and spring from flare header backpressure.              |
|   - Backpressure Limit: Pb up to 30%–50% of Pset.                           |
|   - Capacity derating factor Kb applies when Pb / P1 > 30%.                 |
+-----------------------------------------------------------------------------+
| Pilot-Operated Relief Valve (POPRV):                                        |
|   - External sense tube maintains full seating force right up to set point. |
|   - Backpressure Limit: Pb up to 50%–70% of Pset without capacity loss.     |
+-----------------------------------------------------------------------------+

6. Comprehensive Step-by-Step Industrial Worked Example

Design Problem Statement:

Size an API 526 safety relief valve for the overhead reflux drum of a deethanizer fractionation column: * Governing Relief Scenario: Blocked overhead vapor outlet due to inadvertently closed control valve (Operational case, $10\%$ allowable overpressure). * Vessel Design Rating: MAWP = $P_{set} = 18.0\text{ bar gauge}$ ($1,800\text{ kPa g} = 261.1\text{ psig}$). * Required Relief Mass Flow Rate: $W = 28,500\text{ kg/h}$ ($62,832\text{ lb/h}$). * Relieving Fluid: Saturated light hydrocarbon vapor (predominantly ethane/propane mixture): * Molecular Weight: $M = 36.5\text{ kg/kmol}$ * Specific Heat Ratio: $k = 1.22$ * Relieving Temperature: $T = 45.0^\circ\text{C}$ ($318.15\text{ K}$) * Vapor Compressibility Factor: $Z = 0.84$ * Flare Header Backpressure: Built-up backpressure in flare header $P_b = 2.20\text{ bar gauge}$ ($3.21\text{ bar a}$). * Specified Valve Construction: Balanced Bellows Valve ($K_c = 1.0, K_d = 0.975$).

Determine: 1. Upstream relieving pressure ($P_1$) and critical flow pressure ($P_{cf}$). 2. API expansion coefficient ($C$). 3. Backpressure ratio and bellows correction factor ($K_b$). 4. Required effective orifice area ($A_{req}$) in $mm^2$ and $in^2$. 5. Selected standard API 526 orifice letter designation and installed capacity margin.

Step 1: Upstream Relieving Pressure ($P_1$) & Critical Flow Limit

For a single valve non-fire scenario, allowable overpressure is $10\%$:

$$P_{1,\text{barg}} = P_{set} \times 1.10 = 18.0 \times 1.10 = 19.80\text{ bar g}$$

$$P_{1,\text{bara}} = 19.80 + 1.01325 = \mathbf{20.813\text{ bar a}} \quad (2,081.3\text{ kPa a})$$

Critical flow pressure ratio for $k = 1.22$: $$\frac{P_{cf}}{P_1} = \left(\frac{2}{k + 1} \right)^{\frac{k}{k - 1}} = \left(\frac{2}{2.22} \right)^{\frac{1.22}{0.22}} = (0.9009)^{5.545} = \mathbf{0.5602}$$

Critical pressure: $$P_{cf} = 20.813\text{ bar a} \times 0.5602 = \mathbf{11.66\text{ bar a}}$$

Absolute backpressure in the discharge flare header is $P_b = 2.20 + 1.013 = 3.213\text{ bar a}$.

$$\text{Check: } P_b = 3.213\text{ bar a} < P_{cf} = 11.66\text{ bar a} \implies \mathbf{Sonic\ Choked\ Flow\ Verified!}$$

Step 2: Gas Expansion Factor ($C$)

$$C = 0.03948 \cdot \sqrt{1.22 \times \left(\frac{2}{2.22} \right)^{\frac{2.22}{0.22}}}$$ $$\left(\frac{2}{2.22} \right)^{10.091} = (0.9009)^{10.091} = 0.3479$$ $$1.22 \times 0.3479 = 0.4244$$ $$\sqrt{0.4244} = 0.6515$$ $$C = 0.03948 \times 0.6515 = \mathbf{0.02572}$$

Step 3: Backpressure Derating ($K_b$)

Backpressure ratio: $$\frac{P_b}{P_1} = \frac{3.213\text{ bar a}}{20.813\text{ bar a}} = 0.1544 = \mathbf{15.44\%}$$

Per API 520 Figure 30, for a balanced bellows valve in gas service, $K_b$ remains at $1.00$ for all backpressure ratios up to $30\%$. $$K_b = \mathbf{1.00}$$

Step 4: Required Orifice Discharge Area ($A_{req}$)

$$A = \frac{W}{C \cdot K_d \cdot P_1 \cdot K_b \cdot K_c} \cdot \sqrt{\frac{T \cdot Z}{M}}$$

Denominator product: $$\text{Denom} = 0.02572 \times 0.975 \times 2,081.3 \times 1.00 \times 1.00 = 52.193$$

Gas physical properties radical: $$\sqrt{\frac{T \cdot Z}{M}} = \sqrt{\frac{318.15 \times 0.84}{36.5}} = \sqrt{\frac{267.25}{36.5}} = \sqrt{7.3218} = 2.7059$$

Calculating required area: $$A_{req} = \left(\frac{28,500}{52.193} \right) \times 2.7059 = 546.05 \times 2.7059 = \mathbf{1,477.6\text{ mm}^2} \quad (2.290\text{ in}^2)$$

Step 5: API 526 Orifice Selection & Sizing Margin

Comparing $A_{req} = 1,477.6\text{ mm}^2$ against standard API 526 orifice areas: * Orifice K: $A = 1,186\text{ mm}^2$ ($1.838\text{ in}^2$) $\rightarrow$ Undersized ($A < A_{req}$) * Orifice L: $A = 1,841\text{ mm}^2$ ($2.853\text{ in}^2$) $\rightarrow$ Selected!

Installed Orifice Margin: $$\text{Installed Margin} = \frac{1,841 - 1,477.6}{1,477.6} \times 100\% = \mathbf{+24.6\%}$$

Certified Rated Valve Discharge Capacity: $$W_{\text{rated}} = W \times \left(\frac{1,841}{1,477.6} \right) = 28,500 \times 1.246 = \mathbf{35,510\text{ kg/h}}$$

Selected Valve Specification: * Valve Tag: PSV-101 * Size & Rating: $3\text{L}4$ (DN80 inlet $\times$ DN100 outlet), ASME Class 300 inlet $\times$ Class 150 outlet. * Orifice: API 526 Orifice 'L'. * Construction: Balanced Bellows, Carbon Steel body with 316SS trim.

7. Practical Engineering Rules of Thumb & Common Pitfalls

1. The 3% Suction Line Rule (API 520 Part II): The total frictional pressure drop in the piping between the protected vessel and the PSV inlet nozzle must not exceed $3\%$ of the set pressure at rated capacity. Excessive inlet line loss causes destructive rapid valve chatter!
2. The 10% Backpressure Rule for Conventional Valves: Never install a conventional spring-loaded valve if built-up backpressure can exceed $10\%$ of $P_{set}$. Always upgrade to a Balanced Bellows or Pilot-Operated valve.
3. Account for Multi-Component Heavy Ends (API 521 Fire Wetted Surface): For vessels containing liquid mixtures in a pool fire, calculate the wetted surface area $A_{wetted}$ and latent heat of vaporization $\lambda$ at relief temperature: $$Q_{\text{fire}} = 43,200 \cdot F \cdot A_{\text{wetted}}^{0.82} \quad [\text{Watts}], \quad W_{\text{fire}} = \frac{Q_{\text{fire}}}{\lambda}$$
4. Thermal Expansion Relieving (API 521): For blocked liquid lines exposed to solar radiation or heat tracing, standard liquid thermal expansion rates are small ($Q < 1 - 5\text{ m}^3/\text{h}$); a standard $3/4'' \times 1''$ Orifice 'D' valve is universally sufficient.

References & Industry Standards

1. American Petroleum Institute. (2020). API Standard 520: Sizing, Selection, and Installation of Pressure-relieving Devices in Refineries – Part I: Sizing and Selection (10th Edition). API Publishing Services.
2. American Petroleum Institute. (2015). API Standard 520: Part II: Installation (6th Edition).
3. American Petroleum Institute. (2017). API Standard 526: Flanged Steel Pressure-relief Valves (7th Edition).
4. American Petroleum Institute. (2020). API Standard 521: Pressure-relieving and Depressuring Systems (7th Edition).
5. ASME. (2021). ASME Boiler and Pressure Vessel Code, Section VIII, Division 1: Rules for Construction of Pressure Vessels. American Society of Mechanical Engineers.