17.2 Pressure Relief Valve (PRV) and Rupture Disk Sizing

17.2 Pressure Relief Valve (PRV) and Rupture Disk Sizing

Pressure relief systems represent the definitive line of defense protecting pressurized chemical equipment against catastrophic mechanical failure. The NCEES PE Chemical Exam tests the application of ASME Boiler and Pressure Vessel Code (BPVC) Section VIII and American Petroleum Institute (API) Standards 520, 521, and 526 to size relief valves, calculate fire heat absorption, evaluate backpressure limitations, and select standard orifice areas.

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1. ASME BPVC Section VIII Overpressure Allowances

The fundamental design reference for any unfired pressure vessel is its Maximum Allowable Working Pressure (MAWP)—the maximum gauge pressure permitted at the top of the vessel in its operating position at the designated design temperature. Overpressure allowances are defined as percentage increases above MAWP:

   Pressure Scale: ASME Section VIII Overpressure Allowances
   =========================================================================
   121% MAWP +-----------------------------+ Max Relieving Pressure (Fire Case)
             | Allowed Accumulation: +21%   |
   116% MAWP +-----------------------------+ Max Relieving Pressure (Multiple Non-Fire PRVs)
             | Allowed Accumulation: +16%   |
   110% MAWP +-----------------------------+ Max Relieving Pressure (Single Non-Fire PRV)
             | Allowed Accumulation: +10%   |
   100% MAWP +=============================+ MAWP = Typical Single PRV Set Pressure
             | Normal Operating Margin      |
    90% MAWP +-----------------------------+ Typical Maximum Operating Pressure
             |                              |
      0 psig +-----------------------------+ Atmospheric Pressure Reference

[!NOTE] Overpressure vs. Accumulation: Accumulation is the pressure rise above MAWP during discharge, expressed as a percentage of MAWP. Overpressure is the pressure rise above the valve set pressure. When a PRV is set exactly at the vessel's MAWP, accumulation and overpressure are numerically identical.

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2. Relief Scenario Analysis & API 521 Fire Heat Absorption

Before sizing a relief device, chemical engineers perform an Overpressure Contingency Analysis to establish the credible relief scenarios, including:

1. Blocked Liquid/Vapor Discharge: Control valve or manual isolation valve closed while feed continues.
2. Utility Failure: Complete loss of cooling water to condensers, power outage to air-fin coolers, or loss of instrument air.
3. Heat Exchanger Tube Rupture: High-pressure fluid back-pressuring the low-pressure exchanger shell. The Two-Thirds Rule (or 10/13 Rule in modern ASME) states that tube rupture need not be considered a relief scenario if the low-pressure side design pressure is at least $10/13$ ($76.9%$) of the high-pressure side design pressure.
4. Thermal Liquid Expansion: Solar radiation or tracing heat expanding blocked liquid in piping or heat exchanger tubes.
5. External Pool Fire Engulfment: Heat transfer from surrounding burning liquid pools.

API 521 Wetted Surface Fire Heat Absorption Equation

When a liquid-containing vessel is engulfed in an external hydrocarbons pool fire, heat absorbed by the wetted interior generates boil-off vapor. API Standard 521 defines the total heat absorption ($Q$):

$\mathbf{US\;Customary:}\quad Q = 21{,}000\, F\, A_w^{0.82} \quad (Q\text{ in Btu/hr, } A_w\text{ in }\text{ft}^2)$ $\mathbf{SI\;Metric:}\quad Q = 43{,}200\, F\, A_w^{0.82} \quad (Q\text{ in Watts, } A_w\text{ in }\text{m}^2)$

Where:

• $Q$ = total heat absorption rate into the wetted liquid inventory.
• $F$ = Environment Factor accounting for thermal insulation and installation type:
  • Bare vessel on grade: $F = 1.0$.
  • Insulated vessel with fire-resistant jacketing: $F = 0.075 \text{ to } 0.30$ (calculated from insulation thermal conductivity $k$ and thickness $t$ per API 521 formula $F = \frac{k (1660 - T_{\text{relief}})}{21{,}000 t}$).
  • Water spray deluge: API 521 does not permit credit ($F=1.0$) unless the drainage system, water supply reliability, and spray pattern are proven under extreme fire conditions.
  • Earth-covered / mounded storage: $F = 0.03$.
• $A_w$ = Wetted Surface Area exposed to fire.

[!IMPORTANT] The 25-Foot Elevation Cutoff Rule: API 521 specifies that pool fire flames engulf equipment only up to an elevation of $25\text{ ft}$ ($7.6\text{ m}$) above the source of flame (grade or intermediate solid decking). Any vessel surface located higher than $25\text{ ft}$ above grade is excluded from the wetted area calculation ($A_w$). For spherical vessels, the cutoff is the elevation of the horizontal equator or $30\text{ ft}$ ($9.1\text{ m}$), whichever is greater.

The required mass relieving rate of vapor ($W$) is calculated by dividing the total heat absorption by the latent heat of vaporization ($\Delta H_{\text{vap}}$) evaluated at relieving conditions:

$W = \frac{Q}{\Delta H_{\text{vap}}}$

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3. Pressure Relief Device Mechanical Classifications

  Conventional PRV               Balanced Bellows PRV               Pilot-Operated PRV
+--------------------+         +--------------------+            +---------------------+
| Bonnet Open to     |         | Sealed Bellows     |            | Process Sensed to   |
| Downstream Flange  |         | Isolates Bonnet;   |            | Dome Chamber;       |
| (Backpressure adds |         | Vented to Atm.     |            | Tight Seal up to    |
| directly to spring)|         | (Handles Backpress)|            | Set Point           |
+--------------------+         +--------------------+            +---------------------+

1. Conventional Spring-Loaded PRV

• Mechanics: The process pressure exerts an upward force on the disk ($F = P \times A_{\text{seat}}$). A calibrated mechanical spring exerts a downward closing force. The bonnet is vented directly to the valve discharge nozzle.
• Backpressure Behavior: Superimposed backpressure (pressure in the discharge manifold before the valve lifts) and built-up backpressure (pressure generated in the outlet piping by turbulent flow during relief) act directly on top of the disk.
• Limitation: Maximum allowable total backpressure is $\le 10%$ of set pressure (gauge). If backpressure exceeds $10%$, the effective opening pressure increases, seat chattering occurs, and discharge flow capacity drops precipitously.

2. Balanced Bellows PRV

• Mechanics: A metallic bellows (typically Inconel or stainless steel) seals the disk guide and isolates the upper disk surface and bonnet cavity from discharge backpressure. The bonnet is vented to the atmosphere (or a safe non-hazardous vent).
• Backpressure Behavior: Backpressure acts equally on opposing equal areas of the bellows and disk, neutralizing backpressure effects on set point.
• Limitation: Handles variable superimposed and built-up backpressure from $30%$ up to $50%$ of set pressure. When backpressure exceeds critical flow velocity, capacity must be derated by the backpressure correction factor ($K_b < 1.0$).

3. Pilot-Operated PRV

• Mechanics: Process fluid is piped to a control pilot and into a dome chamber above a main piston/diaphragm. Because the dome area exceeds the seat nozzle area, higher process pressure drives the seat tighter as set point is approached. At set pressure, the pilot rapidly vents the dome, popping the main valve fully open.
• Advantages: Completely insensitive to backpressure (up to $> 70%$); tight shut-off up to $95-98%$ of set pressure.
• Limitations: Not suitable for polymerizing, highly viscous, fouling, or cryogenic services that plug the small pilot sensing tubes.

4. Rupture Disks (Non-Reclosing)

• Thin calibrated diaphragms (forward-acting tension loaded, or reverse-buckling compression loaded) designed to burst at a precise differential pressure.
• Combination with PRVs: When corrosive, fouling, or toxic chemicals threaten PRV seat leakage, a rupture disk is installed upstream of the PRV. ASME Section VIII mandates that the intermediate cavity between the disk and PRV must have a "tell-tale" assembly (pressure gauge, excess flow valve, or pressure switch) to detect disk pinhole pin leaks, because backpressure accumulation in the cavity would raise the effective burst pressure of the disk!
• Combination Capacity Derating: If an ASME-certified combination test flow factor has not been established, the engineer must apply a combination correction factor of $K_c = 0.90$ ($10%$ capacity reduction).

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4. Orifice Sizing Equations (API 520 Part I)

Standard API 526 Orifice Letter Designations

API Standard 526 standardizes 14 flanged relief valve orifice sizes:

Gas and Vapor Sizing (Critical Sonic Flow)

Critical (choked) flow occurs across the PRV nozzle when the downstream backpressure is less than or equal to the critical flow pressure ($P_{cf}$):

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

Where $k = c_p/c_v$ is the ideal gas specific heat ratio. (For $k=1.4$, $P_{cf}/P_1 = 0.528$; for $k=1.13$, $P_{cf}/P_1 = 0.579$).

Under critical flow conditions, API 520 specifies the required effective discharge area ($A$):

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

Where:

• $A$ = effective discharge area ($\text{in}^2$).
• $W$ = required relieving mass flow rate ($\text{lbm/hr}$).
• $C$ = gas expansion coefficient, defined as: $C = 520 \sqrt{k \left( \frac{2}{k+1} \right)^{\frac{k+1}{k-1}}}$ (Representative values: for $k=1.40$, $C = 356.1$; for $k=1.30$, $C = 346.0$; for $k=1.13$, $C = 329.0$).
• $K_d$ = effective discharge coefficient ($K_d = 0.975$ for gas/vapor PRVs with or without rupture disks per API 520).
• $P_1$ = absolute upstream relieving pressure ($\text{psia}$) = Set Pressure + Overpressure + $14.7\text{ psi}$.
• $K_b$ = backpressure capacity correction factor ($K_b = 1.0$ for conventional PRVs under critical flow; for balanced bellows, $K_b$ is read from manufacturer curves when backpressure exceeds $30%$).
• $K_c$ = combination correction factor ($K_c = 1.0$ standalone; $K_c = 0.90$ with upstream rupture disk).
• $T$ = relieving temperature ($^\circ\text{R} = ^\circ\text{F} + 459.67$).
• $Z$ = gas compressibility factor at relieving conditions.
• $M$ = gas molecular weight ($\text{lbm/lbmol}$).

Liquid Sizing (API 520)

For non-viscous liquids, the orifice area is governed by Bernoulli's orifice relation:

$A = \frac{Q}{38\, K_d\, K_w\, K_c\, K_v} \sqrt{\frac{SG}{\Delta P}}$

Where:

• $Q$ = liquid flow rate ($\text{gpm}$).
• $\Delta P = P_1 - P_b$ = differential pressure across the valve ($\text{psi}$), where $P_1$ is gauge relieving pressure and $P_b$ is gauge total backpressure.
• $K_d$ = certified liquid discharge coefficient (standard default $K_d = 0.65$ per API 520).
• $K_w$ = backpressure correction factor for balanced bellows in liquid service ($K_w = 1.0$ for conventional valves).
• $K_v$ = viscosity correction factor (evaluated via Reynolds number $\text{Re} = \frac{Q (2800 \cdot SG)}{\mu \sqrt{A}}$; $K_v = 1.0$ for water and light hydrocarbons where $\mu < 20\text{ cP}$).
• $SG$ = liquid specific gravity at relieving temperature relative to water at $60^\circ\text{F}$.

Two-Phase Flashing Relief: DIERS Methodology

When runaway reactions, boiling liquid foaming, or flashing liquids relieve through a PRV, two-phase vapor-liquid flow occurs. The Design Institute for Emergency Relief Systems (DIERS) developed the Homogeneous Equilibrium Model (HEM) and coupling models. Because entrained dense liquid drastically slows sonic velocity across the nozzle, two-phase mass flux ($G$) is typically $2$ to $10$ times lower than single-phase vapor flux, requiring substantially larger relief areas (often $2$ to $3$ standard orifice sizes larger).

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5. Comprehensive Worked Numerical Example

Problem Statement

A horizontal cylindrical bullet tank contains saturated liquid propane ($M = 44.10\text{ lbm/lbmol}$, $k = 1.13$). Solve the complete relief system sizing evaluation:

1. Vessel Geometry & Wetted Area: The tank has an inside diameter of $D = 8.0\text{ ft}$ and tangent-to-tangent length of $L = 30.0\text{ ft}$, with two 2:1 semi-elliptical heads. The bottom of the vessel is elevated $3.0\text{ ft}$ above grade. Operating procedures specify maximum fill at $80%$ volume, yielding a calculated wetted surface area of $A_w = 720.0\text{ ft}^2$. Verify whether the $25\text{ ft}$ elevation cutoff applies.
2. Fire Case Heat Absorption: The vessel is uninsulated bare steel ($F = 1.0$) on grade with no certified water deluge. Calculate the total fire heat input $Q$ in $\text{Btu/hr}$ per API 521.
3. Relieving Rate & Orifice Sizing: The vessel MAWP is $250.0\text{ psig}$, and the PRV is set at $250.0\text{ psig}$. At the permitted fire overpressure, saturated propane relieves at $T = 136.5^\circ\text{F}$ ($596.17^\circ\text{R}$) with latent heat $\Delta H_{\text{vap}} = 125.0\text{ Btu/lbm}$ and compressibility $Z = 0.82$. The PRV discharges to atmosphere ($P_b = 0\text{ psig} = 14.7\text{ psia}$). Determine the relieving pressure $P_1$, mass relieving rate $W$, gas expansion factor $C$, required effective orifice area $A$ in $\text{in}^2$, and select the appropriate API 526 orifice letter designation.

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Step 1: Vessel Elevation Cutoff Check

• Vessel bottom elevation = $3.0\text{ ft}$ above grade.
• Vessel top elevation = $3.0\text{ ft} + D = 3.0 + 8.0 = 11.0\text{ ft}$ above grade.
• Since the entire vessel resides well below the $25.0\text{ ft}$ elevation threshold ($11.0\text{ ft} < 25.0\text{ ft}$), all wetted surface area is subject to pool fire engulfment.
• Wetted area for sizing: $A_w = 720.0\text{ ft}^2$.

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Step 2: Total Fire Heat Input (API 521)

Calculate $Q$ using the US Customary API 521 formulation with $F = 1.0$:

$Q = 21{,}000\, F\, A_w^{0.82} = 21{,}000 \times 1.0 \times (720.0)^{0.82}$

Evaluate the exponential term:

$\ln(720.0) = 6.57925$ $0.82 \times 6.57925 = 5.394986$ $(720.0)^{0.82} = \exp(5.394986) = 220.297\text{ ft}^{1.64}$ $Q = 21{,}000 \times 220.297 = \mathbf{4{,}626{,}237\text{ Btu/hr}} \approx 4.626 \times 10^6\text{ Btu/hr}$

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Step 3: Required Relieving Rate and Orifice Sizing

Compute mass relieving rate from latent heat:

$W = \frac{Q}{\Delta H_{\text{vap}}} = \frac{4{,}626{,}237\text{ Btu/hr}}{125.0\text{ Btu/lbm}} = \mathbf{37{,}010\text{ lbm/hr}}$

Determine upstream relieving pressure ($P_1$) for external fire:

• ASME Section VIII allows $21%$ overpressure above MAWP for fire contingencies: $P_{\text{rel,gauge}} = 1.21 \times \text{MAWP} = 1.21 \times 250.0\text{ psig} = 302.5\text{ psig}$
• Convert to absolute pressure: $P_1 = P_{\text{rel,gauge}} + P_{\text{atm}} = 302.5 + 14.7 = \mathbf{317.2\text{ psia}}$

Verify critical choked flow across nozzle:

$\frac{P_{cf}}{P_1} = \left( \frac{2}{k+1} \right)^{\frac{k}{k-1}} = \left( \frac{2}{2.13} \right)^{\frac{1.13}{0.13}} = (0.938967)^{8.6923} = 0.579$ $P_{cf} = 0.579 \times 317.2\text{ psia} = 183.7\text{ psia}$ $\text{Actual backpressure } P_b = 14.7\text{ psia} \ll 183.7\text{ psia} \implies \mathbf{Critical\;Flow\;Applies} \;(K_b = 1.0)$

Calculate gas expansion factor $C$ for $k = 1.13$:

$C = 520 \sqrt{k \left( \frac{2}{k+1} \right)^{\frac{k+1}{k-1}}} = 520 \sqrt{1.13 \times (0.938967)^{\frac{2.13}{0.13}}} = 520 \sqrt{1.13 \times 0.35416} = 520 \sqrt{0.4002} = \mathbf{329.0}$

Calculate required orifice area ($K_d = 0.975, K_b = 1.0, K_c = 1.0$):

$\sqrt{\frac{T Z}{M}} = \sqrt{\frac{(596.17) \times (0.82)}{44.10}} = \sqrt{\frac{488.86}{44.10}} = \sqrt{11.0853} = 3.32946$ $A = \frac{W}{C K_d P_1 K_b K_c} \sqrt{\frac{T Z}{M}} = \frac{37{,}010}{(329.0) \times (0.975) \times (317.2) \times 1.0 \times 1.0} \times 3.32946$ $\text{Denominator} = 329.0 \times 0.975 \times 317.2 = 101{,}752.5$ $A = \frac{37{,}010 \times 3.32946}{101{,}752.5} = \frac{123{,}223.3}{101{,}752.5} = \mathbf{1.211\text{ in}^2}$

Select standard API 526 orifice:

• Orifice H ($A = 0.785\text{ in}^2$) is undersized ($0.785 < 1.211$).
• Orifice J ($A = 1.287\text{ in}^2$) is the next standard size and provides sufficient relieving capacity ($1.287 > 1.211$).

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6. Critical PE Exam Traps & Pitfalls

[!WARNING] Trap 1: Applying Fire Overpressure (21%) to Non-Fire Scenarios
ASME Section VIII permits $21%$ overpressure only for fire contingencies. Using $21%$ overpressure for a blocked discharge or utility failure case is a major error; standard non-fire process contingencies strictly limit overpressure to $10%$ for a single valve or $16%$ for multiple valves.

[!WARNING] Trap 2: Gauge vs. Absolute Pressure in Gas Sizing
The upstream relieving pressure $P_1$ in the API 520 gas sizing formula MUST be in absolute pressure ($\text{psia}$). Forgetting to add atmospheric pressure ($14.7\text{ psi}$) to the gauge relieving pressure is one of the most frequent mathematical errors on the exam.

[!WARNING] Trap 3: Conventional PRVs with Excessive Backpressure
Specifying a conventional spring-loaded PRV when backpressure exceeds $10%$ of set pressure will lose points. When total backpressure fluctuates between $10%$ and $50%$, a balanced bellows PRV is required. If backpressure exceeds $50%$, a pilot-operated PRV must be selected.

[!WARNING] Trap 4: Omitting Rupture Disk Combination Factor ($K_c = 0.90$)
When a rupture disk is piped upstream of a PRV without certified ASME flow testing data, the relief area calculation must include $K_c = 0.90$ in the denominator, effectively requiring an $11.1%$ larger flow area.