12.2 Safety Valve Relieving Capacity, Orifice Sizing & Accumulation Test Calculations

12.2 Safety Valve Relieving Capacity, Orifice Sizing & Accumulation Test Calculations

Quick Summary: The safety valve is the final non-negotiable safeguard protecting a steam boiler against overpressure catastrophe. Under ASME Section I (PG-67 through PG-70) and Massachusetts 522 CMR, the safety valve capacity on a power boiler must be sized such that the valves will discharge all the steam the boiler can generate at maximum firing rate without allowing pressure to rise more than 6% above the Maximum Allowable Working Pressure (MAWP). Engineers determine statutory minimum relieving capacity using either ASME Table A-44 heating surface empirical constants, the maximum fuel heat input calculation, or the statutory 3.5 lb/hr per kW rule for electric boilers. Sizing orifice flow areas relies on Napier's formula ($W = 51.45 \times A \times P$). Any boiler having more than 500 square feet of water-heating surface or an electric boiler exceeding 1,100 kW input legally mandates two or more safety valves, verified by physical accumulation testing.

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1. ASME Section I Relieving Capacity Mandates (PG-67)

The fundamental philosophy of boiler overpressure protection dictates that mechanical safety valves must be completely independent of electrical controls, burner management systems, or operator intervention. Even if burner operating pressure controls, high-limit switches, and modulating motors fail simultaneously, keeping the burner firing at 100% maximum thermal firing rate while all steam stop valves are slammed shut, the safety valves must relieve all generated steam safely to atmosphere.

The 6% Accumulation Rule (ASME PG-67.2)

Under ASME Section I, Paragraph PG-67.2:

$\text{Maximum Accumulation Limit} = \text{MAWP} \times 1.06$

The combined relieving capacity of all safety valves mounted on an ASME Section I power boiler must be sufficient to prevent boiler operating pressure from exceeding 6% above the Maximum Allowable Working Pressure (MAWP) during the most severe continuous firing conditions.

Mandatory Multi-Valve Thresholds (ASME PG-67.1)

Under ASME Section I (PG-67.1) and Massachusetts 522 CMR, a single safety valve is permitted only on very small boilers. At least two or more safety valves are legally required when either of the following thresholds is met:

1. The boiler has more than 500 square feet of water-heating surface ($> 500\text{ sq ft}$).
2. The boiler is an electric steam boiler with a power input exceeding 1,100 kW ($> 1,100\text{ kW}$).

When two or more valves are installed, each valve must be sized such that the failure of any one valve does not leave the boiler completely unprotected, and the primary valve must be set to relieve at or below MAWP.

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2. Napier's Formula for Safety Valve Flow and Orifice Sizing

To calculate the theoretical and certified steam flow through a safety valve nozzle or orifice, ASME Section I applies Napier's Formula for Steam Flow. Formulated by Scottish engineer James Robert Napier in the 19th century, this equation governs the choked (critical acoustic velocity) flow of dry saturated steam discharging through an orifice from high pressure into the atmosphere.

Theoretical Formulation

When steam discharges from a high upstream pressure ($P_1$) to atmospheric pressure ($P_2$), if the downstream absolute pressure is less than the critical pressure ratio ($P_2 / P_1 \le 0.577$ for saturated steam), acoustic choking occurs at the nozzle throat. Under choked flow, the mass flow rate is directly proportional to upstream absolute pressure and orifice flow area:

$W = \frac{A \times P \times 3,600}{70} = 51.428 \times A \times P$

ASME Section I Code Equation (PG-69)

ASME Section I formalizes Napier's empirical relationship with a certified discharge coefficient for code safety valves:

$W = 51.45 \times A \times P \times K$

Where:

• $W = \text{Rated relieving capacity of the safety valve in pounds of steam per hour (lb/hr)}$.
• $A = \text{Actual net orifice flow area of the safety valve nozzle throat (square inches, sq in)}$.
• $P = \text{Absolute relieving pressure at 3% accumulation (psia)}$. $P = (1.03 \times P_{\text{set}}) + 14.7$ (Under ASME PG-67.3, safety valves are certified and rated at 3% overpressure above their setpoint).
• $K = \text{Certified coefficient of discharge}$ (determined by National Board flow testing; typically $0.90 \times K_d$ as mandated by ASME). When using standard simplified screening equations, $K$ is factored into rated tables or taken as unity ($1.0$).

Calculating Required Orifice Area ($A$)

Rearranging Napier's formula to determine the required valve orifice throat area for a known steaming capacity:

$A = \frac{W}{51.45 \times P \times K}$

Standard API 526 / ASME Orifice Letter Designations

Safety valves are manufactured with standardized orifice throat sizes designated by letters from D through T:

If an engineering calculation determines that a required valve orifice area is $1.45\text{ sq in}$, the engineer must select the next larger standard orifice size: an orifice letter 'K' ($1.838\text{ sq in}$). Selecting an undersized orifice (such as 'J' at 1.287 sq in) would violate ASME Section I.

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3. Minimum Relieving Capacity Based on Heating Surface (ASME Table A-44)

ASME Section I Table A-44 (referenced by PG-67.4.3) establishes empirical minimum steam generation constants in pounds of steam per hour per square foot of heating surface (lb/hr/sq ft). These constants represent the minimum legal baseline for safety valve capacity based on physical boiler dimensions:

The Radiant vs. Convective Engineering Distinction

• Convective Generating Tubes: In a watertube boiler, the generating bank absorbs heat primarily via convective gas flow, producing 10 lb/hr/sq ft on gas/oil.
• Radiant Waterwalls: In contrast, the furnace waterwalls directly enclose the combustion fireball. Because radiant heat transfer increases with the fourth power of absolute temperature ($q \propto T^4$), waterwalls generate a much higher 16 lb/hr/sq ft on oil/gas firing.
• Governing Equation for Total Heating Surface Relieving Capacity ($W_{\text{surface}}$): $W_{\text{surface}} = (A_{\text{boiler}} \times C_{\text{boiler}}) + (A_{\text{waterwall}} \times C_{\text{waterwall}})$

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4. Minimum Relieving Capacity Based on Maximum Fuel Heat Input

While Table A-44 provides an empirical baseline, modern high-efficiency boilers frequently utilize forced-draft power burners capable of firing rates that generate steam far in excess of Table A-44 values. Under ASME Section I (PG-67.2.1), the minimum safety valve relieving capacity must never be less than the maximum designed steaming capacity of the boiler based on maximum fuel heat input.

The Fuel Heat Input Formula

The thermodynamic relationship determining maximum possible steam generation from fuel firing is:

$W_{\text{fuel}} = \frac{Q_{\text{max}} \times \eta}{h_s - h_f}$

Where:

• $W_{\text{fuel}} = \text{Minimum required safety valve relieving capacity (lb/hr)}$.
• $Q_{\text{max}} = \text{Maximum burner heat input at full continuous firing rate (Btu/hr)}$.
  • Natural gas: $\text{Gas Flow (scfh)} \times \text{HHV (Btu/scf)}$.
  • Fuel oil: $\text{Oil Flow (gal/hr)} \times \text{HHV (Btu/gal)}$.
• $\eta = \text{Boiler thermal efficiency (expressed as a decimal, e.g., 0.80 to 0.85)}$.
• $h_s = \text{Enthalpy of steam at relieving pressure (Btu/lb)}$.
• $h_f = \text{Enthalpy of feedwater entering boiler (Btu/lb)}$.

Simplified ASME Conservative Screening Rule

When detailed steam and feedwater enthalpy data are not stamped on the boiler nameplate, ASME Section I permits a conservative screening formula where the net heat required to produce one pound of steam is taken as approximately 1,000 Btu/lb:

$W_{\text{screen}} = \frac{Q_{\text{max}} \times \eta}{1,000}$

The Governing Code Mandate: The stationary engineer or plant designer must calculate safety valve capacity using BOTH Table A-44 (heating surface) and the maximum fuel heat input method. The LARGER of the two calculated capacities legally governs the installation!

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5. Electric Steam Boilers: The 3.5 lb/hr/kW Rule

Under ASME Section I (PG-67.2) and Massachusetts 522 CMR, electric resistance and electrode steam boilers have no combustion gases or flame radiation. Their maximum steaming capacity is directly constrained by the electrical energy converted into heat within the water volume.

Derivation of the Statutory Constant

Electrical power converts to thermal energy at the exact physical constant: $1\text{ kW} = 3,412.14\text{ Btu/hr}$

At typical power boiler operating pressures, converting liquid feedwater (preheated to ~200°F) into saturated steam requires approximately $970$ to $1,020\text{ Btu/lb}$: $\text{Steam Production per kW} = \frac{3,412.14\text{ Btu/kWh}}{970.3\text{ Btu/lb}} = 3.5166\text{ lb of steam per hour per kW}$

ASME Section I formalizes this physical ceiling into a mandatory statutory rating constant:

$W_{\text{electric}} = \text{Kilowatt Rating (kW)} \times 3.5\text{ lb/hr per kW}$

Dual Safety Valve Mandate for Electric Boilers

Under ASME Section I (PG-67.1): $\text{If } \text{Power Input} > 1,100\text{ kW} \implies \text{Minimum of TWO (2) safety valves required}$ For electric boilers with inputs of $1,100\text{ kW}$ or less, a single safety valve meeting the 3.5 lb/hr/kW requirement is permitted, provided the boiler has 500 sq ft or less of heating surface.

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6. Multi-Valve Sizing and Staggered Setpoint Rules

When a boiler requires two or more safety valves, their opening setpoints and individual relieving capacities must adhere to strict ASME Section I rules (PG-67.4 and PG-70):

  +-------------------------------------------------------------------------+
  |             ASME SECTION I SAFETY VALVE SETPOINT RULES                  |
  |                                                                         |
  |  1. Primary Valve Setting:      Must pop AT or BELOW the MAWP.          |
  |  2. Secondary Valve Setting:    May pop up to 3% ABOVE the highest set  |
  |                                 valve, provided highest setting does    |
  |                                 not exceed MAWP (unless staged).        |
  |  3. Staged Superheater Spread:  Superheater valve must pop FIRST to     |
  |                                 maintain cooling steam flow!            |
  |  4. Total Capacity:             Sum of all valves must equal or exceed  |
  |                                 W_min at 3% overpressure.               |
  +-------------------------------------------------------------------------+

Setting Rules Under ASME PG-67.4

1. Primary Valve (Lowest Set Valve): At least one safety valve must be set at or below the Maximum Allowable Working Pressure (MAWP) stamped on the boiler.
2. Secondary Valves (Staggered Pressure Spread): If additional valves are used, the highest set valve must not be set higher than 3% above the MAWP ($P_{\text{set}} \le 1.03 \times \text{MAWP}$). The complete pressure spread between the lowest set valve and the highest set valve cannot exceed 10% of the highest set pressure.
3. Why Stagger Setpoints? If two or three identical safety valves were set to pop at the exact same pressure (e.g., 200 psig), minor system pressure oscillations would cause all valves to pop simultaneously. This discharges a violent surge of steam that causes dramatic drum water swell, carries water over into the steam headers, and subjects the boiler structure to severe mechanical shock. Staggering the valves (e.g., Valve 1 at 200 psig, Valve 2 at 204 psig) ensures that minor overpressure events are handled smoothly by the primary valve alone.

Superheater Valve Priority Rule (ASME PG-68.2)

In boilers equipped with superheaters, the safety valve on the superheater outlet must be set to pop at a lower pressure than the drum safety valves (accounting for piping pressure drop between drum and superheater outlet). This ensures that steam flow is maintained through the superheater tubes during an overpressure relief event, preventing the alloy superheater tubes from instantly overheating and burning out.

• Capacity Credit Limitation: ASME Section I mandates that superheater safety valve capacity can be credited toward total boiler relieving capacity up to a maximum of 25% of the total required capacity, but must provide at least 10% of total required capacity.

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7. The Accumulation Test: Calculations, Protocol & Pass/Fail Criteria

An accumulation test is the physical verification procedure conducted to prove conclusively that the installed safety valves will protect the boiler under the most catastrophic firing conditions.

                     ACCUMULATION TEST OPERATING CONFIGURATION

        +---------------------------------------------------------------+
        |  • Main Steam Stop Valve:        LOCKED FULLY CLOSED          |
        |  • All Steam Distribution Lines: ISOLATED                     |
        |  • Burner Controls:              OVERRIDDEN TO 100% HIGH FIRE |
        |  • Boiler Feedwater Supply:      OPERATIONAL (Water Level OK) |
        |  • Test Duration:                15 min (Firetube)            |
        |                                  7 min (Watertube)            |
        |  • PASS CRITERION:               Peak P <= 1.06 x MAWP        |
        +---------------------------------------------------------------+

Mandatory Pass/Fail Threshold

$\text{Maximum Allowable Accumulation Pressure} = \text{MAWP} \times 1.06$

• Passing Criterion: If the safety valves pop and discharge all generated steam such that drum pressure stabilizes at or below $1.06 \times \text{MAWP}$, the test is an official PASS. For example, on a 200 psig MAWP boiler, the pressure must never exceed $200 \times 1.06 = 212.0\text{ psig}$.
• Failure Criterion: If boiler pressure climbs past $1.06 \times \text{MAWP}$ (e.g., reaching 213 psig on a 200 psig boiler), the test is an immediate FAIL. The operator must instantly kill the burner and open manual free-blow drains. The safety valves are legally declared undersized, and the boiler cannot be operated until additional or larger relieving valves are installed.

Test Duration Under ASME Section I and Massachusetts 522 CMR

• Firetube Boilers: The accumulation test must be maintained continuously for 15 minutes.
• Watertube Boilers: The accumulation test must be maintained continuously for 7 minutes.
• Why the difference? Watertube boilers contain a much smaller water volume relative to their steaming rate. Running an accumulation test for longer than 7 minutes on a high-steaming watertube boiler risks severe thermal distress or running the feed pumps out of water capacity.

Accumulation Test Exemption (ASME PG-67.4.3)

If an accumulation test is impractical because the boiler operating pressure exceeds 400 psig, or if high-temperature superheaters would be damaged during the test, ASME Section I permits calculation-based capacity certification by an Authorized Inspector using manufacturer certified relieving data in lieu of a physical accumulation test.

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8. Step-by-Step Worked Problems

Worked Problem 1: Napier's Formula Orifice Area Sizing

Problem: A gas-fired steam boiler with an MAWP of 150 psig requires a dedicated safety valve to relieve a steam capacity of 12,500 lb/hr. The valve set pressure is 150 psig. Assume atmospheric pressure is 14.7 psia. Using Napier's formula for ASME Section I valves with an ASME certified discharge coefficient $K = 0.90$, determine:

1. The absolute relieving pressure at 3% accumulation ($P$).
2. The required nozzle throat orifice area ($A$).
3. The standard API 526 orifice letter designation required for this installation.

Step 1: Calculate absolute relieving pressure ($P$) Under ASME Section I, rated capacity is established at 3% overpressure: $P_{\text{gage, relieving}} = P_{\text{set}} \times 1.03 = 150\text{ psig} \times 1.03 = 154.5\text{ psig}$ $P = P_{\text{gage, relieving}} + P_{\text{atm}} = 154.5 + 14.7 = 169.2\text{ psia}$

Step 2: Apply Napier's formula to solve for required orifice area ($A$) $W = 51.45 \times A \times P \times K$ $12,500 = 51.45 \times A \times 169.2 \times 0.90$ $12,500 = 7,834.79 \times A$ $A = \frac{12,500}{7,834.79} = 1.5954\text{ sq in} \approx 1.60\text{ sq in}$

(Note: If calculated using the standard unadjusted screening formula $W = 51.45 \times A \times P$ without separate K factor: $A = 12,500 / [51.45 \times 169.2] = 1.436\text{ sq in}$).

Step 3: Select standard orifice size from API 526 table Looking at standard orifice areas:

• Orifice 'J': $1.287\text{ sq in}$ (Undersized: $1.287 < 1.60$, rejects).
• Orifice 'K': $1.838\text{ sq in}$ (Sufficient: $1.838 > 1.60$, complies).

The engineer must select an orifice letter 'K' safety valve ($1.838\text{ sq in}$). An orifice 'K' valve will provide a certified relieving capacity of: $W_{\text{certified}} = 51.45 \times 1.838 \times 169.2 \times 0.90 = 14,399\text{ lb/hr}$ This provides ample safety margin over the 12,500 lb/hr requirement.

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Worked Problem 2: Sizing Safety Valves for an Industrial Watertube Boiler

Problem: An industrial gas-fired watertube boiler operates at an MAWP of 250 psig.

• Convective boiler generating tube heating surface: 3,200 square feet.
• Furnace radiant waterwall heating surface: 1,200 square feet.
• Maximum burner heat input ($Q_{\text{max}}$): 65,000,000 Btu/hr.
• Boiler thermal efficiency at peak firing: 82% ($0.82$).
• Enthalpy of saturated steam at 250 psig: $h_s = 1,202.1\text{ Btu/lb}$.
• Feedwater temperature entering boiler: 228°F ($h_f = 196.2\text{ Btu/lb}$). Net heat added = $1,202.1 - 196.2 = 1,005.9\text{ Btu/lb}$.

Determine:

1. Minimum relieving capacity from ASME Table A-44 (Heating Surface).
2. Minimum relieving capacity from Maximum Fuel Heat Input.
3. The governing code capacity.
4. The minimum number of safety valves required.
5. Appropriate setpoints and individual capacities.

Step 1: Calculate minimum capacity from Table A-44 For a gas-fired watertube boiler:

• Convective generating surface constant: 10 lb/hr/sq ft.
• Waterwall radiant surface constant: 16 lb/hr/sq ft. $W_{\text{generating}} = 3,200\text{ sq ft} \times 10\text{ lb/hr/sq ft} = 32,000\text{ lb/hr}$ $W_{\text{waterwall}} = 1,200\text{ sq ft} \times 16\text{ lb/hr/sq ft} = 19,200\text{ lb/hr}$ $W_{\text{surface}} = 32,000 + 19,200 = 51,200\text{ lb/hr}$

Step 2: Calculate minimum capacity from Maximum Fuel Heat Input $W_{\text{fuel}} = \frac{Q_{\text{max}} \times \eta}{h_s - h_f} = \frac{65,000,000\text{ Btu/hr} \times 0.82}{1,005.9\text{ Btu/lb}} = \frac{53,300,000}{1,005.9} = 52,987.37\text{ lb/hr}$ (Simplified ASME screening formula: $W = 53,300,000 / 1,000 = 53,300\text{ lb/hr}$).

Step 3: Identify governing code capacity Comparing $51,200\text{ lb/hr}$ (surface) vs $52,987\text{ lb/hr}$ (fuel input): Because $52,987\text{ lb/hr} > 51,200\text{ lb/hr}$, the fuel heat input method governs. The total stamped safety valve capacity must be at least 53,000 lb/hr.

Step 4: Determine number of valves required Total heating surface = $3,200 + 1,200 = 4,400\text{ sq ft} > 500\text{ sq ft}$. Under ASME PG-67.1, at least two (2) safety valves are legally required.

Step 5: Establish set pressures and valve capacities Selecting two valves splitting the capacity equally ($53,000 / 2 = 26,500\text{ lb/hr}$ each):

• Valve 1 (Primary): Set at 250 psig (MAWP). Capacity: 27,000 lb/hr.
• Valve 2 (Secondary): Maximum setting is $\text{MAWP} \times 1.03 = 250 \times 1.03 = 257.5\text{ psig}$. Set at 255 psig. Capacity: 27,000 lb/hr.
• Total Installed Capacity: $27,000 + 27,000 = 54,000\text{ lb/hr} \ge 52,987\text{ lb/hr}$. Fully compliant with ASME Section I and Massachusetts 522 CMR.

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Worked Problem 3: Accumulation Test Evaluation

Problem: An ASME Section I firetube Scotch Marine boiler has a stamped MAWP of 150 psig. During an official accumulation test witnessed by a Massachusetts District Engineering Inspector, the main steam stop valve is locked shut and the burner is locked in manual high-fire at 100% capacity. During the 15-minute test period:

• The primary safety valve pops at 149 psig.
• The secondary safety valve pops at 153 psig.
• Boiler drum pressure continues to rise, peaking at 156.5 psig, where it stabilizes for the remainder of the 15-minute test.

Evaluate: Did the safety valves pass or fail the accumulation test under Massachusetts 522 CMR and ASME Section I?

Solution:

1. Calculate the maximum permissible accumulation pressure: $P_{\text{limit}} = \text{MAWP} \times 1.06 = 150\text{ psig} \times 1.06 = 159.0\text{ psig}$
2. Compare peak recorded pressure to the limit: $P_{\text{peak}} = 156.5\text{ psig}$ $\Delta P_{\text{above MAWP}} = 156.5 - 150.0 = 6.5\text{ psi}$ $\%\text{ Rise} = \left(\frac{6.5}{150.0}\right) \times 100\% = 4.33\%$
3. Conclusion: Because the peak pressure of 156.5 psig is well below the statutory maximum limit of 159.0 psig (a 4.33% rise vs. the allowable 6.0%), and both valves opened at or below their stamped setpoints, the boiler successfully PASSES the accumulation test.