API Std 521 (2020) — Section 4.4.12 — Three calculation approaches — SI units
Select Calculation Approach
Case 1 — Ambient / Solar Heat Gain
Heat input from ambient temperature rise or solar radiation acting on exposed, blocked-in piping.
Φ = U × Aext × ΔT → used directly in API 521 Eq.(1).
Pipe Geometry
NPS
SCH
From ANSI B36.10 / B36.19. OD and ID auto-filled below.
m
m
m²
Computed as A = π × D × L
Heat Input Method
preset
W/m²K
Still air 5–10 | natural conv. 8–15 | windy 15–30 W/(m²·K)
°C
°C
K
Absorbed solar heat: Φ = A × G × α
where G = solar flux [W/m²], α = pipe surface absorptivity [—]
preset
Moderate climate 500–700 | Conservative design 800–1000 | API thermal relief practice often 800–1000 W/m²
W/m²
preset
Polished metal 0.2–0.3 | Weathered steel 0.6–0.8 | Black painted 0.9–0.95
—
kW
Fluid Properties (for API 521 Eq. 1)
kg/m³
°API
1/°C
look-up
Source: Table A-3 Properties of common liquids (Cengel). Selecting a fluid also updates ρ above.
J/kg·K
Results — Case 1
Heat input methodConvection or Solar—
External area Aextπ × D × L—
Solar: G × αEffective flux—
ΔTT_amb − T_liq—
Heat input Φ = U·A·ΔT—
Specific gravity d—
Relief flow qAPI 521 Eq.(1) — m³/s—
Relief flow qm³/h—
Relief flow qkg/h—
Relief flow qUSGPM—
Calculation Trace — Case 1
Heat input method—
Step 1 — Surface area A = π×D×L—
Step 2 — Solar flux G—
Step 3 — Absorptivity α—
Step 2 — ΔT = T_amb − T_liq—
Step 3/4 — Φ = A·G·α or U·A·ΔT—
Specific gravity d = ρ / 1000—
Step 4/5 — q = (αv × Φ) / (1000×d×c)—
q → m³/h (× 3600)—
q → USGPM (× 264.172 × 60)—
Case 2 — Hot Source Heat Input
The controlling heat input is the maximum credible heat transfer from the hot source — e.g. full exchanger duty, steam tracing capacity, or fired heater output. Enter Φ directly.
Heat Source
Enter the maximum exchanger duty (design heat duty at worst operating case). This is the full Φ to use in API 521 Eq.(1).
Exchanger Duty Helper (optional)
Fill in hot-side data to compute Φ = ṁhot × chot × ΔThot, or enter Φ directly below.
kg/h
J/kg·K
°C
°C
Steam Tracing Helper (optional)
Approximate: Φ = Utrace × L × (Tsteam − Tpipe). Or enter Φ directly.
W/m·K
Typical 10–25 W/(m·K) for bare tubing; lower with insulation
m
°C
°C
Heat Input to API 521 Eq.(1)
W
Auto-filled from helper above, or enter directly. 1 kW = 1000 W | 1 MW = 1 000 000 W
Fluid Properties
kg/m³
°API
1/°C
look-up
Source: Table A-3 Properties of common liquids (Cengel). Selecting a fluid also updates ρ above.
J/kg·K
Results — Case 2
Heat source type—
Heat input ΦW—
Heat input ΦkW—
Specific gravity d—
Relief flow qAPI 521 Eq.(1) — m³/s—
Relief flow qm³/h—
Relief flow qkg/h—
Relief flow qUSGPM—
Calculation Trace — Case 2
Heat input Φ (entered / auto-filled)—
Specific gravity d = ρ / 1000—
αv used—
Numerator: αv × Φ—
Denominator: 1000 × d × c—
q [m³/s] = num / den—
q → m³/h (× 3600)—
q → USGPM (× 264.172 × 60)—
Alternative — Volume Expansion Rate
Used when the temperature rise rate dT/dt is known directly — typical for long pipelines with solar heating. Does not require calculating heat input Φ.
Q = V × β × (dT/dt)
Pipe / Vessel Geometry
NPS
SCH
From ANSI B36.10 / B36.19. ID auto-filled below.
m
m
m³
For vessels, override with actual liquid volume below
m³
Thermal Data
kg/m³
Used to convert volumetric flow to mass flow (kg/h)
°API
1/°C
Same as αv in API 521 Table 2; typical hydrocarbons 0.0007–0.0016 1/°C
°C/h
Typical solar heating: 2–10 °C/h. For steam tracing use Case 2 instead.
Results — Alternative
Trapped liquid volume Vm³—
β (expansion coefficient)—
dT/dt (temperature rise rate)—
Relief flow Q = V·β·(dT/dt)m³/h—
Relief flow Qkg/h—
Relief flow Qm³/s—
Relief flow QUSGPM—
Calculation Trace — Alternative
Pipe ID used—
V = (π/4) × ID² × L—
V used (override if set)—
β expansion coeff.—
dT/dt temperature rise—
Q [m³/h] = V × β × dT/dt—
Q → kg/h (× ρ)—
Q → USGPM (× 4.403)—
Theory & Reference — API Std 521 Section 4.4.12
When is thermal expansion (hydraulic expansion) relief required?
A thermal relief valve is required when liquid-full piping or equipment can be blocked in and subsequently heated, causing pressure build-up from liquid expansion. The three most common sources:
Ambient / solar heating — exposed piping blocked in at low temperature, then heated by sun or ambient air
Hot source — cold side of heat exchanger blocked in with hot side still flowing; steam tracing; near fired heater
Pipeline temperature rise — long above-ground line with known solar temperature rise rate
⚠ API 521 §4.4.12.1 — Critical Safety Cautions
Caution 1 — Bubble-point / BLEVE risk: This calculator sizes the PRD for sub-cooled liquid expansion only. If the trapped liquid can be heated above its bubble-point temperature at the relief pressure, vaporisation can occur while the fluid is still contained. This results in far higher relief loads and a potential BLEVE unless a significantly larger PRD is installed. See API 521 §4.4.13.2.5.3 for guidance on vapour generation sizing.
Caution 2 — Superheat / SLT: If the contained fluid can be heated above its Superheat Limit Temperature (SLT), equipment failure due to thermal hydraulic expansion can result in a BLEVE — not a minor flange release. See API 521 §4.4.6 for the SLT discussion. Always verify the maximum credible fluid temperature against both the bubble-point at relief pressure and the SLT before accepting the thermal relief sizing from this calculator.
Case 1 — Heat input from ambient (Φ = U·A·ΔT)
Φ = U × Aext × ΔT [W]
Symbol
Description
Unit
U
Overall heat transfer coefficient
W/(m²·K)
Aext
External pipe surface area = π × D × L
m²
ΔT
Temperature difference ambient − liquid
K
Typical U values:
Condition
U [W/(m²·K)]
Still air
5 – 10
Outdoor natural convection
8 – 15
Windy conditions
15 – 30
Solar radiation included
significantly higher
The resulting Φ is then fed into API 521 Eq.(1) to find the required relief rate q.
Case 2 — Hot source (exchanger / steam tracing / fired heater)
The controlling heat input is the maximum credible heat transfer rate from the hot source. These cases typically produce much larger relief loads than ambient heating:
Steam tracing: tens of kW typical
Blocked heat exchanger: full exchanger duty, hundreds of kW to MW
Fired heater: always use maximum absorbed duty
For exchangers: Φ = ṁhot × chot × (Tin − Tout) [W] (convert from kg/h using ÷ 3600).
This Φ is entered directly into API 521 Eq.(1).
API 521 Eq.(1) — Relief flow rate (SI)
q [m³/s] = (αv × Φ) / (1000 × d × c)
Symbol
Description
Unit
αv
Cubic expansion coefficient
1/°C
Φ
Total heat transfer rate
W
d
Specific gravity vs water at 15.6°C
—
c
Specific heat of trapped liquid
J/(kg·K)
Alternative — Volume expansion rate (Q = V·β·dT/dt)
Q [m³/h] = V × β × (dT/dt)
Symbol
Description
Unit
V
Trapped liquid volume = (π/4)×ID²×L
m³
β
Volumetric expansion coefficient (= αv)
1/°C
dT/dt
Maximum temperature rise rate
°C/h
Note: units must be consistent — if dT/dt is in °C/h, Q is in m³/h.
Typical αv values — API 521 Table 2 (at 15.6 °C)
Gravity (°API)
αv [1/°C]
αv [1/°F]
3 – 34.9
0.00072
0.0004
35 – 50.9
0.0009
0.0005
51 – 63.9
0.00108
0.0006
64 – 78.9
0.00126
0.0007
79 – 88.9
0.00144
0.0008
89 – 93.9
0.00153
0.00085
94 and lighter
0.00162
0.0009
Water
0.00018
0.0001
Valve selection — API 521 Sec. 4.4.12.2
Because thermal expansion flows are very small, a DN 20 × DN 25 (NPS ¾" × NPS 1") thermal relief valve is commonly sufficient and is the standard minimum per API 521. Only when there is reason to believe this size is inadequate should the procedure above be used to size a larger valve.