KEVOS® MAKEOS · Hydraulics
Blocked-In Liquid Thermal Pressure Rise Check (Trapped Liquid Heated in a Pipe or Vessel)
If a pipe or vessel completely full of liquid is closed off and then warms up, how fast does the pressure rise, what pressure does it reach, and how much warming can it take before it passes the design or relief pressure?
Pressure rise in a liquid trapped between closed valves (or in a sealed vessel) as it warms: the liquid’s volumetric expansion coefficient less the container’s, divided by the liquid’s compressibility (1 ÷ bulk modulus) plus the stretch of a thin pipe wall under pressure (inside diameter ÷ modulus × wall, with the restraint factor for free ends, anchored pipe or expansion joints), gives the pressure rise per kelvin; from it the pressure after the warming you enter, the warming that would reach your limit, and the hoop stress at the final pressure.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Liquid volumetric expansion coefficient | ×10⁻⁶ /K | Volumetric expansion coefficient of the liquid over the temperature range, from the fluid supplier’s data sheet or property data (water near 20 °C about 210, mineral hydraulic oil several times larger). |
| Liquid bulk modulus (isothermal) | MPa | Isothermal bulk modulus of the liquid at the working pressure and temperature, from the fluid data sheet (compressibility is its inverse). |
| Temperature rise of the trapped liquid | K | How much the trapped liquid can warm after it is blocked in: sun, tracing, a hot process or a warm room. |
| Pressure when the liquid was blocked in | bar g | Gauge pressure of the liquid at the moment the valves were closed. |
| Design pressure or relief set pressure (the limit) | bar g | The lower of the design pressure of the weakest component in the blocked section and the set pressure of any relief device. |
| Container | ||
| Container wall | choice | How the container responds: rigid (no wall stretch, conservative) or a thin-wall pipe with its axial restraint. |
| Pipe inside diameter | mm | Inside diameter of the pipe (not used for a rigid container). |
| Pipe wall thickness | mm | Wall thickness of the pipe, less any corrosion allowance you want to ignore (not used for a rigid container). |
| Wall material elastic modulus | GPa | Elastic modulus of the pipe material at the working temperature (steel about 200 GPa). |
| Wall material Poisson’s ratio | Poisson’s ratio of the pipe material (steel about 0.3). | |
| Wall material linear expansion coefficient | ×10⁻⁶ /K | Linear expansion coefficient of the pipe or vessel material (carbon steel about 12); the container’s volume grows three times as fast. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Liquid volumetric expansion coefficient
- What it is
- How much the trapped liquid’s volume grows for each kelvin it warms, per unit of volume: the expansion that the closed space has to resist.
- Units and format
- Enter a number in ×10⁻⁶ /K: ×10⁻⁶ /K. The form accepts 1 ×10⁻⁶ /K to 5000 ×10⁻⁶ /K. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 210 ×10⁻⁶ /K as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The pressure rise is proportional to this coefficient less three times the wall coefficient.
- Get it right
- Liquids expand faster as they warm, so use the average over the temperature range, not the value at the starting temperature.
What this means: Liquid bulk modulus (isothermal)
- What it is
- The liquid’s resistance to being squeezed: the pressure needed to shrink its volume by a given share at constant temperature. Its inverse is the compressibility κ.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 1 MPa to 100000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2200 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A stiffer liquid gives a larger pressure rise for the same warming, up to the limit set by the pipe wall.
- Get it right
- Entrained air lowers the effective bulk modulus sharply; the data-sheet value applies to air-free liquid.
What this means: Temperature rise of the trapped liquid
- What it is
- How many kelvin the liquid warms after the section is closed off, from sun, heat tracing, a hot process or a warm room.
- Units and format
- Enter a number in K: kelvin (a difference of 1 K equals 1 °C). The form accepts 0.01 K to 500 K. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10 K as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The pressure rise grows in direct proportion to the warming.
- Get it right
- Allow for the highest surface temperature in direct sun, which is well above the air temperature.
What this means: Pressure when the liquid was blocked in
- What it is
- The gauge pressure of the liquid at the moment both valves were shut, the starting point of the pressure rise.
- Units and format
- Enter a number in bar g: bar g. The form accepts 0 bar g to 2000 bar g. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 5 bar g as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Adds directly to the final pressure and reduces the warming the section can take before the limit.
- Get it right
- Use the highest pressure the section can be left at, for example the pump shut-off pressure if it is isolated while running.
What this means: Design pressure or relief set pressure (the lim…
- What it is
- The pressure the blocked-in section must not pass: the design pressure of its weakest component or the set pressure of a relief device that protects it.
- Units and format
- Enter a number in bar g: bar g. The form accepts 0.01 bar g to 5000 bar g. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 16 bar g as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Valve bodies, gauges, hoses and instruments in the section may have lower ratings than the pipe; take the lowest.
Container
What this means: Container wall
- What it is
- How the container responds to the pressure: a rigid container that does not stretch, or a thin-walled pipe whose axial restraint sets how much its bore swells.
- Units and format
- Choose one of 4 options from the list: Rigid container (wall stretch ignored); Thin-wall pipe anchored against axial movement; Thin-wall pipe with closed ends free to move; Thin-wall pipe with expansion joints.
- Example
- The form opens with “Thin-wall pipe anchored against axial movement” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- Rigid ignores wall stretch and gives the largest rise; the three pipe options differ only slightly.
- Get it right
- Choose rigid for heavy vessels, short thick sections or when unsure; it gives the highest, safest estimate.
What this means: Pipe inside diameter
- What it is
- The bore of the pipe holding the liquid; with the wall thickness it sets how much the pipe swells under pressure.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 52.5 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger bore for the same wall stretches more and slightly lowers the pressure rise.
- Get it right
- Use the actual inside diameter for the pipe schedule, not the nominal size.
What this means: Pipe wall thickness
- What it is
- The thickness of the pipe wall; a thinner wall stretches more under pressure and carries a higher hoop stress.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 3.91 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A thicker wall stretches less, so the pressure rise is slightly higher and the hoop stress lower.
- Get it right
- Ignore corrosion allowance for a new pipe, but use the remaining wall for a corroded one.
What this means: Wall material elastic modulus
- What it is
- The stiffness of the pipe material in tension, which with the bore and wall sets how much the pipe swells per unit of pressure.
- Units and format
- Enter a number in GPa: gigapascals (1 GPa = 1000 MPa). The form accepts 0.1 GPa to 1000 GPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 200 GPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A stiffer wall material raises the pressure rise towards the rigid-container value.
- Get it right
- Plastic pipe is about a hundred times less stiff than steel, so its stretch takes up far more of the expansion.
What this means: Wall material Poisson’s ratio
- What it is
- The ratio of sideways contraction to stretch in the pipe material, used in the restraint factor that converts hoop and axial strain into a change of volume.
- Units and format
- Enter a plain number with no unit. The form accepts 0 to 0.5. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.3 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Changes the restraint factor slightly; its effect on the pressure rise is small.
- Get it right
- Metals are close to 0.3 and most plastics between 0.35 and 0.45.
What this means: Wall material linear expansion coefficient
- What it is
- How much the pipe or vessel material grows in length per kelvin; the enclosed volume grows three times as fast, giving the liquid a little more room.
- Units and format
- Enter a number in ×10⁻⁶ /K: ×10⁻⁶ /K. The form accepts 0 ×10⁻⁶ /K to 200 ×10⁻⁶ /K. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 12 ×10⁻⁶ /K as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Three times this value is subtracted from the liquid’s coefficient, so a larger value lowers the pressure rise.
- Get it right
- Use the coefficient of the container material, not of the liquid; stainless steel grows faster than carbon steel.
Method and formulas
- For a liquid, V(T, p) = V₀[1 + β(T − T₀) − κ(p − p₀)] with κ = 1 ÷ K, so at constant volume (∂p/∂T)ᵥ = β ÷ κ (Greiner eqs 1.37–1.39 and 4.142–4.143).
- The container also grows: its volume rises by 3αₛ per kelvin, and a thin pipe’s volume grows by c₁ D ÷ (E e) per unit pressure, the volumetric strain 2ε_θ + ε_z from Hooke’s law with hoop stress pD ÷ (2e) and the axial stress set by the restraint: c₁ = 1 − ν² anchored (no axial strain), 5/4 − ν with closed ends free (axial stress pD ÷ 4e) and 1 − ν/2 with expansion joints (no axial stress); a rigid container takes c₁ = 0.
- Pressure rise per kelvin = (β − 3αₛ) ÷ (1 ÷ K + c₁ D ÷ (E e)); final pressure = p₀ + rate × ΔT; warming to the limit = (limit − p₀) ÷ rate; hoop stress = p D ÷ (2e).
- Example: the pressure rises 3.3745 bar for every kelvin, so 10 K of warming takes it to 38.745 bar g against a limit of 16 bar g; the limit is reached after 3.2598 K.
Limitations
- The section must be completely full of liquid: a gas or vapour pocket absorbs the expansion and the real rise is far smaller; even a small pocket matters.
- Coefficients are taken as constant over the range; for large warming or pressures of several hundred bar, use the average β and K over the range.
- Thin-wall elastic pipe; valves, flanges, hoses and fittings in the section stretch differently, and yielding or leakage past valve seats is ignored.
- Screening aid only: relief requirements for real pressure systems are set by the applicable code and a qualified engineer.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Decide whether a section of pipe that can be isolated needs a thermal relief valve
- Check a sealed hydraulic line, heat-transfer oil line or water-filled vessel left in the sun
- Explain to operators why a full, closed line must not be heated
Who it is for
- Piping and process engineers
- Hydraulic and maintenance engineers
- Commissioning and operations staff
What you get
- Pressure rise per kelvin of warming
- Pressure rise and final pressure for the warming entered
- Warming that reaches the limit
- Hoop stress at the final pressure
Interpreting the result
- Because liquids are nearly incompressible, a few kelvin of warming can add tens of bar; this is why a blocked-in liquid section normally needs thermal relief or a procedure that never leaves it full and closed.
- The “liquid share of the stiffness” shows whether the liquid or the pipe wall controls the rise: for steel pipe the liquid usually dominates, so making the pipe stronger does not remove the problem.
References
- Greiner W., Neise L. and Stöcker H., Thermodynamics and Statistical Mechanics (Springer-Verlag, New York, 1995), pp. 22, 111: linear equation of state of a liquid (expansion coefficient α and compressibility κ, eqs 1.37–1.39) and (∂p/∂T)ᵥ = α ÷ κ (eqs 4.142–4.143). Method reference; no text, tables or worked examples are reproduced.
- Timoshenko S. P. and Gere J. M., Mechanics of Materials (PWS Publishing): thin-walled cylindrical pressure vessels and the generalised Hooke’s law used for the volumetric strain of the pipe.
- API Standard 521, Pressure-relieving and Depressuring Systems: hydraulic (thermal) expansion of blocked-in liquid as an overpressure cause; requirements are not reproduced here.
Learn the Concepts Behind This Tool
These KEVOS® Library readings explain the method this tool uses, with worked examples.
Explains the concept · KEVOS® original book
Liquids in closed systems
In Thermal Expansion by the Numbers
What it covers: Calculate how much water a closed heating loop expels as it warms, size a diaphragm expansion vessel from its precharge and maximum pressure, and see what a wrong precharge or no vessel does.
Useful alongside · KEVOS® original book
Wave speed in a pipe: water, wall and air
In Water Hammer by the Numbers
What it covers: Calculate pressure-wave speed from water's bulk modulus, the pipe wall's stiffness and its restraint, and see how a little entrained air or a plastic pipe changes it.
KEVOS® tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.