KEVOS® MAKEOS · Hydraulics
Trapped Liquid Thermal Pressure Check (Blocked-In Line Heating, Pipe Stretch, Relief Need)
If liquid is shut in between closed valves and warms up, how fast does its pressure climb, does it pass the design pressure, and how much liquid must a thermal relief let out?
Pressure rise of a liquid trapped full in a closed pipe or vessel when it warms: the liquid’s volumetric expansion less the growth of the pipe, divided by the liquid’s compressibility plus the stretch of the pipe wall under pressure (thin-walled pipe with closed ends, or a rigid container as the upper bound), giving the pressure rise per kelvin, the final pressure, the largest temperature rise the design pressure allows, the hoop stress at the final pressure and the volume a thermal relief must release, against the design pressure or relief setting.
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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 its property data. |
| Liquid bulk modulus (isothermal) | MPa | Isothermal bulk modulus of the liquid (the reciprocal of its isothermal compressibility). |
| Temperature rise of the trapped liquid | K | How much the trapped liquid warms after it is shut in. |
| Pressure when the section was shut in | bar g | Gauge pressure at the moment the valves were closed. |
| Pipe or vessel | ||
| Wall model | choice | Thin-walled pipe with closed ends that can stretch, or a rigid container for the upper bound. |
| Bore | mm | Bore of the pipe. |
| Wall thickness | mm | Wall thickness of the pipe, less any corrosion allowance. |
| Wall modulus of elasticity | GPa | Modulus of elasticity of the pipe material. |
| Wall Poisson’s ratio | Poisson’s ratio of the pipe material. | |
| Wall linear expansion coefficient | µm/m·K | Linear thermal expansion coefficient of the pipe material. |
| Length of the trapped section | m | Length of pipe between the closed valves. |
| Limit | ||
| Design pressure or relief setting | bar g | Design pressure of the weakest component, or the relief valve setting. |
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 a given volume of the liquid grows for each degree it warms, as a fraction of its volume, at the pressure and temperature it is trapped at.
- Units and format
- Enter a number in ×10⁻⁶ /K: ×10⁻⁶ /K. The form accepts 0 ×10⁻⁶ /K to 5000 ×10⁻⁶ /K. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 207 ×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 grows in proportion to this coefficient less three times the pipe’s linear coefficient.
- Get it right
- Use the volumetric coefficient, about three times a linear one; it rises with temperature, so take a mean over the warming range.
What this means: Liquid bulk modulus (isothermal)
- What it is
- The liquid’s resistance to being squeezed at constant temperature: the pressure needed per unit fractional loss of volume, the reciprocal of its isothermal compressibility.
- 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 raises the pressure rise, up to the limit set by the wall stretch.
- Get it right
- Use the isothermal (secant) value, not the higher adiabatic or sonic value; entrained air lowers it sharply.
What this means: Temperature rise of the trapped liquid
- What it is
- How many degrees the trapped liquid warms after the valves are closed, for example from sun on an exposed line, heat tracing or a hot process nearby.
- Units and format
- Enter a number in K: kelvin (a difference of 1 K equals 1 °C). The form accepts 0 K to 300 K. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 3 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 directly proportional to it.
- Get it right
- Take the worst case: the highest liquid temperature the section can reach less the temperature when it was shut in.
What this means: Pressure when the section was shut in
- What it is
- The gauge pressure inside the section at the moment it was isolated, from which the thermal rise starts.
- Units and format
- Enter a number in bar g: bar g. The form accepts -1 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; the rise itself does not change.
- Get it right
- Use the highest pressure the section can be shut in at, such as the pump shut-off or supply pressure.
Pipe or vessel
What this means: Wall model
- What it is
- Whether the container stretches and grows as a thin-walled pipe with closed ends free to move lengthwise, or is treated as perfectly rigid to give the highest possible rise.
- Units and format
- Choose one of 2 options from the list: Thin-walled pipe, closed ends, free to stretch; Rigid container (liquid alone, upper bound).
- Example
- The form opens with “Thin-walled pipe, closed ends, free to stretch” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- The rigid model removes the wall stretch and growth, so the rise is higher.
- Get it right
- Use the pipe model for straight steel or copper pipe; use the rigid model as a quick upper bound or for very thick-walled parts.
What this means: Bore
- What it is
- The inside diameter of the pipe or vessel holding the trapped liquid, which sets the trapped volume and how much the wall stretches under pressure.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.1 mm to 5000 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 stretches more per bar, which lowers the rise, and holds more liquid.
- Get it right
- Use the actual bore for the pipe schedule or tube size, not the nominal size.
What this means: Wall thickness
- What it is
- The thickness of the pipe or vessel wall resisting the pressure, which with the bore sets how much the wall stretches.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.01 mm to 500 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 rises faster.
- Get it right
- Use the thickness after any corrosion allowance and manufacturing tolerance if you want the least-stiff case.
What this means: Wall modulus of elasticity
- What it is
- The stiffness of the pipe wall material in tension: stress divided by the elastic strain it causes.
- 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 stretches less, so the pressure rises faster.
- Get it right
- Steels are about 200 GPa and copper about 117 GPa; plastics are hundreds of times lower and change with temperature.
What this means: Wall Poisson’s ratio
- What it is
- The ratio of the sideways contraction of the wall material to its lengthwise stretch, used in the wall’s change of volume under pressure.
- 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
- A higher value slightly reduces the wall stretch and raises the pressure rise.
- Get it right
- Metals are close to 0.3; it cannot exceed 0.5.
What this means: Wall linear expansion coefficient
- What it is
- How much the wall material grows in length per degree of warming; the pipe’s volume grows at three times this rate as it warms with the liquid.
- Units and format
- Enter a number in µm/m·K: micrometres per metre per kelvin. The form accepts 0 µm/m·K to 300 µm/m·K. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 12 µm/m·K as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger value takes up more of the liquid’s expansion and lowers the rise.
- Get it right
- Carbon steel is about 12 and stainless steel about 16 µm/m·K; check the material data sheet.
What this means: Length of the trapped section
- What it is
- The length of pipe between the closed valves or other isolation points that holds the trapped liquid.
- Units and format
- Enter a number in m: metres. The form accepts 0.001 m to 100000 m. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20 m as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Changes only the trapped volume and the relief volume; the pressure rise does not depend on it.
- Get it right
- Measure along the pipe between the two closed valves, including bends.
Limit
What this means: Design pressure or relief setting
- What it is
- The highest pressure the section is allowed to reach: the design pressure of its weakest component, or the setting of a relief valve already fitted.
- Units and format
- Enter a number in bar g: bar g. The form accepts 0 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.
- Effect on the result
- Moves the pass line, the largest allowed rise and the relief volume; the rise itself does not change.
- Get it right
- Check flanges, valves, instruments and hoses as well as the pipe; the lowest rating governs.
Method and formulas
- At constant volume the pressure rises at (∂p/∂T)_V = α ÷ κ, the volumetric expansion coefficient over the isothermal compressibility (Guggenheim §1.46); κ = 1 ÷ K.
- A thin-walled pipe with closed ends grows by 3α_w per kelvin and stretches by (D ÷ E t)(5/4 − ν) per unit pressure (hoop stress pD ÷ 2t, axial stress pD ÷ 4t), so Δp = (β − 3α_w) ΔT ÷ (1/K + (D ÷ E t)(5/4 − ν)). A rigid container drops the wall terms.
- Largest rise = (design pressure − initial pressure) ÷ rise per kelvin; relief volume = V [(β − 3α_w) ΔT − (p_d − p_0)(1/K + wall stretch)].
- Example: water shut in at 5 bar g in 20 m of DN50 steel pipe gains 3.2991 bar per kelvin, so 3 K takes it to 14.897 bar g against 16 bar g; 3.3342 K is the most it can take.
Limitations
- The liquid must fill the section with no gas or vapour; a gas pocket lowers the rise a great deal.
- Liquid properties are taken as constant over the rise; for large rises use mean values over the range.
- Thin-walled elastic pipe with free ends; flanges, fittings, valves, hoses and restrained buried pipe stretch differently.
- This is a screening aid: pressure piping design and relief are governed by AS 4041 and a qualified engineer’s review.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Decide whether a pipe section shut in between valves needs a thermal relief valve
- Check a hydraulic line or hose left blocked in while it warms in the sun
- Size the volume a thermal relief has to pass, and show how little that is
Who it is for
- Piping and process engineers
- Hydraulic engineers
- Maintenance and commissioning engineers
What you get
- Pressure rise per kelvin and final pressure
- Largest temperature rise within the limit
- Trapped volume and volume a relief must release
- Hoop stress at the final pressure
Interpreting the result
- Liquids are nearly incompressible, so a few kelvin of warming can raise the pressure by tens of bar; the relief volume needed is only millilitres, which is why a small thermal relief valve is enough.
- The rigid-container result is the upper bound; pipe stretch and growth lower it (by about 28 % for the example’s water in steel pipe).
Before relying on the result
- Take β and K for the actual liquid at the working temperature; both change with temperature and, for oils, with grade.
References
- E. A. Guggenheim, Thermodynamics: An Advanced Treatment for Chemists and Physicists, 5th revised edition (North-Holland, 1967), §1.46 (thermal expansivity and isothermal compressibility). Method reference; no text, tables or examples are reproduced.
- Thin-walled cylinder strains from Hooke’s law with hoop stress pD ÷ 2t and axial stress pD ÷ 4t (standard elasticity).
- AS 4041, Pressure piping (Standards Australia): design pressure and pressure relief of piping. Scope reference only.
Learn the Concepts Behind This Tool
These KEVOS® Library readings explain the method this tool uses, with worked examples.
Useful alongside · Library collection
Pressure vessels and thin-walled cylinders: stresses, design codes, relief valves and inspection
In Machine Elements and Mechanical Design
What it covers: Why pressurised cylinders split lengthwise, how hoop and longitudinal stresses are calculated, how vessels are designed to codes, and how relief valves and inspection keep them safe.
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.