Summary
For long sections of pipe, the thermal expansion of trapped liquid can be significant. It is often required that the increase in volume of the fluid be determined in order to select suitable thermal relief valves to protect the integrity of the pipework. This article details how to calculate the required relief flow rate to prevent over pressure due to thermal expansion.
Definitions
| $C_{e}$ | : | Coefficient of thermal expansion (volumetric) |
| $Q$ | : | Thermal relief flow rate |
| $t$ | : | Time over which expansion occurs |
| $T_0$ | : | Liquid initial temperature |
| $T_1$ | : | Liquid final temperature |
| $V_0$ | : | Liquid volume at initial temperature |
| $V_{\Delta}$ | : | Liquid volume change due to thermal expansion |
Introduction
When a volume of liquid is isolated in a section of piping, consideration needs to be given to volume change due to thermal expansion. The liquid volume will generally expand at a greater rate than the volume of the piping, and therefore the pressure in the pipe will rise rapidly. Thermal relief valves or check valves may be used to alleviate the pressure built up in the line.
For small sections of pipe the thermal relief rate is generally small and therefore the calculation of the volume is not necessary as a small relief valve or check valve will have sufficient capacity to prevent pressure build up. For larger pipelines consideration needs to be given to the volume of liquid that will need to be relieved under the highest expected heating conditions to ensure that the thermal relief valve is adequately sized.
Heating Rate
The thermal relief rate will predominately be governed by the heat transfer rate to the locked in fluid. Typical sources of heat transfer to the blocked in pipe may be listed as follows:
- The sun
- Ambient temperature
- Hot nearby process units e.g. furnace or reactor
- Heat tracing
Once the heat transfer rate to the fluid has been calculated, the temperature change of the fluid for the duration of time the fluid is blocked in and exposed to the heat source may be determined. The temperature change of the fluid can then be used to calculate the thermal relief rate as described in the following section.
Calculation of Thermal Relief Rate
The volume change of a liquid for a given temperature changes is calculated as follows:
$$ \displaystyle V_{\Delta} = C_{e} \times \left(T_1 - T_0 \right) \times V_0 $$
To convert the calculated volume into a flow rate divide the volume by the expected time for expansion.
$$ \displaystyle Q = \frac{V_{\Delta}}{t} $$
This calculated flow rate may be conservatively taken as the required capacity of a thermal relief valve as it is the thermal expansion rate at maximum expected heat transfer into the trapped fluid.
Coefficient of Thermal Expansion
Estimating the Coefficient of Thermal Expansion
When a thermal expansion coefficient is not readily available in the literature, it may be estimated from density data at two different temperature points as shown below.
$$ \displaystyle C_{e} = \frac{\rho_0 / \rho_1 - 1}{t_1 - t_0} $$
To ensure the accuracy of this estimate it is recommended that the two data points cover the range of temperatures at over which the volume change will occur. Alternatively if expansion over a large temperature range is being considered, the range can be discretised and the expansion coefficient and subsequent volume expansion of fluid be calculated for each temperature interval.
Typical Values of Coefficient of Thermal Expansion
| Substance | Reference Temperature $^{\circ}C$ | Expansion Coefficient $1/^{\circ}C$ | Reference Temperature $^{\circ}F$ | Expansion Coefficient $1/^{\circ}F$ |
|---|---|---|---|---|
| Acetic Acid | 20 | $1.07\times10^{-3}$ | 68 | $5.94\times10^{-4}$ |
| Acetone | 20 | $1.43\times10^{-3}$ | 68 | $7.94\times10^{-4}$ |
| Aniline | 20 | $8.50\times10^{-4}$ | 68 | $4.72\times10^{-4}$ |
| Benzene | 20 | $1.21\times10^{-3}$ | 68 | $6.72\times10^{-4}$ |
| Bromine | 20 | $1.12\times10^{-3}$ | 68 | $6.22\times10^{-4}$ |
| Carbon Disulphide | 20 | $1.19\times10^{-3}$ | 68 | $6.61\times10^{-4}$ |
| Carbon Tetrachloride | 20 | $1.22\times10^{-3}$ | 68 | $6.78\times10^{-4}$ |
| Chloroform | 20 | $1.27\times10^{-3}$ | 68 | $7.06\times10^{-4}$ |
| Ethanol | 20 | $1.12\times10^{-3}$ | 68 | $6.22\times10^{-4}$ |
| Ether | 20 | $1.63\times10^{-3}$ | 68 | $9.06\times10^{-4}$ |
| Ethyl Bromide | 20 | $1.41\times10^{-3}$ | 68 | $7.83\times10^{-4}$ |
| Ethylene Glycol | 20 | $5.70\times10^{-4}$ | 68 | $3.17\times10^{-4}$ |
| Gasoline | 20 | $9.50\times10^{-4}$ | 68 | $5.28\times10^{-4}$ |
| Glycerol | 20 | $4.90\times10^{-4}$ | 68 | $2.72\times10^{-4}$ |
| Jet Fuel, Kerosene | 20 | $9.90\times10^{-4}$ | 68 | $5.50\times10^{-4}$ |
| Mercury | 20 | $1.82\times10^{-4}$ | 68 | $1.01\times10^{-4}$ |
| Methanol | 20 | $1.18\times10^{-3}$ | 68 | $6.56\times10^{-4}$ |
| Methyl iodide | 20 | $1.20\times10^{-3}$ | 68 | $6.67\times10^{-4}$ |
| Pentane (n) | 20 | $1.58\times10^{-3}$ | 68 | $8.78\times10^{-4}$ |
| Sulphuric Acid | 20 | $5.60\times10^{-4}$ | 68 | $3.11\times10^{-4}$ |
| Toluene | 20 | $1.07\times10^{-3}$ | 68 | $5.94\times10^{-4}$ |
| Turpentine | 20 | $9.60\times10^{-4}$ | 68 | $5.33\times10^{-4}$ |
| Xylene (m) | 20 | $9.90\times10^{-4}$ | 68 | $5.50\times10^{-4}$ |
| Water | 20 | $2.07\times10^{-4}$ | 68 | $1.15\times10^{-4}$ |
| Water | 40 | $3.85\times10^{-4}$ | 104 | $2.14\times10^{-4}$ |
| Water | 60 | $5.22\times10^{-4}$ | 140 | $2.90\times10^{-4}$ |
| Water | 80 | $6.40\times10^{-4}$ | 176 | $3.56\times10^{-4}$ |
Further Reading
- The Safety Relief Valve Handbook: Design and Use of Process Safety Valves to ASME and International Codes and Standards (Butterworth-Heinemann/IChemE)
- Metal Fatigue in Engineering, Second Edition
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