Fluid response to heat transfer without a phase change

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Fluid response to heat transfer without a phase change

Thursday, June 20, 2019

When heat is transferred to or from a single phase fluid, that fluid will expand or contract, and the isobaric cubic expansion coefficient is used to characterize that expansion. Usually this expansion coefficient is taken to be constant; however, there can be a pressure/temperature dependence, especially in the thermodynamic supercritical region, where a method for calculating the peak relieving rate can be used. The fluid response to heat transfer discussed below is one of three elements in the evaluation of relief requirements for overpressure scenarios based on heating or cooling of a constant volume container:1

  1. the characterization of the heat transfer to the container
  2. the fluid behavior in response to that heat transfer
  3. the hydrodynamics within the container

Fluid response without a phase change. The use of the isobaric cubic expansion coefficient is a common method for single-phase systems1,2. The isobaric cubic expansion coefficient, βv, which is a function of temperature and pressure, is defined as:

β_v=1/ρ (d&rho/dT)_P

Where ρ is density, T is temperature, and the derivative is evaluated at constant pressure. The CCPS Guidelines1 and API Standard 5212 provide the means for estimating a cubic expansion coefficient for a liquid given densities at two temperatures (labeled as equation 3.3.1-3 in CCPS Guidelines 2nd edition):

β_v=(ρ_1^2-ρ_2^2)/(2(T_2-T_1 ) ρ_1 ρ_2 )

The CCPS Guidelines 2nd edition recommends a small temperature increment for evaluation; 5°F at the relief temperature was suggested in the CCPS Guidelines 1st edition.

The CCPS Guidelines 2nd edition removed some of the 1st edition cautionary language about tables of cubic expansion coefficients (they are dependent on temperature) as well as the cubic expansion coefficient for an ideal gas. For an ideal gas, the cubic expansion coefficient is the reciprocal of the absolute temperature.

β_v=1/T

For systems containing a fluid having a relief pressure above the thermodynamic critical pressure, the pressure dependence of the cubic expansion coefficient can become more significant. Dividing these supercritical fluids into two categories is useful – those having temperatures low enough to behave similarly to liquids (dense supercritical fluids) and those that do not (sparse supercritical fluids). The thermodynamic critical temperature is a convenient criterion for distinguishing between these types of fluids. The dense supercritical fluids can be modeled similarly to the liquids. An appropriate cubic expansion coefficient for sparse supercritical fluids is difficult to determine; therefore, evaluating the relieving requirements at various temperatures to determine the peak volumetric relieving rate is recommended. See the relief requirements for single phase systems as well as Ouderkirk’s methodology3 for a detailed description.

Blog series information. This blog is part of a series on the proposed updates to the CCPS Guidelines 2nd edition §3.3Venting Requirements for Nonreacting Cases that were removed during final editing. See the general CCPS Guidelines for Pressure Relief and Effluent Handling 2nd Edition review for more information.


[1] AIChE Center for Chemical Process Safety. “CCPS Guidelines for Pressure Relief and Effluent Handling Systems”. 2nd Edition, 2017; New Jersey: John Wiley & Sons, Inc.
[2] American Petroleum Institute. “API Standard 521-Pressure-relieving and Depressuring Systems”. 6th Edition, April 2014.
[3] Ouderkirk R. “Rigorously Size Relief Valves for Supercritical Fluids”. Chemical Engineering Progress. 2002 August; 98(8): 34-43.

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