Bernoulli Principle applied to relief devices

The bernoulli equation, applied to general fluids, both incompressible or compressible is:

g(z₂ - z₁)+ 1/2(V₂² - V₁²) + (H₂ - H₁) = W + Q
It assumes the flow is steady, and represents the change in energy per unit of mass of fluid as it travels from point 1 to point 2.

• z represents the vertical dimension and the first term therefore represents the increase in potential energy for a unit mass of fluid,
• V is the speed of the fluid, and therefore the second term represents the increase in kinetic energy per unit of mass of the fluid between point 1 and point 2
• H is the enthalpy of the fluid (expressed as energy per unit of mass) and the third term is therefore the change in enthalpy of the fluid. Enthalpy is : U + P/ρ
  U is the internal energy
  ρ is the fluid density
  P is the pressure
• W is the work per unit of mass of fluid entering the system as the unit mass of fluid travels from 1 to 2 (e.g. transferred from a pump perhaps)
• Q is the amount of heat per unit of mass transferred to the fluid between point 1 and point 2

This is a very important equation and sadly we seem to have forgotten it despite the fact it has been around in this form well over a century. Jacob Bernoulli was already working on it 400 years ago!

In the past, until a couple of decades ago, it wasn't easy to apply it directly and simplifications were made. For liquids which are not compressible, the internal energy is due entirely to the heat, Q, so the equation can be split into two separate equations, one dealing with heat, and the other dealing with work:

Liquids

1. C (T₂ - T₁) = Q
2. ρ g (z₂ - z₁) + 1/2 ρ (V₂² - V₁²) + (P₂ - P₁) = ρ W

The first is left to be dealt with by heat exchanger designers, and the second one becomes the "simple" Bernoulli's equation for liquids that remain liquid all the way from point 1 to point 2

Gases and Vapours

Fluids that behave like perfect gases, or at least come close, are dealt with by using perfect gases formulas. The two most in use are:

1. P/ρ = Z R/M_W T
   Z is a factor equal to 1 for perfect gases, and slightly different from 1 for real ones. This is knowns as the Equation of State
2. P/ρ^k = constant when the gas expands reversably and adiabatically, in other words isentropically. This is known as the isentropic expansion law

Z is a function of pressure and temperature and can be estimated by various methods, some more accurate than others. Where we really get into strife is with the exponent k. For ideal gases k=C_p/C_v but for real gases all bets are off, and the k we really want is the one linking P and ρ during an isentropic compression or expansion. The problem is not helped by standards such as API 520 which defines k as C_p/C_v. A far better method is to take two points at different pressures but with the entropy being the same at both points and use k=Log(P₂/P₁)/Log(ρ₂/ρ₁).

Universal Formula for All Compressible Fluids

However this was 20 years ago, nowadays we have Hysys, Aspen and lots of other process modelling tools, the values we use for Z, and k are usually derived from those models, the question I want answered is why do we insist on still using the approximations and assumptions about Z and k when we can use the Bernoulli formula directly without any assumption save that the process model is correct.

So here is a method on how to size relief valves without needing Z or k and which handles both gases, vapours, and flashing liquids

The flow through the nozzle of a relief valve, all the way from the vessel (where the velocity is negligible) to the throat, can be considered as an isentropic expansion. It also occurs with little elevation difference so altitude differences can often be ignored. No work is produced or received from the external world so the generalised Bernoulli equation, when applied to a relief valve becomes:

1/2 V₂²= (H₁ - H₂)

The conditions at the throat are those that maximise the flux G=ρ₂V₂ so we make the pressure drop slowly from condition 1 to condition 3 (set by the backpressure of the relief valve) and we calculate the flux at each point. If the maximum is reached for the downstream condition (3) then the flow is subsonic, but if a maximum flux is obtained between P₁ and P₃ then this is the condition at the throat which for a gas corresponds to sonic flow.

Now, to help convince you that I'm right, I'm adding two Excel sheets (trust me, I really hate using Excel or any other Microsoft products, but I know everybody else uses it, so I accepted to dirty my hands a little to help convince you, I've since washed them thoroughly) which show how it is done. The first one uses steam at saturation point, the second one uses water at boiling point. So here is the steam one and here is the water one.

Before you all come and ask questions, I'll explain why I used "2000" as the multiplication factor for the enthalpy difference, 2 is for converting enthalpy to V², and 1000 is for converting kiloJoules to Joules. My American friends can convert the enthalpy units for their own system, let me tell them that if they use pound for mass unit, then they would be wise to use ""poundal" for unit of force and pdl.ft for units of energy (not BTU!!!). Or they could use "slug" (NASA does) as unit of mass and pound as unit of force and pound.ft (Still not BTU!!!) as unit of energy and with the speed being ft/s in both cases, the Bernoulli equation will require no unit correction factors, but if they insist on using pound for both mass and force and BTU/Lb for enthalpy, then I'm afraid I'm not prepared to help them extricate themselves from their own mess. I sure wish they'd come to their senses and start using the metric system.

Any comments or questions about the spreadsheets? Then send me a short message with the link below and be sure to put a correct email address where I can reply to.

Return