Bode state equation

Bode state equation One important point is the use of Bode’s state equation.

In 1990, Bode wrote the article “Verfahren zur Extrapolation wichtiger Stoffeigenschaften von Flüssigkeiten unter hohem Druck”,

which means:

”Procedure for the extrapolation of important material properties of liquids under high pressure”.

Bode introduced an equation for the density of fluids that will be used by Bosch GmbH as the reference state equation for fuel properties models at high pressure. Siemens kept the Bode state equation as the reference state of equation for HYD Bosch Fuel Properties after studying other possible state equations with Bosch GmbH.

Bode’s article is useful for understanding the meaning of the A, B1, B2 coefficients that are used in the Bosch GmbH’s FORTRAN code and Siemens C-code, as well as the way the temperature is considered.

Bode’s article focused on different fluids for temperatures between +20°C and +120°C, and for pressures between 0 bar and 4000 bar.

Bode then proposed an equation of state adapted for these fluids at high operating pressure with temperature dependency.

Below are reproduced some figures to highlight the types of fluids studied by Bode with their operating pressures and temperatures:

Figure 2: Extracts from Bode’s article [3] – illustration_1

Figure 3: Extracts from Bode’s article [3] – illustration_1

Figure 4: extracts from Bode’s article [3] - illustration_3

Figure 5: extracts from Bode’s article [3] - illustration_4

The proposed Bode’s state equation has the following form :

Figure 6: Bode’s equation of state from Bode’s article [3]

The HYD Bosch Fuel Properties are based on similar approach as the one developed by Bode.

The Bosch GmbH’s FORTRAN code, and finally the HYD Bosch Fuel Properties industrialized by Siemens, used a modified state equation:

Figure 7: Bode’s equation of state from Bosch GmbH’s FORTRAN code or Siemens’ C code

The coefficients (a, A, B1, B2) have been determined experimentally for 9 different fuels.

The added coefficients B3 and B4 are finally kept to 0.0.

It is known that Bosch GmbH spent a lot of time comparing different state equations. Some studied state equations at Bosch GmbH were: Bode, Bessières r(P,T), Bessières T(r,P), Simcenter Amesim optimized for density, Bode optimized for density, Bessières optimized for density, Simcenter Amesim optimized for speed of sound, Bode optimized for speed of sound, Bessières optimized for speed of sound.

Bosch GmbH & Siemens spent a lot of time studying the deviations. The focus was on the thermodynamics variables themselves as well as their derivatives with pressure and temperature

Note that interest was much less than 1% of deviation on quantities such as density , as well as derivatives such as and .

The deviation for speed of sound c [m/s] or bulk modulus B [bar] are sometimes more important.

Finally , Siemens & Bosch GmbH have selected the Bode’s state equation as being the most predictive for fuel at high pressure.