Full Numerical Solution for a Single-Stage Flash

The "bubble T" calculation method described in Chapter 4 can be extended to solve the problem of a single stage flash process where the pressure in the flash chamber and product enthalpy are specified.  To accomplish this, the enthalpy and material balance relations are included as equations to be solved.  More specifically, if the feed composition, feed enthalpy (or, equivalently, the temperature and pressure of the feed), product enthalpy (or, equivalently, the heat transferred in the heat exchanger) and the pressure in the flash chamber are specified, then we have the following 12 unknowns (assuming five components are present):

    5 liquid phase mole fractions

    5 vapor phase mole fractions

    The fraction of the feed that ends up in the vapor, i.e., V/F.

    Temperature in the flash chamber

These are solved for using the following 12 equations

    5 equilibrium equations, one for each component

    Sum of mole fraction in vapor is unity (one equation)

    5 material balances, one equation for each component

    1 Enthalpy balance

Note that the equation where the sum of the liquid phase mole fractions is unity is not an independent equation here since this equation can be derived from the 5 component material balances along with the equation corresponding to the sum of the vapor-phase mole fractions equaling unity.

The above 12 equations and 12 unknowns can be solved using MS Excel or MATLAB. An example of how to use MS Excel to solve this problem is given in the links below. Alternatively, the above 12 equations and 12 unknowns can be solved more simply by ignoring the composition dependence of the equilibrium ratios K_i (which are equal to y_i/x_i) and then using the numerical methods based on solving the Rachford Rice equation followed by back substitution described in Section 2.6 of the textbook Separation Process Engineering (3rd ed.) by Wankat.

Graphical Solutions Using the Lever Rule

Material balances for various single stage processes can also be solved graphically using various versions of the "lever rule," which can be stated as follows:  On a diagram where one (or more) of the axes correspond to a conserved quantity (like mass fraction, which corresponds to mass), if you locate the feed and product streams corresponding to a steady state process on the diagram, then the product streams will be located on a straight line that includes the feed stream, and distance between the feed and one product stream on the diagram is proportional to the amount of the other product stream.

Versions of the lever rule can be applied graphically to accomplish the following:

    1.  Simultaneous solution of material balance and equilibrium relations on a T-(x,y) phase diagram for the case of a two component system where the feed composition and product temperature and pressure are specified.

    2.  Simultaneous solution of material balance, enthalpy balance, and equilibrium relations on a  H-(x,y) phase diagram for the case of a two component system where the feed composition and product enthalpy and pressure are specifed.     

    3.  Simultaneous solution of two independent material balance relations and equilibrium relations on a x_a, x_b, x_c triangular phase diagram for the case of a three component system liquid system where the feed composition and the temperaure and pressure are specified.

• Detailed notes for Chapter 5.

 

• Example Excel spreadsheet for solving a 5 component single-stage flash problem. This is a printed version of an example Excel spreadsheet for solving a 5 component flash problem where the feed parameters along with the flash chamber pressure and the product enthalpy are specificed. The spreadsheet uses the virial equation of state for the vapor, the Pitzer and Curl correlation for determining the virial coefficients, the Lewis fugacity rule to determine multicomponent fugacity coefficients in the vapor, and regular solution theory for the liquid phase activity coefficients. It is also annotated with text to explain all the formulas and methods used. This spreadsheet can be used to help debug student-written software for solving single-stage flash problems.