Conventional Approach to the Degree of Freedom Analysis

Consider again the 5 component single-stage flash process described in Part 1 of this chapter where all the variables associated with the feed steam are assumed to be known. To illustrate that there are several approaches to listing the variables and independent equations for the fully defined process, consider the case where for any stream only 4 mole fractions are considered as variables (in which case the equation stating that the mole fractions sum to unity cannot be used as an equation), and also where the vapor and liquid flow rates (V and L) are considered as variables (instead of the ratio V/L). In this case the variables that apply can be listed as follows:

    4 mole fractions in the liquid product

    4 mole fractions in the vapor product

    The vapor product flow rate

    The liquid product flow rate

    Temperature in the flash chamber

    Pressure in the flash chamber

    The amount of heat transferred in the heat exchanger

These variables can be solved for using the following set of independent equations

    5 phase equilibrium equations (one for each component)

    Four species material balances

    One overall material balance

    One enthalpy balance

Since there are 13 variables and 11 equations given above, it follows that two of the variables listed above (for example temperature and pressure or pressure and the heat transferred in the heat exchanger) must be assigned known values so that the remaining 11 variables can be solved for using the 11 equations described above. Consequently there are two degrees of freedom for this system. More generally, provided that the variables associated with the feed are fixed, there will be 2 degrees of freedom for a single stage flash process regardless of the number of components present since the specific number of components cancels out from the degree of freedom analysis. Note also that the determination of the degrees of freedom just given is described by the following figure:

Fig. 1. The conventional approach to the degree of freedom analysis.

Using the Description Rule to Determine the Number of Degrees of Freedom

Although the method just described and illustrated in Fig. 1 gives the proper number of degrees of freedom, the task of listing the variables and equations is prone to errors, especially since there is not a unique list of variables or a unique list of equations for any given problem. This can be seen by comparing the list of variables and equations in the above approach with that in the approach in Part 1 of this chapter for the single-stage flash problem. It is worthwhile therefore to explore the use of a more practical approach which takes the viewpoint of the process unit builder and operator.

Consider again the case of a 5 component single-stage flash process where all the variables associated with the feed have known values, and where the following process flow diagram therefore applies:

Fig. 2. Adding control loops to ensure steady state operation.

The left side of the above figure shows the process for the case where manual control applies (except for the completely specified feed stream where a flow control loop is present). Clearly, the three manual valves shown must be given specific settings to achieve steady state operation, and for random settings of these valves it is likely that the inventory of liquid in the flash chamber will either be increasing or descreasing with time (so that steady state is not achieved). In practice, automatic control loops need to be added to ensure steady state operation, as shown in the right side of the above figure where a pressure control loop, a temperature control loop, and a liquid level control loop have been added.

When steady state has been reached for the situation on the right side of the above figure, the number of equations that apply must be equal to the number of variables that can be solved for so that the steady state values of the variables can be solved for. Furthermore, for each control loop that has been added to ensure steady state operation, a corresponding control equation has been added to the system of governing equations, and possibly a new variable (i.e., the controlled variable of the control loop) has also been added depending on how the control loop operates. Therefore, by determining the number of control loops that need to be added to the system and then subtracting from this number the number of new variables (if any) added by the control loops, the degrees of freedom for the original original system can be determined. The Description Rule calculation method therefore corresponds to Fig. 3 shown below, which can be compared to the Fig. 1 shown above.

Fig. 3. The description rule approach to the degree of freedom analysis.

As shown in the figures above, three control loops are needed to ensure steady state operation (again assuming the feed stream variables are known), but one of these control loops (the liquid level control system) does not affect the steady state values of the variables being considered except for the liquid level itself, which is not a variable of interest here. This indicates that the degrees of freedom for the original system is 2. This result agrees that given above when the total numbers of equations and variables were considered. Notice that in the description rule approach to the degree of freedom analysis no attempted is made to list the entire set of variables and equations so that no attempt is made to determine I or D + E + F in Fig. 3.

In most cases the number of control loops which must be added to the system to achieve steady state can be easily obtained by noting the number of control loop actuators present in the system since every control loop must have an actuator, and all potential actuators in the system need to be part of a control loop. The most common control loop actuator is a valve, and as a general rule a valve must be present on every exit stream from the process as well as on each each steam line entering the system. Note that a special valve termed a steam trap (as shown the second figure above) is generally located on the condensate line from a steam heater. Steam traps operate automatically to ensure only condensed liquid exits a steam heating system, and consequently a steam trap cannot be an element in a control loop. Note also that tanks containing two phases, such as the tank containing liquid and vapor in this example, require a level control loop of some type to ensure that steady state is achieved.

Selection of Specified Variables to Completely Define a Problem

Note that although the above development indicates that temperature and pressure specifically need to be specified for the single state flash process to become a completely specified problem, the key conclusion is that there are two degrees of freedom for the problem as stated such that any two variables can be given fixed values to completely specify the problem. However, not all possible pairs of variables, when specified, will yield a problem that can be solved. For example, if the mole fractions of two different components are arbitrarily specified in the liquid phase, it may be that no solution is possible for the overall system of equations. Alternatively, if the vapor product flow rate is specified at an arbitrary value to use one of the degrees of freedom available, then it is not possible to arbitrarily select the liquid product flow rate (i.e., to use the liquid product flow rate as the second degree of freedom available) since with the feed flow rate specified there is only one possibly liquid product flow rate possible is the vapor product flow rate is specified.

Consideration of Construction Variables

In addition to the controlled variables described aove, in many cases construction variables also need to be specified in order to have a fully defined process. In the above example, the dimensions of the flash chamber and the heat exchanger area can be considered contruction variables. However, in the case considered here these construction variables do not affect the steady state values of the other variables beind considered, and therefore there are no construction variables that need to be considered in the present case.

• Description rule for a single stage flash. This material, which comes from Chapter 2 of “Separation Processes” by C. J. King, describes a simple method to determine the number of degrees of freedom for a single stage flash process.