Steady-state and dynamic degrees of freedom

When a process engineer works with a detailed steady-state simulation of a distiUation column, a certain number of variables have to be specified in order to converge to a solution. The number of variables that need to be specified, or degrees of freedom, can be determined through the concept of the description mle as stated by King [3]  

In order to describe a separation process uniquely, the number of independent variables 

Applying the desaiption mle to a distillation column with a total condenser and two product streams gives two steady-state degrees of freedom In this case the column would require two specifications, i.e. a composition and a component recovery. The steady-state simulator will then manipulate two variables, such as reboiler and condenser duties, in order to satisfy the specifications and close the steady-state material and energy balances. If a partial condenser is added to the column, then another degree of freedom is added to the steady-state column. Likewise, for each additional side draw added to the column, a new degree of freedom is added, requiring another specification. 

When the same two-product distillation column is viewed in dynamics, the number of degrees of freedom increases from two to five. These three new dynamic degrees of freedom correspond to three new manipulated variables needed to control the integrating, inventory variables within the column that are not fixed by tbe steady-state material and energy balances alone. The inventory variables for this column are condenser level, reboiler level, and the column pressure. 

There are restrictions on the control of a distillation column. The overall enthalpy balance limits the heat removed by the condenser and added by the reboiler. The rate of distillate produced may not exceed the feed rate. The number of stages in the column and the reflux ratio must be greater than or equal to the number required for the desired separation [3]. 

---

Although the previous paragraph describes the manipulated variables as control valves, there are many choices available other than just the individual valves. For example, many columns have reflux ratio as a manipulated variable for either inventory or composition control. When ratios and linear combinations of variables are included, the choice of a manipulator for a given loop broadens considerably for a simple two-product column. However, the steady-state and dynamic degrees of freedom remain unchanged as two and three respectively, totalling five. 

One must take care in determining the number of steady-state and dynamic degrees of freedom for more complex columns. Tyreus [4] describes the determination of the degrees of freedom for an extractive distillation system and for an azeotropic column with an entrainer. In the case of an extractive distillation system, recycle streams reduce the dynamic degrees of freedom through an increase in the steady-state degrees of freedom if the recycle contains a component that neither enters nor leaves the process. Also, if it is important to control the inventory of a trapped component, such as an entrainer for azeotropic distillation, then it is necessary to provide extra control valves to account for the loss of degrees of freedom. The loss comes from the addition of a side stream. 

The column has seven control valves and requires four degrees of freedom for steady-state control. The remaining three dynamic degrees of freedom are used to control the column inventories. Column pressure is controlled by manipulating the condenser duty. However, if there were non-condensables in the column, then the overhead vapour stream would have been a more suitable choice as a manipulated variable. Non-condensables in a column tend to accumulate in the condenser and significantly reduce the dew point of incoming vapours. The low dew point reduces heat transfer because of small temperature driving forces. Because the vent stream is rich in... 

Control of the reflux drum is fairly straightforward. Because the reflux ratio is very high, with a steady state value of 145, reflux flow is the only reasonable manipulator for dmm level. However, there is a potential loss of one dynamic degree of freedom unless it is ensured that the material balance for the distillate product is satisfied. This can be achieved by ratioing the distillate flow to the reflux flow. The effective manipulator is now the distillate flow and the reflux flow combined instead of just reflux flow. Control of the base level in the column is basically restricted to the use of reboiler steam due to the large vapour boil-up to bottoms ratio. 

The extra degree of freedom introduced into the present theory by the nonzero electric field divergence gives rise to new classes of phenomena such as bound steady electromagnetic equilibria and free dynamic states, including wave phenomena. These possibilities are demonstrated by Fig. 1. 

Control is the manipulation of a degree of freedom (e.g., heater, cooler or exchanger load, stream split fraction) in order to make a process feasible and/or economically optimal in the steady state. In this chapter, control is used in a static sense only process dynamics are not considered. 

After satisfying all of the basic regulatory requirements, we usually have additional degrees of freedom involving control valves that have not been used and setpoints in some controllers that can be adjusted. These can be utilized either to optimize steady-state economic process performance (e.g., minimize energy, maximize selectivity) or to improve dynamic response. 

In this chapter size effects in encounter and reaction dynamics are evaluated using a stochastic approach. In Section IIA a Hamiltonian formulation of the Fokker-Planck equation (FPE) is develojjed, the form of which is invariant to coordinate transformations. Theories of encounter dynamics have historically concentrated on the case of hard spheres. However, the treatment presented in this chapter is for the more realistic case in which the particles interact via a central potential K(/ ), and it will be shown that for sufficiently strong attractive forces, this actually leads to a simplification of the encounter problem and many useful formulas can be derived. These reduce to those for hard spheres, such as Eqs. (1.1) and (1.2), when appropriate limits are taken. A procedure is presented in Section IIB by which coordinates such as the center of mass and the orientational degrees of freedom, which are often characterized by thermal distributions, can be eliminated. In the case of two particles the problem is reduced to relative motion on the one-dimensional coordinate R, but with an effective potential (1 ) given by K(l ) — 2fcTln R. For sufficiently attractive K(/ ), a transition state appears in (/ X this feature that is exploited throughout the work presented. The steady-state encounter rate, defined by the flux of particles across this transition state, is evaluated in Section IIC. 

Find the degrees of freedom for the system at its dynamic state and steady state. Are they equal If not, why What are the implications on control in this case ... 

High resolution spectroscopic measurements in the gas phase yield the most detailed structural information possible. For example, measurements of weakly-bound complexes in the far-IR [24, 25] and IR [26, 27] have provided the most exact information on their structure and steady-state dynamics. Of course, a much higher level of theory must be used than was presented in section Bl.2.2.1. Quite often the modes are so strongly coupled that all vibrational degrees of freedom must be treated simultaneously. Coriolis coupling and symmetry-allowed interactions among bands, i.e. Fermi resonances, are also quite significant, and must be treated explicitly. Direct measurement of the low frequency van der... 

Count the control valves. The number of control valves available in a process equals the degrees of freedom for control. Most of the valves will be allocated to the basic control features, as production rate, control of inventories (gas and liquid), product quality control, as well as safety and environmental constraints. The remaining valves can be used to enhance steady state economics or improve dynamic controllability. Additional manipulations can be obtained by bypassing heat exchangers or some separators. 

Based on the steady-state optimization for each module, only one active control constraint is identified for the HDA process - the reactor inlet temperature. The nine-step procedure to generate a plantwide control structure developed by Luyben et al.[7] is now applied to each module. These steps are (i) establish the control objectives, (ii) determine the control degrees of freedom, (iii) establish energy management, (iv) set the production rate, (v) control the product quality, (vi) fix a flow in every recycle loop and control inventories, (vii) check component balances and (viii) control individual imit operations, and (ix) optimize the economics or improve the dynamic controllability. The number of control degrees of freedom identified for each module (referred to by their respective dominant unit operation) are as follows reactor 10, product column 10, and recycle column 5. 

With all feed conditions and the column configuration specified (number of trays in each section, tray holdup in the reactive section, feed tray locations, pressure, and desired conversion), there is only one remaining degree of freedom. The reflux flowrate is selected. It is manipulated by a distillate composition controller to drive the distillate composition to 95 mol% C. The vapor boilup is manipulated to control the liquid level in the base. Note that the distillate and bottoms flowrates are known and fixed as the dynamic model is converged to the steady state that gives a distillate composition of 95 mol% C. The composition of the bottoms will be forced by the overall component balance to be 95 mol% D. 

---