Unsteady systems

This is a second-order ODE with independent variable z and dependent variable k C t,z), which is a function of z and of the transform parameter k. The term C(t, 0) is the initial condition and is zero for an initially relaxed system. There are two spatial boundary conditions. These are the Danckwerts conditions of Section 9.3.1. The form appropriate to the inlet of an unsteady system is a generalization of Equation (9.16) to include time dependency ... 

Finally, if Eq. (1.13) gives < 0, it does not necessarily mean the absence of the film in the system. The situation is also possible (e.g., very small a or k) when the film is formed however, its positive stationary thickness is impossible. These non-stationary unsteady systems can exist either at partial covering of the surface or in the regime of periodic formation and dissolution of the film (type 3). The behaviour of such systems is somewhat similar to the unsteady electrochemical systems (see Chap. 5). 

By its nature process control is concerned with the dynamic behaviour of systems. It is no longer sufficient to make the steady-state assumption. Material and energy balances for unsteady systems must include the accumulation terms so far omitted. Because of the extra mathematical complexity involved in a quantitatve treatment of control this section will, instead, concentrate on general concepts rather than detailed analysis of... 

The numerical method uses centered finite differences for spatial derivatives and time integrations are performed using the ADI method. The ADI scheme splits each time step into two and a semi-implicit Crank-Nicolson scheme is used treating implicitly the r-direction over half a time step and then the -direction over the second half. In addition, a pseudo-unsteady system is solved which includes a term d tl ldt on the left hand side of (121) and integrating forward to steady state (see Peyret and Taylor [55]). The physical domain is mapped onto a rectangular computational domain by the transformation r = 0 = Try,... 

There are special numerical analysis techniques for solving such differential equations. New issues related to the stabiUty and convergence of a set of differential equations must be addressed. The differential equation models of unsteady-state process dynamics and a number of computer programs model such unsteady-state operations. They are of paramount importance in the design and analysis of process control systems (see Process control). 

The nature of batch operations (unsteady-state), frequently involving manual intervention, creates significant issues pertaining to the design of control systems, design of operating procedures, and the interaction between the... 

Develop and implement system and operating procedure designed to allow for unsteady evaporation rates... 

Matching the flow between the impeller and the diffuser is complex because the flow path changes from a rotating system into a stationary one. This complex, unsteady flow is strongly affected by the jet-wake of the flow leaving the impeller, as seen in Figure 6-29. The three-dimensional boundary layers, the secondary flows in the vaneless region, and the flow separation at the blades also affects the overall flow in the diffuser. 

Historical data management—This includes the data acquisition and storage capabilities. Present-day prices of storage mediums have been dropping rapidly, and systems with 80 gigabyte hard disks are available. These disks could store a minimum of five years of one-minute data for most plants. One-minute data is adequate for most steady state operation, while start-ups and shutdowns or other non steady state operation should be monitored and stored at an interval of one second. To achieve these time rates, data for steady state operation can be obtained from most plant-wide D-CS systems, and for unsteady state conditions, data can be obtained from control systems. 

In some cases where condensing loads are high, or where it is required to recover condensed liquid blowdown material for pollution, toxicity or economic reasons, an unsteady state condensing system may be appropriate. Examples or such applications are as rollows ... 

Another example of an unsteady state condensible blowdown system is the design for a phenol condensible blowdown tank. A blowdown tank is used in phenol treating plants to handle streams containing phenol and heavy hydrocarbons (lubricating oil stocks). The blowdown tank is illustrated in Figure 4. The design basis is as rollows ... 

In predicting the time required to cool or heat a process fluid in a full-scale batch reactor for unsteady state heat transfer, consider a batch reactor (Figure 13-2) with an external half-pipe coil jacket and non-isothermal cooling medium (see Chapter 7). From the derivation, the time 6 to heat the batch system is ... 

The general case is that of steady-state flow, and the thermal conductivity factor is a function of the temperature. In the unsteady state the temperature of the system changes with time, and energy is stored in the system or released from the system reduced. The storage capacity is... 

The exhaust flow rate influences the flow of the jets and some reports recommend a ratio of supply airflow rate to exhaust airflow rate of approximately 0.3. A ratio of 0.2 is unsteady and ratios larger than 0.4 have not been studied. In the cases that have been studied, the exhaust opening was 80 mm in diameter, the distance between the horizontal planes was 750 mm, the tubes were placed in a square w ith side length equal to 670 mm, and the inward angles of the jets were 10 degrees. This configuration resulted in better capture of hot gases than use of an exhaust system alone. 

Given expressions for the crystallization kinetics and solubility of the system, the population balance (equation 2.4) can, in principle, be solved to predict the performance of both batch and of continuous crystallizers, at either steady- or unsteady-state... 

A steady-state process is one in wliich there is no change in conditions (temperature, pressure, etc.) or rates of flow with time at any given point in die system. The accumulation term in Eq. (4.5.1) is dien zero. If diere is no cheniieid reaetion, the generation tenn is also zero. All other processes are unsteady state. 

All processes may be classified as batch, continuous, or semibatch depending on how materials are transferred into and out of the system. Also, the process operation may be characterized as unsteady state (i.e., transient) or steady state, depending on whether the process variables (e.g., pressure, temperature, compositions, flowrate, etc.) are changing with time or not, respectively. In a batch process, the entire feed material (i.e., charge) is added instantaneously to the system marking the beginning of the process, and all the contents of the system including the products are removed at a later time, at the end of the process. In a continuous process, the materials enter and leave the system as continuous streams, but not necessarily at the same rate. In a semibalch process, the feed may be added at once but the products removed continuously, or vice versa. It is evident that batch and semibatch processes are inherently unsteady state, whereas continuous processes may be operated in a steady or unsteady-state mode. Start-up and shut-down procedures of a steady continuous production process are examples of transient operation. 

Iim88] Lim, H.A., Lattice gas automata of fluid dynamics for unsteady flow, Complex Systems 2 (1988) 45-58. 

At unsteady-state conditions, the change of concentration with respect to time is detectable, ASIAt + 0 but for steady-state conditions the leaving substrate may be constant. For a plug flow bioreactor we can treat it like a batch system. 

In the theoretical treatment, the heat- and mass-transfer processes shown in Fig. 6 were considered. Simultaneous solution of the equations describing the behavior of the unsteady-state reaction system permits the temperature history of the propellant surface to be calculated from the instant of oxidizer propellant contact to the runaway reaction stage. 

The equations represented by (230) reduce to the familiar equations for the special ideal case of unsteady-state mass transfer without coupling in a binary system if we let... 

---