Approach to steady state

For laboratory purposes it is sometimes convenient to recycle the foamate directly to the pool in a manner analogous to an equihbrium still. This eliminates the feed and bottoms streams and makes for a more reliable approach to steady-state operation. However, this recycling may not be advisable for colligena measurements in the presence of slowly dissociating collector micelles. 

Time Dependence—The Transient Approach to Steady-State and Saturation Kinetics... 

Figure 9.3. Illustration of the approach to steady-state conditions for the populations of states So, Si, and Ti under the continuous excitation conditions of Example 9.5...

Using the digital simulation approach to steady-state design, the design calculation is shown to proceed naturally from the defining component balance and energy balance equations, giving a considerable simplification to conventional text book approaches. 

Carry out simulations for differing tank volumes, flow rates and feed concentrations, in which the inlet and outlet flow rates are set equal, (F + P2 = F3) and observe the approach to steady state. Relate the time taken to approach steady state to the mean tank residence time (t = V/F3). 

The analytical solution shows that the approach to steady state is very rapid when Vq is small and that the concentration in the tank is always constant, when starting with a relatively empty tank. 

Observe the oscillatory approach to steady-state for different initial liquid depths. Plot the depths and the flow rate versus time. 

Figure 5.183. Approach to steady state, showing how equilibrium is maintained.

Fig. 6 Plot showing the approach to steady state for a reservoir device that has been stored for an extended period (the burst effect curve) and for a device that has been freshly made (the lag time curve). (From Ref. 29.)...

Examination of Figures 2 to 4 clearly indicates flow enhancement of the desorption as seen by the substantial decreases in adsorbance once flow is applied. The overall trend of the desorption curves for the three molecular weight samples is similar and is characterized by a fairly rapid initial desorption followed by an approach to steady state. The desorption rate increases with the velocity gradient whereas the steady state adsorbance decreases as the flow is increased. 

This problem may also be solved by numerical integration using the E-Z Solve software (file exl4-3.msp). This simulation is well-suited to the investigation of the effect of initial conditions on the time for a specified approach to steady-state. To optimize... 

The times for various approaches to steady state are figured from Eq (2) and tabulated. 

The approach to steady-state can be extremely slow (see Figure 9), and would lead to considerable error in the determination of the characteristic parameters of the system if an inappropriate transient value is taken as the final... 

The analytical solution shows that the approach to steady state is very rapid when V0 is small and that the concentration in the tank is always constant, when starting with a relatively empty tank, ft also indicates that the rate of change of volume in the tank is equal to the net volumetric flow rate, but only for a linear density concentration relationship. Check the above analytical conclusions numerically and test the case of a non-linear density-concentration relationship by simulation. 

Compare the residence time for the L phase, Vl/L, with the time constant for mass transfer, l/KLa. Maintain a constant ratio of these two time constants but vary the individual parameters. How is the approach to steady state influenced by these changes ... 

Fig. 2 Approach to steady state [KLa = l and 5 1 /h]. Note that equilibrium steady state is apparently not attained.

Start the reactor with the initial concentration conditions of S=0, X=2 and D steady state. Did washout occur ... 

Two parameters have been demonstrated to be fundamental in calculating the performance characteristics of a continuous analyser, the lag phase and the half-wash time they afford a correlation between the approach to steady state, fraction of steady state reached in a given time and the interaction between samples. The half-wash time (Wy ) is the time for the detector response to change from any value to half that value, the lag phase L is defined in the ensuing discussion. 

The duration of the unsteady period, denoted by C, the time required for Sh to come within 100x% of the steady value, is different for rigid and fluid spheres. For a rigid sphere at high Pe, T ocPe "/". From Stokes s law, Eq. (3-18), Uj oc a" hence C is independent of particle size for a given fluid. However, for a fluid sphere, oc (1 + K )/Pe thus Ujt /a is a constant, and a given fractional approach to steady state is achieved when the particle has moved a fixed number of radii, e.g.. 

Heinrichs, M. and Schneider, F. W. (1980). On the approach to steady states of reacting systems in the continuous strirred tank reactor. Ber. Bunsenges. Phys. Chem., 84, 857-65. 

Raines, R.T. Hansen, D.E. (1988) An intuitive approach to steady-state kinetics. J. Chem. Educ. 65, 757-759. 

This equation is integrated from the initial condition C = 0.7765 to the final condition C = 0.8720. The tabulation shows the numerical solution. Interpolating, t = 6.38 at 95% approach to steady state. 

H. Kramers Approach to Steady-State Rates of Reaction and Its Extension to Non-Markovian Processes... 

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