Material balance unsteady state

The general approach to the solution of unsteady-state problems is illustrated in Example 2.15. Batch distillation is a further example of an unsteady-state material balance (see Volume 2, Chapter 11). 

Most real situations are, however, such that conditions change with respect to time. Under these circumstances, a steady-state material balance is inappropriate and must be replaced by a dynamic or unsteady-state material balance, expressed as... 

Note that since there are two independent variables of both length and time, the defining equation is written in terms of the partial differentials, dC/dt and dC/dZ, whereas at steady state only one independent variable, length, is involved and the ordinary derivative function is used. In reality the above diffusion equation results from a combination of an unsteady-state material balance, based on a small differential element of solid length dZ, combined with Fichs Law of diffusion. 

After inoculation, assuming no lag phase, the resultant growth can be analysed by considering the unsteady-state material balances for the substrate and biomass. The general form of this balance for a fermenter is ... 

The unsteady state material balance of component A is uCAO =uCA +(-rA)VR + VR Rearranging Equation 5-335 gives... 

EXAMPLE 6.1 Unsteady-State Material Balance without Generation... 

Step 5 Solve the unsteady-state material balance on the salt. By separating the independent and dependent variables, we get... 

The general unsteady state material balance for each reactor is ... 

Unsteady-state material balances can be written for each phase. For the general case of unsteady operation and variable physical properties, the liquid phase balance is... 

In the most general case, the following unsteady-state material balance equations are written for the vascular, interstitial, and intracellular compartments, respectively ... 

In a tubular reactor, the concentration and temperature may vary both in time and space. One speaks about a distributed system. The ideal plug flow reactor (PFR) model is the most used. Because of the flat velocity profile, the concentrations and temperature varies only along the length. Consider for simplification a homogeneous reaction. The unsteady state material balance of the reactive species leads to the following equation ... 

Consider the reaction Ag -I- B C taking place in a semibatch reactor with a concentration Cg and an active volume V. Assume the reaction is first order to A and to B and that the gas-phase resistance is negligible. The unsteady-state material balance for A is... 

Figure 2.4 Unsteady-state material balance on a stirred vessel...

In the section on unsteady-state material balances you were asked to draw your own conclusions regarding when it might be appropriate, for all practical purposes, to say that the change in concentration was complete. Will for all practical purposes be dependent upon circumstances ... 

Adopting the same symbols as those used earlier for unsteady-state material balances, the above equation is symbolically ... 

If = 0, equation (8.5) is very similar to equation (2.5) which came in the section on unsteady-state material balances. Solutions for K =, K = wfCl and K = SmpCL are given in figure 8.2. 

Unsteady state material balance on the receiving compartment (B) yields ... 

Rayleigh equation An equation that relates the quantity to the concentration of the more volatile component in batch distillation. The equation is formulated from an unsteady-state material balance in which vapour leaving the still is in equilibrium wrtth the liquid. As the vapour is richer in the more volatile component, the composition of the Rquid and vapour are not constant ... 

The unsteady-state material balance on a nonreactive, nonadsorbing tracer that obeys the Dispersion model is... 

---