ProcessDynamics1 - Process Dynamics 1 Copyright Brian G...

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Process Dynamics 1 ü Copyright Brian G. Higgins (2004) Overview In these notes we show how to derive the governing equations for a perfectly mixed flow through tank, also known as a CST. A schematic of the tank (taken from Ref 1) is shown below Figure 1 The reactor displayed in Figure 1 has a single entrance stream at 1 and an exit stream at 2. The overall mass balance for this vessel is (1) ÅÅÅÅÅÅÅ t V H t L r „ V = A 1 r v ÿ n A - A 2 r v ÿ n A Here v is the velocity of the fluid, r is the density of the fluid, and n is the outward directed unit normal at the entrance and exits of the control volume V H t L . The cross-sectional areas at positions 1 and 2 are denoted by A 1 and A 2 . Note that r v ÿ n A is the mass flux passing through the differential area A. Thus we can write (1) as (2) M ÅÅÅÅÅÅÅ t = F ° 1 - F ° 2 where M is the mass of fluid in the tank at time t and F ° i is the mass flow rate across exit/entrance i. In liquid systems it is reasonably (at low to moderate pressures) to assume that the density r is constant so that (3) M = r V, F ° i = r Q ° i where now Q ° i is the volumetric flow rate across exit/entrance i. Thus (2) becomes (4) V ÅÅÅÅÅÅÅ t = Q ° 1 - Q ° 2 Now consider a solute in the inlet stream with concentration C H kg ê m 3 L . A solute balance over the tank gives (5) ÅÅÅÅÅÅÅ t V H t L C V = A 1 C v c ÿ n A - A 2 C v c ÿ n A
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Now v c is the solute velocity which at entrances and exits can be approximated by the mass average velocity v (since diffusion effects are negligible at entrances and exits). Thus we can write (6) ÅÅÅÅÅÅÅ t H CV L = Q ° 1 C 1 - Q ° 2 C 2 º Q °
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