3_blend_tank - Spring 2006 Process Dynamics, Operations,...

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Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 3: The Blending Tank 3.0 context and direction A particularly simple process is a tank used for blending. Just as promised in Section 1.1, we will first represent the process as a dynamic system and explore its response to disturbances. Then we will pose a feedback control scheme. We will briefly consider the equipment required to realize this control. Finally we will explore its behavior under control. DYNAMIC SYSTEM BEHAVIOR 3.1 math model of a simple continuous holding tank Imagine a process stream comprising an important chemical species A in dilute liquid solution. It might be the effluent of some process, and we might wish to use it to feed another process. Suppose that the solution composition varies unacceptably with time. We might moderate these swings by holding up a volume in a stirred tank: intuitively we expect the changes in the outlet composition to be more moderate than those of the feed stream. F, C Ai F, C Ao volume V Our concern is the time-varying behavior of the process, so we should treat our process as a dynamic system. To describe the system, we begin by writing a component material balance over the solute. Ao Ai Ao FC FC VC dt d = (3.1-1) In writing (3.1-1) we have recognized that the tank operates in overflow: the volume is constant, so that changes in the inlet flow are quickly duplicated in the outlet flow. Hence both streams are written in terms of a single volumetric flow F. Furthermore, for now we will regard the flow as constant in time. Balance (3.1-1) also represents the concentration of the outlet stream, C Ao , as the same as the average concentration in the tank. That is, the tank is a perfect mixer: the inlet stream is quickly dispersed throughout the tank volume. Putting (3.1-1) into standard form, revised 2005 Jan 13 1
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Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 3: The Blending Tank Ai Ao Ao C C dt dC F V = + (3.1-2) we identify a first-order dynamic system describing the response of the outlet concentration C Ao to disturbances in the inlet concentration C Ai . The speed of response depends on the time constant, which is equal to the ratio of tank volume and volumetric flow. Although both of these quantities influence the dynamic behavior of the system, they do so as a ratio. Hence a small tank and large tank may respond at the same rate, if their flow rates are suitably scaled. System (3.1-2) has a gain equal to 1. This means that a sustained disturbance in the inlet concentration is ultimately communicated fully to the outlet. Before solving (3.1-2) we specify a reference condition: we prefer that C Ao be at a particular value C Ao,r . For steady operation in the desired state, there is no accumulation of solute in the tank. r , Ao r , Ai r Ao C C 0 dt dC F V = = (3.1-3) Thus, as expected, steady outlet conditions require a steady inlet at the same concentration; call it C A,r . Let us take this reference condition as an initial condition in solving (3.1-2). The solution is dt ) t ( C e e e C ) t ( C Ai t 0 t t t r , A Ao τ τ τ τ + = (3.1-4)
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This note was uploaded on 11/27/2011 for the course CHEMICAL E 20.410j taught by Professor Rogerd.kamm during the Spring '03 term at MIT.

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3_blend_tank - Spring 2006 Process Dynamics, Operations,...

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