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Dynamic Energy Balance

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Last time: well-mixed CSTR w/flow & reaction for multiple reactions: rxn #

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Simplifications ACC Term: For moderate T changes use average Cp i assume Cp i are constant

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Simplifications Also: Simplified Form: well-mixed CSTR w/flow, reaction & avg. Cp

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In-Class Exercise Liquid flows continuously into an initially empty tank, which contains a full-depth heating coil. As the tank fills, an increasing proportion of the coil is covered by liquid. Once the tank is full, the liquid starts to overflow into the discharge pipe, but heating is maintained. How long does it take the system to reach steady state, what is the final exit temperature, and how long does it take before the tank “overflows”? vovo ToTo vTvT Steam in T s Condensate

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In-Class Exercise

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Energy Balance on Vessel Jacket If Constant T j : Jacket energy balance: assume j1 = j2 T j2 = constant Assumptions: jacket well insulated liquid in jacket is well mixed V j (Volume of jacket fluid) is constant j is constant no work nor reaction in jacket

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Energy Balance on Vessel Jacket (especially if T j varies significantly) (for small systems, low flow, or not well mixed) If T j not constant: assume j1 = j2 i.e. constant [1]

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Energy Balance on Vessel Jacket (T j not constant) Energy balance on heat transferred: Rate of energy loss by jacket fluid Rate of energy transfer from jacket to the reactor = [1] [2] from [1]then into [2]

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Energy Balance on Vessel Jacket (T j not constant) KjKj

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Effect of Flow on K j so as j K j ?? jj

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Energy Balance on Vessel Jacket (T j not constant) Energy balance on jacket: Assumptions: jacket well insulated liquid in jacket is more like plug flow (i.e cooling coil) V j (Volume of jacket fluid) is constant j is constant no work nor reaction in jacket and:

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Small Vessel Jackets For small vessels or high pressure systems the thickness of the vessel wall can be significant…thus one needs to consider the thermal capacity of the wall. Energy balance for jacket: Energy balance on wall: T j constant

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In-Class Exercise Liquid flows continuously into continuous stirred tank reactor, which is fully-jacketed and well-mixed. At a certain time, reactant A is introduced into the feed liquid, such that the volumetric flowrate remains constant. 1) Show that the steady state solution to the problem gives a reactor T of ~331 K. 2) With no control (Kc=0), use the dynamic model to find the SS solution. 3) Add proportional control to the dynamic model. Examine the effect of varying Tset from K, and Kc from -1 to 5. 4) Add integral control to the controller equation. What effect does this have? o, T o, C ao TT TE TC jo, T jo V, T C a, C b j, T j

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In-Class Exercise From before: Let: So:

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In-Class Exercise P only controller: with: K c = controller gain PI controller: with: I = integral time Let: or: Then: initial condition?

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In-Class Exercise

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In-Class Exercise : SS Solution

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In-Class Exercise: dynamic - SS

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In-Class Exercise

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