Dynamic Energy Balance Dynamic Energy Balance Last time

Dynamic Energy Balance

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

Simplifications • ACC Term: For moderate T changes use average Cpi assume Cpi are constant

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

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”? vo To Steam in Ts v T Condensate

In-Class Exercise

In-Class Exercise

In-Class Exercise

In-Class Exercise

In-Class Exercise

Energy Balance on Vessel Jacket • If Constant Tj: Assumptions: • jacket well insulated • liquid in jacket is well mixed • Vj (Volume of jacket fluid) is constant • j is constant • no work nor reaction in jacket • Jacket energy balance: assume j 1 = j 2 Tj 2 = constant

Energy Balance on Vessel Jacket (especially if Tj varies significantly) • If Tj not constant: (for small systems, low flow, or not well mixed) [1] assume j 1 = j 2 i. e. constant

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

Energy Balance on Vessel Jacket (Tj not constant) Kj

Effect of Flow on Kj so as j Kj ? ? j

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

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

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 300 - 375 K, and Kc from -1 to 5. 4) Add integral control to the controller equation. What effect does this have? TC o, To, Cao j, Tj TE V, T Ca, Cb T jo, Tjo

In-Class Exercise • From before: • Let: • So:

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

In-Class Exercise

In-Class Exercise

In-Class Exercise

In-Class Exercise

In-Class Exercise : SS Solution

In-Class Exercise: SS Solution

In-Class Exercise: dynamic - SS

In-Class Exercise : dynamic - SS

In-Class Exercise

In-Class Exercise