5_heated_tank - Spring 2006 Process Dynamics Operations and...

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Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 5: Heated Tank 5.0 context and direction From Lesson 3 to Lesson 4, we increased the dynamic order of the process, introduced the Laplace transform and block diagram tools, took more account of equipment, and discovered how control can produce instability. Now we change the process: our system models have previously depended on material balances, but now we will write the energy balance. We will also introduce the integral mode of control in the algorithm. DYNAMIC SYSTEM BEHAVIOR 5.1 a heated tank We consider a tank that blends and heats two inlet streams. The heating medium is a condensing vapor at temperature T c in a heat exchanger of surface area A. F 1 T 1 F 2 T 2 F T o T c V A For the present, we continue to assume constant mass in an overflow tank. Writing the material balance, F F F 2 1 ρ = ρ + ρ (5.1-1) This is not yet the time for complications: we will approximate the physical properties of the liquid (density, heat capacity, etc.) as constants. We will also simplify the problem by assuming that the flow rates remain constant in time. The energy balance is ( ) ) T T ( FC ) T T ( UA ) T T ( C F ) T T ( C F ) T T ( VC dt d ref o p o c ref 2 p 2 ref 1 p 1 ref o p ρ + ρ + ρ = ρ (5.1-2) revised 2006 Mar 31 1
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Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 5: Heated Tank where the overall heat transfer coefficient is U and the thermodynamic reference is T ref . We identify a steady-state operating reference condition with all variables at their desired values. ( ) ) T T ( FC ) T T ( UA ) T T ( C F ) T T ( C F 0 ) T T ( VC dt d ref or p or cr ref r 2 p 2 ref r 1 p 1 ref or p ρ + ρ + ρ = = ρ (5.1-3) We subtract (5.1-3) from (5.1-2), define deviation variables, and rearrange to standard form. ' c p ' 2 p p 2 ' 1 p p 1 ' o ' o p p T UA FC UA T UA FC C F T UA FC C F T dt dT UA FC VC + ρ + + ρ ρ + + ρ ρ = + + ρ ρ (5.1-4) To make some sense of the equation coefficients, define the tank residence time F V R = τ (5.1-5) and a ratio of the capability for heat transfer to the capability for enthalpy removal by flow. p FC UA ρ = β (5.1-6) β thus indicates the importance of heat transfer in the mixing of the fluids. We now use (5.1-5) and (5.1-6) to define the dynamic parameters: time constant and gains. β + τ = τ 1 R (5.1-7) Thus the dynamic response of the tank temperature to disturbances is faster as heat transfer capability ( β ) becomes more significant. For no heat transfer ( β = 0) the time constant is equal to the residence time. β + = 1 F F K 1 1 (5.1-8) β + = 1 F F K 2 2 (5.1-9) revised 2006 Mar 31 2
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Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 5: Heated Tank Gains K 1 and K 2 show the effects of inlet temperatures T 1 and T 2 on the outlet temperature. For example, a change in T 1 will have a small effect on T o if the inlet flow rate F 1 is small compared to overall flow F.
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