Causes of delay in process systems Dynamic processes Causes of delay in process systems Whenever material or energy flows into or out of certain system, it takes time. Thus, the level of a liquid, the temperature of the vessel or the position of the solid mass cannot change suddenly. They are subjects to delays which are dependent on the magnitude of the capacity and the resistance to the inflow of material or energy. Heater process: Electric hot water boiler has electric power Pin (w). We assume that the metal parts of the systems (resistor and casing) are small compared to mass of water. Thermal power Q = c * m * ΔT (1) c = heat capacity of water m = mass ΔT = Twater - Tambient = temperature difference Picture 1 Twater Tambient Pin ΔT

First let's assume that insulation of the boiler is very good First let's assume that insulation of the boiler is very good. There are no heat transfer from boiler. Heater resistor is switched on with constant power. Next block diagram describes the case STEP Pin ΔTout WATER Temperature difference Power/w Picture 2 Equation (1) gives ΔT = Q/(c * m) (2) Energy of vessel = Q = W = ∫Pdt (3) Equations (2) and (3) gives: ΔT = ∫Pdt/(c * m) (4) We have got temperature difference to the process, if input power is step function. With constant power equation (4) gives: ΔT = P * t/(c * m) Time-domain plot is growing strait line.

1/S ∫Pdt = W integrator Pin Laplace transform: Laplace transforms are used to solve dynamic systems. Time-domain signals are transferred to s-domain signals. s = jω, ω = angular velocity Computer based transient analyses are using L-transform. Integral change to multiplication and gives easy handling for equations. Matlab simulink can utilize L-transforms. Time-domain Time-domain Time-domain Pin ∫Pdt = W integrator 1/S s-domain STEP 1/c*m GAIN ΔT Picture 3 Time P ΔT Transient Analyse

- + 1/S Pout Pout Tambient Pin ∫Pdt = W integrator Twater Pin Let´s assume that isolation of the boiler is not perfect. Heat is transferred through the walls of vessel. Q = U * A * ΔT (5) Q = Transferred heat w. U = Conductance w/m2K ΔT = Tout - Tin = Temperature difference A = Area of casing m2 Pin ∫Pdt = W integrator 1/S s-domain STEP 1/c*m GAIN ΔT + - U * A = 0.2 U * A * ΔT Picture 4

Temperature difference of the boiler is not any more strait line Temperature difference of the boiler is not any more strait line. It is exponential and it reaches certain maximum level after a long time. In this example power loses are huge and temperature inside of the vessel doesn't get over 100 degrees. ΔT Pin Time Transient analyse We can edit the block diagram and substitute the feedback:

Pin Block diagram in picture 4 can be more simple. Feed-back block diagram with integrator is substituted to one transfer function. Step response will remain the same figure and magnitude. Pin STEP 5/(5s+1) ΔT Transfer function Picture 5 Transfer function formula can be written: A/(τs + 1) A = Gain of the transfer function = ΔT/Pin , K/w τ = Time constant , second

Next graph describes common step response Next graph describes common step response. Any process with one "energy store" is indentical. The process has 1st order delay. Output = A * (1 - e-t/τ) τ = Time constant tangent A Time/s Step function 0...100 63,2 Exponent function of 1 st order delay

in out - mass R + C out in out in RC-circuit, τ = R * C Heat power Next processes have similar responses: Heat power in mass ΔT =Tmass - Tambient out + - RC-circuit, τ = R * C R C in out Velocity +Force - Force, example wind force out in

water Cattle Electric heater plate Heat resistor Resistor Heater plate SECOND ORDER DELAYS Always we cannot describe the process with one delay. Example of heat process: water Cattle Electric heater plate Heat resistor Resistor Heater plate Cattle Water Power/w τ1 τ2 τ3 τ4 Tresist Tplate Tcattle Twater

Step response 1...4 delays, τ1 = τ2 = τ3 = τ4 = 1 s input step = 100 time in out

velocity = v Could water TRANSPORT DELAY Delays are also caused by the time taken for material to travel along pipes or conveyors from one place to another. This type delay, shown in next figure. Temperature mixed water Hot water velocity = v Distance to measuring point = L Time delay = L/v Could water Step response of the prosess Transport delay = 2 s, Capacitive delay = 3s

RELATIVE DIFFICULTY OF THE PROCESS Step response of the process is run to the open-loop system: steepest tangent Step Step response in output signal Relative change Tt Tr A B Controllability S = Tr/Tt S < 4 difficult to control S > 4 easy to control Sketch deepest tangent to the response and mark the points A and B. A = minimum level of response. B = maximum level of response (after a long period)

Dynamic Process Simulation Information What is the difference between mathematical models and simulations? On this site we will use the term "mathematical models" to represent sets of equations that mathematically describe the process. The term "simulator" refers to a computer program or a digital system running a computer program that implements the mathematical model. The simulator may be connected to the control system or may be embedded within the control system. What types of simulations used in process engineering? Static Static simulations, typically used in process design, simulate the process at steady state conditions, usually at the design operating conditions. Time is not a variable. Dynamic Dynamic models consider time as a variable and simulate the process over a period of time. A dynamic simulation can be used to estimate or illustrate the response, over time, to a change in the process. This primary concern of this site is dynamic models.