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PDC - Coggle Diagram

- - - - Component mass balance
      - Total mass balance
    - - Use substitution to write DAE's to differential equation
      - Determine variables
        
        Density and volume may change in time
        
        Control/state, manipulated/input and disturbance variables
        
        If can make a DV from it, it is probably a state variable. x,v
        
        If you have no control over them = dist. Feed sometimes
      - DOF
        
        Total variables - amount of independent equations
        
        DOF = 0, determined system
        
        DOF < 0 = overspecified system
        
        DOF > 0 = controlled/underspecified system, hebt er dus invloed op
        
        DOF = 0 does not mean only 1 answer
        
        Als je n control valves hebt, heb je dus n operational DOF
      - Use numerical solution if nessacary
        
        forward/explicit euler
        
        backward/implicit euler
        
        RK
  - - - 8 trays, feed in tray 4
      - Bottom inventory (heat element)
      - Top inventory (condensor)
    - - 3 Trays are in equilibrium (alpha is constant within range)
      - 4 Constant molar holdup
      - 2 VE's are well mixed
      - 5 Constant molar overflow (Ln = Ln-1 etc) + dT = 0
      - 1 Neglect vapour phase Pliq >> Pvap
    - - Alpha = relative volatility = (ym/xm)/(ye/xe)
      - Daltons law Pi = yi*P
      - Raoults law Pi = xi * Pi0
      - Only the most volatile component is denoted with x or y
      - Antoine equation = non-linear
      - Nemen aan dat P constant is
      - a can be used to remove pressure from the xm/ym relation
    - - Is in moles
      - Is not solved by hand
- - - - Multiple steady state points
        
        Stable
        
        Unstable
        
        Small change permenately takes system away from steady state
    - - State space format
        
        Using the difference variables (x'=x-x0) and taylor expansion
        
        d/dt(x) = Ax+Bu+Bdd
        
        Relates the change of the state/ouput variable to a change in the state/input/disturbance variable for a function f(x,u,d)
        
        Only valid near the operating point
        
        y = Cx
        
        y = output variable
        
        D and Dd can be assumed to be zero because the input and disturbance variable have no direct effect on the output. They change the state variable who changes the output.
        
        C will be given
        
        Linearised form of non-linear equations
      - Transfer functions
        
        General
        
        P(s)=Y(s)/U(s)
        
        Uses laplace
        
        Properties stay the same in t/s domain
        
        Systematic method of solve DEs
        
        easier
        
        Relates the change in the proces output to changes in the proces input
        
        Tranfer funtion is improper als order nominator > order denominator
        
        Dan beschrijft het geen fysisch proces
        
        Method
        
        Single equation
        
        If non-linear first linearise using Taylor
        
        Take the laplace transformation f(0)=0
        
        Rearrange to obtain transfer function
        
        1 more item...
        
        State space format
        
        Take the laplace transform (sX=AX+BU+Bdd)
        
        Rearrange (X=(SI-A)^-1*BU)
        
        Fill in y=CX (C(SI-A)^-1*BU))
        
        2 more items...
        
        Result
        
        Info transfer function P(s)=Y(s)/U(s)
        
        Poles
        
        Determines
        
        Stability
        
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        Order system
        
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        Speed
        
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        Root of denominator polynomial U(s)
        
        Pole = 0 is charasteristic equation of system
        
        Zeros
        
        Root of nominator polynomial Y(s)
        
        Determines
        
        Steady state gain kp
        
        Variants transfer function
        
        Second order system
        
        t^2d^2y/dt^2 + 2tzdy/dt+y(t)=Kp*u(t)
        
        z = damping coefficient
        
        3 more items...
        
        P(s) = K/(t^2s^2+2zts+1)
        
        Dead time
        
        P(s) = e^-ds
        
        Integrator
        
        dy/dt = Kp*u(t)
        
        P(s) = Kp/s
        
        Not stable, s=0
        
        Processes in series
        
        P(s) = P1(s)*P2(s)
        
        First order systems
        
        tdy/dt + y(t) = Kp*u(t)
        
        Tau
        
        3 more items...
        
        Kp
        
        3 more items...
        
        P(s) = Kp/ts+1
        
        Processes in parralel
        
        P(s) = P1(s) + P2(s)
        
        Recyle stream
        
        Verkleint de poles en vertraagt dus veranderingen
        
        Inverse response
        
        Bij een proces in parralel één tf veel sneller reageerd is maar tegengesteld en kleiner dan de ander gaat bijvoorbeeld eerst de graph omlaag om daarna weer omhoog te gaan.
    - - To hard to solve analytical
  - - - Properties (antoine equation)
      - Reaction kinetics Ca^n*Cb^m
      - Saturation terms
      - Cross terms phi(t)*T(t)
- - - - Control valves
      - Variable speed drives
    - - Converts proces variables to electrical variable
      - Variables
        
        T
        
        Flow rate
        
        Pressure
        
        quality control
- - - - Basic
        
        PID
        
        Proportional
        
        Present errors
        
        Combinations
        
        Always include proportional
        
        PID
        
        PI
        
        PD
        
        Determination coefficients
        
        IMC tuning = less overshoot, lower values, less u change
        
        With dead time: cohen coon, ziegler nicholas and IMC tunig
        
        Derivative
        
        Gevoelig voor noise
        
        Future errors
        
        Integral
        
        Bij saturation:external reset nodig
        
        Past errors
        
        On/of
        
        Direct synthesis method
        
        Controller is designed with closed loop transfer function
        
        If chosen for first order proces than C(s) = 1/(P(s)stau)
        
        Only works if P(s) is invertible, not is P(s) has dead time
        
        C(s) = P(s)-1*R(s)/(1-R(s)
        
        Internal model control (IMC)
        
        Addition of plant model Pe
        
        Pe-= invertible part
        
        Pe+ = non-invertible part
        
        Accurate if P(s) = Pe(s)
        
        Derivation
        
        Write loop in standard form to obtain C(s)
        
        fill in in normal function
        
        Assume P(s) = Pe(s)
        
        Ideal if y=r, so C(s) = 1/P(s)
        
        Add filter 1/(lamda*s+1)^n to properialise function
      - Enhanced
        
        Ratio control
        
        One flow is controlled in ratio with other flow
        
        Time delay/ dead time compensation
        
        Addition of theoretical model, to counter the effect of dead time
        
        smith model
        
        Cascade control
        
        Addition of secondary loop + intermeadiate measurement point to have a quicker respons on disturbances
        
        Inferential
        
        Output cannot be directly measured: use of simulation model
        
        Feedforward
        
        Disturbance of D is measered
        
        Does not change stability system
        
        Respons quicker than FB control, but is less stable
        
        Selective
        
        More controlled variables than manipulated variables
        
        Others: split range 1cv but 2 dof, overide control
      - Chose of strategie
        
        Money
        
        Type of process
    - - Stability
        
        Hoe groter het imaginaire deel, hoe meer oscillaties
        
        Bij fysische processen heb je altijd een complex conjugate
        
        If charesteristic equation has no positive poles, also determines on chosen Kc and other controller variables
      - Steady state closed loop respons * s
        
        Vaak nooit ysp = y, voor elke k bij proportional
      - Closed loop transfer function Y(s)/R(s)
        
        Obtained when all transfer functions are combined
      - Open loop means without control
  - - - Comes with Interaction
        
        Can be controlled with
        
        1 Multiple independent controllers
        
        Right mv and cv have to be paired
        
        2 Relative gain array (RGA)
        
        Lamda (relative gain) = open loop gain / closed loop gain
        
        4 more items...
        
        Pairing
        
        4 more items...
        
        1 Reasoning
        
        Or use decoupling
        
        Addition of controllers to eliminate interaction in the closed loop
        
        But uses inverses of the process
        
        2 One multivariable controller
        
        Manipulated variable affects multiple controlled variables
      - More than one DOF and or controlled variable --> Matrix Y(s) = Gp(s)U(s)
    - - Not in the s-domain but in t-domain
      - Given dx/dt = Ax+BU
        
        Eigenvalues matrix A are the poles of the open loop system
        
        Determination eigenvalues/poles --> det(SI-A) = 0
        
        How can u be changed to affect those poles in closed loop
        
        u = -Kx
        
        det(SI-Acl) = 0
        
        Determine own poles (pole placement)
        
        Solve for right k values to get those poles
        
        Must be close to open loop poles
        
        Acl = (A-BK)
  - - - Stable
      - Offset is minimum
      - Control action is not excessive
- - - - Needed for
        
        2 Keep between operational constraints
        
        Safety
        
        Safety relieve valves
        
        Instrument protective function
        
        Inherent
        
        3 Economic operation
        
        1 Regulation of quantity and quality
    - - Batch
        
        Prefferd for small quantities
        
        Less efficient and safe
        
        Flexible and cheaper initial cost
      - Continious