Distributed Parameter Systems Introduction All the apparatus dynamic experiments in the laboratory exhibit the effect known as "minimum phase dynamics". Process control loops are often based on simulations or assumptions of first-order and second-order processes. Perfect mixing is assumed in each unit operation. The text discusses these as lumped parameter type processes. What do we mean when we say a process exhibits "Non Minimum Phase" characteristics? Phase is one of the terms plotted with the Bode plots. If the process has dead time the phase is calculated as: Φ = 57.3ωτ Where phi is in degrees for a time delay of τ seconds. ω = 2πf. One can see that as the frequency increases the phase lag becomes more negative, hence "Non Minimum Phase" When the dynamic elements in the process do not shift phase beyond 90 (first-order), the resulting closed loop cannot exhibit a damped oscillation under proportional only control. The proportional gain in this case is purely arbitrary. This is the case of the lambda method, the dynamics are approximated as a single first order and the reset setting is based on that time constant. Stable control is assured in this case, but not necessarily the optimum control. True second order systems do not occur frequently in chemical and process applications, and process dead time is usually assumed to be attributed to transportation delay, such as the fluid transport time we experienced in the heat exchanger experiment with the added tubing. In our lab, the experiments all have lumped parameters. Lumped parameters occur where these is back mixing. Even the heat exchanger has back mixing because of the passes and tube baffles. However, most industrial control systems exhibit more complex behavior. (Sorry about that, but the real world functions quite different than most texts describe.) Many industrial processes have distributed parameters, the process variable are a function of distance as well as time. This requires us to view the dynamics in an additional dimension of length. There are two ways of viewing this, one as multiple interacting lags, and the second as non-interacting lags. The lags can be either discrete or operate as a continium. Examples of these from the fuel ethanol plants are, double pipe exchangers cool the mash before fermentation, falling film evaporators concentrate the syrup, distillation column to separate ethanol from water and molecular sieves; which are packed columns used to reduce the water content from the ethanol beyond the azeotrope concentration.
All these continuum processes result in the dynamic effects being expressed as a function of distance as well as time in the form of a partial differential equation or PDE. k d + k 2 t A process could have both interactive and non-interactive lags, as an example, a distillation column tray levels are non-interactive while the composition in interactive. This has been confirmed. The hydraulic time constant (non-interactive) is faster than the composition (interactive). Discrete Multiple Interacting Lags Although dead time added to a first-order lag provides a dynamic model that demonstrates realistic behavior for many industrial control simulations, some processes require more accurate dynamic modeling. An example is a distillation column, where many equilibrium stages or trays interact; the exiting compositions are a function of the interacting stages. Each tray can be represented by a first-order lag whose time constant is the familiar V/F or volume divided by the flow rate. The column cannot be represented by nth order lags where n is the number of column trays. The compositions on the trays interact with each other, and frequently the feed point is somewhere in the mid section of the column, further complicating the problem. These processes are frequently simulated as a series on first order function blocks, the resultant step response is frequently assumed to be a second order plus dead time or SOPDT. The following interactive lag example is a simulation of 4 first order lags where the V/F or volume divided by the flow time constants increase as the flow increases.
Multiple Interactive Lags 0.9 0.8 0.7 0.6 PV 0.5 0.4 0.3 0.2 0. 0 0 0 20 30 40 50 60 70 80 90 00 time Distributed Lags Adding additional first order blocks does not noticeably affect the shape of the response curve; the 20-lag simulation can even be used to represent a packed column that has no discrete stages, i.e., a distributed lag. Another common process of this type is the heat exchanger, consisting of a distribution of heat-transfer surface and heat capacity. However, heat exchangers have an added complication: with a fixed volume but variable flow, their totaled time constants vary inversely with flow-rate, the familiar V/F term. Let us look at the dynamics of these processes from the viewpoint of heat transfer.
First Order Example T L Tamb Consider the thermo well example. In this case we had heat flowing from one end of the well to the other as well as heat being transferred along the well length. We can model the fundamental equation is Fourier's law of heat transfer where: A - area of heat transfer surface h heat transfer coefficient P well perimeter surface k - thermal conductivity of the material 2 T ka = hpt 2 x While this process does exhibit distributed dynamics, the temperature along the well is a function of time as well as distance, the solution of the dynamic behavior of the tip of the well, where the temperature is measured is a simple first order, calculated based on the well dimensions, thermal conductivity of the material and the heat transfer coefficient at the well surface. The thermo well first order time constant can be calculated by the following formula: τ = 560GC Uf d P f d D = 2 3( D d 2 ) Where t is the time constant in minutes G is the specific gravity of the thermowell Cp is the specific heat of the thermowell material BTU/lb-degF U is the heat transfer coefficient in BUT/hr-ft^2-degF fd is a dimension factor D is the outside diameter in inches k thermal conductivity d is the inside diameter in inches a well cross sectional area
Thermowell Response 60 50 40 Temp, DegF 30 20 0 00 90 80 70 0 2 3 4 5 time, minutes This is to show that even processes that have PDEs the resultant dynamic behavior can be simplified.
Process Identification For non-interactive lags, the total lag is the sum of each lag or Στ = nτ, 2 n + n For interactive lags the total response is τ = τ 2 We can identify the process dynamics by using the Ziegler-Nichols open loop testing method. Two times are noted once the slope is drawn on the output step response, τ de and τ e. τ e is the time for the PV to respond to the same percentage change, along the slope, as the output step change, m, in percent. These two times can be used to define the controller time constants for either interactive or non-interactive distributed process. The process gain is calculated as K p = de.5 and Σ τ = 7. 0τ de e 7 τ τ Tuning constants can be calculated for a non-interacting PID controller: for the interacting version are: Popt = 0K p Iopt = 0.30Σt Dopt = 0.09 t Popt = 5K p Iopt = 0.25 t Dopt = 0.0 t Popt is Proportional Band in percent, Iopt and Dopt are in time units. Using the ultimate period method will produce better results than the graphical method.
Ziegler and Nichols found that closed-loop testing under proportional control can produce still more-accurate estimates of controller settings. In this method, they reduced the controller to effectively proportional-only action by setting derivative time to zero and integral time to maximum (using a DeltaV system this will require a PD controller with no reset). Increase the gain until a uniform oscillation was produced. From this test obtained two pieces of information are found: the gain that produced the sustained oscillation and the period of the oscillation. Under these conditions, a distributed process will oscillate uniformly with the proportional band in percent set at 8.5Kp, and the resulting period of oscillation is observed to be 0.643Στ. The process parameters are then: Pu K p = Σ τ =. 55τ u 8.5 Where Pu is the un-damped proportional band and τu is the period of the un-damped cycle. It is often possible to calculate these parameters from known process information, when available, which can avoid testing, and even give valuable insight into the potential for parameter variations.
Distributed Lag Example A double pipe heat exchanger and a packed bed are examples of these distributed lags. The transient response is influenced by the nature of the heat transfer fluid as well as the relative flow patterns with respect to the transferred material. For the double pipe exchanger, we will discuss the heat being transferred in three ways. The first is assuming the outer jacket heat media is operating at a constant temperature while the fluid is flowing through the second pipe. The heating media could be either condensing steam or Dowtherm. In either case the supply temperature is assumed to be constant. This mass energy balance is written as partial differential equation or PDE. t L = v z L + τ ( T T ) W L τ = πdihi A ρc i i t W = τ 22 ( T T ) ( T T ) S W τ 2 W The wall temperature, T W and the liquid temperature, T L, vary across the length of the exchanger. If we assume the fluid flows through the outer jacket pipe and looses heat through the pipe to the inner fluid, the outer fluid can flow in either co-current or counter current flow. The co and counter flows refer to the flow pattern relative to the inner flow. In either of these cases, the supply temperature will also change both as a function of time and length through the exchanger. This is the simulated process you will control in Experiment 6 L τ 2 πdihi = A ρ C w w w τ 22 πdo ho = A ρ C w w w
Double Pipe Response 290 270 Temp 250 230 20 constant supply counter current 90 70 50 cocurrent 0 50 00 50 200 250 Time Double Pipe Heat Exchanger Simulated Outlet Temperature Response Note that in the case of the constant supply temperature, the result is the most responsive design. This is because the supply temperature is constant along the entire length of the outlet jacket. The dead time is the transportation time. Except for the knee at the bottom, the response appears to behave like a first order with dead time. Note the comparison with the interactive lag response. The constant supply and interactive lags examples are processes that have a smaller dead time to time constant ratio, these are easier to control. Processes multiple interactive variable time constants are more difficult to control because the apparent dead time will vary as the load. Tuning is difficult because different tuning settings will be required as the load varies Cascade Control If steam is supplied to a jacket or outer pipe in the double pipe exchanger, it is better to cascade to a pressure controller than a flow controller. Consider the following example: Assume a double pipe exchanger is temperature controlled by cascading a flow control loop. If a disturbance is introduced, that is an increase in load, the pressure in the steam cavity will decrease. This will cause the flow to increase. However that flow loop, typically 5 times faster than the temperature loop, will sense the increased flow and begin
to close the valve. This is not the right direction for valve travel. If, on the other hand, the inner loop is a pressure controller, the increased load will decrease the pressure in the cavity. The pressure controller will open the steam valve, which is the correct action for an increased load. Interactive Lags Distributive Lags 290 270 250 PV 230 20 90 70 constant supply interactive lags 50 0 50 00 50 200 time In the case of the counter current design, the apparent dead time is increased and the response curve appears to behave more like a second order behavior. The counter current design had more dead time and it too appears to have a shape that appears to be second order. Simulated second order behavior response will not truly demonstrate this behavior. Double Pipe Exchanger Flow Patterns Constant Temperature Supply Ts Tin T L
Counter Current Flow Tin T L Ts Co-current Flow Ts Tin T L
Packed Bed Distribution Pattern Tin T L g t g s s = c c2 = c3 ( Tg Ts ) t L t In this case a fluid flows through a packed bed, interaction occurs with the particles in the bed. This interchange can be either thermal, such as heating or cooling the bed contents or adsorption of some material in the fluid stream, such as drying a compressed air stream. The above equations show that the exiting gas temperature is a function of the length as well as the solids temperature. The resultant response curve is shown below. There is an obvious dead time due to the transportation delay through the bed. However the transient response is obviously much more complicated than a second order. The shape is similar to the co-current double pipe heat exchanger. 250 200 50 00 50 0 0 50 00 50 200
ODE PDE Simulations Frequently distributed parameter systems are simulated as multiple interactive lags. This can lead to modeling errors and if these models are used to predict control behavior, incorrect results will be obtained. Consider the double pipe exchanger, with steam as the outer heating media and Therminol-55 as the inner fluid. Two separate simulations were developed, one simulating 00 feet of jacketed pipe as 0 first order segments, two differential equations were used to simulate each segment. The term L v z in the partial differential equation L L = v + ( TW TL ) t z τ is approximated as and the partial derivatives with respect to time are TL v z evaluated as full time derivatives. This model is considered an ordinary differential equation model or ODE. An exact solution requires the integration of partial differential equations or PDE. The differences between these methods can be illustrated by comparing a step response for both solutions. A comparison of the two methods is shown as: ODE-PDE Step Response 220 20 200 Temperature, DegF 90 80 ODE PDE 70 60 50 0 5 0 5 20 25 30 35 40 45 time, seconds
Note that during the first few seconds both methods appear to agree, however during the final approach, the PDE model appears to lag the ODE estimation, even though the both appear to reach the final value at about the same time. The exact inverse transform, to the time domain, of the Laplace transform of the equations has a delay element, the familiar exponential term exp(-av/l) where v is the inner fluid velocity, L is the pipe length and a is a Laplace transform. In this case the exact transform is not easily inverted. The function does take on the form of a Bessel function of the first kind, zeroth order. This follows closely with the findings of the original transport work done by Schumann and Furnas. While this ODE assumption appears to show the approximate behavior, this modeling error is magnified when controlled in a closed loop. In these simulations, both the ODE and PDE models are used to simulate the exit fluid temperature. A proportional plus integral controller is used to control the exiting fluid temperature by varying the steam condensation temperature. The same controller and settings was used for both models and yielded the following result: ODE PDE Control Comparisons 30 290 270 Temperature, DegF 250 230 20 ODE PDE 90 70 50 0 50 00 50 200 250 Time, seconds Note that it takes much longer to stabilize the control even though the initial overshoots are almost equal. One way to avoid overshoot with a distributed parameter system is to use an external reset function, with the reset term delayed. Refer to the Shinskey article, The power of external-reset feedback.
Conclusions Control engineers should avoid making control simulations of distributed systems as a series of first order systems. The process dynamics for distributed parameter systems should include terms to simulate the distance dimension as well as the time dimension. References: Francis G. Shinskey, "Process Control: As Taught vs as Practiced" Industrial & Engineering Chemistry Research, 4(6), pp.3745-3750 Francis G. Shinskey, The power of external reset feedback Copyright 2004-2007 Control Global Coughanower, D. R., Process System Analysis and Control, Boston, MA: McGraw-Hill, 99. Furnas, C. C., "Heat Transfer from a Gas Stream to a Bed of Broken Solids" Trans. AIChE 24:42 (930). Schumann, T. E. W., "Heat Transfer: a Liquid Flowing Through a Porous Prism," Jour. Franklin Inst., vol 208, 929, pp. 305-36.