Instrumentation and Process control

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This section covers the introduction to process control, mathematical modelling, the dynamic behaviour of chemical processes, the controller modes P, PI and PID, and control valves.

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Why Control Is Needed • The objectives of process control are safety first, then environmental and equipment protection, then maintaining product quality and production rate, and finally economic optimisation.

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Safety always takes precedence, and this ordering is examined directly. • Variables: the controlled variable is the one to be held at the desired value; the manipulated variable is the one the controller adjusts; the disturbance or load variable is an outside influence that upsets the process and cannot be manipulated; and the set point is the desired value of the controlled variable. • Servo versus regulatory problem: in the servo problem the set point changes and the output must follow it; in the regulatory problem the set point is fixed and the controller must reject disturbances.

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Most chemical plant control is regulatory. • The elements of a feedback loop, in order: process → measuring element (sensor and transmitter) → comparator → controller → final control element (usually a valve) → back to the process.

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Mathematical Modelling • A model is a set of equations describing how the process behaves, built from conservation of mass, of energy and of momentum, together with constitutive relations such as rate laws and equilibrium relations. • Degrees of freedom = number of variables − number of independent equations.

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For a properly specified problem the degrees of freedom must be zero, and the number of control loops that can be placed on a unit equals its degrees of freedom. • Linearization: most process models are non-linear, but control theory is built on linear systems, so the model is linearized about a steady state by a Taylor expansion truncated after the first order term.

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The approximation is good near the operating point and deteriorates as the deviation grows — a standard examination statement. • Deviation variables, defined as the difference from the steady-state value, are used throughout, because they make all initial conditions zero and so simplify the Laplace transform. • Transfer function G(s) = output(s)/input(s), defined in deviation variables with zero initial conditions.

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It describes only the input-output behaviour and applies only to linear systems. • Key Laplace results: step of magnitude A → A/s; impulse → 1; ramp of slope a → a/s²; dead time of θ → e−θs.

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The final value theorem, limt→∞ y(t) = lims→0 sY(s), gives the ultimate value without inverting the transform and is used constantly — but it is valid only if the system is stable.

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Controller Modes Mode Equation Effect Drawback Proportional (P) p = p̄ + Kce Simple, fast, stabilising Always leaves a steady-state offset Integral (I) p = p̄ + (Kc/τI)∫e dt Eliminates offset completely Slows the response and reduces stability; can wind up Derivative (D) p = p̄ + KcτD(de/dt) Anticipates, adds stability, speeds response Amplifies noise; useless alone, since it responds only to change PI P + I terms Removes offset; the commonest industrial controller More oscillatory than P alone PID All three terms Best overall performance on slow, noise-free loops Three parameters to tune; poor on noisy flow loops • The proportional band PB = 100/Kc expresses the gain as the percentage change in error required to drive the output over its full range.

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A narrow band corresponds to a high gain, and a wide band to a low gain — a relation that is inverted so often in examinations that it is worth writing out. • Offset, the permanent difference between set point and controlled variable left by a proportional controller, exists because a proportional controller needs a non-zero error to produce any change in its output away from the bias value.

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For a unit step change in set point, offset = 1/(1 + KcKp): increasing the gain reduces the offset but can never remove it, and eventually destabilises the loop. • Integral (reset) action removes offset because the integral of a non-zero error grows without limit, so the controller output keeps changing until the error is exactly zero.

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This is the single most important statement in the chapter. • Reset windup occurs when a sustained error saturates the final control element while the integral term continues to accumulate; recovery is then delayed because the accumulated term must unwind.

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It is prevented by anti-reset windup logic that stops integration once the output saturates. • Derivative action responds to the rate of change of error, so it acts before a large error develops.

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It cannot be used alone because it produces no output at all when the error is constant, however large that error is.

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It is unsuitable for noisy measurements such as flow, where differentiation amplifies the noise, and is usually applied to the measurement rather than the error to avoid derivative kick on a set-point change. • Typical loop practice: flow and pressure loops use PI (fast and noisy, so no derivative); temperature loops use PID (slow, with significant lag, and relatively clean signals); level loops often use P alone, since tight level control is usually unnecessary and averaging control is preferred.

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Control Valves • The control valve is the commonest final control element, and typically comprises a valve body, a plug and seat, a stem, and a pneumatic diaphragm actuator opposed by a spring, with a positioner to overcome friction and ensure the stem reaches the demanded position. • Fail-safe action — the most examined point: an air-to-open valve is fail-closed, since loss of air lets the spring close it; an air-to-close valve is fail-open.

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The choice is made entirely on safety grounds: a steam valve to a reactor is normally fail-closed, while cooling water to an exothermic reactor is fail-open. • Valve characteristic is the relation between flow and stem position at constant pressure drop: • Quick opening — most of the flow is passed in the first part of the travel; used for on-off service. • Linear — flow proportional to lift; suitable where the valve pressure drop is a large and nearly constant fraction of the system drop. • Equal percentage — each equal increment of lift changes the flow by an equal percentage of the existing flow, giving a small change at low lift and a large change at high lift.

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It is the commonest industrial characteristic, because it compensates for the falling valve pressure drop as flow rises and so gives a nearly linear installed characteristic. • Valve sizing uses the flow coefficient Cv, the flow of water in US gallons per minute that passes at a pressure drop of 1 psi.

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Valves are normally sized to operate between about 20 and 80 per cent open at normal flow, leaving room to correct in both directions. • Cavitation and flashing occur when the pressure at the vena contracta falls below the vapour pressure of the liquid; if the downstream pressure recovers above the vapour pressure the bubbles collapse, which is cavitation and causes noise and erosion; if it stays below, the liquid flashes and stays two phase.

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This section covers advanced control schemes including feedback, feedforward, cascade and ratio control, together with selective and split-range control, and their application to equipment such as distillation columns and reactors.

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Feedback and Feedforward Feature Feedback control Feedforward control Acts on The measured error in the controlled variable The measured disturbance, before it affects the output Timing Corrective action begins only after a deviation appears Corrective action is taken in anticipation Model needed No process model required An accurate process and disturbance model is essential Measurement needed The controlled variable Every disturbance to be compensated Unmeasured disturbances Handled, since any deviation is corrected Not handled at all Model error Tolerated; the loop self-corrects Leads to permanent error Stability Can become unstable Cannot itself cause instability, being open loop • The conclusion that is examined: feedback is universal and robust but always acts late; feedforward is fast and anticipatory but blind to anything it does not measure and dependent on model accuracy.

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In practice the two are combined — feedforward handles the major measured disturbance and a feedback trim corrects the residual error, which gives the advantages of both. • The ideal feedforward controller is Gf = −Gd/Gp, the ratio of the disturbance transfer function to the process transfer function with a negative sign.

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It is often physically unrealisable — for example if it would require prediction, or a derivative of higher order than the process allows — and is then approximated by a lead-lag unit with a gain. • Negative feedback is what makes a control loop work; positive feedback reinforces the deviation and is destabilising.

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A loop with an even number of sign reversals around it is effectively positive feedback, which is why the controller's direct or reverse action must be set correctly.

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A reverse-acting controller decreases its output as the measurement rises, and is the correct choice when the process gain is positive.

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Cascade Control • Cascade control uses two controllers in series: the output of the primary (master, outer) controller becomes the set point of the secondary (slave, inner) controller, whose output drives the valve.

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There is one manipulated variable but two measurements. • The classic example is a jacketed reactor: the master controls reactor temperature and sets the set point of the slave, which controls jacket or coolant temperature.

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A change in coolant supply temperature or pressure is then corrected by the fast inner loop before it can disturb the reactor at all. • The design rules, which are asked directly: the inner loop must be significantly faster than the outer loop — a factor of three to five in time constants is the usual guideline — and the inner loop must contain the principal disturbance.

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The inner controller is usually P or PI with a high gain; only the outer controller needs integral action to remove offset in the true controlled variable. • The benefit: cascade control rejects disturbances entering the inner loop far faster than a single loop could, and reduces the effect of non-linearity in the valve and the inner process.

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Ratio, Selective and Split-Range Control • Ratio control maintains a fixed ratio between two flows.

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The wild (uncontrolled) stream is measured and the controlled stream is manipulated to keep the desired ratio.

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It is used for the air-to-fuel ratio in a furnace, the reflux ratio in a column, and the reactant ratio to a reactor.

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The preferred implementation multiplies the measured wild flow by the desired ratio to compute the set point of a conventional flow loop, rather than dividing two measurements, because division amplifies noise at low flow. • Selective (override) control allows several controllers to compete for one valve, with a high or low selector passing the most urgent signal.

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It is used to protect against constraint violation — for example, a pump discharge pressure override on a flow controller. • Split-range control lets one controller drive two or more final control elements over different parts of its output range, as when 0-50 per cent of the signal opens a cooling valve and 50-100 per cent opens a heating valve. • Inferential control is used when the true controlled variable cannot be measured on line, so a secondary measurement is used to infer it — the standard case being the use of a tray temperature to infer composition in a distillation column. • Adaptive control adjusts the controller settings as conditions change, and gain scheduling is its simplest form, switching between tuning sets according to the operating region.

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Application to Distillation Columns • A distillation column is highly interacting, non-linear and slow, and is the standard example of a difficult control problem. • Five degrees of freedom are typically available on a simple two-product column: reflux flow, distillate flow, boil-up (steam), bottoms flow and condenser duty.

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Two are consumed by inventory control — condenser drum level and column base level — and one by pressure control, leaving two for composition. • Pressure control comes first, because relative volatility, and therefore the entire separation, depends on pressure; it is usually held by manipulating condenser cooling or the vapour vent. • Common composition control configurations are named by the two variables chosen for composition: the L-V scheme (reflux and boil-up), the D-V scheme, the L-B scheme and the ratio schemes.

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The L-V scheme is the commonest but shows strong interaction between the two loops. • Tray temperature is used to infer composition, since analysers are slow and expensive; the tray is chosen where the temperature is most sensitive to composition change and least sensitive to pressure, and pressure-compensated temperature is used where pressure varies. • Interaction between the two composition loops is the central difficulty, and is assessed by the relative gain array (RGA): a relative gain of 1 means no interaction, a value between 0 and 1 means the loops help each other but interact, a value greater than 1 means they fight each other, and a negative value means the pairing is unstable and must never be used.

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Pair the loops on relative gains closest to 1 and never on a negative value — the standard rule.

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Application to Reactors • Temperature is almost always the critical controlled variable in a reactor, because rate, selectivity and safety all depend on it and an exothermic reaction can run away. • Cascade control of reactor temperature onto jacket temperature is the standard arrangement, for the reasons given above. • An exothermic reactor is open-loop unstable at the middle steady state (the result from Chapter 4), so control is not merely a matter of performance but of stability — the loop is what holds the operating point. • Additional measures: feed-flow ratio control for reactant stoichiometry, split-range control between heating and cooling for start-up and running, and an independent safety instrumented system with its own sensors and final elements, separate from the basic process control system.

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The independence of the protective layer from the control layer is a fundamental safety principle, taken up again in Chapter 9.

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This section covers first, second and higher order systems, linearization, the response to step, pulse, impulse and ramp inputs, the level tank and the U-tube manometer as examples, interacting and non-interacting systems, and dead time.

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First-Order Systems • Standard form:

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G(s) = Kp/(τs + 1), where Kp is the steady-state gain and τ the time constant. • Step response of magnitude A: y(t) = AKp(1 − e−t/τ), an exponential approach to the final value with no overshoot and no oscillation whatever.

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A first-order system can never oscillate — a point asked repeatedly. • The percentages that must be memorised: the response reaches 63.2 per cent of its final change in one time constant, 86.5 per cent in two, 95.0 per cent in three, 98.2 per cent in four and 99.3 per cent in five.

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The system is conventionally taken as settled after four to five time constants. • The initial slope, if maintained, would reach the final value in exactly one time constant — the graphical way of reading τ from a recorder chart. • Ramp response: the output eventually follows the ramp but lags it permanently by exactly τ.

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Impulse response: an immediate jump followed by exponential decay.

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Pulse response: a rise followed by exponential decay once the pulse ends. • Physical examples: a stirred tank with a first-order reaction; a mercury thermometer in a well-stirred bath; a liquid level tank with a linear outlet resistance. • The level tank: for a tank of area A with outlet resistance R, τ = AR and Kp = R.

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If the outflow is set by a pump rather than by head, the tank becomes a pure integrator, G(s) = 1/As, with no self-regulation at all — a standard contrast.

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Second-Order Systems • Standard form:

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G(s) = Kp/(τ²s² + 2ζτs + 1), with ζ the damping coefficient or damping ratio.

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Damping Value of ζ Roots Step response Overdamped ζ > 1 Two distinct real negative roots Sluggish, no overshoot; equivalent to two first-order lags in series Critically damped ζ = 1 Two equal real roots Fastest possible response without any overshoot Underdamped 0 < ζ < 1 Complex conjugate pair with negative real part Oscillatory with decaying amplitude; overshoots the final value Undamped ζ = 0 Purely imaginary pair Sustained oscillation at the natural frequency Unstable ζ < 0 Positive real part Oscillation of growing amplitude • Characteristics of the underdamped response, which supply most of the numerical questions: • Overshoot = exp(−πζ/√(1 − ζ²)), expressed as a fraction of the final value. • Decay ratio = (overshoot)² = exp(−2πζ/√(1 − ζ²)).

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The widely used quarter-decay tuning criterion corresponds to a decay ratio of 0.25 and hence to ζ ≈ 0.215. • Period of oscillation T = 2πτ/√(1 − ζ²), always longer than the natural period 2πτ. • Rise time falls and overshoot rises as ζ decreases, which is the essential trade-off in tuning: speed is bought at the cost of oscillation. • Physical examples: the U-tube manometer, a pneumatic control valve with its spring and diaphragm, and any two first-order systems in series. • The U-tube manometer is the standard illustration of a genuine second-order system: it possesses inertia (the mass of the liquid column), a restoring force (gravity) and damping (viscous friction at the wall).

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A narrow tube or a viscous fluid gives high damping and a sluggish, non-oscillatory reading; a wide tube or a mobile fluid such as mercury gives a lightly damped, oscillatory response. • Two first-order systems in series are always overdamped, with ζ ≥ 1, so they can never oscillate — a frequently examined consequence.

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Interacting and Non-Interacting Systems • Non-interacting tanks in series: the outflow of the first tank depends only on its own level, so the downstream tank has no effect upstream.

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The overall transfer function is the simple product of the individual first-order transfer functions, G = K/[(τ₁s + 1)(τ₂s + 1)]. • Interacting tanks: the flow between them depends on the difference in the two levels, so the second tank does influence the first.

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The transfer function then has the form K/(τ₁τ₂s² + (τ₁ + τ₂ + A₁R₂)s + 1), the extra term in the coefficient of s being the signature of interaction. • The consequence, which is what the examination asks: interaction increases the effective damping and makes the system more sluggish — the response of an interacting system is always slower than that of the corresponding non-interacting system.

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Both remain overdamped. • Higher-order systems, formed by n first-order lags in series, become progressively more sluggish and develop an S-shaped response with an apparent delay.

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This is why a high-order process is commonly approximated by a first-order plus dead time (FOPDT) model, which captures the essential behaviour with only three parameters and is the basis of most tuning rules.

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Dead Time • Dead time (transport lag, time delay) θ is the interval during which the output shows no response at all to an input change.

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Its transfer function is e−θs. • Physical origins: the time taken for fluid to travel along a pipe from the point of change to the point of measurement, the transport of material on a belt, the analysis time of a chromatograph, and the sampling and transmission delays of a digital system.

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For pipeline flow, θ = length/velocity = volume/volumetric flow rate. • Why dead time matters so much: it contributes phase lag that increases without limit as frequency rises, while attenuating the amplitude not at all.

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It is therefore the most destabilising element that can appear in a control loop, and a loop with a large ratio of dead time to time constant must be detuned to a low gain, giving poor performance. • The controllability ratio θ/τ is the usual measure: below about 0.3 the loop is easy to control with conventional PID; above about 1 it is difficult and a dead-time compensator such as the Smith predictor should be considered. • The Smith predictor uses a model of the process to remove the dead time from the effective feedback path, so the controller can be tuned as though the delay were absent; it is very sensitive to error in the assumed dead time, which limits its use. • Approximating dead time: the Padé approximation, e−θs ≈ (1 − θs/2)/(1 + θs/2), converts the delay to a rational transfer function for analysis.

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Note that it introduces a right-half-plane zero, which produces an inverse response — an initial movement in the wrong direction, which is itself a real phenomenon in processes such as reboiler swell in a distillation column and shrink-and-swell in a boiler drum.

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This section covers controllers and final control elements, control valves, block diagram algebra, the closed-loop transfer functions, the transient response of a simple control system, offset, and stability including the characteristic equation and the Routh test.

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Block Diagrams and Closed-Loop Transfer Functions • The standard feedback loop contains Gc (controller), Gv (valve or final control element), Gp (process), Gd (disturbance or load transfer function) and Gm (measuring element). • Open-loop transfer function GOL = GcGvGpGm, the product of everything around the loop.

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It is the quantity on which all stability analysis is based. • Servo (set point) response:

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Y/Ysp = GcGvGp/(1 + GOL). • Regulatory (load) response:

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Y/D = Gd/(1 + GOL). • The general rule for any single loop: the closed-loop transfer function is the product of the blocks in the forward path divided by one plus the product of all the blocks around the loop.

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Learning this one sentence replaces the need to derive each case. • Block diagram algebra: blocks in series multiply; blocks in parallel add; and a feedback loop reduces to forward path over one plus loop gain.

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Transient Response and Offset • Consider a first-order process under proportional control.

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The closed loop is also first order, but with: • Closed-loop gain K′ = KcKp/(1 + KcKp) and closed-loop time constant τ′ = τ/(1 + KcKp). • Two conclusions follow immediately and are examined directly: feedback control makes the process faster, since τ′ is smaller than τ, and it leaves an offset, since K′ is less than one so the output never quite reaches the set point. • Offset for a unit step change in set point = 1 − K′ = 1/(1 + KcKp).

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Raising the gain shrinks the offset and shortens the time constant, which is why high gain is desirable — and why stability sets the limit on how far it can be pushed. • Adding integral action raises the order of the closed-loop system by one, removes the offset entirely, and makes the response more oscillatory.

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A first-order process under PI control becomes a second-order closed loop, which can overshoot even though the process alone could not. • A second-order process under proportional control becomes more oscillatory as the gain is raised: the damping coefficient falls, overshoot and decay ratio increase, and the period shortens, until at a critical gain the loop oscillates continuously.

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Stability • Definition: a system is stable if its response to a bounded input remains bounded. • The characteristic equation is 1 + GOL = 0, and the roots of this equation are the poles of the closed-loop transfer function. • The fundamental criterion, which must be known exactly: the closed-loop system is stable if and only if every root of the characteristic equation has a negative real part — that is, all roots lie in the left half of the complex plane.

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A single root with a positive real part makes the system unstable, and roots on the imaginary axis give sustained oscillation. • Routh-Hurwitz test determines stability without actually finding the roots.

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The necessary condition is that all coefficients of the characteristic polynomial be present and of the same sign — if any coefficient is zero or of opposite sign the system is certainly unstable.

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If that test is passed, the Routh array is constructed, and the number of sign changes in its first column equals the number of roots in the right half plane; the system is stable only if there is no sign change at all. • Ultimate gain and ultimate period: the ultimate gain Kcu is the proportional gain at which the loop just oscillates continuously, and the ultimate period Pu is the period of that oscillation.

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They are found by setting s = jω in the characteristic equation, or from the Routh array by finding the gain that makes a whole row vanish.

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They are the basis of the Ziegler-Nichols tuning rules. • What affects stability: increasing the controller gain is destabilising; adding integral action is destabilising; adding derivative action is stabilising; increasing dead time is strongly destabilising; and increasing the number of lags in the loop is destabilising.

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This list answers a large share of the conceptual questions in this chapter. • A first-order process with a proportional controller is stable at any gain, because the single root simply moves further into the left half plane; likewise a second-order process under proportional control cannot be made unstable.

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Instability requires at least three lags, or two lags plus integral action, or dead time — a conclusion worth holding on to.

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Final Control Elements • The final control element converts the controller signal into an action on the process.

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Besides the control valve of 8.1, it may be a variable-speed drive, a damper, a metering pump, or an electrical heater with a thyristor drive. • Signal standards: pneumatic 3-15 psi (0.2-1.0 bar) and electronic 4-20 mA.

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The live zero — 4 mA or 3 psi rather than zero — is deliberate: it allows a broken wire or lost air supply to be distinguished from a genuine zero reading, and for a two-wire transmitter the 4 mA also powers the instrument.

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This is a standard examination question. • The valve is usually the slowest element in the loop apart from the process itself, and stiction (static friction in the stem packing) is the commonest cause of persistent cycling in an otherwise well-tuned loop.

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A positioner reduces stiction and hysteresis by acting as a fast inner position loop.

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This section covers frequency domain analysis, the design of control systems by frequency response, Bode and Nyquist plots, the Bode stability criterion, gain and phase margins, and the different methods of tuning controllers.

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Frequency Response • The basic result: if a stable linear system is subjected to a sustained sinusoidal input, the output is eventually a sinusoid of the same frequency but different amplitude and shifted in phase.

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The frequency never changes — a point asked directly. • Amplitude ratio AR = output amplitude/input amplitude, and phase angle φ is the shift, negative for a lag. • Both are obtained by substituting s = jω into the transfer function:

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AR = |G(jω)| and φ = arg G(jω). • Normalised amplitude ratio ARN = AR/Kp is used so that the plot is independent of the steady-state gain.

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Element Amplitude ratio Phase angle Behaviour Gain K K, constant 0° No phase shift at any frequency First-order lag 1/√(1 + ω²τ²) −tan−1(ωτ) AR falls from 1 to 0; phase from 0° to −90° Second-order 1/√[(1 − ω²τ²)² + (2ζωτ)²] 0° to −180° Resonant peak if ζ < 0.707 Dead time 1, at every frequency −ωθ, without limit No attenuation but unbounded phase lag Integrator 1/s 1/ω −90°, constant Infinite AR at zero frequency Derivative s ω +90°, constant AR rises with frequency, so noise is amplified • The first-order lag has an asymptotic Bode plot: the low-frequency asymptote is AR = 1 (0 dB) and the high-frequency asymptote has a slope of −20 dB per decade, the two meeting at the corner (break) frequency ω = 1/τ, where the exact AR is 0.707 (−3 dB) and the phase is exactly −45°.

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These values should be memorised. • Dead time is the critical case: its amplitude ratio is exactly 1 at every frequency, so it never attenuates the signal, while its phase lag −ωθ grows without limit.

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This combination is what makes it so destabilising, as already noted. • Bode plot presents log AR and phase angle against log frequency on two separate graphs; the Nyquist (polar) plot presents the same information as a single curve of the tip of G(jω) in the complex plane as frequency varies.

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Stability in the Frequency Domain • Bode stability criterion: a closed-loop system is stable if the amplitude ratio of the open-loop transfer function is less than 1 at the crossover frequency — the frequency at which the open-loop phase lag equals −180°.

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If AR exceeds 1 at that frequency the system is unstable, and if it equals exactly 1 the loop oscillates continuously.

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The criterion applies only to open-loop stable systems whose phase crosses −180° just once. • The physical reasoning is worth holding: at −180° the feedback signal returns exactly inverted, so negative feedback has effectively become positive feedback; if the signal also returns undiminished or amplified, the oscillation sustains itself or grows. • Nyquist criterion is the general form and is stated in terms of encirclements of the point (−1, 0) in the complex plane; for a system with no open-loop right-half-plane poles, stability requires that the plot of GOL(jω) not encircle (−1, 0). • Gain margin GM = 1/AR at the crossover frequency — the factor by which the gain may be increased before instability.

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Phase margin PM = 180° + φ at the frequency where AR = 1 — the additional phase lag the loop can tolerate. • Recommended design values: a gain margin of about 1.7 to 2.0 and a phase margin of about 30 to 45 degrees.

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Larger margins mean a more robust but more sluggish loop.

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Controller Tuning Method Basis Procedure and remarks Ziegler-Nichols closed loop (continuous cycling) Ultimate gain Kcu and ultimate period Pu With I and D off, raise Kc until sustained oscillation; then P:

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0.45Kcu, τI = Pu/1.2;

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0.6Kcu, τI = Pu/2, τD = Pu/8.

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Aggressive, quarter-decay; the plant must be driven to instability Ziegler-Nichols open loop (process reaction curve) Step test giving K, τ and θ of an FOPDT fit Only one step test needed, loop stays in manual; poor for large θ/τ Cohen-Coon Same FOPDT parameters Developed for processes with significant dead time; gives a faster but less stable response than Z-N IMC / lambda tuning A process model and one tuning parameter λ Gives smooth, robust, non-oscillatory response; λ trades speed against robustness; widely preferred in modern practice Integral criteria (ISE, IAE, ITAE) Minimise an integral of the error ISE penalises large errors, IAE all errors equally, ITAE penalises persistent errors and gives the least oscillatory settings Trial and error / field tuning Operator experience Set I and D minimal, adjust gain for acceptable response, then add integral, then derivative • The Ziegler-Nichols continuous cycling method is the one most often examined, and its drawback should be stated with it: it requires the loop to be driven to the point of instability, which is unacceptable on many plants, and it produces aggressive quarter-decay settings that many operators find too oscillatory. • The practical field procedure: start with a low gain, no integral and no derivative; increase the gain until the response is acceptably fast with a modest overshoot; then decrease the integral time until offset is removed without excessive oscillation; then add derivative only if the loop is slow and the measurement clean. • Diagnosing a badly tuned loop: a slow, sluggish approach with no oscillation suggests too low a gain or too long an integral time; continuous oscillation of growing or constant amplitude suggests too high a gain or too short an integral time; erratic, jerky valve movement suggests too much derivative on a noisy signal; and a slow cycle that no tuning improves suggests valve stiction rather than a tuning problem at all.

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This section covers the classification and elements of measuring instruments, their static and dynamic characteristics, the working principles of transducers and instruments used for the measurement of temperature, pressure, flow and liquid level, moisture and humidity analysis, pH measurement, and high performance liquid chromatography.

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Elements and Characteristics of Measuring Instruments • The three functional elements of a measuring system: the primary sensing element (detector or transducer), which responds to the measured quantity; the variable conversion and manipulation element, which converts and amplifies the signal; and the data presentation element, which displays, records or transmits it. • Classification: active (drawing energy from the measured quantity, such as a thermocouple) versus passive (requiring an external supply, such as a resistance thermometer); analogue versus digital; contact versus non-contact; and deflection versus null type — a null instrument being inherently more accurate because it draws no energy from the measured system at balance. • Static characteristics, whose definitions are examined directly: • Accuracy — closeness to the true value.

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Precision — closeness of repeated readings to one another.

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The two are independent: an instrument can be precise but consistently wrong, and this distinction is a certainty in the examination. • Sensitivity — the ratio of the change in output to the change in input, that is the slope of the calibration curve. • Resolution — the smallest change in input that produces a detectable change in output. • Range and span — the limits of measurement, and the difference between them. • Hysteresis — a different reading for the same input depending on whether it is approached from above or below. • Drift — a gradual change of output with time at constant input. • Dead zone (dead band) — the range of input over which no change in output occurs. • Linearity — closeness of the calibration curve to a straight line. • Dynamic characteristics are speed of response, measuring lag, fidelity and dynamic error, and are described by the time constant and dead time of Section 8.3. • Errors are systematic (consistent and correctable by calibration) or random (scattered and reducible only by repeated measurement).

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Temperature Measurement Device Principle Range and characteristics Thermocouple Seebeck effect: a voltage arises at the junction of two dissimilar metals Very wide range (−200 to 1700 °C); cheap, rugged, fast; low output and non-linear; needs cold-junction compensation RTD (usually Pt100) Electrical resistance of a pure metal rises with temperature −200 to 650 °C; the most accurate and stable industrial sensor; nearly linear; slower and costlier; needs three- or four-wire connection Thermistor Resistance of a semiconductor, usually falling steeply with temperature −50 to 300 °C; very high sensitivity but strongly non-linear and limited range Bimetallic strip Differential expansion of two bonded metals Local indication and simple thermostats; no electrical output Filled system thermometer Expansion of a liquid, gas or vapour in a bulb and capillary Self-powered, suitable for hazardous areas Radiation pyrometer Stefan-Boltzmann law applied to emitted radiation Non-contact; very high temperatures and moving or inaccessible targets; needs the emissivity to be known Optical pyrometer Visual match of a filament against the target Above about 700 °C; manual • The thermocouple-versus-RTD comparison is the most examined point of the section: the thermocouple wins on range, cost, ruggedness and speed; the RTD wins decisively on accuracy, stability and linearity.

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Common thermocouple types are K (chromel-alumel, general purpose), J (iron-constantan), T (copper-constantan, low temperature) and S or R (platinum-rhodium, high temperature). • A thermowell protects the sensor and allows replacement without breaking containment, at the cost of adding a substantial thermal lag — an important practical point for control loops.

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Pressure, Flow and Level • Pressure: manometers for low pressures;

8

Bourdon tubes, bellows and diaphragms as elastic elements; strain gauge, capacitance and piezoelectric transducers for electrical output — a piezoelectric element responds only to changing pressure and cannot measure a steady pressure, which is a standard point.

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Very low pressures use the McLeod gauge (the vacuum standard), the Pirani gauge (thermal conductivity) and the ionisation gauge (for the highest vacuum). • Flow, adding to the devices of Chapter 3: the electromagnetic flowmeter works by Faraday's law and requires a conductive liquid, but offers no obstruction and handles slurries; the Coriolis meter measures true mass flow directly and also gives density, which is its distinguishing feature; the vortex shedding meter counts vortices shed from a bluff body, the frequency being proportional to velocity; the ultrasonic meter uses transit-time difference or Doppler shift; and the turbine meter counts rotor revolutions.

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Positive displacement meters are the choice for custody transfer of viscous liquids. • Level: sight glass and float for direct indication; differential pressure (hydrostatic head), the industrial workhorse, which needs the density to be known and constant; displacer, working on Archimedes' principle; capacitance, ultrasonic and radar (the last being non-contact, unaffected by vapour, and the modern choice for difficult service); and nucleonic gauges for extreme conditions.

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Zero and span of a DP level transmitter must be corrected for wet legs and elevated or suppressed zeros.

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Moisture, Humidity and pH • Humidity of a gas: the psychrometer (wet and dry bulb) reads humidity from the wet-bulb depression, using the Lewis relation of Chapter 6; the dew-point hygrometer chills a mirror until condensation is detected, which is the most fundamental method; capacitive and resistive polymer sensors are the usual industrial transmitters; and the hair hygrometer is the classic mechanical device. • Moisture in solids and liquids: loss on drying (the reference method), Karl Fischer titration (specific to water, sensitive to trace levels and the standard analytical technique), infrared absorption and microwave or capacitance methods for on-line use. • pH measurement uses a glass electrode as the measuring electrode and a silver/silver chloride or calomel reference electrode, often combined into one body.

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The Nernst equation gives about 59.16 mV per pH unit at 25 °C, a figure worth remembering, and because the slope is temperature dependent, automatic temperature compensation is essential. • Practical points, which are examined: the glass membrane must be kept hydrated and never allowed to dry out; calibration uses at least two standard buffers bracketing the expected range; the reference junction fouls and is the commonest cause of drift; and at very high pH and high sodium concentration the electrode reads low — the alkaline or sodium error. pH control is notoriously difficult because the titration curve is extremely non-linear near neutrality, so the process gain changes by orders of magnitude over the range, which is why staged neutralisation vessels and non-linear or gain-scheduled controllers are used.

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High Performance Liquid Chromatography • HPLC separates the components of a liquid mixture by their differing distribution between a liquid mobile phase pumped at high pressure and a solid or bonded stationary phase packed in a column. • Components in order: solvent reservoir → high-pressure pump → injector → column (often in an oven) → detector → data system, with a degasser and guard column commonly fitted. • Reversed phase is the commonest mode: a non-polar stationary phase, typically C18 bonded silica, with a polar mobile phase such as water-acetonitrile or water-methanol.

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Polar compounds elute first and non-polar compounds are retained longest.

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Normal phase reverses both the phases and the elution order. • Isocratic elution holds the mobile phase composition constant; gradient elution changes it during the run to resolve a wide range of retention without an excessively long analysis. • Detectors:

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UV-visible and diode array (the commonest), refractive index (universal but insensitive and incompatible with gradients), fluorescence (highly sensitive and selective), electrochemical, and mass spectrometric (LC-MS, giving identification as well as quantification). • Terms: retention time identifies a compound under fixed conditions; peak area quantifies it; resolution measures the separation of two adjacent peaks; and the number of theoretical plates measures column efficiency, exactly as in distillation. • HPLC versus gas chromatography:

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HPLC handles non-volatile, thermally labile and high-molecular-weight compounds — proteins, pharmaceuticals, sugars — that GC cannot, because GC requires the sample to be volatile and thermally stable.

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This comparison is a standard question.