Q: Speeding up an expansion (filled-system) pressure-spring thermometer: which bulb design/material choices increase speed of response?

Introduction / Context: Expansion (filled-system) thermometers transmit temperature-induced pressure/volume changes from a bulb through a capillary to a pressure spring and indicator. Their time constant depends on how quickly the bulb equilibrates with the process. Given Data / Assumptions: We aim to reduce the thermometer time constant (faster response). Changes to geometry and materials are allowed. Concept / Approach: Time constant roughly scales with thermal mass / heat-transfer coefficient / area. Thus, increasing area, reducing mass, using high-conductivity materials, and lowering specific heat all reduce the energy required and the thermal resistance between process and sensing fluid, yielding a faster response. Step-by-Step Solution: Increase bulb surface area to boost heat flux.Choose high-k materials (e.g., copper alloys) to minimize internal temperature gradients.Reduce specific heat/mass so less energy is needed to change bulb temperature.Combine all → fastest practical response. Verification / Alternative check: Manufacturers offer “fast response” bulbs with thin walls, fins (area), and high-k materials; response testing confirms reduced time constants. Why Other Options Are Wrong: Thick, heavy bulbs slow response due to high thermal mass and reduced area/volume ratio. Common Pitfalls: Over-thinning walls to the point of mechanical failure; ensure compatibility with pressure rating and erosion/corrosion conditions. Final Answer: All of the above (a), (b), and (c)

Q: Measuring about 0.0001 psi(abs) (≈ 5.2 × 10^−3 torr) continuously: which gauge is typically used for this low-vacuum region?

Introduction / Context: At a few millitorr (≈ 5 × 10^−3 torr), many industrial systems require continuous indication rather than occasional reference measurements. Thermal conductivity gauges (Pirani/thermocouple) are commonly used in this vacuum region. Given Data / Assumptions: Target: ~0.0001 psi(abs) ≈ 0.0052 torr. Continuous, on-line indication is desired. Gas composition is reasonably constant (thermal gauges are gas-dependent). Concept / Approach: Pirani gauges operate by relating filament heat loss to gas pressure, covering roughly 10^−3 to 10 torr (instrument-dependent). Thermocouple gauges are similar. While a McLeod gauge can measure in this region, it is an intermittent, manual device and not suited for continuous monitoring. Thus, the practical, textbook answer for continuous measurement near 5 mTorr is the Pirani gauge. Step-by-Step Solution: Convert units: 0.0001 psi → ~5 mTorr.Match ranges: Pirani covers ~10^−3–10 torr.Select Pirani as the continuous-reading instrument. Verification / Alternative check: Vacuum practice charts place Pirani/thermocouple gauges in the “low to medium vacuum” window that includes a few millitorr; ionization gauges are used for high/ultra-high vacuum below ~10^−3 torr. Why Other Options Are Wrong: McLeod: Accurate but intermittent and unsuitable for continuous control.Mercury manometer: Far too insensitive at millitorr.“None”: Not true—thermal gauges are standard here. Common Pitfalls: Ignoring gas-dependence and contamination; calibration should match the actual gas mixture to minimize reading bias. Final Answer: Pirani (thermal conductivity) gauge

Q: Closed-loop with dead time and first-order blocks (typical process + measurement + controller): which characteristic equation is consistent with two first-order lags ((τm s + 1), (τp s + 1)) in unity feedback and a controller gain Kc with overall pr…

Introduction / Context: Writing the characteristic equation from a block diagram is a core skill in control engineering. When a loop contains first-order dynamics for the process and the measuring element, a controller gain, and a pure dead time, the loop equation must reflect these factors in the denominator of the closed-loop transfer function. Given Data / Assumptions: Unity feedback with measurement dynamics Km/(τm s + 1). Process dynamics Kp/(τp s + 1) with a dead time e^(−τd s). Controller gain Kc (no integral/derivative terms shown). No extra transport lag besides those stated. Concept / Approach: For unity feedback, the characteristic equation is 1 + L(s) = 0 where L(s) is the loop transfer function. With L(s) = Kc * [Km/(τm s + 1)] * [Kp/(τp s + 1)] * e^(−τd s), multiply both sides by (τm s + 1)(τp s + 1) to obtain (τm s + 1)(τp s + 1) + Kc Kp Km e^(−τd s) = 0. Step-by-Step Solution: Write L(s) = Kc Km Kp e^(−τd s)/[(τm s + 1)(τp s + 1)].Characteristic equation: 1 + L(s) = 0.Multiply through by (τm s + 1)(τp s + 1): (τm s + 1)(τp s + 1) + Kc Kp Km e^(−τd s) = 0. Verification / Alternative check: Setting τd → 0 recovers the familiar polynomial (τm s + 1)(τp s + 1) + Kc Kp Km = 0, confirming consistency without dead time. Why Other Options Are Wrong: Options with τl terms introduce an extra lag not stated.Options missing Kc (just Km Kp) omit the controller gain.Option with product multiplied by e^(−τd s) on the polynomial part misrepresents the structure of 1 + L(s) = 0. Common Pitfalls: Placing e^(−τd s) on the wrong factor, forgetting to clear denominators, or misreading unity feedback versus sensor-in-loop placement. Final Answer: (τm s + 1)(τp s + 1) + Kc Kp Km e^(−τd s) = 0

Q: Ideal PID controller Gc(s) = Kc [ 1 + 1/(τi s) + τd s ] with Kc = 1, τi = 0.5 s, τd = 0.2 s. At what angular frequency ω (rad/s) is the controller's magnitude |Gc(jω)| equal to 1?

Introduction / Context: For an ideal PID Gc(s) = 1 + 1/(τi s) + τd s (with Kc = 1), the frequency response determines how the controller shapes loop gain and phase. A useful reference point is the frequency where the magnitude of the controller is unity, which helps anticipate closed-loop crossover behavior. Given Data / Assumptions: Kc = 1, τi = 0.5 s, τd = 0.2 s. We work with Gc(jω) = 1 + 1/(jω τi) + jω τd. We seek |Gc(jω)| = 1. Concept / Approach: Write Gc(jω) = 1 + j(ω τd − 1/(ω τi)). The real part is 1; the imaginary part is ω τd − 1/(ω τi). The magnitude |Gc| = sqrt{ 1^2 + [ω τd − 1/(ω τi)]^2 }. Setting |Gc| = 1 forces the imaginary part to zero: ω τd − 1/(ω τi) = 0. Hence ω = 1/√(τi τd). This is the frequency where derivative and integral contributions balance, leaving the real part equal to 1 exactly. Step-by-Step Solution: Set imaginary part to zero: ω τd = 1/(ω τi).Solve for ω: ω^2 = 1/(τi τd) ⇒ ω = 1/√(τi τd).Compute with τi = 0.5, τd = 0.2: τi τd = 0.1 ⇒ ω ≈ 3.162 rad/s. Verification / Alternative check: At ω = 1/√(τi τd), Gc(jω) = 1 + j(0) ⇒ |Gc| = 1 exactly for Kc = 1; small numerical checks confirm the balance. Why Other Options Are Wrong: τi/τd or τd/τi: Wrong functional form and units.1/(τi τd): Misses the square root, overestimates frequency.1/τi: Ignores derivative time constant entirely. Common Pitfalls: Forgetting that |a + jb| = 1 implies b = 0 when a = 1; otherwise magnitude exceeds 1. Also, keep units consistent—ω is in rad/s. Final Answer: ω = 1/√(τi τd) ≈ 3.16 rad/s

Q: Paper industry measurement: continuous (on-line) measurement of moisture content in paper webs is commonly achieved using which sensing approach?

Introduction / Context: Paper moisture strongly affects quality, strength, and runnability. Mills use continuous, non-destructive sensors to monitor moisture across the web and adjust dryer steam or machine speed accordingly. Given Data / Assumptions: Continuous, on-line monitoring is required. Contact or non-contact electrical methods correlate sheet moisture with dielectric or resistive properties. Concept / Approach: As paper moisture changes, its electrical resistance and dielectric constant change markedly. High-resistance Wheatstone bridge circuits (or RF/dielectric sensors) detect these changes in situ, enabling continuous moisture profiling. Psychrometers or hair hygrometers read ambient humidity, not sheet moisture; weighing is offline and discontinuous. Step-by-Step Solution: Select an electrical property sensitive to moisture (resistance/dielectric).Implement as a high-resistance bridge with electrodes/capacitance heads.Calibrate output versus % moisture using lab references. Verification / Alternative check: Modern scanners combine microwave/infrared absorption with electrical methods; classic curricula cite high-resistance bridge techniques as foundational. Why Other Options Are Wrong: Psychrometer/hair hygrometer: Measure air humidity, not sheet moisture.Weighing: Discrete lab method, not continuous.Thermocouples: Monitor temperature, not moisture. Common Pitfalls: Poor electrode contact, contamination, and temperature variation can bias readings; temperature compensation and proper electrode design are essential. Final Answer: High-resistance Wheatstone bridge across the sheet

Q: Filled-system thermometers: which property of the fluid inside the temperature-sensing bulb changes with temperature and provides the measurement signal?

Introduction / Context: Filled-system thermometers (gas-filled, liquid-filled, or vapor-pressure types) couple a bulb and capillary to a pressure element (Bourdon tube, bellows). Temperature changes in the bulb alter the condition of the filling fluid, producing a measurable mechanical motion via pressure. Given Data / Assumptions: Standard industrial filled-system thermometer with bulb, capillary, and pressure spring element. Signal variable transmitted to the indicator is mechanical displacement proportional to internal pressure. Concept / Approach: Although the fluid’s volume and viscosity may vary with temperature, the instrument’s operating principle is that temperature changes cause a pressure change in the sealed system. The Bourdon/bellows converts this pressure into pointer movement. In gas-filled types, P ∝ T (approximately), while in liquid-filled and vapor-pressure types, volumetric expansion and phase equilibrium still manifest as a pressure change at the spring element. Step-by-Step Solution: Recognize the mechanical transduction path: bulb → pressure change → spring motion.Identify pressure as the variable that is transmitted through the capillary to the gauge element.Select “Pressure.” Verification / Alternative check: Datasheets specify range in terms of pressure–temperature filling characteristics and list capillary length limits because the signal is a pressure wave, not free expansion volume. Why Other Options Are Wrong: Volume/viscosity changes may occur locally but are not the transmitted signal; the pointer responds to pressure at the spring element.“All”: Overstates what the measurement relies upon. Common Pitfalls: Confusing vapor-pressure thermometers (which use saturation pressure vs. temperature) with volumetric glass thermometers; the filled-system industrial design reads pressure. Final Answer: Pressure

Q: Convert amplitude ratio to decibels (dB): if the amplitude ratio (output/input) is 0.1 for a sinusoidal steady-state signal, compute the corresponding value in decibels using 20*log10(AR).

Introduction / Context: In frequency-response analysis and instrumentation, gains are often expressed on a decibel scale for convenient visualization over many orders of magnitude. Converting amplitude ratio (AR) to decibels (dB) uses a logarithmic formula tied to power ratios and sinusoidal magnitudes. Given Data / Assumptions: Amplitude ratio AR = 0.1 (i.e., output is one-tenth of input). Decibel conversion for magnitudes is dB = 20 * log10(AR). Base-10 logarithms are used. Concept / Approach: Because power is proportional to magnitude squared, the factor “20” appears for converting magnitude ratios, while “10” is used for power ratios. Thus, to convert AR to dB, multiply 20 by the common logarithm of the ratio. Step-by-Step Solution: Compute log10(0.1) = −1.Multiply by 20: dB = 20 * (−1) = −20 dB.Interpretation: a −20 dB gain indicates strong attenuation. Verification / Alternative check: As a quick mental check, −6 dB roughly halves amplitude; −20 dB corresponds to 0.1 amplitude, consistent with the logarithmic scale. Why Other Options Are Wrong: 20 or 10 dB: positive values imply gain > 1, not 0.1.−10 dB: corresponds to AR ≈ 0.316, not 0.1. Common Pitfalls: Using 10*log10 for magnitude ratios (that is for power), or mixing natural logs with base-10 logs without conversion. Final Answer: -20

Question 1

Ultra-low absolute pressure measurement: a pressure of 0.01 psi(abs) (about 0.517 torr) is best measured using which vacuum gauge type?

Accepted Answer

Introduction / Context: Different vacuum gauges cover different ranges. Picking the correct device depends on expected absolute pressure, gas type, and whether continuous reading is needed. Converting units helps map the problem to a gauge’s effective range. Given Data / Assumptions: Target pressure: 0.01 psi(abs). Unit conversion: 1 atm = 14.696 psi = 760 torr. Process gas is non-condensable for McLeod applicability. Concept / Approach: Compute torr: P = 0.01/14.696 * 760 ≈ 0.517 torr (≈ 0.517 mmHg). The McLeod gauge is a primary, compressing manometer typically used from about 10^−3 to several torr, overlapping the 0.5 torr region with good accuracy, provided the gas is not condensable. Ionisation gauges are intended for much lower pressures (≈ 10^−3 torr and below), while Pirani gauges span roughly 10^−3 to 10 torr but are secondary and gas-dependent; McLeod is the textbook choice for accurate measurement near 0.5 torr. Step-by-Step Solution: Convert 0.01 psi to torr → ~0.517 torr.Match to gauge ranges → McLeod comfortably covers ~0.5 torr.Select “McLeod gauge.” Verification / Alternative check: Handbooks list McLeod ranges overlapping 10^−3–10 torr. A Pirani could indicate this pressure, but it is gas-dependent and typically used for continuous indication rather than reference accuracy. Why Other Options Are Wrong: Ionisation: Better suited for < 10^−3 torr.Capacitance manometer/Bourdon: Either high-precision but different range selection, or inadequate sensitivity near 0.5 torr for standard types. Common Pitfalls: Using McLeod on condensable vapors introduces large errors; dry the gas or choose a thermal conductivity/Capacitance diaphragm gauge accordingly. Final Answer: McLeod gauge

Question 2

Speeding up an expansion (filled-system) pressure-spring thermometer: which bulb design/material choices increase speed of response?

Accepted Answer

Introduction / Context: Expansion (filled-system) thermometers transmit temperature-induced pressure/volume changes from a bulb through a capillary to a pressure spring and indicator. Their time constant depends on how quickly the bulb equilibrates with the process. Given Data / Assumptions: We aim to reduce the thermometer time constant (faster response). Changes to geometry and materials are allowed. Concept / Approach: Time constant roughly scales with thermal mass / heat-transfer coefficient / area. Thus, increasing area, reducing mass, using high-conductivity materials, and lowering specific heat all reduce the energy required and the thermal resistance between process and sensing fluid, yielding a faster response. Step-by-Step Solution: Increase bulb surface area to boost heat flux.Choose high-k materials (e.g., copper alloys) to minimize internal temperature gradients.Reduce specific heat/mass so less energy is needed to change bulb temperature.Combine all → fastest practical response. Verification / Alternative check: Manufacturers offer “fast response” bulbs with thin walls, fins (area), and high-k materials; response testing confirms reduced time constants. Why Other Options Are Wrong: Thick, heavy bulbs slow response due to high thermal mass and reduced area/volume ratio. Common Pitfalls: Over-thinning walls to the point of mechanical failure; ensure compatibility with pressure rating and erosion/corrosion conditions. Final Answer: All of the above (a), (b), and (c)

Question 3

Measuring about 0.0001 psi(abs) (≈ 5.2 × 10^−3 torr) continuously: which gauge is typically used for this low-vacuum region?

Accepted Answer

Introduction / Context: At a few millitorr (≈ 5 × 10^−3 torr), many industrial systems require continuous indication rather than occasional reference measurements. Thermal conductivity gauges (Pirani/thermocouple) are commonly used in this vacuum region. Given Data / Assumptions: Target: ~0.0001 psi(abs) ≈ 0.0052 torr. Continuous, on-line indication is desired. Gas composition is reasonably constant (thermal gauges are gas-dependent). Concept / Approach: Pirani gauges operate by relating filament heat loss to gas pressure, covering roughly 10^−3 to 10 torr (instrument-dependent). Thermocouple gauges are similar. While a McLeod gauge can measure in this region, it is an intermittent, manual device and not suited for continuous monitoring. Thus, the practical, textbook answer for continuous measurement near 5 mTorr is the Pirani gauge. Step-by-Step Solution: Convert units: 0.0001 psi → ~5 mTorr.Match ranges: Pirani covers ~10^−3–10 torr.Select Pirani as the continuous-reading instrument. Verification / Alternative check: Vacuum practice charts place Pirani/thermocouple gauges in the “low to medium vacuum” window that includes a few millitorr; ionization gauges are used for high/ultra-high vacuum below ~10^−3 torr. Why Other Options Are Wrong: McLeod: Accurate but intermittent and unsuitable for continuous control.Mercury manometer: Far too insensitive at millitorr.“None”: Not true—thermal gauges are standard here. Common Pitfalls: Ignoring gas-dependence and contamination; calibration should match the actual gas mixture to minimize reading bias. Final Answer: Pirani (thermal conductivity) gauge

Question 4

Closed-loop with dead time and first-order blocks (typical process + measurement + controller): which characteristic equation is consistent with two first-order lags ((τm s + 1), (τp s + 1)) in unity feedback and a controller gain Kc with overall pr…

Accepted Answer

Introduction / Context: Writing the characteristic equation from a block diagram is a core skill in control engineering. When a loop contains first-order dynamics for the process and the measuring element, a controller gain, and a pure dead time, the loop equation must reflect these factors in the denominator of the closed-loop transfer function. Given Data / Assumptions: Unity feedback with measurement dynamics Km/(τm s + 1). Process dynamics Kp/(τp s + 1) with a dead time e^(−τd s). Controller gain Kc (no integral/derivative terms shown). No extra transport lag besides those stated. Concept / Approach: For unity feedback, the characteristic equation is 1 + L(s) = 0 where L(s) is the loop transfer function. With L(s) = Kc * [Km/(τm s + 1)] * [Kp/(τp s + 1)] * e^(−τd s), multiply both sides by (τm s + 1)(τp s + 1) to obtain (τm s + 1)(τp s + 1) + Kc Kp Km e^(−τd s) = 0. Step-by-Step Solution: Write L(s) = Kc Km Kp e^(−τd s)/[(τm s + 1)(τp s + 1)].Characteristic equation: 1 + L(s) = 0.Multiply through by (τm s + 1)(τp s + 1): (τm s + 1)(τp s + 1) + Kc Kp Km e^(−τd s) = 0. Verification / Alternative check: Setting τd → 0 recovers the familiar polynomial (τm s + 1)(τp s + 1) + Kc Kp Km = 0, confirming consistency without dead time. Why Other Options Are Wrong: Options with τl terms introduce an extra lag not stated.Options missing Kc (just Km Kp) omit the controller gain.Option with product multiplied by e^(−τd s) on the polynomial part misrepresents the structure of 1 + L(s) = 0. Common Pitfalls: Placing e^(−τd s) on the wrong factor, forgetting to clear denominators, or misreading unity feedback versus sensor-in-loop placement. Final Answer: (τm s + 1)(τp s + 1) + Kc Kp Km e^(−τd s) = 0

Question 5

Ideal PID controller Gc(s) = Kc [ 1 + 1/(τi s) + τd s ] with Kc = 1, τi = 0.5 s, τd = 0.2 s. At what angular frequency ω (rad/s) is the controller's magnitude |Gc(jω)| equal to 1?

Accepted Answer

Introduction / Context: For an ideal PID Gc(s) = 1 + 1/(τi s) + τd s (with Kc = 1), the frequency response determines how the controller shapes loop gain and phase. A useful reference point is the frequency where the magnitude of the controller is unity, which helps anticipate closed-loop crossover behavior. Given Data / Assumptions: Kc = 1, τi = 0.5 s, τd = 0.2 s. We work with Gc(jω) = 1 + 1/(jω τi) + jω τd. We seek |Gc(jω)| = 1. Concept / Approach: Write Gc(jω) = 1 + j(ω τd − 1/(ω τi)). The real part is 1; the imaginary part is ω τd − 1/(ω τi). The magnitude |Gc| = sqrt{ 1^2 + [ω τd − 1/(ω τi)]^2 }. Setting |Gc| = 1 forces the imaginary part to zero: ω τd − 1/(ω τi) = 0. Hence ω = 1/√(τi τd). This is the frequency where derivative and integral contributions balance, leaving the real part equal to 1 exactly. Step-by-Step Solution: Set imaginary part to zero: ω τd = 1/(ω τi).Solve for ω: ω^2 = 1/(τi τd) ⇒ ω = 1/√(τi τd).Compute with τi = 0.5, τd = 0.2: τi τd = 0.1 ⇒ ω ≈ 3.162 rad/s. Verification / Alternative check: At ω = 1/√(τi τd), Gc(jω) = 1 + j(0) ⇒ |Gc| = 1 exactly for Kc = 1; small numerical checks confirm the balance. Why Other Options Are Wrong: τi/τd or τd/τi: Wrong functional form and units.1/(τi τd): Misses the square root, overestimates frequency.1/τi: Ignores derivative time constant entirely. Common Pitfalls: Forgetting that |a + jb| = 1 implies b = 0 when a = 1; otherwise magnitude exceeds 1. Also, keep units consistent—ω is in rad/s. Final Answer: ω = 1/√(τi τd) ≈ 3.16 rad/s

Question 6

Paper industry measurement: continuous (on-line) measurement of moisture content in paper webs is commonly achieved using which sensing approach?

Accepted Answer

Introduction / Context: Paper moisture strongly affects quality, strength, and runnability. Mills use continuous, non-destructive sensors to monitor moisture across the web and adjust dryer steam or machine speed accordingly. Given Data / Assumptions: Continuous, on-line monitoring is required. Contact or non-contact electrical methods correlate sheet moisture with dielectric or resistive properties. Concept / Approach: As paper moisture changes, its electrical resistance and dielectric constant change markedly. High-resistance Wheatstone bridge circuits (or RF/dielectric sensors) detect these changes in situ, enabling continuous moisture profiling. Psychrometers or hair hygrometers read ambient humidity, not sheet moisture; weighing is offline and discontinuous. Step-by-Step Solution: Select an electrical property sensitive to moisture (resistance/dielectric).Implement as a high-resistance bridge with electrodes/capacitance heads.Calibrate output versus % moisture using lab references. Verification / Alternative check: Modern scanners combine microwave/infrared absorption with electrical methods; classic curricula cite high-resistance bridge techniques as foundational. Why Other Options Are Wrong: Psychrometer/hair hygrometer: Measure air humidity, not sheet moisture.Weighing: Discrete lab method, not continuous.Thermocouples: Monitor temperature, not moisture. Common Pitfalls: Poor electrode contact, contamination, and temperature variation can bias readings; temperature compensation and proper electrode design are essential. Final Answer: High-resistance Wheatstone bridge across the sheet

Question 7

Filled-system thermometers: which property of the fluid inside the temperature-sensing bulb changes with temperature and provides the measurement signal?

Accepted Answer

Introduction / Context: Filled-system thermometers (gas-filled, liquid-filled, or vapor-pressure types) couple a bulb and capillary to a pressure element (Bourdon tube, bellows). Temperature changes in the bulb alter the condition of the filling fluid, producing a measurable mechanical motion via pressure. Given Data / Assumptions: Standard industrial filled-system thermometer with bulb, capillary, and pressure spring element. Signal variable transmitted to the indicator is mechanical displacement proportional to internal pressure. Concept / Approach: Although the fluid’s volume and viscosity may vary with temperature, the instrument’s operating principle is that temperature changes cause a pressure change in the sealed system. The Bourdon/bellows converts this pressure into pointer movement. In gas-filled types, P ∝ T (approximately), while in liquid-filled and vapor-pressure types, volumetric expansion and phase equilibrium still manifest as a pressure change at the spring element. Step-by-Step Solution: Recognize the mechanical transduction path: bulb → pressure change → spring motion.Identify pressure as the variable that is transmitted through the capillary to the gauge element.Select “Pressure.” Verification / Alternative check: Datasheets specify range in terms of pressure–temperature filling characteristics and list capillary length limits because the signal is a pressure wave, not free expansion volume. Why Other Options Are Wrong: Volume/viscosity changes may occur locally but are not the transmitted signal; the pointer responds to pressure at the spring element.“All”: Overstates what the measurement relies upon. Common Pitfalls: Confusing vapor-pressure thermometers (which use saturation pressure vs. temperature) with volumetric glass thermometers; the filled-system industrial design reads pressure. Final Answer: Pressure

Question 8

Dynamics of two identical tanks in series: for two first-order tanks of the same size and the same flow resistance connected in series (non-interacting), state the qualitative nature of the overall step response of the combined system.

Accepted Answer

Introduction / Context: Two perfectly mixed process tanks in series, each behaving as a first-order lag, together form a second-order dynamic system. When both tanks have the same capacity and the same hydraulic resistance (i.e., equal time constants), the closed-loop free response to a step input shows a characteristic shape that control engineers classify as over damped, critically damped, or under damped. This question tests recognition of that qualitative behavior. Given Data / Assumptions: Two non-interacting, identical first-order tanks in series. Each tank time constant: τ (same for both). Linear, constant-density mixing; no recirculation. Concept / Approach: The transfer function of one tank is 1/(τs + 1). Two in series give G(s) = 1/(τs + 1)^2, which is a standard second-order system with a double real pole at s = −1/τ. A double pole on the negative real axis has no oscillatory component and corresponds to an over damped response (slower, with an inflection point), not to critical damping (distinct condition ζ = 1 with complex-to-real transition) and not to under damping (which would require complex conjugate poles). Step-by-Step Solution: Write single-tank model: G1(s) = 1/(τs + 1).Series connection: G(s) = G1(s)^2 = 1/(τs + 1)^2.Identify pole locations: s = −1/τ (double pole, real).Real repeated poles imply non-oscillatory, over damped behavior. Verification / Alternative check: The step response of 1/(τs + 1)^2 is y(t) = 1 − (1 + t/τ) * e^(−t/τ), a monotonic, sigmoid-like rise without overshoot, confirming over damping. Why Other Options Are Wrong: Under damped: requires complex poles and overshoot; not present here.Critically damped: occurs for ζ = 1 but not with a perfect pole multiplicity at the same location from two identical lags.None of the above: incorrect because over damped fits. Common Pitfalls: Assuming “second order” means oscillatory by default. Real repeated poles still yield monotone, over damped responses. Final Answer: Over damped

Question 9

Convert amplitude ratio to decibels (dB): if the amplitude ratio (output/input) is 0.1 for a sinusoidal steady-state signal, compute the corresponding value in decibels using 20*log10(AR).

Accepted Answer

Introduction / Context: In frequency-response analysis and instrumentation, gains are often expressed on a decibel scale for convenient visualization over many orders of magnitude. Converting amplitude ratio (AR) to decibels (dB) uses a logarithmic formula tied to power ratios and sinusoidal magnitudes. Given Data / Assumptions: Amplitude ratio AR = 0.1 (i.e., output is one-tenth of input). Decibel conversion for magnitudes is dB = 20 * log10(AR). Base-10 logarithms are used. Concept / Approach: Because power is proportional to magnitude squared, the factor “20” appears for converting magnitude ratios, while “10” is used for power ratios. Thus, to convert AR to dB, multiply 20 by the common logarithm of the ratio. Step-by-Step Solution: Compute log10(0.1) = −1.Multiply by 20: dB = 20 * (−1) = −20 dB.Interpretation: a −20 dB gain indicates strong attenuation. Verification / Alternative check: As a quick mental check, −6 dB roughly halves amplitude; −20 dB corresponds to 0.1 amplitude, consistent with the logarithmic scale. Why Other Options Are Wrong: 20 or 10 dB: positive values imply gain > 1, not 0.1.−10 dB: corresponds to AR ≈ 0.316, not 0.1. Common Pitfalls: Using 10*log10 for magnitude ratios (that is for power), or mixing natural logs with base-10 logs without conversion. Final Answer: -20

Question 10

Identify the device that is not a speed-measuring instrument: compare psychrometer, stroboscope, and tachometer based on their primary measurement function.

Accepted Answer

Introduction / Context: Industrial and laboratory instruments are often identified by the measurand they target. Distinguishing among devices that measure rotational or periodic speed versus those measuring fluid properties is basic instrumentation literacy. Given Data / Assumptions: Psychrometer: a humidity instrument (dry-bulb and wet-bulb thermometers). Stroboscope: visualizes periodic motion and can infer rotational speed. Tachometer: directly measures rotational speed (rpm). Concept / Approach: Only one of the listed devices does not measure speed by design. A psychrometer measures relative humidity via temperature depression; a stroboscope and a tachometer both relate to speed—one optically by matching flash frequency, the other via mechanical/electrical pickup. Step-by-Step Solution: Classify each device by measurand.Psychrometer → humidity; Stroboscope → speed by optical synchronization; Tachometer → speed directly.Select the humidity instrument as “not speed-measuring.” Verification / Alternative check: Operation manuals and standards categorize psychrometers under hygrometry, whereas stroboscopes/tachometers fall under speed/rotational measurement. Why Other Options Are Wrong: Stroboscope and tachometer both measure or infer speed.“All (a), (b) and (c)” is invalid because two of the three are speed-related. Common Pitfalls: Confusing a stroboscope’s optical method with purely visual effects; it is a legitimate speed measurement tool when calibrated. Final Answer: Psychrometer

Question 11

Bimetallic thermometer behavior: the deflection of the free end of a bimetal strip versus temperature change is, over its normal operating range, approximately which functional form?

Accepted Answer

Introduction / Context: A bimetallic thermometer uses two metals with different coefficients of thermal expansion bonded together. Temperature change creates differential expansion, bending the strip and producing a pointer deflection. Understanding the approximate deflection-temperature relationship helps with calibration and scale layout. Given Data / Assumptions: Small to moderate temperature spans within design limits. Linear thermal expansion coefficients assumed constant over the range. Uniform strip properties and constant curvature approximation. Concept / Approach: When the expansion coefficients are treated as constant and strains are within elastic limits, the curvature of a bimetal strip is proportional to the temperature difference, leading to a nearly linear relation between tip deflection and temperature. While exact mechanics involve composite-beam theory, for practical thermometer ranges the scale is laid out linearly, and deviations are minor. Step-by-Step Solution: Model differential expansion: ΔL ≈ (α1 − α2) * L * ΔT.Relate curvature to temperature through composite bending relations.For typical ranges, deflection ∝ ΔT → near-linear scale. Verification / Alternative check: Calibration charts for commercial bimetallic thermometers show straight-line scales over their rated ranges; strong nonlinearity appears only at extremes or with nonuniform materials. Why Other Options Are Wrong: Non-linear, parabolic, hyperbolic: these can describe edge cases, but the standard operating range is designed to be almost linear. Common Pitfalls: Assuming perfect linearity at all temperatures; materials change properties with temperature, causing slight deviations near limits. Final Answer: Linear

Question 12

Thermal–electrical analogy: in lumped-parameter modeling, which thermal quantity is analogous to electrical current in a circuit representation?

Accepted Answer

Introduction / Context: Analogies map thermal systems to electrical circuits to leverage well-developed analysis methods. Correct mapping of variables is essential for building equivalent RC networks for heat transfer problems. Given Data / Assumptions: Resistive conduction or convection modeled as a thermal resistance. Thermal capacitance models stored heat (mass * specific heat). One-dimensional lumped-parameter representation. Concept / Approach: In the most common analogy (temperature ↔ voltage), temperature difference ΔT corresponds to voltage difference ΔV. The time rate of heat flow q̇ (W) corresponds to electrical current i (A). Thermal resistance R_th corresponds to electrical resistance R, and thermal capacitance C_th corresponds to electrical capacitance C. Using this mapping, Fourier/Newton laws mirror Ohm’s law: q̇ = ΔT / R_th just as i = ΔV / R. Step-by-Step Solution: Identify electrical current i as the flow variable driven by potential difference.Map potentials: temperature difference ↔ voltage.Conclude the flow analog is heat flow rate q̇. Verification / Alternative check: Energy storage equations align: C_th * dT/dt ↔ C * dV/dt, reinforcing the mapping consistency. Why Other Options Are Wrong: Temperature difference: analog of voltage, not current.Heat content: corresponds to electric charge storage (C*V), not current.None of these: incorrect because heat flow rate is correct. Common Pitfalls: Mixing the less common analogy (current ↔ temperature difference) used in some texts; the voltage–temperature mapping is more widely taught in process control. Final Answer: Heat flow rate

Question 13

General first-order unit-step response: for a stable, positive-gain first-order transfer function, describe how the output approaches its final steady state (qualitative trajectory shape).

Accepted Answer

Introduction / Context: First-order systems appear everywhere: tanks, thermal masses, RC circuits. Their hallmark is an exponential approach to a new steady state following a step input, with trajectory governed by the time constant τ and steady-state gain K. Given Data / Assumptions: Single real pole at s = −1/τ, τ > 0. Positive steady-state gain K > 0. Stable system (pole in left half plane). Concept / Approach: The standard step response is y(t) = K * (1 − e^(−t/τ)) for t ≥ 0 (assuming zero initial bias). This rises smoothly from 0 toward K without overshoot or oscillation. Its slope is maximum at t = 0 and decays exponentially; the output never decreases at any time when K > 0. Step-by-Step Solution: Recognize first-order canonical form: τ dy/dt + y = K u.Apply step input u = 1: solution is y(t) = K (1 − e^(−t/τ)).Note y′(t) = (K/τ) e^(−t/τ) ≥ 0 → strictly increasing to the asymptote. Verification / Alternative check: At t = τ, the response reaches about 63.2% of final; at t ≈ 5τ, it is essentially at steady state, consistent with a monotonic rise. Why Other Options Are Wrong: Monotonic decrease: would require negative gain or a negative step.Increase then decrease, or oscillatory patterns: those imply higher-order dynamics or zeros causing non-minimum-phase behavior, not present here. Common Pitfalls: Confusing first-order behavior with second-order underdamped responses that can overshoot. Final Answer: A monotonic increase

Question 14

Control valve positioners (“positioning controllers”): identify the typical application context where a valve positioner is preferred for reliable stem positioning.

Accepted Answer

Introduction / Context: A control valve positioner compares the controller’s output signal with the actual stem position and boosts actuator pressure/force to eliminate positioning error. It is commonly installed where direct I/P (current-to-pressure) action cannot overcome friction, unbalanced forces, or hysteresis. Given Data / Assumptions: Pneumatic or electro-pneumatic actuators are used. Mechanical linkages may introduce backlash and friction. Process conditions can create large unbalanced torques/forces. Concept / Approach: Positioners amplify the control signal locally at the valve, increasing stiffness and reducing dead band. They are particularly valuable for high-load applications: large valves, high differential pressures, sticky packing, long strokes, or fast response requirements. In light-load systems, direct I/P may suffice and a positioner may be unnecessary. Step-by-Step Solution: Identify scenarios where actuator force is insufficient or friction is significant.Recognize that a positioner provides high gain around stem position, improving accuracy.Select “High loads” as the most appropriate description. Verification / Alternative check: Valve vendor guidelines recommend positioners for valves with high shutoff differentials, large diameters, or where stroking speed and accuracy must be ensured. Why Other Options Are Wrong: Low loads: unnecessary complexity; direct I/P often adequate.Temperature changes: a process variable context, not a load descriptor.Small flow rate changes: not the deciding factor; mechanical load is. Common Pitfalls: Assuming positioners are only for split-range or cascaded loops; their primary role is accurate positioning under demanding mechanical loads. Final Answer: High loads (large actuators, high friction or long strokes)

Question 15

Accuracy indicator in static calibration: among dead zone, drift, and static error, identify the measure that directly judges the accuracy of an instrument at a stated operating point.

Accepted Answer

Introduction / Context: Instrument accuracy is assessed by how closely the indicated value matches the true value under specified conditions. Several static characteristics influence readings, but only one directly quantifies the deviation at a point in time and input. Given Data / Assumptions: Calibration conditions are steady (no transients). Error is defined as indicated value minus true value. Repeatability and hysteresis are considered separately. Concept / Approach: Static error (bias) = indicated value − true value. This directly expresses accuracy. Dead zone (dead band) describes an insensitivity region where small changes do not alter reading. Drift is the slow change of indication with time when input is constant. Both affect usability but are not the accuracy metric itself at a given point. Step-by-Step Solution: Define accuracy via absolute or percentage error relative to full-scale or reading.Recognize static error as the direct quantitative judge.Classify dead zone and drift as secondary characteristics. Verification / Alternative check: Calibration certificates report allowable accuracy as ±(percentage of full-scale or of reading), which is a bound on static error. Why Other Options Are Wrong: Dead zone: affects sensitivity around operating point but does not quantify accuracy at that point.Drift: time-dependent change; not the instantaneous accuracy measure.None of these: incorrect because static error is correct. Common Pitfalls: Equating precision (repeatability) with accuracy; an instrument can be precise yet biased (large static error). Final Answer: Static error

Question 16

Sinusoidal response magnitude: for a pure transportation lag (dead time) element, identify the amplitude ratio (magnitude) of the frequency response.

Accepted Answer

Introduction / Context: A transportation lag (dead time) element delays a signal by a fixed time L without changing its shape. In the frequency domain, dead time contributes phase lag that increases with frequency, but it does not attenuate amplitude in the ideal mathematical model. Given Data / Assumptions: Ideal dead time with transfer function e^(−sL). Sinusoidal steady-state input of frequency ω. No additional dynamics in series. Concept / Approach: The frequency response of e^(−sL) evaluated at s = jω is e^(−jωL) whose magnitude is |e^(−jωL)| = 1 and whose phase is −ωL radians. Thus, the amplitude ratio (output amplitude / input amplitude) is unity for all frequencies, while the phase lag grows linearly with frequency. Step-by-Step Solution: Start with G(jω) = e^(−jωL).Compute magnitude: |G(jω)| = 1.Compute phase: ∠G(jω) = −ωL.Conclude amplitude ratio is 1 for a pure dead time. Verification / Alternative check: Bode plots for dead time show a flat 0 dB magnitude line (AR = 1) and a phase that decreases linearly with frequency. Why Other Options Are Wrong: First/second order systems: their magnitudes vary with frequency (roll-off or resonance); not constant 1.None of these: incorrect because transportation lag matches the property. Common Pitfalls: Confusing practical dead time (with filters and measurement noise) with the ideal mathematical element; in practice, some attenuation can appear, but in theory AR = 1. Final Answer: Transportation lag

Question 17

Paramagnetism gas analysis: identify the gas for which the paramagnetic method is not appropriate when determining composition in gas mixtures.

Accepted Answer

Introduction / Context: Paramagnetic gas analyzers exploit the attraction of paramagnetic molecules to magnetic fields. The technique is widely used for oxygen measurement and, with specialized designs, for certain other paramagnetic species. Diamagnetic gases, however, are poorly suited to this method. Given Data / Assumptions: Paramagnetism implies unpaired electrons and positive magnetic susceptibility. Oxygen (O2) is strongly paramagnetic. Some nitrogen oxides (e.g., NO) exhibit paramagnetism. Carbon dioxide (CO2) is diamagnetic. Concept / Approach: In a paramagnetic analyzer, the force on paramagnetic molecules in a non-uniform magnetic field leads to measurable effects (e.g., pressure or thermal changes) proportional to concentration. Oxygen’s strong paramagnetism makes it ideal. Nitric oxide also has unpaired electrons and can be detected with certain designs. Carbon dioxide lacks unpaired electrons and is diamagnetic, yielding negligible response, so paramagnetism is not used for CO2 composition measurements. Step-by-Step Solution: Classify candidate gases by magnetic susceptibility.Note O2 (paramagnetic) → suitable; some NOx such as NO (paramagnetic) → potentially suitable.CO2 (diamagnetic) → not suitable for paramagnetic detection. Verification / Alternative check: Instrument catalogs list O2 paramagnetic analyzers; CO2 is measured by NDIR (non-dispersive infrared) or other optical methods, not by paramagnetism. Why Other Options Are Wrong: Oxygen: canonical application of paramagnetic analyzers.Oxides of nitrogen: some are paramagnetic; specialized analyzers exist.Any of these: incorrect since CO2 is not suited. Common Pitfalls: Assuming “all gases” can be measured by one principle; analyzer selection depends on molecular properties such as magnetism or IR absorption. Final Answer: Carbon dioxide

Question 18

Second-order step response characteristics: For the linear time-invariant system with transfer function G(s) = 1 / (s^2 + 2s + 3), which statement best describes the unit step response?

Accepted Answer

Introduction / Context: Recognising the qualitative time response of a given transfer function is a core skill in control engineering. Here, the denominator s^2 + 2s + 3 defines a second-order system. From its coefficients we can infer damping ratio, natural frequency, stability, and whether the unit step response oscillates or not. The goal is to select the statement that best matches the true response behaviour. Given Data / Assumptions: Transfer function: G(s) = 1 / (s^2 + 2s + 3). Input: unit step r(t) with Laplace transform 1/s. System starts from rest (zero initial conditions) and is linear time-invariant. Concept / Approach: Compare the standard second-order form: G(s) = ω_n^2 / (s^2 + 2ζω_n s + ω_n^2). For s^2 + 2s + 3 we have ω_n^2 = 3 ⇒ ω_n = √3, and 2ζω_n = 2 ⇒ ζ = 2/(2√3) = 1/√3 ≈ 0.577. Since 0 < ζ < 1, the system is underdamped and stable, which means the step response will overshoot and oscillate with exponentially decaying envelope. Step-by-Step Solution: Identify ω_n: ω_n = √3 ≈ 1.732.Compute damping ratio: ζ = 1/√3 ≈ 0.577 (between 0 and 1).Conclude stability: all poles have negative real parts (−1 ± j√2).Infer response: stable, damped oscillations with finite overshoot.Check initial slope using the initial value theorem on ẏ(t): ẏ(0⁺) = lim_{s→∞} s[sY(s)] = 0 → slope at t = 0⁺ is zero, not non-zero. Verification / Alternative check: Poles at −1 ± j√2 confirm an exponentially decaying sinusoid in the homogeneous response. The particular solution reaches a finite steady value, so the total step response is underdamped and stable. Why Other Options Are Wrong: Non-zero initial slope — false; here the initial slope is zero for this strictly proper form.Overdamped — would require ζ ≥ 1, not satisfied.Unstable — poles have negative real parts; system is stable.Zero-overshoot/zero-oscillation — contradicts ζ ≈ 0.577. Common Pitfalls: Mixing up damping categories or assuming any second-order denominator implies overdamping. Always compute ζ and inspect pole locations. Final Answer: It exhibits a damped oscillatory response (stable underdamped).

Question 19

Thermometer applicability at sub-zero conditions: Which thermometer type cannot reliably measure sub-zero temperatures (below 0°C) in standard practice without special modifications?

Accepted Answer

Introduction / Context: Selecting a thermometer for cold-region or cryogenic-adjacent service hinges on whether the sensing medium remains liquid and responsive below 0°C. Many industrial and laboratory questions stress that simple mercury-in-glass instruments are not suitable for routine sub-zero measurements, whereas alcohol or gas systems are commonly preferred. Given Data / Assumptions: “Standard practice” implies typical, unmodified instruments. Sub-zero means temperatures below 0°C. Mechanical integrity and readability are part of suitability, not merely theoretical range. Concept / Approach: Mercury freezes at around −39°C, and even above that, mercury-in-glass thermometers require special design (e.g., nitrogen-filled) for reliable sub-zero operation. In practical exam contexts, the accepted guidance is that mercury-in-glass is unsuitable for sub-zero work; alcohol-in-glass, vapour-pressure, gas-filled, or RTD devices are used instead because their working medium performs better below 0°C and the readout can be remote or electrical as needed. Step-by-Step Solution: Identify the limiting factor: mercury’s behaviour below 0°C.Contrast with alternatives: alcohols and gas systems remain useful well below 0°C; RTDs are electrical and cover wide ranges.Conclude the standard choice that “cannot measure sub-zero” in this context is mercury-in-glass. Verification / Alternative check: Instrument handbooks prescribe alcohol-filled or gas-filled devices for low temperatures and caution about mercury separation and readability issues below 0°C unless specially constructed. Why Other Options Are Wrong: Bimetallic — workable below 0°C; metal strip deformation continues at low temperatures.Vapour-pressure — designed for remote indication across low ranges.RTD — excellent low-temperature performance with proper wiring.Gas-filled bulb — suited to low-temperature measurement. Common Pitfalls: Assuming any mercury thermometer works fine below freezing; in practice, specialised designs are needed, hence the general rule used in exams. Final Answer: Mercury-in-glass thermometer

Question 20

Sinusoidal response magnitude check: For which element is the amplitude ratio |Output/Input| strictly less than 1 for all nonzero frequencies (assuming unity static gain)?

Accepted Answer

Introduction / Context: Frequency-response intuition allows quick predictions without solving time-domain equations. This question asks which listed element guarantees an amplitude ratio less than 1 at all nonzero frequencies when normalised to unity DC gain. Knowing these signatures helps in controller design and filter selection. Given Data / Assumptions: Steady-state sinusoidal input at frequency ω. Unity static (DC) gain for fair comparison. Standard textbook forms for each element. Concept / Approach: A unity-gain first-order lag G(jω) = 1 / (1 + jωτ) has magnitude |G(jω)| = 1 / sqrt(1 + (ωτ)^2) < 1 for any ω > 0. A pure time delay e^(−jωL) has magnitude exactly 1 at all frequencies (it only adds phase). A unity-gain second-order system can exhibit resonance with |G(jω)| > 1 when damping ζ < 1/√2, so it is not guaranteed to stay below 1. Therefore, only the first-order lag always satisfies the “< 1 for all ω > 0” condition under the stated assumptions. Step-by-Step Solution: Evaluate delay: |e^(−jωL)| = 1 → not < 1.Evaluate first-order lag: |1/(1 + jωτ)| = 1/√(1 + (ωτ)^2) → always < 1 for ω > 0.Evaluate second-order: resonance may occur → |G(jω)| can exceed 1. Verification / Alternative check: Bode magnitude plots confirm a constant 0 dB for delay, a monotonic roll-off for first-order lag, and a possible resonant peak for underdamped second-order systems. Why Other Options Are Wrong: Transportation lag — magnitude equals 1; only phase changes.Second-order — resonance can produce |G| > 1.None of these — incorrect because first-order lag qualifies.Lead compensator — may amplify at certain bands depending on zero/pole placement. Common Pitfalls: Forgetting that delay only shifts phase; it does not attenuate amplitude. Final Answer: First-order system

Process Control and Instrumentation Questions