Transparent Acrylic Vertical Cylindrical Surge Tank
Uniform 25.23 cm ID column (A = 500.0 cm^2, height 100 cm) equipped with overflow weir and graduated scale.
Chemical & Process Engineering / Gravity-Drained Tank Dynamics
FREE TO EXPLORELABORATORY 40 · CHEMICAL & PROCESS ENGINEERING
Dynamic Modeling, Step Response & Hydraulic Resistance of a First-Order Gravity-Drained Tank
Model the unsteady-state liquid level dynamics of a vertical cylindrical gravity-drained tank with laminar linear discharge resistance, apply step changes in influent volumetric flow rate F_in from 2.0 to 12.0 L/min, monitor liquid level transients h(t) using an ultrasonic level transmitter, evaluate the process time constant τ_p, and determine the linear hydraulic discharge resistance R. Explore interactive transport phenomena, unit operations simulation, and chemical engineering telemetry.
TRANSPORT PHENOMENA · UNIT OPERATIONS · PROCESS CONTROL · REACTION ENGINEERING
Your virtual chemical and process engineering laboratory.
STEADY-STATE LIQUID LEVEL H_SS—
PROCESS & PILOT PLANT TELEMETRY
Ready to sample with Influent Flow Rate F_in = 6.0 L/min.
Adjust Influent Flow Rate F_in, then execute chemical process trial.
Chemical and process engineering telemetry is evaluated rigorously using 64-bit IEEE 754 precision. Residuals reflect ordinary least-squares regression across successive observations. Continuous data structures support high-fidelity transport phenomena and unit operations modeling.
Process engineering assumptions and boundary condition limits
The transport and thermodynamic models apply within the specified calibrated operating window (2.0 to 12.0 L/min). Evaluated using a precision transparent acrylic column (ID = 25.23 cm, cross-sectional area A = 500.0 cm^2 = 0.0500 m^2) with laminar capillary discharge resistor. Ultrasonic transit-time level transmitter (±0.05 cm) and magnetic-drive gear pump calibrated at 20.0 °C.
Process apparatus & industrial sensors.
Laminar Capillary Discharge Restriction Manifold
Multi-capillary bundle providing strictly linear Hagen–Poiseuille laminar discharge resistance over 0 to 60 cm head.
Microprocessor Ultrasonic Transit-Time Level Transmitter
Measures liquid surface height h(t) from top flange with ±0.05 cm resolution and 10 Hz analog output.
Variable-Speed Magnetic-Drive Inflow Gear Pump
Delivers pulseless, positive-displacement water feed rates from 1.0 to 15.0 L/min with ±0.02 L/min precision.
Precision In-Line Turbine Flowmeter & PID Controller
Monitors and regulates influent water volumetric flow rate F_in with fast closed-loop servo tracking.
Recirculating Sump Reservoir with Deaeration Baffles
Supplies air-free, temperature-conditioned water (20.0 ± 0.2 °C) to prevent bubble formation in discharge tubes.
FOLLOW THE METHOD
A reason behind every step.
Model the unsteady-state liquid level dynamics of a vertical cylindrical gravity-drained tank with laminar linear discharge resistance, apply step changes in influent volumetric flow rate F_in from 2.0 to 12.0 L/min, monitor liquid level transients h(t) using an ultrasonic level transmitter, evaluate the process time constant τ_p, and determine the linear hydraulic discharge resistance R. Standardized parameter settings: 2.0, 4.0, 6.0, 8.0, 10.0, 12.0 L/min.
01
Sensor Calibration & Zero Datum Alignment
Calibrate ultrasonic level transmitter against manual vernier sight gauge across 0 to 80 cm; align transmitter 4 mA signal with bottom datum.
02
Pump Prime & Flowmeter Verification
Prime the magnetic gear pump; verify delivery flow rate at 6.0 L/min using gravimetric timed-collection catch-and-weigh method.
03
Initial Step Inflow Setpoint (F_in = 2.0 L/min)
Set pump controller to deliver F_in = 2.0 L/min into empty tank; log level trajectory h(t) at 1 Hz until steady state is reached.
04
Transient Time Constant Evaluation
Extract time constant τ_p from the time required to achieve 63.2% of the total level step rise; record final steady-state level h_ss.
05
Step Flow Progression Protocol
Apply sequential step changes to influent flow rate F_in = 4.0, 6.0, 8.0, 10.0, and 12.0 L/min, recording dynamic curves and steady levels.
06
Hydraulic Resistance Linear Regression
Plot steady-state level h_ss versus influent flow rate F_in; perform linear regression through origin to evaluate hydraulic resistance R.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into chemical process engineering insight.
Points are your recorded observations. The solid regression line validates Gravity-Drained Tank Dynamics mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
h_ss = R · F_in = 3.8500 · F_in
| # | Influent Volumetric Flow Rate F_in (L/min) | Steady-State Liquid Level h_ss (cm) | Influent volumetric flow rate F_in (L/min) | Steady-state liquid level h_ss (cm) | Remove |
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No readings yet. Execute a process trial and record it.
Calculate it yourself.
CHECK YOUR UNDERSTANDING
Explain what the process engineering metrics mean.
For a gravity-drained tank with linear discharge resistance R, the steady-state liquid level h_ss equals:
The time required for a first-order system to reach 63.2% of its ultimate step response is equal to:
If tank cross-sectional area A is doubled while discharge resistance R remains constant, the time constant τ_p:
A gravity-drained tank is considered self-regulating because:
THEORY AND UNIT OPERATIONS CONTEXT
Process engineering background & physical principles.
Liquid level control in surge tanks, flash drums, reflux accumulators, and chemical storage reservoirs is a ubiquitous unit operation in process engineering. Maintaining stable tank levels prevents pump cavitation from run-dry conditions while avoiding hazardous overfills. A vertical, uniform cross-sectional surge tank (cross-sectional area A = 0.050 m^2) drained by gravity through a laminar restriction tube is the canonical physical realization of a first-order dynamic process. Applying the unsteady-state mass conservation principle to incompressible liquid (constant density ρ) gives: ρ · A · (dh/dt) = ρ · F_in(t) - ρ · F_out(t), which simplifies to: A · (dh/dt) = F_in(t) - F_out(t). When discharge occurs through a capillary tube or packed porous plug under laminar flow (Re < 2000), the Hagen–Poiseuille equation dictates that effluent flow rate is directly proportional to hydrostatic pressure head: F_out = h / R, where R is the linear hydraulic discharge resistance (with dimensions of head per volumetric flow rate, cm/(L/min)). Substituting this linear outflow relation yields the first-order ordinary differential equation: A · R · (dh/dt) + h(t) = R · F_in(t). Defining the process time constant τ_p = A · R, this standard first-order transfer function in Laplace deviation variables is: G(s) = H'(s) / F_in'(s) = K_p / (τ_p · s + 1), where steady-state process gain K_p equals hydraulic resistance R. In response to a step increase in inflow ΔF_in, the level response is an exponential relaxation: h'(t) = R · ΔF_in · [1 - exp(-t / τ_p)]. At long times (t -> inf), the new steady-state liquid level is h_ss = R · F_in. In this laboratory investigation, an instrumented transparent acrylic surge tank equipped with an ultrasonic level transmitter and variable-speed magnetic gear pump is operated across inflow setpoints from 2.0 to 12.0 L/min to evaluate transient step responses, calculate τ_p, and determine the linear hydraulic resistance R = 3.8500 cm/(L/min).
h_ss = R · F_in = 3.8500 · F_in
Evaluated using a precision transparent acrylic column (ID = 25.23 cm, cross-sectional area A = 500.0 cm^2 = 0.0500 m^2) with laminar capillary discharge resistor. Ultrasonic transit-time level transmitter (±0.05 cm) and magnetic-drive gear pump calibrated at 20.0 °C.