Fluid Mechanics · Unit 1 · Lesson 11 of 13
Follow The Fluid Or Watch A Fixed Box?
Air rushes through a jet engine at hundreds of kilograms per second. Would you try to follow every kilogram, or just watch what goes in and comes out?
1. Real-World Problem
A pump fills a rooftop tank while the residents draw water from it. Will the tank overflow, run dry, or settle at a steady level? You cannot track every drop, but you can watch what crosses the tank's boundary.
2. Predict
Make your guess before you try the experiment. You will see the answer after the next step.
Predict
3. Experiment
Start with Control mass: follow one blob of water through the tank. Its boundary moves and stretches, but its mass never changes.
Switch to Control volume. Set the pump to 1 L/s with the valve fully open and watch the level rise and settle. In Engineering mode, try Household demand, load the Project preset, and half-close the valve to see the overflow.
4. Understand
- A control mass is a fixed amount of matter followed wherever it goes. No mass crosses its boundary, which can move and change shape. It is the tool for analysing a closed system.
- A control volume (CV) is a region in space chosen for analysis. Its boundary is the control surface. A CV can be fixed (a box around a pump), moving (a box riding with a jet engine on a flying aircraft) or deforming (a boundary that follows a rising water surface, or a balloon being filled). Mass may cross the control surface, which makes CVs the natural tool for open systems, but it does not have to: a CV with no flow across it is still a valid CV.
| Control mass | Control volume | |
|---|---|---|
| What is fixed | The amount of matter | The region you choose (it may be fixed, moving or deforming as you specify) |
| Mass crossing the boundary | Never | Allowed, and usually happens |
| Energy crossing the boundary | Yes (heat, work) | Yes (heat, work, and energy carried by the flow) |
| Boundary | Moves with the matter | The control surface |
| Usually used to analyse | Closed systems | Open systems |
| Examples | Gas in a sealed piston and cylinder, air in a tied balloon | Pump, nozzle, jet engine, car radiator, water tank |
Fluids keep moving while machines stay in one place, so fluid mechanics mostly uses control volumes: draw a box around the machine and track what crosses it.
5. Derive
Step 1. Mass flow rate. In a short time dt, fluid crossing a section of area A moves a distance V dt, so the volume crossing is A V dt and its mass is ρ A V dt. Dividing by dt:
\dot{m} = \rho A Vṁ is mass flow rate (kg/s), ρ density (kg/m³), A the area of the section (m²). V is the average velocity normal (perpendicular) to the section, and ρ is taken as uniform across it. In a real pipe the speed is zero at the wall and highest at the centre, so V is the average that gives the true flow, V = Q/A. If the flow crosses the section at an angle, only the component perpendicular to it counts.
This form assumes a fixed control surface, which is what every numerical exercise in this unit uses. For a moving or deforming control surface, see the Deep Dive at the end of this step.
Step 2. Mass balance for a control volume:
\frac{dm_{CV}}{dt} = \sum \dot{m}_{in} - \sum \dot{m}_{out}In steady flow nothing builds up, so total mass in equals total mass out.
The tank's level equation. Steps 3 and 4 turn the mass balance into an equation for a tank with an orifice and solve it for the steady level you predict in Experiment mode.
Step 3. For the tank, m_CV = ρAₜh, so:
A_t \frac{dh}{dt} = Q_{in} - C_d\,a\,\sqrt{2 g h}Step 4. Steady level: set dh/dt = 0 and solve for h. This applies to the orifice outflow only; with a constant household demand there is no single steady depth:
h_{steady} = \frac{1}{2g}\left(\frac{Q_{in}}{C_d\,a}\right)^2When the control surface itself moves, what decides how much mass crosses it is the fluid's velocity relative to the surface, not its absolute velocity:
\[\dot{m}_{net,\,out} = \int_{CS} \rho\,(\vec{V} - \vec{V}_{CS}) \cdot \vec{n}\; dA\]
V is the fluid velocity, V_CS the velocity of the control surface at that point, n the outward unit normal, and the integral runs over the whole control surface. Two checks: for the deforming tank CV, the top surface moves with the water, so V − V_CS = 0 there and no water crosses it. For a jet engine on an aircraft flying at 250 m/s through still air at about 10 km altitude (ρ ≈ 0.4 kg/m³), the engine captures a stream tube of free-stream air 2 m² in cross-section, approaching at 250 m/s relative to the engine (the air slows before the inlet face, but the mass flow is the same), so ṁ = 0.4 × 2 × 250 = 200 kg/s, the "hundreds of kilograms per second" in this lesson's curiosity question. For a fixed surface V_CS = 0 and, with uniform flow, the integral reduces to ṁ = ρAV.
6. Calculate
- Water at an average speed of 2 m/s in a 20 mm pipe: A = π × 0.02² ÷ 4 = 3.14 × 10⁻⁴ m², so ṁ = 1000 × 3.14 × 10⁻⁴ × 2 = 0.63 kg/s.
- Tank with Q_in = 1 L/s and the valve fully open: C_d a √(2g) = 0.6 × 4.91 × 10⁻⁴ × 4.43 = 1.30 × 10⁻³. So h_steady = (0.001 ÷ 1.30 × 10⁻³)² = 0.59 m.
- When the depth is 0.30 m: Q_out = 1.30 × 10⁻³ × √0.30 = 0.71 L/s, so dm/dt = 1000 × (0.001 − 0.000715) = 0.29 kg/s and the level rises at 0.29 mm/s.
7. Apply
- A water pump is studied with a fixed CV: water in, water out, power in.
- A pressure cooker with its lid locked and whistle down is a control mass: heat enters, no mass leaves.
- A jet engine on a flying aircraft: a CV that moves with the engine; air and fuel in, hot gas out, thrust from the difference.
- A car tyre being filled: a CV whose mass is rising.
- A balloon being inflated: a deforming CV.
8. Validate
In Engineering mode set Q_in = 1 L/s, valve 100%, start empty: the depth must level off at 0.59 m. Pause when the depth reads 0.30 m: dm/dt must read about 0.29 kg/s. Load the Project preset (10,000 L, pump 1.5 L/s, demand 0.5 L/s): dm/dt must read 1.0 kg/s and the tank must fill from empty in about 10,000 s (2.8 h). Back in the Lesson tank, halve the valve opening: the steady depth would be 2.35 m, above the 2 m tank, so the depth must stop at 2.00 m, the Overflowing banner must appear and the overflow readout must show about 0.08 L/s (inflow 1 L/s minus the orifice outflow of 0.92 L/s at 2 m). Where the model does not apply: very small openings or very shallow water (C_d changes), a pump whose flow drops as the tank fills, and fast valve movements (unsteady effects).
Easy To Mix Up
- A control mass can change shape and size; only its mass is fixed.
- A control volume is chosen by you. It can be fixed, moving or deforming; choose the one that makes the crossings easy to measure.
- "Control volume" and "open system" are not the same thing. Open and closed describe what crosses a boundary; a control volume is the analysis tool, most often used for open systems.
- In ṁ = ρAV, V is the average velocity perpendicular to the section, not the fastest speed in the pipe.
9. Quiz
Four quick questions: a concept, a calculation and an interpretation. Each answer comes with an explanation. Score 3 or more to mark this lesson completed.
Check Your Understanding
Four questions: a concept, a calculation and an interpretation. Every answer comes with an explanation. Score 3 of 4 to mark the lesson completed.
Pressure in bar, viscosity in poise, flow in litres per minute. How do engineers keep all these units straight?
Go to Lesson 12: How Do Engineers Measure Everything In Fluids? →Read More
- Cengel and Cimbala, "Mass, Bernoulli and Energy Equations" (conservation of mass); Bansal, "Orifices and Mouthpieces" (time to empty a tank).
- NPTEL, Introduction to Fluid Mechanics (Prof. Suman Chakraborty, IIT Kharagpur): Weeks 9 and 10.