Questions: Control Volume and Mass Balance

Question 1

Water flows through a horizontal pipe that narrows from a cross-sectional area of 0.04 m² to 0.01 m². If the velocity at the wide section is 2 m/s, what is the velocity at the narrow section for steady, incompressible flow?

• A. 0.5 m/s — velocity decreases as the pipe narrows because pressure increases
• B. 2 m/s — velocity is unchanged because flow rate is conserved
• C. 4 m/s — the area halved, so velocity doubles to conserve mass
• D. 8 m/s — the area quartered, so velocity must quadruple to conserve mass

Answer: (D)

For steady incompressible flow, the continuity equation gives A₁V₁ = A₂V₂. Here A₁ = 0.04 m², V₁ = 2 m/s, A₂ = 0.01 m² — the area decreased by a factor of 4. Therefore V₂ = A₁V₁/A₂ = (0.04 × 2)/0.01 = 8 m/s. The area ratio is 4, so velocity increases by a factor of 4. Option A gets the direction wrong — velocity increases, not decreases, when area decreases, to conserve mass flow. Option B confuses mass flow rate conservation with velocity conservation. Option C uses the wrong area ratio.

Question 2

An engineer must find the flow rate in one branch of a pipe that splits into two. She places the control volume boundaries at the single inlet and the two outlet faces, where conditions are known or measurable. Why is this CV choice effective?

• A. It eliminates the need for conservation laws by making the geometry simple
• B. The boundaries are placed where conditions are known, making the single unknown directly accessible from the conservation equation
• C. A control volume that includes the split point must by definition have equal flow in each branch
• D. Control volume analysis is only valid when inlet and outlets are at the same elevation

Answer: (B)

The power of the control volume method lies in the choice of where to draw the boundary. Placing boundaries where conditions are known (inlet flow rate known; one outlet flow rate known) leaves only one unknown — the other outlet's flow rate — which is directly solved by ṁ_in = ṁ_out. This 'black box' approach is the fundamental insight: you do not need to know what happens inside the control volume, only what crosses its boundaries. A poor CV choice places boundaries across regions of unknown conditions, making the conservation equation unsolvable.

Question 3

For steady flow of an incompressible fluid through a control volume with one inlet and one outlet, the mass flow rate entering must equal the mass flow rate leaving — regardless of how complex or convoluted the flow path is in between.

• T. True
• F. False

Answer: True

This is the integral form of mass conservation for steady incompressible flow. 'Steady' means no accumulation of mass inside the control volume over time, so whatever mass enters per unit time must exit per unit time: ṁ_in = ṁ_out, or ρA₁V₁ = ρA₂V₂. For incompressible flow (constant density), this simplifies to A₁V₁ = A₂V₂. The internal geometry — bends, diameter changes, turbulence — does not matter for mass conservation as long as steady state holds and no mass is created or destroyed inside.

Question 4

The control volume approach requires tracking each individual fluid particle through the flow field in order to correctly apply conservation of mass at the boundary.

• T. True
• F. False

Answer: False

This exactly describes the Lagrangian approach, which is what the control volume method is designed to avoid. The control volume (Eulerian) approach fixes attention on a region of space and asks what enters and leaves that region. You never need to follow individual fluid particles; you only need to know conditions at the boundary surfaces (velocity, density) to apply conservation of mass. This converts an impossibly complex particle-tracking problem into a tractable boundary accounting problem — which is the whole point of the method.

Question 5

Why does the control volume approach make fluid mechanics problems tractable, and what is the key engineering judgment it requires?

Model answer: The control volume approach makes problems tractable by replacing particle tracking (Lagrangian, intractable for real flows) with applying conservation laws at a fixed boundary. Since mass cannot be created or destroyed inside the CV, whatever enters must account for whatever leaves (plus accumulation for unsteady problems). The key engineering judgment is choosing where to draw the CV boundary: effective placement puts boundaries where conditions are already known or can be measured — pipe inlets, outlets, nozzle faces — so the conservation equation has as few unknowns as possible. A poor CV choice places boundaries across regions of unknown conditions, making the equation unsolvable.

The 'cleverness' of CV placement is both the art and the lesson of this topic. The same physical system can yield an immediately solvable equation or one with multiple unknowns, depending entirely on where you draw the boundary. The pattern generalizes: mass, momentum, and energy conservation all apply to the same CV, giving a system of equations that together characterize the complete flow. Learning to place the CV boundary, identify what crosses it, and apply the appropriate conservation law is the foundational skill all subsequent fluid mechanics builds on.