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# Computational Fluid Dynamics – Continuity Equation – Finite Control Volume

1 - Question

The physical principle behind the continuity equation is __________

a) Mass conservation

b) Zeroth law of thermodynamics

c) First law of thermodynamics

d) Energy conservation

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Answer: aExplanation: Continuity equation is derived from the mass conservation principle. It states that for an isolated system, the mass of the system must remain constant.

2 - Question

Which of these models directly gives this equation?

∂∂t∭vρdV+∬sρV⃗ .dS→=0

a

b

c

d

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Answer: bExplanation: The equation is in conservative integral form. So, the model must be finite control volume fixed in space.

3 - Question

Which of these models directly gives this equation?

DDt∭vρdV=0

a)

b

c

d

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Answer: cExplanation: The equation is the non-conservative form of the integral continuity equation. This is obtained from a finite control volume moving along with the flow.

4 - Question

Consider a model of finite control volume (volume V and surface area S) fixed in space with elemental volume dV, vector elemental surface area dS⃗ , density ρ and flow velocity V⃗ . What is the net mass flow rate out of the surface area?

a) ∬VρV⃗ .dV

b) ρV⃗ .dS⃗

c) ∭VρV⃗ .dS⃗

d) ∬VρV⃗ .dS⃗

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Answer: dExplanation: In general, mass flow rate=density × velocity × area For this case, elemental mass flow rate = ρV⃗ .dS⃗ total mass flow rate=∬VρV⃗ .dS⃗

5 - Question

Consider a model of finite control volume (volume V and surface area) fixed in space with elemental volume dV, vector elemental surface area dS⃗ , density ρ and flow velocity V⃗ . What is the mass inside the control volume?

a) ∬sρV⃗ .dS⃗

b) ∭VρdV

c) ρdV

d) ∂∂t∭VρdV

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Answer: bExplanation: Mass=density × volume mass inside dV=ρdV mass inside V=∭VρdV.

6 - Question

Consider a model of finite control volume (volume V and surface area) moving along the flow with elemental volume dV, vector elemental surface area dS⃗ , density ρ and flow velocity V⃗ . What is the time rate of change of mass inside the control volume?

a) ∭VρdV

b) ∂∂t∭VρdV

c) DDt∭VρdV

d) ρdV

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Answer: cExplanation: Substantial derivative is used as the model is moving. mass=density × volume mass inside dV=ρdV mass inside V=∭VρdV time rate of change of mass inside DDt∭VρdV.

7 - Question

To convert the non-conservative integral equation of mass conservation into the conservative integral form, which of these theorems is used?

a) Stokes theorem

b) Kelvin-Stokes theorem

c) Gauss-Siedel theorem

d) Gauss Divergence Theorem

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Answer: dExplanation: The expansion of non-conservative integral equation gives two volume integral terms. One of these terms representing the mass flow is converted into surface integral using the Gauss Divergence theorem.

8 - Question

Consider a model of finite control volume (volume V and surface area) fixed in space with elemental volume dV, vector elemental surface area dS⃗ , density ρ and flow velocity V⃗ .

ρV⃗ .dS⃗ is positive when _____________

a) The mass flow is outward

b) The mass flow is inward

c) The mass flow is positive

d) The mass flow is negative

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Answer: aExplanation: dS⃗ always points outwards to the control volume. So, the product ρV⃗ .dS⃗ is positive when the mass flow is outwards.

9 - Question

What is the physical statement of mass conservation equation for a finite control volume fixed in space?

a) Net mass flow through the control surface = constant

b) Rate of change of mass inside the control volume = constant

c) Net mass flow through the control surface = Rate of change of mass inside the control volume

d) Net mass flow through the control surface≠Rate of change of mass inside the control volume

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Answer: cExplanation: Statement of mass conservation equation for a finite control volume fixed in space: Net mass flow through the control surface is equal to the rate of change of mass inside the control volume.

10 - Question

What is the physical statement of mass conservation equation for a finite control volume moving along with the flow?

a) Rate of change of mass inside the control volume = 0

b) Rate of change of mass inside the control volume = constant

c) Net mass flow through the control surface = Rate of change of mass inside the control volume

d) Net mass flow through the control surface≠Rate of change of mass inside the control volume

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Answer: bExplanation: Statement of mass conservation equation for a finite control volume moving along with the flow: Mass inside the control volume = constant Rate of change of mass inside the control volume = 0.