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 d\(\vec{S}\), density ρ and flow velocity \(\vec{V}\). What is the net mass flow rate out of the surface area?

(a) \(\iint_V\rho \vec{V}.dV\)

(b) \(\rho \vec{V}.d \vec{S}\)

(c) \(\iiint_V\rho \vec{V}.d\vec{S}\)

(d) \(\iint_V\rho \vec{V}.d\vec{S}\)

I got this question in examination.

This question is from Continuity Equation topic in section Governing Equations of Fluid Dynamics of Computational Fluid Dynamics

Answer 1

The correct answer is (d) \(\iint_V\rho \vec{V}.d\vec{S}\)

The explanation is: In general,

mass flow rate=density × velocity × area

For this case,

elemental mass flow rate = \(\rho \vec{V}.d \vec{S}\)

total mass flow rate=\(\iint_V\rho \vec{V}.d\vec{S}\)