Question

Statement of Reynolds Transport Theorem is given below with three blanks: The rate of change of ____________ extensive property can be calculated by summing the rate of change of the amount of same property in the ____________ and the rate at which the property is ____________ the surface of the control volume. Which one of the following options correctly fills the blanks by using its comma-separated phrases in sequence?

• A. system, control volume, exiting
• B. control volume, system, exiting
• C. system, control volume, entering
• D. control volume, control volume, entering

Hint:

When using RTT, always remember: \emph{system} $\rightarrow$ \emph{control volume} $+$ \emph{flux across boundary}. The flux is taken positive in the outward (exiting) direction.

Answer 1

The Correct Option is A

Step 1: Recall Reynolds Transport Theorem (RTT).
The Reynolds Transport Theorem is the fundamental bridge between the \emph{system approach} (tracking a fixed set of fluid particles) and the \emph{control volume approach} (fixed region in space). It tells us how to express the time rate of change of a property of a system in terms of what happens inside a control volume and across its boundaries.

Step 2: Write the RTT in mathematical form.
For an extensive property $B$ (with intensive property $\beta = \tfrac{B}{m}$), RTT states: \[ \frac{dB_{system}}{dt} = \frac{\partial}{\partial t}\int_{CV} \rho \beta \, dV + \int_{CS} \rho \beta \, (\vec{V}\cdot \vec{n}) \, dA. \] Interpretation: - LHS = rate of change of property for the system. - First RHS term = rate of change of property inside the control volume. - Second RHS term = net rate at which the property is leaving (exiting) through the control surface.

Step 3: Match to blanks.
- Blank 1: We are talking about a system property (on the LHS). - Blank 2: Inside the integral, it is over the control volume. - Blank 3: The last term corresponds to the property exiting through the surface.

Step 4: Verify other options.
- (B) starts with “control volume”, but RTT begins with “system” — wrong. - (C) says “entering”, but RTT considers net flow \emph{out of} the control volume (exiting) — wrong. - (D) has “control volume, control volume” — mismatched with RTT — wrong. So only (A) matches perfectly. Final Answer:
\[ \boxed{\text{system, control volume, exiting}} \]