Which of the following equations describes the work done by environment, as it pushes mass elements through the boundary? (qm is mass flow rate, p is pressure, v is specific volume, w is velocity)

Question

Which of the following equations describes the work done by environment, as it pushes mass elements through the boundary? (qm is mass flow rate, p is pressure, v is specific volume, w is velocity)

BlackTom AI · Accepted Answer

Question restatement: The prompt asks which equation describes the work done by the environment as it pushes mass elements through a boundary, given qm is mass flow rate, p is pressure, v is specific volume, and w is velocity.

Option analysis not fully provided: The list of answer choices is missing, so I cannot enumerate and compare each specific option. Nevertheless, I can reason about the physics to identify a correct form that represents boundary (flow) work for a control volume.

Key concept: In open-channel or open-system (control volume) analysis, the work associated with pushing fluid through a boundary, also called flow work or boundary work, is due to the pressure acting through the specific volume of the fluid being added or removed at the boundary. For a steady flow, the rate of work associated with the boundary is W_dot = ṁ p v, where ṁ (or qm) is the mass flow rate, p is the pressure, and v is the specific volume (volume per unit mass) of the fluid at the boundary.

Reasoning for the commonly used form: - The term p v represents the specific work needed to push a unit mass across the boundary against the surrounding pressure (it's the work required to create a new control volume boundary around the fluid element).
- Multiplying by the mass flow rate ṁ (or qm) converts this per-mass quantity to a power (work rate), giving the rate at which work is done by the environment on the fluid as it enters or leaves the boundary.

Why the given expression qm · p · v is the appropriate form: - It aligns with the standard flow-work formula W_dot = ṁ p v for steady flow across a boundary, capturing how pressure and specific volume together determine the energy needed to move mass through the boundary over time.
- If the problem explicitly defines qm as mass flow rate, p as pressure, and v as specific volume, the product qm p v cleanly expresses the boundary work rate in consistent units (e.g., [kg/s] [Pa] [m^3/kg] -> [W]).

Potential alternatives and why they are incorrect in this context: - A form like qm · p would represent pressure-volume work per unit mass but lacks the per-mass volume term v, so it would be missing a factor that accounts for how much specific volume is carried by the mass flow.
- A form like qm · v would correspond to mass flow rate times specific volume, which has units of volume flow rate, not work rate; it does not incorporate pressure and thus does not represent boundary work.
- A form involving velocity w would imply kinetic energy or momentum terms; while w can be relevant for kinetic energy or momentum balance, it does not by itself describe the classic boundary (flow) work due to pressure acting through specific volume.

In summary, when describing the work done by the environment as mass flows across a boundary in an open system, the standard and physically consistent expression is W_dot = qm · p · v, which matches the common interpretation of flow work for boundary crossing.

Which of the following equations describes the work done by environment, as it pushes mass elements through the boundary? (qm is mass flow rate, p is pressure, v is specific volume, w is velocity)

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