Question

When is the flow through a control volume steady?

Short answer

Answer: The flow through a control volume can be considered steady when the following conditions are met: 1. The mass flow rate entering the control volume is equal to the mass flow rate leaving the control volume (no accumulation or depletion of mass within the control volume). 2. The momentum flow rate entering and leaving the control volume remains constant (no net force acting on the control volume causing changes in fluid's velocity). 3. The energy flow rate entering and leaving the control volume remains constant (satisfying energy conservation and preventing changes in internal, kinetic, or potential energy in the fluid).

Step-by-step solution

Define the Control Volume

A control volume is simply a region in space through which fluid flows. It can be any shape or size. In general, we analyze the control volume by examining the fluid's properties (such as velocity, pressure, and density) at its boundaries.

Define Steady Flow

Steady flow is a situation where the fluid properties (velocity, pressure, density, etc.) at any given point within the control volume do not change with time. Mathematically, this can be represented as: \[\frac{\partial}{\partial t}\left(\text{property}\right) = 0\] Where "property" represents any fluid property (e.g., velocity, pressure, density) and t is time. In simpler terms, the time-derivatives of all fluid properties are equal to zero under steady flow conditions.

Identify Conditions for Steady Flow

Based on the definition of steady flow, the flow through a control volume will be steady when the following conditions are met: 1. The mass flow rate entering the control volume is equal to the mass flow rate leaving the control volume. This ensures that there is no accumulation or depletion of mass within the control volume. Mathematically, this can be written as: \[\dot{m}_{in} = \dot{m}_{out}\] 2. The momentum flow rate entering and leaving the control volume remains constant. This ensures that there is no net force acting on the control volume, which would cause changes in the fluid's velocity. Mathematically, this can be written as: \[\frac{d}{dt}\int_{CV}\rho \textbf{v} dV = \dot{\textbf{m}}_{in}\textbf{v}_{in} - \dot{\textbf{m}}_{out}\textbf{v}_{out}\] 3. The energy flow rate entering and leaving the control volume remains constant. This ensures that energy conservation is satisfied, preventing changes in internal energy, kinetic energy, or potential energy in the fluid. Mathematically, this can be written as: \[\frac{d}{dt}\int_{CV}\rho E dV = \dot{E}_{in} - \dot{E}_{out}\] When all these conditions are met, the flow through a control volume can be considered steady.