5.1 Conservation of mass and flow work

Mass Conservation in Control Volumes

Conservation of Mass Principle

The conservation of mass principle says that mass can't be created or destroyed within a system. For a control volume (an open system where mass crosses the boundary), this means any change in mass stored inside must equal the difference between what flows in and what flows out.

The mass balance equation for a control volume is:

$\frac{dm_{cv}}{dt} = \sum \dot{m}_{in} - \sum \dot{m}_{out}$

• $dm_{cv}/dt$ is the rate of change of mass inside the control volume
• $\dot{m}_{in}$ and $\dot{m}_{out}$ are the mass flow rates at each inlet and outlet

This equation is the starting point for virtually every control volume problem you'll encounter. Everything else builds on it.

Steady-State Conditions

Most devices you'll analyze (turbines, compressors, nozzles, heat exchangers) operate at steady state during normal conditions. Steady state means properties inside the control volume don't change with time, so the mass stored inside stays constant:

$\frac{dm_{cv}}{dt} = 0$

The mass balance then simplifies to:

$\sum \dot{m}_{in} = \sum \dot{m}_{out}$

Total mass in equals total mass out. For a single-inlet, single-outlet device, this reduces further to $\dot{m}_{in} = \dot{m}_{out}$, which you'll use constantly.

Steady vs Unsteady Flow

Steady-Flow Processes

A process is steady-flow when fluid properties (pressure, temperature, density, velocity) at any fixed location in the control volume don't change over time. That also means mass flow rates at every inlet and outlet stay constant.

Key characteristics:

• $dm_{cv}/dt = 0$, so $\sum \dot{m}_{in} = \sum \dot{m}_{out}$
• Properties can vary from point to point within the device, but at any given point they're constant in time
• Examples: flow through a pipe at constant flow rate, a turbine or pump running at fixed operating conditions

The steady-flow assumption is what makes most textbook problems tractable. Recognize when you can apply it and when you can't.

Unsteady-Flow Processes

When fluid properties or mass flow rates change with time, you have an unsteady-flow (or transient) process. The mass inside the control volume is no longer constant:

$\frac{dm_{cv}}{dt} \neq 0$

You must use the full mass balance:

$\frac{dm_{cv}}{dt} = \sum \dot{m}_{in} - \sum \dot{m}_{out}$

For a finite time interval, you can integrate this to get:

$m_{2} - m_{1} = \sum m_{in} - \sum m_{out}$

where $m_{1}$ and $m_{2}$ are the mass inside the control volume at the start and end of the process.

Examples of unsteady flow:

• Filling or emptying a tank
• Startup or shutdown of a turbine or compressor
• Pulsating flow in an internal combustion engine cylinder

Flow Work and Energy Transfer

Flow Work Concept

Flow work (sometimes called flow energy) is the work required to push a fluid element into or out of a control volume against the pressure at that boundary. It's not shaft work or boundary work; it's the energy needed just to maintain flow across the control surface.

Think of it this way: for fluid to enter a control volume, something upstream must push it through the inlet against the pressure there. That push requires work equal to the pressure times the specific volume of the fluid.

On a per-unit-mass basis, flow work is:

$w_{flow} = Pv$

where $P$ is the pressure at the boundary and $v$ is the specific volume of the fluid at that location. On a rate basis:

$\dot{W}_{flow} = P \dot{V} = P v \dot{m}$

where $\dot{V}$ is the volume flow rate.

Why Flow Work Matters for the Energy Equation

Flow work is the reason enthalpy ($h = u + Pv$) appears in the energy equation for control volumes instead of just internal energy $u$. When you combine the internal energy of the flowing fluid with its flow work, you get enthalpy:

$u + Pv = h$

So every time you see enthalpy in a control volume energy balance, the flow work is already baked in. This is one of the most important conceptual points in this unit: enthalpy naturally accounts for both the internal energy of the fluid and the work needed to push it across the system boundary.

Energy Transfer Analysis

Flow work shows up whenever fluid crosses a control volume boundary. In practice:

• Pumps do shaft work on the fluid, increasing its pressure and therefore its flow work at the outlet
• Turbines extract energy from high-pressure fluid; the drop in flow work (and enthalpy) between inlet and outlet is what drives shaft work output
• Heat exchangers involve flow work at each inlet and outlet, with heat transfer occurring between fluid streams

Understanding flow work helps you set up the full energy balance correctly and interpret where energy goes in these devices.

Mass Balance with Multiple Inlets and Outlets

General Mass Balance Equation

Many real devices have more than one inlet or outlet. A mixing chamber might have two inlet streams and one exit. A feedwater heater might combine extracted steam with liquid water. The mass balance still applies the same way:

$\frac{dm_{cv}}{dt} = \sum \dot{m}_{in} - \sum \dot{m}_{out}$

At steady state:

$\sum \dot{m}_{in} = \sum \dot{m}_{out}$

You just sum over all inlets on one side and all outlets on the other.

Calculating Mass Flow Rates

The mass flow rate at any inlet or outlet is:

$\dot{m} = \rho V A$

• $\rho$ is the fluid density
• $V$ is the average fluid velocity normal to the cross-section
• $A$ is the cross-sectional area

You can also write this using specific volume: $\dot{m} = VA / v$, which is often more convenient when working with steam tables.

Solving Multi-Port Problems

Here's a practical approach for problems with multiple inlets and outlets:

1. Sketch the control volume and label every inlet and outlet with its own subscript
2. Write the steady-state mass balance: $\dot{m}_1 + \dot{m}_2 + \cdots = \dot{m}_3 + \dot{m}_4 + \cdots$
3. Apply any given constraints, such as known flow rate ratios (e.g., "stream 2 is 30% of stream 1") or equal exit velocities
4. Use $\dot{m} = \rho V A$ at ports where you know density, velocity, and area
5. Solve the system of equations for the unknowns

Common applications include pipe junctions, mixing chambers, and heat exchangers with multiple fluid streams. In each case, the mass balance gives you the relationship between flow rates that you'll need before you can apply the energy balance.