The Continuity Equation

## Notes

• cv or CV is an abbreviation for "Control Volume"
• port is a location when fluid flows across the surface of a cv
• cs is an abbreviation for "Control Surface"
• efflux refers to fluid flow out of a CV by crossing the control surface (negative efflux would describe inflow)

## Holistic Meaning (i.e. physics) of the Continuity Equation

As shown in the table below, there are two ways to understand and apply the continuity equation.

Label When Applies? Meaning Primary Dimensions Units
Rate Form instant in time (rate of accumulation of mass in cv) = (rate of mass flow into cv) - (rate of mass flow out of cv) M/T kg/s, lbm/s
Amount form time interval (amount of mass that accumulated during the time interval) = (amount of mass that entered the cv) - (amount of mass that left the cv) M kg, lbm

The figure below shows these ideas in a sketch. As shown, the principle of mass balance is that accumulation plus outflow minus inflow sums to zero.

## Procedural Knowledge (How to Apply the Continuity Equation)

Step 1. Selection. Select the continuity equation when the parameters of the problem include variables such as flow rate $(Q$ or $\dot{m})$, velocity, and area.

If the problem involves an instant in time, select the rate form.
If the problem involves a time interval (time 1 and time 2), select the amount form.

Assumptions. Unlike previous eqns. (e.g. hydrostatic, Bernoulli, etc.) the continuity equation does not have assumptions that need to be checked.

Step 2. Control Volume. Select you CV so the control surfaces are situated where
(a) you know parameters, and
(b) where you want to find parameters.

Sketch your CV with a dotted line. Label it (CV or cv). Label ports (1 = inlet; 2 = outlet is most common). If needed, add a label to describe your CV (Is the cv stationary or moving? Is the cv moving or deforming?).

Step 3. General Equation. Write down the general form of the continuity equation. Most of the time, select Eq. (5.25) because it is easier to apply. If there is velocity variation at the cs or other nuances, select Eq. (5.24) because it is the most general.

Step 4. Term-by-Term Analysis. Analyze the accumulation term, the inflow term(s), the outflow term(s). Reduce the general equation so it applies to the problem at hand. Validate your final result: (a) is it DH (dimensional homogeneous)? (b) does it make sense physically?

## Key Factors for Quality (What to Look For)

• The right CV is selected, sketched, and labeled appropriately.
• Inlet and exit ports are labeled appropriately.
• Key dimensions are labeled appropriately.
• The general equations is
• written down.
• reduced using term-by-term analysis (all key details documented).