Conservation and transport equations
Start with what enters, what leaves and what changes inside a chosen region.
Control volume and conservation
A finite control volume is a region of a specified size that we choose for our calculation. For example, it may be a section of a pipe, the liquid in a tank, or a small part of a solid wall. Its boundary is called the control surface. We calculate how much mass, energy or momentum crosses this surface and how much is present inside the region.
The conservation balance is: rate of accumulation = rate entering − rate leaving + rate of generation − rate of consumption. Accumulation means that the amount inside increases with time. If it decreases, the accumulation rate is negative; this decrease is called depletion. Generation and consumption describe production and removal inside the region. They do not describe transport across its boundary. For total mass, there is no generation or consumption in an ordinary flow process. A chemical reaction can produce or consume an individual species while conserving total mass.
For example, if water enters a tank at 2 kg/s and leaves at 1 kg/s, the mass of water in the tank increases at 1 kg/s. If both rates are 2 kg/s, the mass remains constant. The mass stored has unit kg, whereas each term in its rate balance has unit kg/s. A heat transfer rate has unit W, which is J/s. Dividing a transfer rate by area gives a flux: mass flux has unit kg/(m²·s), and heat flux has unit W/m².
We first write this balance for the whole control volume. We then apply it to a small region and take the limit as its size tends to zero to obtain the local transport equation. “Local” means that the equation describes what happens at each position. Worked examples show how to apply this procedure to total mass, individual species, energy and momentum, with each quantity and its unit defined.