15.5.3. Conservation of Energy

As I said earlier, we need to have three equations. One is conservation of mass, the other is momentum, and another is energy. Most of the times you remember whenever you solve the fluid mechanics problems we have three helping hands: it is not two, it is three helping hands. One is conservation of mass, the other is momentum, and another is energy.

In fluid mechanics, understanding the principles of conservation is essential. There are three fundamental conservation equations we must consider: conservation of mass, conservation of momentum, and conservation of energy. These principles act as guiding rules during fluid analysis, ensuring that we account for all changes in a fluid system. Conservation of mass tells us that mass cannot be created or destroyed; it can only flow in or out of a system. Conservation of momentum addresses the forces acting on the fluid and their effects. Finally, conservation of energy considers all forms of energy in the fluid system and how they change over time. All three work together to provide a complete picture of fluid behavior.

Think of a river system. The water flowing through the river represents the mass. As water flows from one area to another, the total amount of water remains the same (conservation of mass). The speed of the water may increase or decrease due to the landscape (conservation of momentum). The kinetic energy of the water when it flows faster is an expression of energy conservation, as it transforms potential energy (like when flowing downhill) into kinetic energy.

If I put B = m and b = 1, then simplify the problems. D/Dt ∫_CV ρ dV = ∫_CS (ρv · n) dA. This indicates that the net outflow of mass through this control surface is equal to the change of mass storage within the control volume.

To derive the mass conservation equation, we often simplify the parameters by assuming 'B' as mass (m) and 'b' as 1. We then use the integral form, which states that the change in mass within a control volume (CV) over time is equal to the net mass flow across the control surface (CS). The equation shows that any change in mass within a given control volume is proportional to what enters and exits through its surfaces. If the incoming mass is balanced by the outgoing mass, then the mass within that volume remains constant.

A classic example would be a bathtub: when you fill it (mass influx) and the water drains out (mass outflux), the water level changes. Here, the amount of water in the bathtub can be represented as the control volume, and the overflow represents the mass outflux. If you turn off the tap and the drain, the water level stabilizes (no change in mass), demonstrating mass conservation.

The net change of the mass in this control volume should be equal to the change of mass storage within the control volume. If the flow is steady, there is no change in the storage within the control volume. What it indicates is that net outflow of this control volume becomes zero.

In fluid dynamics, when we refer to a steady flow, we mean that the flow parameters (like velocity and density) do not change with time at any point within the control volume. Therefore, when we analyze a steady state, the mass coming into the control volume must equal the mass leaving it. This leads to the conclusion that if we sum up the inflows and outflows in a steady-state condition, the total change in the storage of mass will be zero, reinforcing the conservation principle.

Consider a garden hose with a nozzle: when water flows consistently out of the nozzle while you fill the hose, the amount of water in the hose remains the same as long as you don't change the flow from the tap or block the nozzle. This represents steady flow, where inflow equals the outflow.

If the flow is one-dimensional, instead of doing surface integrals, you can simply sum the inflow and outflow rates, leading to a simplified expression for conservation of mass.

In fluid mechanics, when dealing with one-dimensional flow scenarios—where fluid flows uniformly and straight—the analysis becomes easier. Instead of calculating complex integrals across surfaces to find the total mass flow, we can simply add the mass flow rates at the inlet and outlet. By knowing the density, cross-sectional areas, and velocities, we can derive equations that embody conservation of mass without complex calculations.

Imagine a long tube where water flows in and out from both ends. If you were to determine how much water comes in versus how much goes out, counting the flow rates at each end provides an easier way to see that they balance out, rather than measuring the mass at every point along the pipe.

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