- Joined
- Jan 19, 2012
- Messages
- 9
- Reaction score
- 0
If you're like me, Bernoulli's can be a real pain to remember. Even worse, it seems to make little intuitive sense at first blush. I like physics because it generally just "makes sense," but I often have difficulty feeling that way about this equation. However, I've found a way of thinking about it that, at least for me, seems to make it a bit easier. It may work for you as well. If you've heard it explained this way before, congratulations -- you probably had a better teacher than I did 👍.
Short Version: Bernoulli's is nothing more than an extrapolation of the conservation of energy for a closed system. It basically just says that Pressure + KE + PE remains constant. Pressure is energy density, so we should expect P + KE + PE to remain unchanged in ideal conditions.
Long Version:
First, here's Bernoulli's: P1+1/2ρv^2+ρgy = P2 +1/2ρ^2+pgy
Obviously, the left side represents initial values and the right side represents final values.
If we break it down, however, we can see that this equation is really nothing more than saying Pressure + KE + PE is constant.
First, 1/2ρv^2 is the kinetic energy of a standard unit of fluid because the mass of a standard unit of volume is equal to the density of that fluid (the mass of a cubic meter of water is 1000 kg, and the density of water is 1000 kg/m^3). Thus, 1/2ρv^2 is the kinetic energy of a cubic meter of a fluid.
Second, ρgy is the formula for the PE of a standard unit of fluid. mgh is the formula for the PE due to gravity of any object, and the mass of a standard unit (cubic meter) of a fluid is equal to its density, as mentioned above. "y" in this case is the same as height, or at least the change in height. So, ρgy is nothing more than an expression of the PE of a fluid, if only in terms relative to our starting "y" point.
Finally, it makes intuitive sense that Pressure + KE + PE should be constant under ideal flow conditions. Pressure is nothing more than F/A, so a change in pressure means a change in the energy of the fluid*. No energy is being lost to the outside due to heat under ideal conditions, so the total energy of the system must be constant. Thus, the sum of P + KE + PE must remain constant at all points.
*Since force is not exactly equivalent to energy, and there's no such thing as a "conservation of force" law, it may not initially seem like pressure (force per unit of area) is an expression of energy in a fluid. However, pressure is a statement about energy density. F/A = Fd/Ad = Work/Volume = Energy/Volume. Macroscopically, high pressure means a greater application of force from the molecules of the fluid to the container. In order to exert this force, energy is required. Thus, attaining a KE of 100J at high pressure takes more energy than attaining 100J of KE at low pressure. As a result, as pressure increases or decreases on one side of Bernoulli's equation, there must be a corresponding energy change in KE or PE to ensure energy is conserved.
Short Version: Bernoulli's is nothing more than an extrapolation of the conservation of energy for a closed system. It basically just says that Pressure + KE + PE remains constant. Pressure is energy density, so we should expect P + KE + PE to remain unchanged in ideal conditions.
Long Version:
First, here's Bernoulli's: P1+1/2ρv^2+ρgy = P2 +1/2ρ^2+pgy
Obviously, the left side represents initial values and the right side represents final values.
If we break it down, however, we can see that this equation is really nothing more than saying Pressure + KE + PE is constant.
First, 1/2ρv^2 is the kinetic energy of a standard unit of fluid because the mass of a standard unit of volume is equal to the density of that fluid (the mass of a cubic meter of water is 1000 kg, and the density of water is 1000 kg/m^3). Thus, 1/2ρv^2 is the kinetic energy of a cubic meter of a fluid.
Second, ρgy is the formula for the PE of a standard unit of fluid. mgh is the formula for the PE due to gravity of any object, and the mass of a standard unit (cubic meter) of a fluid is equal to its density, as mentioned above. "y" in this case is the same as height, or at least the change in height. So, ρgy is nothing more than an expression of the PE of a fluid, if only in terms relative to our starting "y" point.
Finally, it makes intuitive sense that Pressure + KE + PE should be constant under ideal flow conditions. Pressure is nothing more than F/A, so a change in pressure means a change in the energy of the fluid*. No energy is being lost to the outside due to heat under ideal conditions, so the total energy of the system must be constant. Thus, the sum of P + KE + PE must remain constant at all points.
*Since force is not exactly equivalent to energy, and there's no such thing as a "conservation of force" law, it may not initially seem like pressure (force per unit of area) is an expression of energy in a fluid. However, pressure is a statement about energy density. F/A = Fd/Ad = Work/Volume = Energy/Volume. Macroscopically, high pressure means a greater application of force from the molecules of the fluid to the container. In order to exert this force, energy is required. Thus, attaining a KE of 100J at high pressure takes more energy than attaining 100J of KE at low pressure. As a result, as pressure increases or decreases on one side of Bernoulli's equation, there must be a corresponding energy change in KE or PE to ensure energy is conserved.
