Energy conservation in fluid mechanics is crucial for understanding how energy moves and changes in flowing systems. It's based on the first law of thermodynamics, which states that energy can't be created or destroyed, only converted between forms.
The energy equation balances different types of energy in a fluid system. It includes kinetic, potential, and internal energy, as well as work and heat transfer. This principle is key for analyzing real-world fluid systems like pipes, turbines, and pumps.
Conservation of Energy in Fluid Mechanics
Conservation of energy equation
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Derived from the first law of thermodynamics which states energy cannot be created or destroyed, only converted from one form to another
General form of the energy equation for a control volume:
∂t∂∫CVeρdV+∫CSeρV⋅dA=Q˙−W˙+∫CSρPρV⋅dA
e represents total energy per unit mass including kinetic, potential, and internal
ρ represents fluid density
V represents velocity vector
Q˙ represents rate of heat transfer into the system (heat added by a combustion process)
W˙ represents rate of work done by the system (work extracted by a turbine)
P represents pressure
Simplified form for steady flow with one inlet and one outlet:
2V12+gz1+ρP1+u1+m˙Q˙=2V22+gz2+ρP2+u2+m˙W˙
V represents velocity
g represents gravitational acceleration
z represents elevation
u represents internal energy per unit mass
m˙ represents mass flow rate
Forms of energy
Kinetic energy associated with the motion of a fluid particle calculated using KE=21mV2 and depends on the velocity of the fluid (water flowing through a pipe)
Potential energy due to the position or elevation of a fluid particle in a gravitational field calculated using PE=mgz and depends on the elevation of the fluid relative to a reference level (water stored in an elevated tank)
Internal energy associated with the molecular motion and intermolecular forces within a fluid, represents the microscopic energy of the fluid particles, and depends on the temperature and properties of the fluid (thermal energy of steam in a power plant)
In the absence of heat transfer and work, the total energy (sum of kinetic, potential, and internal) of a fluid system remains constant
Energy transfer problem-solving
Work defined as energy transfer due to a force acting over a distance calculated using W=∫Fds and often associated with moving boundaries in fluid systems (pumps, turbines)
Power defined as the rate of work or energy transfer calculated using P=dtdW
Problem-solving steps using the energy equation:
Identify the control volume and the relevant energy terms
Determine the appropriate form of the energy equation based on the problem conditions (steady or unsteady flow, number of inlets/outlets)
Substitute known values and solve for the unknown quantity
Mechanical energy conservation
Mechanical energy defined as the sum of kinetic and potential energy calculated using ME=KE+PE=21mV2+mgz
In the absence of dissipative forces (friction, viscosity) and no work or heat transfer, mechanical energy is conserved
Leads to the Bernoulli equation for steady, incompressible, inviscid flow along a streamline: 2V12+gz1+ρP1=2V22+gz2+ρP2
Relates velocity, pressure, and elevation changes along a streamline (flow through a constricted pipe)
Dissipative forces, such as friction, cause a loss of mechanical energy which is converted into internal energy or heat (pressure drop in a pipe due to wall friction)