Bernoulli's Principle - Derivations of Bernoulli Equation

Derivations of Bernoulli Equation

Bernoulli equation for incompressible fluids
The Bernoulli equation for incompressible fluids can be derived by integrating the Euler equations, or applying the law of conservation of energy in two sections along a streamline, ignoring viscosity, compressibility, and thermal effects. The simplest derivation is to first ignore gravity and consider constrictions and expansions in pipes that are otherwise straight, as seen in Venturi effect. Let the x axis be directed down the axis of the pipe. Define a parcel of fluid moving through a pipe with cross-sectional area "A", the length of the parcel is "dx", and the volume of the parcel A dx. If mass density is ρ, the mass of the parcel is density multiplied by its volume m = ρ A dx. The change in pressure over distance dx is "dp" and flow velocity v = dx / dt. Apply Newton's Second Law of Motion Force =mass×acceleration and recognizing that the effective force on the parcel of fluid is -A dp. If the pressure decreases along the length of the pipe, dp is negative but the force resulting in flow is positive along the x axis. In steady flow the velocity is constant with respect to time, v = v(x) = v(x(t)), so v itself is not directly a function of time t. It is only when the parcel moves through x that the cross sectional area changes: v depends on t only through the cross-sectional position x(t). With density ρ constant, the equation of motion can be written as by integrating with respect to x where C is a constant, sometimes referred to as the Bernoulli constant. It is not a universal constant, but rather a constant of a particular fluid system. The deduction is: where the speed is large, pressure is low and vice versa. In the above derivation, no external work-energy principle is invoked. Rather, Bernoulli's principle was inherently derived by a simple manipulation of the momentum equation. Another way to derive Bernoulli's principle for an incompressible flow is by applying conservation of energy. In the form of the work-energy theorem, stating that the change in the kinetic energy E_kin of the system equals the net work W done on the system; Therefore, the work done by the forces in the fluid = increase in kinetic energy. The system consists of the volume of fluid, initially between the cross-sections A₁ and A₂. In the time interval Δt fluid elements initially at the inflow cross-section A₁ move over a distance s₁ = v₁ Δt, while at the outflow cross-section the fluid moves away from cross-section A₂ over a distance s₂ = v₂ Δt. The displaced fluid volumes at the inflow and outflow are respectively A₁ s₁ and A₂ s₂. The associated displaced fluid masses are – when ρ is the fluid's mass density – equal to density times volume, so ρ A₁ s₁ and ρ A₂ s₂. By mass conservation, these two masses displaced in the time interval Δt have to be equal, and this displaced mass is denoted by Δm: $\begin{align} \rho A_1 s_1 &= \rho A_{1} v_{1} \Delta t = \Delta m, \\ \rho A_2 s_2 &= \rho A_{2} v_{2} \Delta t = \Delta m. \end{align}$ The work done by the forces consists of two parts: The work done by the pressure acting on the areas A₁ and A₂ The work done by gravity: the gravitational potential energy in the volume A₁ s₁ is lost, and at the outflow in the volume A₂ s₂ is gained. So, the change in gravitational potential energy ΔE_pot,gravity in the time interval Δt is Now, the work by the force of gravity is opposite to the change in potential energy, W_gravity = −ΔE_pot,gravity: while the force of gravity is in the negative z-direction, the work—gravity force times change in elevation—will be negative for a positive elevation change Δz = z₂ − z₁, while the corresponding potential energy change is positive. So: And the total work done in this time interval is The increase in kinetic energy is Putting these together, the work-kinetic energy theorem W = ΔE_kin gives: or After dividing by the mass Δm = ρ A₁ v₁ Δt = ρ A₂ v₂ Δt the result is: or, as stated in the first paragraph: (Eqn. 1), Which is also Equation (A) Further division by g produces the following equation. Note that each term can be described in the length dimension (such as meters). This is the head equation derived from Bernoulli's principle: (Eqn. 2a) The middle term, z, represents the potential energy of the fluid due to its elevation with respect to a reference plane. Now, z is called the elevation head and given the designation z_elevation. A free falling mass from an elevation z > 0 (in a vacuum) will reach a speed when arriving at elevation z = 0. Or when we rearrange it as a head: The term v2 / (2 g) is called the velocity head, expressed as a length measurement. It represents the internal energy of the fluid due to its motion. The hydrostatic pressure p is defined as , with p₀ some reference pressure, or when we rearrange it as a head: The term p / (ρg) is also called the pressure head, expressed as a length measurement. It represents the internal energy of the fluid due to the pressure exerted on the container. When we combine the head due to the flow speed and the head due to static pressure with the elevation above a reference plane, we obtain a simple relationship useful for incompressible fluids using the velocity head, elevation head, and pressure head. (Eqn. 2b) If we were to multiply Eqn. 1 by the density of the fluid, we would get an equation with three pressure terms: (Eqn. 3) We note that the pressure of the system is constant in this form of the Bernoulli Equation. If the static pressure of the system (the far right term) increases, and if the pressure due to elevation (the middle term) is constant, then we know that the dynamic pressure (the left term) must have decreased. In other words, if the speed of a fluid decreases and it is not due to an elevation difference, we know it must be due to an increase in the static pressure that is resisting the flow. All three equations are merely simplified versions of an energy balance on a system.

Bernoulli equation for incompressible fluids

The Bernoulli equation for incompressible fluids can be derived by integrating the Euler equations, or applying the law of conservation of energy in two sections along a streamline, ignoring viscosity, compressibility, and thermal effects.

The simplest derivation is to first ignore gravity and consider constrictions and expansions in pipes that are otherwise straight, as seen in Venturi effect. Let the x axis be directed down the axis of the pipe.

Define a parcel of fluid moving through a pipe with cross-sectional area "A", the length of the parcel is "dx", and the volume of the parcel A dx. If mass density is ρ, the mass of the parcel is density multiplied by its volume m = ρ A dx. The change in pressure over distance dx is "dp" and flow velocity v = dx / dt.

Apply Newton's Second Law of Motion Force =mass×acceleration and recognizing that the effective force on the parcel of fluid is -A dp. If the pressure decreases along the length of the pipe, dp is negative but the force resulting in flow is positive along the x axis.

In steady flow the velocity is constant with respect to time, v = v(x) = v(x(t)), so v itself is not directly a function of time t. It is only when the parcel moves through x that the cross sectional area changes: v depends on t only through the cross-sectional position x(t).

With density ρ constant, the equation of motion can be written as

by integrating with respect to x

where C is a constant, sometimes referred to as the Bernoulli constant. It is not a universal constant, but rather a constant of a particular fluid system. The deduction is: where the speed is large, pressure is low and vice versa.

In the above derivation, no external work-energy principle is invoked. Rather, Bernoulli's principle was inherently derived by a simple manipulation of the momentum equation.

Another way to derive Bernoulli's principle for an incompressible flow is by applying conservation of energy. In the form of the work-energy theorem, stating that

the change in the kinetic energy E_kin of the system equals the net work W done on the system;

Therefore,

the work done by the forces in the fluid = increase in kinetic energy.

The system consists of the volume of fluid, initially between the cross-sections A₁ and A₂. In the time interval Δt fluid elements initially at the inflow cross-section A₁ move over a distance s₁ = v₁ Δt, while at the outflow cross-section the fluid moves away from cross-section A₂ over a distance s₂ = v₂ Δt. The displaced fluid volumes at the inflow and outflow are respectively A₁ s₁ and A₂ s₂. The associated displaced fluid masses are – when ρ is the fluid's mass density – equal to density times volume, so ρ A₁ s₁ and ρ A₂ s₂. By mass conservation, these two masses displaced in the time interval Δt have to be equal, and this displaced mass is denoted by Δm:

$\begin{align} \rho A_1 s_1 &= \rho A_{1} v_{1} \Delta t = \Delta m, \\ \rho A_2 s_2 &= \rho A_{2} v_{2} \Delta t = \Delta m. \end{align}$

The work done by the forces consists of two parts:

The work done by the pressure acting on the areas A₁ and A₂

The work done by gravity: the gravitational potential energy in the volume A₁ s₁ is lost, and at the outflow in the volume A₂ s₂ is gained. So, the change in gravitational potential energy ΔE_pot,gravity in the time interval Δt is

Now, the work by the force of gravity is opposite to the change in potential energy, W_gravity = −ΔE_pot,gravity: while the force of gravity is in the negative z-direction, the work—gravity force times change in elevation—will be negative for a positive elevation change Δz = z₂ − z₁, while the corresponding potential energy change is positive. So:

And the total work done in this time interval is

The increase in kinetic energy is

Putting these together, the work-kinetic energy theorem W = ΔE_kin gives:

After dividing by the mass Δm = ρ A₁ v₁ Δt = ρ A₂ v₂ Δt the result is:

or, as stated in the first paragraph:

(Eqn. 1), Which is also Equation (A)

Further division by g produces the following equation. Note that each term can be described in the length dimension (such as meters). This is the head equation derived from Bernoulli's principle:

(Eqn. 2a)

The middle term, z, represents the potential energy of the fluid due to its elevation with respect to a reference plane. Now, z is called the elevation head and given the designation z_elevation.

A free falling mass from an elevation z > 0 (in a vacuum) will reach a speed

when arriving at elevation z = 0. Or when we rearrange it as a head:

The term v2 / (2 g) is called the velocity head, expressed as a length measurement. It represents the internal energy of the fluid due to its motion.

The hydrostatic pressure p is defined as

, with p₀ some reference pressure, or when we rearrange it as a head:

The term p / (ρg) is also called the pressure head, expressed as a length measurement. It represents the internal energy of the fluid due to the pressure exerted on the container.

When we combine the head due to the flow speed and the head due to static pressure with the elevation above a reference plane, we obtain a simple relationship useful for incompressible fluids using the velocity head, elevation head, and pressure head.

(Eqn. 2b)

If we were to multiply Eqn. 1 by the density of the fluid, we would get an equation with three pressure terms:

(Eqn. 3)

We note that the pressure of the system is constant in this form of the Bernoulli Equation. If the static pressure of the system (the far right term) increases, and if the pressure due to elevation (the middle term) is constant, then we know that the dynamic pressure (the left term) must have decreased. In other words, if the speed of a fluid decreases and it is not due to an elevation difference, we know it must be due to an increase in the static pressure that is resisting the flow.

All three equations are merely simplified versions of an energy balance on a system.

Bernoulli equation for compressible fluids
The derivation for compressible fluids is similar. Again, the derivation depends upon (1) conservation of mass, and (2) conservation of energy. Conservation of mass implies that in the above figure, in the interval of time Δt, the amount of mass passing through the boundary defined by the area A₁ is equal to the amount of mass passing outwards through the boundary defined by the area A₂: . Conservation of energy is applied in a similar manner: It is assumed that the change in energy of the volume of the streamtube bounded by A₁ and A₂ is due entirely to energy entering or leaving through one or the other of these two boundaries. Clearly, in a more complicated situation such as a fluid flow coupled with radiation, such conditions are not met. Nevertheless, assuming this to be the case and assuming the flow is steady so that the net change in the energy is zero, where ΔE₁ and ΔE₂ are the energy entering through A₁ and leaving through A₂, respectively. The energy entering through A₁ is the sum of the kinetic energy entering, the energy entering in the form of potential gravitational energy of the fluid, the fluid thermodynamic energy entering, and the energy entering in the form of mechanical p dV work: where Ψ = gz is a force potential due to the Earth's gravity, g is acceleration due to gravity, and z is elevation above a reference plane. A similar expression for may easily be constructed. So now setting : which can be rewritten as: Now, using the previously-obtained result from conservation of mass, this may be simplified to obtain which is the Bernoulli equation for compressible flow.

Bernoulli equation for compressible fluids

The derivation for compressible fluids is similar. Again, the derivation depends upon (1) conservation of mass, and (2) conservation of energy. Conservation of mass implies that in the above figure, in the interval of time Δt, the amount of mass passing through the boundary defined by the area A₁ is equal to the amount of mass passing outwards through the boundary defined by the area A₂:

Conservation of energy is applied in a similar manner: It is assumed that the change in energy of the volume of the streamtube bounded by A₁ and A₂ is due entirely to energy entering or leaving through one or the other of these two boundaries. Clearly, in a more complicated situation such as a fluid flow coupled with radiation, such conditions are not met. Nevertheless, assuming this to be the case and assuming the flow is steady so that the net change in the energy is zero,

where ΔE₁ and ΔE₂ are the energy entering through A₁ and leaving through A₂, respectively.

The energy entering through A₁ is the sum of the kinetic energy entering, the energy entering in the form of potential gravitational energy of the fluid, the fluid thermodynamic energy entering, and the energy entering in the form of mechanical p dV work:

where Ψ = gz is a force potential due to the Earth's gravity, g is acceleration due to gravity, and z is elevation above a reference plane.

A similar expression for may easily be constructed. So now setting :

which can be rewritten as:

Now, using the previously-obtained result from conservation of mass, this may be simplified to obtain

which is the Bernoulli equation for compressible flow.

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