Basic Equations for One-Dimensional Flow

Euler’s Equation of Motion

Euler’s equation of motion is derived by analyzing a small cylindrical fluid system along a streamline. The forces acting to accelerate this system include the pressure forces on the ends, expressed as $-dpdA$, and the weight component in the direction of motion, given as $-\rho g dAdz$. Applying Newton’s second law, $dF = dm \times a$, where the differential mass $dm = \rho dsdA$, leads to the one-dimensional Euler equation: $\frac{dp}{\rho} + V dV + g dz = 0$. When divided by $g$, the equation is written as $d\left(\frac{p}{\gamma} + \frac{V^2}{2g} + z\right) = 0$.

Bernoulli’s Equation and Flow Visualization

Integrating the Euler equation between any two arbitrary points on a streamline provides Bernoulli’s equation: $\frac{p_1}{\gamma} + \frac{V_1^2}{2g} + z_1 = \frac{p_2}{\gamma} + \frac{V_2^2}{2g} + z_2$. The sum of these terms is defined as the total head, $H = \frac{p}{\gamma} + \frac{V^2}{2g} + z$, which remains constant along the streamline. This relationship involves the pressure head $\frac{p}{\gamma}$, the velocity head $\frac{V^2}{2g}$, and the potential head $z$. The energy line (E.L.) represents the total head, while the hydraulic grade line (H.G.L.) or piezometric head line tracks the variation in static pressure and potential energy. The vertical distance between the E.L. and the H.G.L. is proportional to the velocity head.

Mechanical Energy and Flow Work

The energy of a flowing fluid consists of potential energy per unit weight $z$, kinetic energy per unit weight $\frac{V^2}{2g}$, and pressure energy or flow work per unit weight $\frac{p}{\gamma}$. Pressure energy represents the work done by pressure forces as the fluid moves through a cross-section of area $A$. In a uniform pipe where velocity cannot change, potential energy is converted into pressure energy rather than kinetic energy, effectively making pressure energy a form of potential energy in transit. Bernoulli’s equation serves as a statement of the conservation of energy for a steady flow of a frictionless fluid along a streamline, where total energy per unit weight remains constant despite variations in the distribution between its three forms.

Energy Gains and Losses in Flow Systems

In practical applications, Bernoulli’s equation is expanded to account for energy added to the system via a pump, energy lost due to friction in pipes, or work extracted by a machine like a turbine. The general balance states that the total energy per unit weight at an initial point plus energy supplied equals the total energy per unit weight at a secondary point plus losses and work done. For a fire engine system from Example 4.1, a pump adds a head of $50\,m$ to water drawn through a $150\,mm$ pipe with friction loss $h_1 = 5 \times \frac{V_1^2}{2g}$. The water is discharged through a $75\,mm$ nozzle at height $z_3 = 32\,m$ via a $100\,mm$ delivery pipe with loss $h_2 = 12 \times \frac{V_2^2}{2g}$.

Calculations and Results from Example 4.1

Solving for the system using Bernoulli’s equation between the sump surface (point A) and the nozzle discharge (point C) involves the continuity equation where $V A$ is constant. Applying the energy balance provides the jet velocity $V_3 = 8.31\,m/s$. Further application of the equation between the sump (A) and the pump inlet (B), where $z_2 = 2\,m$ and the suction pipe velocity $V_1 = 2.08\,m/s$, allows for the determination of the pressure in the suction pipe. The resulting pressure at point B is calculated as $p_B = -3.32\,t/m^2$, which indicates a pressure below atmospheric levels.