The Mechanical Energy Balance (Bernoulli with Friction)

From ideal energy conservation to real pipes with pumps and losses

core-transportenergy-balancepipes

⏱️ About 20 min

If the pressure drops along a pipe, where did that mechanical energy go? The mechanical energy balance answers that—quantitatively.

💡

The big idea: Ideal Bernoulli assumes no friction and no machines. Real piping systems require adding head losses and shaft work to connect pressures, velocities, and elevations.

🎯 By the end, you'll be able to

write Bernoulli’s equation for ideal flow
identify friction head loss and pump head terms
solve for unknown pressure, head loss, or pump head
use consistent units (head form or pressure form)

📎 Helpful to know first

Reynolds Number & Flow Regimes

Ideal Bernoulli (no friction, no machines)

Bernoulli’s equation is a mechanical energy balance for steady, incompressible flow along a streamline. In its common form per unit mass, it states that pressure energy, kinetic energy, and potential energy trade off but sum to a constant.

\[ \frac{P}{\rho}+\frac{v^2}{2}+gz=\text{const} \]

Ideal Bernoulli equation per unit mass (steady, incompressible, no losses).

⚠️ Bernoulli is not automatically valid everywhere

Bernoulli’s ideal form neglects viscous dissipation (friction), assumes steady incompressible flow, and is best applied between points in the same flow path. Real pipes need loss terms.

Head form (per unit weight) for engineering work

Engineers often divide by g to write terms as a head (meters of fluid):

Pressure head: P/(ρg)
Velocity head: v²/(2g)
Elevation head: z

Losses (like friction) also appear naturally as head terms.

\[ \frac{P_1}{\rho g}+\frac{v_1^2}{2g}+z_1+h_p=\frac{P_2}{\rho g}+\frac{v_2^2}{2g}+z_2+h_f+h_m \]

Mechanical energy balance (head form) including pump head hp, friction loss hf, and minor losses hm.

🔑 Interpretation of loss terms

h_f (friction loss) is the irreversible conversion of mechanical energy into internal energy due to viscosity. h_m (minor losses) captures fittings/entrances/exits. A pump adds head (positive), a turbine removes head (negative).

When velocities are the same

Many practical pipe problems use the same diameter at points 1 and 2, so v₁=v₂. Then the velocity-head terms cancel, simplifying the balance to mainly pressure, elevation, and losses.

📝 Worked example: A pump must move water at 20°C through a horizontal pipe run where the total head loss (friction + minor) is h_L = 12.0 m of water. The inlet and outlet are at the same elevation and the pipe diameter is constant, so v1=v2. What pump head hp is required? Use g=9.81 m/s².

Write head balance with z1=z2 and v1=v2: P1/(ρg) + hp = P2/(ρg) + h_L.
If the pump just needs to overcome losses with no net pressure change desired between endpoints, set P1≈P2, so hp = h_L.
Thus hp = 12.0 m (of water).

✓ hp = 12.0 m

✏️ Practice: Water (ρ=998 kg/m³) flows steadily between two points in the same diameter pipe with no pump and no elevation change. The friction + minor head loss is h_L = 5.0 m. Compute the pressure drop ΔP = P1 − P2. Use g=9.81 m/s².

Solution

With no pump and z1=z2 and v1=v2: P1/(ρg) = P2/(ρg) + h_L → (P1−P2)/(ρg)=h_L.
Compute ΔP = ρ g h_L = (998)(9.81)(5.0) = 48951.9 Pa.

Check your understanding

1. In the mechanical energy balance (head form), friction losses appear as:

Friction is included as a head loss term (hf), representing irreversible mechanical energy dissipation.

2. If a pipe has constant diameter and z1=z2, the term that often cancels between points 1 and 2 is:

With constant diameter, v1=v2 so v²/(2g) cancels; hf does not cancel because it accumulates along the length.

3. Pressure head P/(ρg) has units of:

Dividing pressure by ρg converts it to an equivalent fluid column height (meters).

✅ Key takeaways

Ideal Bernoulli ignores friction and machines
Mechanical energy balance adds pump/turbine head and loss terms
Head form uses P/(ρg), v²/(2g), and z, all in meters
Many pipe problems simplify when v1=v2 and/or z1=z2

➡️ Next, we’ll connect friction head loss to velocity using the Darcy–Weisbach equation and friction factor f.

Want to test yourself on this? Try the Chemical Aptitude test →