$$ \begin{aligned} &\frac{P_{\mathrm{o}}}{\gamma_{\mathrm{o}}}+\frac{V_{\mathrm{o}, \text { avg }}^{2}}{2 g}+z_{\mathrm{o}}=\frac{P_{\mathrm{i}}}{\gamma_{\mathrm{i}}}+\frac{V_{\mathrm{i}, \text { avg }}^{2}}{2 g}+z_{\mathrm{i}}-h_{\mathrm{L}}+h_{\mathrm{P}}-h_{\mathrm{T}} \ &h_{\mathrm{L}}=h_{\mathrm{L}, \text { major }}+h_{\mathrm{L} \text {, minor }} \end{aligned} $$ major losses: due to viscous effects in straight pipes minor losses: due to viscous effects in pipe components Note: "minor" does not mean $h_{\mathrm{L}, \text { minor }}\lth_{\mathrm{L}, \text { major }}$

Components affect flow by:

• Changing the direction of flow
• Obstructing the flow
• Changing flow speed through varying cross sectional area

Some typical components in pipe systems:

• Bends, elbows, and tees
  • Valves
• Sudden expansions or contractions
  • Gradual expansions or contractions (diffusers, nozzles)
  • Entrance into pipe system from reservoir
• Exit from pipe system into reservoir

If flow direction changes too rapidly, regions of separated flow are generated where eddies form, enhancing losses.

The geometries of most components are too complicated to predict $h_{\mathrm{L}, \text { minor }}$ theoretically. Minor losses are calculated using the following equation, $$ h_{\mathrm{L}, \text { minor }}=K \frac{V_{\text {avg }}^{2}}{2 g} $$ where $K$ is a dimensionless loss coefficient obtained experimentally. Factors that impact $K$ values: - size of component - geometry of the component - Reynolds number - type of connection with pipe - proximity to other components - age of component $K$ values are reported using tables, equations, charts, or nomographs. For components that change cross sectional area (e.g., nozzle), the source should indicate whether the inlet or outlet $V_{\text {avg }}$ is used to calculate $h_{\mathrm{L}, \text { minor }}$. The overall $h_{\mathrm{L}, \text { minor }}$ between the inlet and outlet in a pipe system is the sum of the minor losses of all components. $$ h_{\mathrm{L}, \text { minor }}=\sum\left(K \frac{V_{\text {avg }}^{2}}{2 g}\right)=K_{\text {elbow1 }} \frac{V_{\text {avg, } 1}^{2}}{2 g}+K_{\text {elbow2 }} \frac{V_{\text {avg, } 2}^{2}}{2 g}+K_{\text {valve }} \frac{V_{\text {avg, valve }}^{2}}{2 g} $$ If the pipe diameter does not change, $$ h_{\mathrm{L}, \text { minor }}=\left(\sum K\right) \frac{V_{\mathrm{avg}}^{2}}{2 g} $$