Equation Quick Reference
Number |
Name |
Equation |
|---|---|---|
Particle terminal velocity |
\(\bar v_t = \frac{D_{particle}^2 g}{18 \nu} \frac{\rho_p - \rho_w}{\rho_w}\) |
|
Continuity |
\(Q = \bar v A\) |
|
Reynold’s Number |
\({\rm Re} = \frac{\bar vD}{\nu} = \frac{4Q}{\pi D\nu} = \frac{\rho \bar vD}{\mu}\) |
|
Bernoulli |
\(\frac{p_1}{\rho g} + {z_1} + \frac{v_1^2}{2g} = \frac{p_2}{\rho g} + {z_2} + \frac{v_2^2}{2g}\) |
|
Control Volume Energy |
\(\frac{p_{1}}{\rho g} + z_{1} + \frac{\bar v_{1}^2}{2g} = \frac{p_{2}}{\rho g} + z_{2} + \frac{\bar v_{2}^2}{2g} + h_L\) |
|
Darcy-Weisbach |
\(h_{\rm{f}} = {\rm{f}} \frac{L}{D} \frac{\bar v^2}{2g}\) |
|
Swamee-Jain |
\({\rm{f}} = \frac{0.25} {\left[ \log \left( \frac{\epsilon }{3.7D} + \frac{5.74}{{\rm Re}^{0.9}} \right) \right]^2}\) |
|
Hagen-Poiseuille |
\(h_{\rm{f}} = \frac{128\mu L Q}{\rho g\pi D^4}\) |
|
Orifice |
\(Q = \Pi_{vc} A_{or} \sqrt{2g\Delta h}\) |
|
Minor loss |
\({ {\rm{ \mathbf{Third form:} }} \quad h_e = \left( \frac{A_{out}}{A_{in}} -1 \right)^2 \frac{\bar v_{out}^2}{2g} = K_e \frac{\bar v_{out}^2}{2g} \quad {\rm where} \quad K_e = \left( \frac{A_{out}}{A_{in}} - 1 \right)^2 }\) |
|
Fractal floc volume |
\(\forall_{floc} = \forall_{cp} n_{cp}^\frac{3}{\Pi_{fractal}}\) |
|
Floc terminal velocity |
\(v_t = \frac{D_{cp}^2g}{18\nu}\frac{\rho_{cp} - \rho_{H_2O}}{\rho_{H_2O}} \left( \frac{D_{floc}}{D_{cp}} \right) ^{\Pi_{fractal}-1}\) |
|
max floc velocity gradient |
\(G_{max} = \frac{4F_{bond}}{3 \pi \mu D_{floc_{max}}^2}\) |
|
Camp-Stein velocity |
\(\tilde{G} = \sqrt{\frac{P}{\rho \nu \forall}}\) |
|
Einstein’s diffusion |
\(D_{Diffusion} = \frac{k_B T}{3 \pi \mu D_P}\) |
|
Linear CDC |
\(Q_{max Tube} = \frac{\pi D^2}{4} \sqrt{\frac{2 h_L g \Pi_{Error}}{\sum{K} }}\) |
|
Tank with a valve |
\(\frac{Q}{Q_0} = 1 - \frac{1}{2} \frac{t}{t_{Design}} \frac{h_{Tank}}{h_0}\) |
|
Mechanical power |
\(P = \rho g Q \Delta h\) |
|
Flow per chemical injection port |
\(Q_{mixer} = g h_e t_{eddy}^2 \bar v_{exp}\) |
|
Collision potential |
\(\tilde{G} \theta = \sqrt{\frac{g h_L \theta}{\nu}}\) |
|
Channel width |
\(W_{min \Pi_{H_eS}} = \frac{\Pi_{H_eS}Q}{H_e}\left( \frac{K}{2 H_e \nu \tilde{G}^2} \right)^\frac{1}{3}\) |
|
Distance between expansions |
\(H_{e_{max}} = \left[ \frac{K}{2 \nu \tilde{G}^2} \left( \frac{Q \Pi_{{HS}_{max}}}{W_{channel}} \right)^3 \right]^\frac{1}{4}\) |
|
Baffle spacing |
\(S = \left( \frac{K}{2 H_e \tilde{G}^2 \nu } \right)^\frac{1}{3} \frac{Q}{W_{channel}}\) |
|
Maximum jet velocity |
\(\bar v_{Jet_{max}} = \left(\frac{q\nu G_{max}^2}{\Pi_{JetPlane} }\right)^\frac{1}{4}\) |
|
Tube settler flow |
\(Q_{Tube}=\frac{\bar v_{c}\pi D^2}{4} \left(\frac{L}{D} \cos \alpha +\sin \alpha \right)\) |
|
Minimum plate settler spacing |
\(S_{min} \approx \frac{3 \bar v_{z_{Plate}}}{\sin^2 \alpha} \left( \frac{18 \nu}{g D_{cp}} \frac{\rho_{H_2O}}{\rho_{cp} - \rho_{H_2O}} \right)\) |
|
Plate settler head loss |
\(h_L = 2 \frac{\mu}{\rho g} \left( \frac{6 \bar v_{z_{Plate}}}{S sin^2 \alpha cos\alpha} \right) \left( \frac{ \bar v_{z_{Plate}}}{\bar v_c} -1 \right)\) |
|
Floc filter head loss |
\(h_L = H_{ff} \left( \frac{\rho_{clay}}{\rho_{H_2O}} - 1 \right) \frac{C_{clay}}{\rho_{clay}}\) |
|
Clean bed head loss |
\(\frac{h_l}{H_{FiSand}} = 36 k \frac{\left( 1 - \phi_{FiSand} \right)^2}{\phi_{FiSand}^3} \frac{\nu \bar v_a}{g D_{60}^2}\) |
|
Backwash head loss |
\(h_{l_{FiBw}} = H_{FiSand} \left( 1 - \phi_{FiSand} \right) \left( \frac{\rho_{Sand}}{\rho_{Water}} - 1 \right)\) |
|
Fluidization velocity |
\(\bar v_{MinFluidization} = \frac{\phi_{FiSand}^3 g D_{60}^2}{36 k \nu \left( 1 - \phi_{FiSand} \right)} \left( \frac{\rho_{Sand}}{\rho_{Water}} - 1 \right)\) |
|
Sharp crested weir |
\(Q = \Pi_{vc}\frac{2}{3} \sqrt{2g} w \left(H_{channel}\right)^\frac{3}{2}\) |