diff --git a/src/polykin/transport/flow.py b/src/polykin/transport/flow.py index 48b2e68..e81bc14 100644 --- a/src/polykin/transport/flow.py +++ b/src/polykin/transport/flow.py @@ -87,7 +87,7 @@ def DP_Darcy_Weisbach(v: float, where $f_D$ is the Darcy friction factor, $v$ is the velocity, $D$ is the pipe diameter, $L$ is the pipe length, and $\rho$ is the fluid density. This equation is valid for both laminar and turbulent flow. In laminar - flow, $f_D=64/\mathrm{Re}$. For turbulent flow, $f_D$ can be estimated + flow, $f_D=64/Re$. For turbulent flow, $f_D$ can be estimated using either Colebrook's or Haaland's equation. Parameters @@ -151,7 +151,7 @@ def DP_tube(Q: float, This method acts as a convenience wrapper for [`DP_Darcy_Weisbach`](DP_Darcy_Weisbach.md). It determines the flow regime and estimates the Darcy friction factor using the appropriate equation. For - laminar flow, it applies $f_D=64/\mathrm{Re}$. For turbulent flow, it uses + laminar flow, it applies $f_D=64/Re$. For turbulent flow, it uses [`fD_Haaland`](fD_Haaland.md). Finally, the method calls [`DP_Darcy_Weisbach`](DP_Darcy_Weisbach.md) with the correct parameters. @@ -209,9 +209,9 @@ def DP_packed_bed(G: float, density. The packing friction factor $f_p$ is estimated using the Sato and Tallmadge correlation: - $$ f_p = \frac{150}{\mathrm{Re}_p} + \frac{4.2}{\mathrm{Re}_p^{1/6}} $$ + $$ f_p = \frac{150}{Re_p} + \frac{4.2}{Re_p^{1/6}} $$ - where $\mathrm{Re}_p=D_p G/(\mu (1-\epsilon))$ is the packing Reynolds + where $Re_p=D_p G/(\mu (1-\epsilon))$ is the packing Reynolds number. **References** @@ -255,11 +255,11 @@ def DP_packed_bed(G: float, def fD_Colebrook(Re: float, er: float) -> float: r"""Calculate the Darcy friction factor using Colebrook's equation. - For turbulent flow, i.e., $\mathrm{Re} \gtrsim 2500$, the friction factor + For turbulent flow, i.e., $Re \gtrsim 2500$, the friction factor is given by the following implicit expression: $$ \frac{1}{\sqrt{f}}= -2 \log \left( \frac {\epsilon/D} {3.7} + - \frac {2.51} {\mathrm{Re} \sqrt{f}} \right) $$ + \frac {2.51} {Re \sqrt{f}} \right) $$ This equation is a historical landmark but has the disadvantage of being implicit, requiring an iterative solution. @@ -309,11 +309,11 @@ def fnc(f): def fD_Haaland(Re: float, er: float) -> float: r"""Calculate the Darcy friction factor using Haaland's equation. - For turbulent flow, i.e., $\mathrm{Re} \gtrsim 2500$, the friction factor + For turbulent flow, i.e., $Re \gtrsim 2500$, the friction factor is given by the following implicit expression: $$ \frac{1}{\sqrt{f}}= -1.8 \log \left[\left(\frac{\epsilon/D}{3.7}\right)^{1.11} - + \frac{6.9}{\mathrm{Re}} \right] $$ + + \frac{6.9}{Re} \right] $$ This equation is as accurate as Colebrook's but has the advantage of being explicit. @@ -360,10 +360,10 @@ def Cd_sphere(Re: float) -> float: For laminar as well as for turbulent flow, the drag coefficient is given by the following expression: - $$ C_{d} = \frac{24}{\mathrm{Re}} \left(1 + 0.173 \mathrm{Re}^{0.657}\right) - + \frac{0.413}{1 + 16300 \mathrm{Re}^{-1.09}} $$ + $$ C_{d} = \frac{24}{Re} \left(1 + 0.173 Re^{0.657}\right) + + \frac{0.413}{1 + 16300 Re^{-1.09}} $$ - where $\mathrm{Re}$ is the particle Reynolds number. + where $Re$ is the particle Reynolds number. **References** @@ -408,7 +408,7 @@ def terminal_velocity_Stokes(D: float, ) -> float: r"""Calculate the terminal velocity of an isolated sphere using Stokes' law. - In laminar flow ($\mathrm{Re} \lesssim 0.1$), the terminal velocity of an + In laminar flow ($Re \lesssim 0.1$), the terminal velocity of an isolated particle is given by: $$ v_t = \frac{D^2 g (\rho_p - \rho)}{18 \mu} $$