The pressure drop in pipe flow due to fluid friction with pipe walls depends on mass flux density and friction factor distribution along the pipe.
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\left( \frac{dp}{dl} \right)_f = - \frac{ j_m^2}{2 d} \cdot \frac{f(l)}{\rho(l)} |
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where
| pipe length |
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| body | j_m = \dot m \, / A = \rho_0 \, q_0 \, / A |
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| mass flux |
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| body | \dot m (l) = \dot m = \rm const |
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| mass flowrate |
| intake flowrate |
| intake fluid density |
| pipe diameter |
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| body | --uriencoded--A = 0.25 \, \pi \, d%5e2 |
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| pipe cross-section area |
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| body | --uriencoded--f= f(%7B\rm Re%7D, \epsilon) |
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| Darcy friction factor |
| inner pipe walls roughness |
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| body | --uriencoded--\displaystyle %7B\rm Re%7D = \frac%7Bj_m \, d%7D%7B\mu%7D |
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| Reynolds number |
| dynamic viscosity as function of fluid temperature and pressure |
The accurate calculations require solving of a self-consistent equation of Pressure Profile in Homogeneous Quasi-Isothermal Steady-State Pipe Flow @model.
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| \Delta p(L)=- \frac{j_m^2}{\rho_0} \cdot \frac{f_0 \, L}{2 \, d } = - \frac{8}{\pi^2} \frac{\rho_0 \, q_0^2}{d^5} \, f_0 \, L |
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| body | f(l)= f_0 = \rm const |
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| body | \rho(l)=\rho_0= \rm const |
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| Incompressible fluid |
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| \Delta p (L) =- \frac{\rho_0}{c^*} \cdot \left[
1 - \sqrt{ 1 - j_m^2 \cdot \frac{c^* \rho^*}{\rho_0^2}
\cdot \frac{f_0 L}{d}}
\right] |
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| body | f(l)= f_0 = \rm const |
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| body | --uriencoded--\rho(l)=\rho%5e* \cdot (1 + c%5e* \, p) |
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| body | --uriencoded--c%5e* \, p \ll 1 |
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| Slightly compressible fluid |
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| \Delta p (L) =- p_0 \cdot \left[ 1- \sqrt{
1 - \frac{j_m^2}{\rho_0 \, p_0} \cdot \frac{f_o0 L}{d}
} \right] |
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| body | f(l)= f_0 = \rm const |
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| body | --uriencoded--\displaystyle \rho(l)= \frac%7B\rho_0%7D%7Bp_0%7D \cdot p |
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| Ideal gas |
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| \Delta p (L) =- \frac{j_m^2}{\rho_0} \cdot \frac{f_0}{2 \, d} \cdot
\frac{ 1- \exp \left( - c^* \rho^* G \, L \right)}{c^* \rho^* G} |
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| body | f(l)= f_0 = \rm const |
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| body | --uriencoded--\rho(l)=\rho_0 \cdot \exp (c%5e* \rho%5e* G \, l) |
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| Gravity dominated fluid density distribution |
See also
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[ Darcy friction factor ] [ Darcy friction factor @model ] [ Reynolds number in Pipe Flow ]
[ Fluid friction with pipeline walls ][ Darcy friction factor in water producing/injecting wells @model ]
