Motivation
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One of the key challenges in Pipe Flow Dynamics is to predict the pressure distribution along the pipe during the stationary steady-state fluid transport.
In many practical cases the stationary pressure distribution can be approximated by Isothermal or Quasi-isothermal homogenous fluid flow model.
Pipeline Flow Pressure Model is addressing this problem with account of the varying pipeline trajectory, gravity effects and fluid friction with pipeline walls.
Outputs
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Inputs
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sIntake temperature sIntake pressure sIntake Fluid Assumptions
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| Stationary flow | Homogenous Isothermal or conditions| flow |
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| body | --uriencoded--\displaystyle \frac%7B\partial p%7D%7B\partial t%7D = 0 |
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| body | --uriencoded--\displaystyle \frac%7B\partial T%7D%7B\partial t%7D =0 \rightarrow T(t,l) = T(l) |
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| Homogenous flow | |
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| body | --uriencoded--\displaystyle \frac%7B\partial p%7D%7B\partial \tau_x%7D =\frac%7B\partial p%7D%7B\partial \tau_y%7D =0 \rightarrow p(t, \tau_x,\tau_y,l) = p(l) |
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Equations
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| \left( 1 - \frac{crho(p) - \, \rhoj_s^2m^2 \, q_s^2}{A^2}cdot c(p) \right) \cdot \frac{dp}{dl} = \rhorho^2(p) \, g \, \frac{dz}{dl}cos \theta(l) - \frac{\rho_s^2 \, q_s^2 j_m^2 }{2 A^2 d} \frac{f({\rm Re}, \, \epsilon)}{\rho}cdot f(p) |
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| qp(l=0) = \frac{\rho_s \cdot qp_0}{\rho} |
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| u(l) = \frac{\rho_s \cdot q_sj_m}{\rho \cdot A(l)} |
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| pq(l=0) = p_s |
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| q(l=0) = q_s |
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where
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| body | --uriencoded--\displaystyle j_m =\frac%7B \rho_0 \, q_0%7D%7BA%7D= \rm const |
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| mass flux |
| Fluid flowrate at inlet point () |
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s s) = \rho_s...
| Fluid density at inlet point () |
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| body | \rho(l) = \rho(T(l), p(l)) |
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| Fluid density at any point |
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| body | --uriencoded--\displaystyle с(p) = \frac%7B1%7D%7B\rho%7D \left( \frac%7B\partial \rho%7D%7B\partial p%7D \right)_T |
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| Fluid Compressibility |
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| body | --uriencoded--f(T, \rho) = f(%7B\rm Re%7D(T, \rho), \, \epsilon) |
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| Darcy friction factor |
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| body | --uriencoded--\displaystyle %7B\rm Re%7D |
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= \frac%7Bu(l) \cdot d%7D%7B\nu(l)%7D = \frac%7B4 \rho_s q_s%7D%7B\pi d%7D \frac%7B1%7D%7B| (T,\rho) = \frac%7Bj_m \cdot d%7D%7B\mu(T, |
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pcharacteristicCharacteristic linear dimension of the pipe (or exactly a pipe diameter in case of a circular pipe) |
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Alternative forms
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where
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| body | --uriencoded--\displaystyle \left( |
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| frac%7Bdp%7D%7Bdl%7D \right)_G = \rho \cdot g \cdot \cos \theta |
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gravity losses which represent pressure losses for upward flow and pressure gain for downward flow |
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| body | --uriencoded--\displaystyle \left( \frac%7Bdp%7D%7Bdl%7D \right)_K = u%5e2 \cdot \frac%7Bd \rho%7D%7Bdl%7D |
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kinematic losses, which grow contribution at high velocities
and high fluid compressibility (like turbulent gas flow) |
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| body | --uriencoded--\displaystyle \left( \frac%7Bdp%7D%7Bdl%7D \right)_f = - \frac%7B j_m%5e2%7D%7B2 d%7D \cdot \frac%7Bf%7D%7B\rho%7D |
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friction losses which are always negative along the flow direction |
Approximations
See derivation at
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| anchor | drho |
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| page | Derivation of Pressure Profile in Stationary Quasi-Isothermal Homogenous Pipe Flow @model |
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.It does not give much benefit in computations comparing to
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but it makes an easy path to some proxy models (like Slightly Compressible Fluid and Ideal Gas). Approximations
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Pressure Profile in Incompressible Isoviscous Stationary Quasi-Isothermal Pipe Flow @model
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| anchor | PPconst |
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| alignment | left |
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p(l) = p_s + \rho_s \, g \, z(l) - \frac{\rho_s \, q_s^2 }{2 A^2 d} \, f_s \, l |
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\frac{dp}{dl} = \rho_s \, g \cos \theta(l) - \frac{\rho_s \, q_s^2 }{2 A^2 d} \, f_s |
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q(l) =q_s = \rm const |
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u(l) = u_s = \frac{q_s}{A} = \rm const |
where
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| body | --uriencoded--f_s = f(%7B\rm Re%7D_s, \, \epsilon) |
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| Darcy friction factor at intake pointdisplaystyle %7B\rm Re%7D_s frac%7Bul d%7D%7B\nu(l)%7D = \frac%7B4 \rho_s q_s%7D%7B\pi d%7D \frac%7B1%7D%7B\mu_s%7DReynolds number at intake point | ...
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| borderColor | wheat |
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| bgColor | mintcream |
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| borderWidth | 7 |
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| body | --uriencoded--u(l) = u_s = \frac%7Bq_s%7D%7BA%7D = \rm const |
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| body | --uriencoded--f(%7B\rm Re%7D, \, \epsilon) = f_s = \rm const |
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Equation
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becomes:| LaTeX Math Block |
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\frac{dp}{dl} = \rho_s \, g \, \frac{dz}{dl} - \frac{\rho_s \, q_s^2 }{2 A^2 d} f_s |
which leads to
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after substituting | LaTeX Math Inline |
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| body | --uriencoded--\displaystyle \cos \theta(l) = \frac%7Bdz(l)%7D%7Bdl%7D |
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and can be explicitly integrated leading to | LaTeX Math Block Reference |
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.The first term in the right side of
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defines the hydrostatic column of static fluid while the last term defines the friction losses under fluid movement:...
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| body | f(l) = f_s = \rm const |
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See also
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| bgColor | papayawhip |
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| title | ARAX |
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