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 steady-state fluid transport.
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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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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( \rho(p) - j_m^2 \cdot c(p) \right) \cdot \frac{dp}{dl} = \rho^2(p) \, g \, \cos \theta(l) - \frac{ j_m^2 }{2 d} \cdot f({\rm Re}, \, \epsilonp) |
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| p(l=0) = p_0 |
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| u(l) = \frac{j_m}{\rho(l)} |
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| q(l) =A \cdot u(l) |
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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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| body | \rho_0 = \rho(T_0, p_0) |
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| 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(T,\rho) = \frac%7Bj_m \cdot d%7D%7B\mu(T, |
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p | Reynolds number in Pipe Flow |
| dynamic viscosity as function of fluid temperature and density |
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| body | --uriencoded--\displaystyle d = \sqrt%7B \frac%7B4 A%7D%7B\pi%7D%7D= \rm const |
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| Characteristic 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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Pressure Profile in Incompressible Isoviscous Stationary Quasi-Isothermal Pipe Flow @model :
| Pressure profile | Pressure gradient profile | Fluid velocity | Fluid rate |
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PPconstp(l)=p_0 + \rho_0 \, g \, z(l)
-\frac{dp}{dl} = \left( \frac{ |
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j_m^2 f_02, \rho_0 \, d} \cdot l| LaTeX Math Block |
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right)_G + \left( \frac{dp}{dl} |
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=rho0\,g\costhetal)-j_m^2 f_02, \rho_0 \, d}| LaTeX Math Block |
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q(l) =q_0 = \rm const |
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u(l) = u_0 = \frac{q_0}{A} = \rm const |
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where
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| body | --uriencoded--\displaystyle \left( \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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f_0 = f(%7B\rm Re%7D_0, \, \epsilon)Darcy friction factor at inlet point (| \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
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l=0 | and high fluid compressibility (like turbulent gas flow) |
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| body | --uriencoded--\displaystyle |
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%7B\rm Re%7D_0= \frac%7Bj_m \, d%7D%7B\mu_0%7DReynolds number at inlet point ()| \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
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| borderColor | wheat |
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| bgColor | mintcream |
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| borderWidth | 7 |
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Equation
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becomes:| LaTeX Math Block |
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\frac{dp}{dl} = \rho_0 \, g \, \frac{dz}{dl} - \frac{j_m^2 f_0}{2 \, \rho_0 \, d} |
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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| LaTeX Math Inline |
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| body | f(l) = f_0 = \rm const |
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Pressure Profile in GF-Proxy Pipe Flow @model
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See also
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