Motivation
Explicit solution of Pressure Profile in Homogeneous Steady-State Pipe Flow @model
Outputs
Inputs
T_0 | Intake temperature | T(l) | Along-pipe temperature profile |
p_0 | Intake pressure | \rho(T, p) | |
q_0 | Intake flowrate | \mu(T, p) | |
z(l) | Pipeline trajectory TVDss | A | Pipe cross-section area |
\theta(l) | Pipeline trajectory inclination, \displaystyle \cos \theta (l) = \frac{dz}{dl} | \epsilon | Inner pipe wall roughness |
Assumptions
Steady-State flow | Quasi-isothermal flow |
\displaystyle \frac{\partial p}{\partial t} = 0 \rightarrow p(t,l) = p(l) | \displaystyle \frac{\partial T}{\partial t} =0 \rightarrow T(t,l) = T(l) |
Homogenous flow | Constant cross-section pipe area A along hole |
\displaystyle \frac{\partial p}{\partial \tau_x} =\frac{\partial p}{\partial \tau_y} =0 \rightarrow p(\tau_x,\tau_y,l) = p(l) | A(l) = A = \rm const |
Constant inclination | Constant friction along hole |
\displaystyle \theta(l) = \theta = {\rm const} \rightarrow \cos \theta = \frac{dz}{dl} = {\rm const} | f(l) = f = \rm const |
Linear density | |
\rho = \rho^* \cdot ( 1 + c^* \cdot p) |
Equations
Pressure profile along the pipe | |||||
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| ||||
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where
\displaystyle j_m = \frac{ \dot m }{ A} | mass flux |
\displaystyle \dot m = \frac{dm }{ dt} | mass flowrate |
\displaystyle q_0 = \frac{dV_0}{dt} = \frac{ \dot m }{ \rho_0} | Intake volumetric flowrate |
\rho_0 = \rho(T_0, p_0) | Intake fluid density |
\Delta z(l) = z(l)-z(0) | elevation drop along pipe trajectory |
f = f({\rm Re}(T,\rho), \, \epsilon) = \rm const | Darcy friction factor |
\displaystyle {\rm Re}(T,\rho) =\frac{j_m \cdot d}{\mu(T,\rho)} | Reynolds number in Pipe Flow |
\mu(T,\rho) | dynamic viscosity as function of fluid temperature T and density \rho |
\displaystyle d = \sqrt{ \frac{4 A}{\pi}} = \rm const | characteristic linear dimension of the pipe |
G = g \, \cos \theta = \rm const | gravity acceleration along pipe |
F = j_m^2 \cdot f/(2d) = F(l) = \rm const |
The equation (3) for horizontal pipelines can be re-written explicitly in terms of pressure:
(5) | \frac{fL}{2d} = (\rho^*/j_m^2) \cdot (p_0-p) \cdot (1+ 0.5 \, c^* \cdot (p+p_0)) - \ln \frac{1 + c^* \cdot p_0}{1 + c^* \cdot p} |
See also
Physics / Fluid Dynamics / Pipe Flow Dynamics / Pipe Flow Simulation / Pressure Profile in Homogeneous Steady-State Pipe Flow @model
[ Pressure Profile in G-Proxy Pipe Flow @model / Pressure Profile in GF-Proxy Pipe Flow @model ]