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
Stationary flow | Homogenous flow | Isothermal or Quasi-isothermal conditions | Constant cross-section pipe area A along hole |
\theta (l) = \theta = \rm const | f(l) = f = \rm const | \rho = \rho_0 \cdot ( 1 + c^* \cdot p/p_0) |
Equations
Pressure profile along the pipe | ||
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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(T,p) = f({\rm Re}(T,p), \, \epsilon) | Darcy friction factor |
\displaystyle {\rm Re}(T,p) = \frac{u(l) \cdot d}{\nu(l)} = \frac{j_m \cdot d}{\mu(T,p)} | Reynolds number in Pipe Flow |
\mu(T,p) | dynamic viscosity as function of fluid temperature T and pressure p |
\displaystyle d = \sqrt{ \frac{4 A}{\pi}} | characteristic linear dimension of the pipe |
G = g \, \cos \theta | |
F = j_m^2 \cdot f/(2d) |
See also