This information is part of the Modelica Standard Library maintained by the Modelica Association.
One important special case for a pressure loss is the friction at the wall of a pipe under the assumption of quasi steady state flow (i.e., the mass flow rate varies only slowly). In this section it is explained how this case is handled in the Modelica.Fluid library for pipes with nonuniform roughness, including the smooth pipe as a special case (see Pipes.BaseClasses.WallFriction). The treatment is non-standard in order to get a numerically well-posed description.
For pipes with circular cross section the pressure drop is computed as:
dp = λ(Re,D)*(L/D)*ρ*v*|v|/2 = λ(Re,D)*8*L/(π^2*D^5*ρ)*m_flow*|m_flow| = λ2(Re,D)*k2*sign(m_flow); with Re = |v|*D*ρ/μ = |m_flow|*4/(π*D*μ) m_flow = A*v*ρ A = π*(D/2)^2 λ2 = λ*Re^2 k2 = L*μ^2/(2*D^3*ρ)
The first form with λ is used and presented in textbooks, see "blue" curve in the next figure:
This form is not suited for a simulation program since λ = 64/Re if Re < 2000, i.e., a division by zero occurs for zero mass flow rate because Re = 0 in this case. More useful for a simulation model is the friction coefficient λ2 = λ*Re^2, because λ2 = 64*Re if Re < 2000 and therefore no problems for zero mass flow rate occur. The characteristic of λ2 is shown in the next figure and is used in Modelica.Fluid:
The pressure loss characteristic is divided into three regions:
dp = 128*μ*L/(π*D^4*ρ)*m_flow
1/sqrt(λ) = -2*lg( 2.51/(Re*sqrt(λ)) + 0.27*D)gives an implicit relationship between Re and λ. Inserting λ2 = λ*Re^2 allows to solve this equation analytically for Re:
Re = -2*sqrt(λ2)*lg(2.51/sqrt(λ2) + 0.27*D)Finally, the mass flow rate m_flow is computed from Re via m_flow = Re*π*D*μ/4*sign(dp). These are the red curves in the diagrams above.
λ2 = 0.25*(Re/lg(D/3.7 + 5.74/Re^0.9))^2The pressure drop is then computed as dp = k2*λ2*sign(m_flow). These are the blue curves in the diagrams above.
Re1 = 745*e^(if D ≤ 0.0065 then 1 else 0.0065/D)These are the blue curves in the diagrams above.
The absolute roughness d has usually to be estimated. In [Idelchik 1994, pp. 105-109, Table 2-5; Miller 1990, p. 190, Table 8-1] many examples are given. As a short summary:
|Smooth pipes||Drawn brass, copper, aluminium, glass, etc.||d = 0.0025 mm|
|Steel pipes||New smooth pipes||d = 0.025 mm|
|Mortar lined, average finish||d = 0.1 mm|
|Heavy rust||d = 1 mm|
|Concrete pipes||Steel forms, first class workmanship||d = 0.025 mm|
|Steel forms, average workmanship||d = 0.1 mm|
|Block linings||d = 1 mm|
The equations above are valid for incompressible flow. They can also be applied for compressible flow up to about Ma = 0.6 (Ma is the Mach number) with a maximum error in λ of about 3 %. The effect of gas compressibility in a wide region can be taken into account by the following formula derived by Voronin [Voronin 1959; Idelchik 1994, p. 97, sect. 2.1.81]:
λ_comp = λ*(1 + (κ-1)/2 * Ma^2)^(-0.47)
where κ is the isentropic coefficient (for ideal gases, κ is the ratio of specific heat capacities cp/cv). An appreciable decrease in the coefficient "λ_comp" is observed only in a narrow transonic region and also at supersonic flow velocities by about 15% [Idelchik 1994, p. 97, sect. 2.1.81]. This effect is not yet included in Modelica.Fluid. Another restriction is that the pressure drop model is valid only for steady state or slowly changing mass flow rate. For large fluid acceleration, the pressure drop depends additionally on the frequency of the changing mass flow rate.
In the "Advanced menu" it is possible via parameter "from_dp" to define in which form the pressure drop equation is actually evaluated (default is from_dp = true):
from_dp = true: m_flow = f1(dp) = false: dp = f2(m_flow)
"from_dp" can be useful to avoid nonlinear systems of equations in cases where the inverse pressure loss function is needed.
A detailed pressure drop model for pipe wall friction is provided in the form m_flow = f1(dp, D) or dp = f2(m_flow, D). These functions are continuous and differentiable, are provided in an explicit form without solving non-linear equations, and do behave well also at small mass flow rates. This pressure drop model can be used stand-alone in a static momentum balance and in a dynamic momentum balance as the friction pressure drop term. It is valid for incompressible and compressible flow up to a Mach number of 0.6.