Package Modelica.​Fluid.​Dissipation.​Utilities.​SharedDocumentation.​PressureLoss.​Bend
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Class Modelica.​Fluid.​Dissipation.​Utilities.​SharedDocumentation.​PressureLoss.​Bend.​dp_curvedOverall
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Calculation of pressure loss in curved bends at overall flow regime for incompressible and single-phase fluid flow through circular cross sectional area considering surface roughness.

Restriction

This function shall be used inside of the restricted limits according to the referenced literature.

• circular cross sectional area
• 0.5 ≤ curvature radius / diameter ≤ 3 [Idelchik 2006, p. 357, diag. 6-1]
• length of bend along curved axis / diameter ≥ 10 [Idelchik 2006, p. 357, diag. 6-1]
• angle of curvature smaller than 180° (delta ≤ 180) [Idelchik 2006, p. 357, diag. 6-1]

Geometry

Calculation

The pressure loss dp for curved bends is determined by:

    dp = zeta_TOT * (rho/2) * velocity^2

with

Curved bends with relative curvature radius R_0/d_hyd ≤ 3 according to [Idelchik 2006, p. 357, diag. 6-1]

The pressure loss of curved bends is similar to its calculation in straight pipes. There are three different flow regimes observed (laminar,transition,turbulent). The turbulent regime is further separated into sections with a dependence or independence of the local resistance coefficient (zeta_LOC ) on Reynolds number. The local resistance coefficient (zeta_LOC) of a curved bend is calculated in dependence of the flow regime as follows:

• Laminar regime (Re ≤ Re_lam_leave):

        zeta_LOC = A2/Re + A1*B1*C1

• Transition regime (Re_lam_leave ≤ 4e4)

  This calculation is done using a smoothing function interpolating between the laminar and the first turbulent flow regime.

• Turbulent regime (4e4 ≤ 3e5) with dependence of local resistance coefficient on Reynolds number:

        zeta_LOC = k_Re * (A1*B1*C1)

  where k_Re depends on the relative curvature radius R_0/d_hyd

        k_Re = 1 + 4400/Re              for 0.50 < r/d_hyd < 0.55
        k_Re = 5.45/Re^(0.118)          for 0.55 ≤ r/d_hyd < 0.70
        k_Re = 11.5/Re^(0.19)           for 0.70 ≤ r/d_hyd < 3.00

• Turbulent regime (Re ≥ 3e5) with independence of local resistance coefficient on Reynolds number

        zeta_LOC = A1*B1*C1

with

The pressure loss coefficient zeta_TOT of a curved bend including pressure loss due to friction is determined by its local resistance coefficient zeta_LOC multiplied with a correction factor CF for surface roughness according to [Miller, p. 209, eq. 9.4]:

    zeta_TOT = CF*zeta_LOC 

where the correction factor CF is determined from the Darcy friction factor of a straight pipe having the bend flow path length

    CF = 1 + (lambda_FRI_rough * pi * delta/d_hyd) / zeta_LOC

and the Darcy friction factors lambda_FRI_rough is calculated with an approximated Colebrook-White law according to [Miller, p. 191, eq. 8.4]:

    lambda_FRI_rough = 0.25*(lg(K/(3.7*d_hyd) + 5.74/Re^0.9))^-2

with

The correction for surface roughness through CF is used only in the turbulent regime, where the fluid flow is influenced by surface asperities not covered by a laminar boundary layer. The turbulent regime starts at Re ≥ 4e4 according to [Idelchik 2006, p. 336, sec. 15]. There is no correction due to roughness in the laminar regime up to Re ≤ 6.5e3 according to [Idelchik 2006, p. 336, sec. 15].

Nevertheless the transition point from the laminar to the transition regime is shifted to smaller Reynolds numbers for an increasing absolute roughness. This effect is considered according to [Samoilenko in Idelchik 2006, p. 81, sec. 2-1-21] as:

    Re_lam_leave = 754*exp(if k ≤ 0.007 then 0.0065/0.007 else 0.0065/k)

with

Note that the beginning of the laminar regime cannot be beneath Re ≤ 1e2 .

Verification

The pressure loss coefficient zeta_TOT of a curved bend in dependence of the Reynolds number Re for different relative curvature radii R_0/d_hyd and different angles of turning delta is shown in the figures below.

There are deviations of the pressure loss coefficient zeta_TOT comparing different references. Usually these deviations in the transition regime have to be accepted due to an uncertainty for the determination of comparable boundary conditions in the different references. Nevertheless these calculations cover the usual range of pressure loss coefficients for a curved bend. The pressure loss coefficient zeta_TOT for the same geometry can be adjusted via varying the average height of surface asperities K for calibration.

The pressure loss in dependence of the mass flow rate of water is shown for different relative curvature radii:

The pressure loss in dependence of the mass flow rate of water is shown for different angles of turning:

Note that there is a small deviation between the compressible and incompressible calculation due to the lack of a direct analytical invertibility.

References

Elmqvist,H., M.Otter and S.E. Cellier:

Inline integration: A new mixed symbolic / numeric approach for solving differential-algebraic equation systems.. In Proceedings of European Simulation MultiConference, Praque, 1995.

Idelchik,I.E.:

Handbook of hydraulic resistance. Jaico Publishing House, Mumbai, 3rd edition, 2006.

Miller,D.S.:

Internal flow systems. volume 5th of BHRA Fluid Engineering Series.BHRA Fluid Engineering, 1984.

Samoilenko,L.A.:

Investigation of the hydraulic resistance of pipelines in the zone of transition from laminar into turbulent motion. PhD thesis, Leningrad State University, 1968.

VDI:

VDI - Wärmeatlas: Berechnungsblätter für den Wärmeübergang. Springer Verlag, 9th edition, 2002.

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Class Modelica.​Fluid.​Dissipation.​Utilities.​SharedDocumentation.​PressureLoss.​Bend.​dp_edgedOverall
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Calculation of pressure loss in edged bends with sharp corners at overall flow regime for incompressible and single-phase fluid flow through circular cross sectional area considering surface roughness.

Restriction

This function shall be used inside of the restricted limits according to the referenced literature.

• circular cross sectional area [Idelchik 2006, p. 366, diag. 6-7]
• edged bend with sharp corners at turning [Idelchik 2006, p. 366, diag. 6-7]
• 0° ≤ angle of turning ≤ 180° [Idelchik 2006, p. 338, sec. 19]
• length of edged bend along edged axis / diameter ≥ 10 [Idelchik 2006, p. 366, diag. 6-7]

Geometry

Calculation

The pressure loss dp for edged bends is determined by:

    dp = zeta_TOT * (rho/2) * velocity^2

with

The pressure loss coefficient zeta_TOT of an edged bend can be calculated for different angles of turning delta by:

    zeta_TOT = A * C1 * zeta_LOC * CF_Fri* CF_Re [Idelchik 2006, p. 366, diag. 6-7]  and [Miller 1984, p. 149, sec. 9.4]

with

The correction factor CF_Fri regarding the influence of surface roughness is determined as ratio of the Darcy friction factor for rough surfaces to smooth surfaces according to [Miller, p. 207, eq. 9.3]:

    CF_Fri = lambda_FRI_rough / lambda_FRI_smooth

and the Darcy friction factors lambda_FRI are calculated with an approximated Colebrook-White law according to [Miller, p. 191, eq. 8.4]:

    lambda_FRI = 0.25*(lg(K/(3.7*d_hyd) + 5.74/Re^0.9))^-2

with

Note that the Darcy friction factor for a smooth surface lambda_FRI_smooth is calculated with the previous equation and an absolute roughness of K = 0 .

The correction for surface roughness through CF_Fri is used only in the turbulent regime, where the fluid flow is influenced by surface asperities not covered by a laminar boundary layer. Here the correction according to friction starts at Re ≥ Re_lam_leave according to [Idelchik 2006, p. 336, sec. 15]. Here the end of the laminar regime is restricted to a Reynolds number smaller than 2e3 w.r.t [VDI, p. Lac 6, fig. 16].

Nevertheless the transition point from the laminar to the transition regime is shifted to smaller Reynolds numbers for an increasing absolute roughness. This effect is considered according to [Samoilenko in Idelchik 2006, p. 81, sec. 2-1-21] as:

    Re_lam_leave = 754*exp(if k ≤ 0.007 then 0.0065/0.007 else 0.0065/k)

with

Note that the beginning of the laminar regime cannot be beneath Re ≤ 5e2 .

In addition the influence or decreasing Reynolds numbers Re on the pressure loss coefficient zeta_TOT in the laminar and turbulent regime is considered through a second correction factor CF_Re according to [Miller 1984, p. 149, sec. 9.4] by:

   CF_Re = B/Re^exp for Re ≤ 2e5

with

Note that the coefficient B considers the influence of the angle of turning delta on the pressure loss coefficient zeta_TOT in the laminar regime according to [Idelchik 2006, p. 340, sec. 28].

Note that the correction of the pressure loss coefficient zeta_TOT is influenced by the correction factor CF_Re only for decreasing Reynolds numbers Re out of the turbulent fluid flow regime at Re ≤ 2e5 into transition and laminar fluid flow regime.

Verification

The pressure loss coefficient zeta_TOT of a edged bend in dependence of the Reynolds number Re for different angles of turning delta is shown in the figures below.

Pressure loss calculation of edged bends is complex and there are large differences in literature data. Nevertheless these calculations cover the usual range of pressure loss coefficients for an edged bend.

The validation of the pressure loss coefficient for an edged bends shows four possible flow regimes:

• laminar regime for Re ≤ 5e2
• transition regime for 5e2 ≤ Re ≤ 1e3 ... 1e4
• turbulent regime dependent on Reynolds number for 2e3 ... 1e4 ≤ Re ≤ 1e5
• turbulent regime independent of Reynolds number for Re ≥ 2e5

Incompressible case [Pressure loss = f(m_flow)]:

The pressure loss in dependence of the mass flow rate of water is shown for different angles of turning:

Compressible case [Mass flow rate = f(dp)]:

The mass flow rate in dependence of the pressure loss of water is shown for different angles of turning:

References

Idelchik,I.E.:

Handbook of hydraulic resistance. Jaico Publishing House,Mumbai,3rd edition, 2006.

Miller,D.S.:

Internal flow systems. volume 5th of BHRA Fluid Engineering Series.BHRA Fluid Engineering, 1984.

Samoilenko,L.A.:

Investigation of the hydraulic resistance of pipelines in the zone of transition from laminar into turbulent motion. PhD thesis, Leningrad State University, 1968.

VDI:

VDI - Wärmeatlas: Berechnungsblätter für den Wärmeübergang. Springer Verlag, 9th edition, 2002.

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