Engineering task and calculation objective
The L1.3 module calculates the pressure drop of fittings and components in piping according to the VDI Heat Atlas (VDI-Wärmeatlas), 12th edition 2019 — the standard German reference for heat transfer and fluid flow: cross-section contractions and expansions, standard orifice plates, standard nozzles and standard Venturi nozzles, right-angled and oblique tees, changes of direction (elbows, bends) as well as valves and gate valves. Such individual resistances are captured via dimensionless resistance coefficients ζ, which relate the pressure drop to the dynamic pressure of a reference velocity.
In real piping networks, the individual resistances frequently dominate the total pressure drop — especially in short lines with many valves and changes of direction. Anyone who wants to calculate the pressure drop of a pipeline must therefore, in addition to pipe friction (module L1.2), account for every elbow, every tee and every valve with its resistance coefficient. For standard orifices and nozzles there is a second application: they serve as standardized differential-pressure devices for flow measurement, where the permanent pressure loss must be distinguished from the differential pressure.
The resistance coefficients apply equally to gases and liquids as long as the flow may be treated as incompressible. They depend on the geometry — diameter ratio, edge shape, deflection angle, bend radius — and partly on the Reynolds number; the module evaluates the corresponding relations and characteristic values of the VDI Heat Atlas.
Standard and calculation basis: VDI-Wärmeatlas, 12. Auflage 2019
Calculation scope
- Cross section narrowing and widening
- Standard orifices, standard nozzles and standard venturi nozzles
- Tees
- Bends
- Valves and slides
Calculation workflow
- Select the component: First, the resistance type is defined: gradual or sudden contraction/expansion, standard orifice, standard nozzle or Venturi nozzle, tee (dividing or combining, right-angled or oblique), change of direction, or valve (globe valve, gate valve).
- Define the geometry and reference cross-section: The governing geometric quantities are entered, such as diameters upstream and downstream of the element, area ratio, deflection angle or bend radius. The reference cross-section is important: the resistance coefficient always refers to the dynamic pressure of a defined velocity, usually that in the narrower or in the inlet cross-section.
- Provide the flow data: From volume or mass flow rate, density and viscosity of the fluid, the velocity and Reynolds number in the reference cross-section are formed; for Reynolds-dependent coefficients (e.g. orifices at low Re, valves), the Reynolds number enters the evaluation.
- Evaluate the resistance coefficient: For the selected element, the resistance coefficient is determined according to the relations of the VDI Heat Atlas — for the sudden expansion, for example, from the area ratio (Borda-Carnot), for orifices from the area ratio, for elbows from angle, bend radius and roughness.
- Calculate the pressure drop and sum up: The pressure drop of the element follows from Δp = ζ·ρ·w²/2. For a piping run, the ζ values of all fittings — correctly converted to the same reference cross-section — are added to the pipe friction to give the total pressure drop.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Density of the fluid | ρ | kg/m³ |
| Velocity of the fluid | wi | m/s |
| Widerstandsbeiwert | ζ ' | - |
| Pipe distance | S | m |
| 4) | ζEg | - |
| Pressure drop | Δp | Pa |
| Rohrleit. | ζE | - |
| Pressure drop | Δpv | Pa |
| Pressure drop | Δpv | Pa |
| Einlauf | ζEv | - |
| Smaller diameter | di | m |
| Larger diameter | dk | m |
| Reynolds number | Rei | - |
| Pressure drop | Δps | Pa |
| Einlauf | ζEs | - |
| Faktor | b | - |
| Drag coefficient | ζEg0 | - |
| Velocity (large) | w2 | m/s |
| Pressure drop | Δp | Pa |
| Pressure drop | Δp | Pa |
| Opening angle | α | ° |
| Number of standard part | N 1 / 2 / 3 | - |
| Effective pressure | pW | Pa |
| Diameter of orifice | d | m |
Calculated results
| Quantity | Symbol | Unit |
|---|---|---|
| Friction factor | ζ | - |
| Pressure drop | Δp | Pa |
Calculation options
Type of bend
90° Elbow · Elbow joint · Miter bend · 90° Bend
Type
1 · 2 · 4 · 8
Roughness
rough · smooth
Bauform
Abrupt reduction · Continuous reduction in cross-sectional area · Abrupt widening of cross-section · Continuous widening of cross-section · Orifices · Tee · Bends · Valves and slides
Worked example
In a water line, the cross-section expands suddenly from DN 50 to DN 80 (inside diameters 50 mm and 80 mm). The velocity in the narrow pipe is 2.0 m/s, the water is at 20 °C. In this worked example, find the resistance coefficient and the pressure drop of the sudden expansion according to Borda-Carnot.
Given values
| Narrow diameter, d1 | 50 mm |
| Wide diameter, d2 | 80 mm |
| Velocity in the narrow pipe, w1 | 2.0 m/s |
| Density of water (20 °C), ρ | 998 kg/m³ |
Solution
Area ratio and velocity in the wide pipe
A1/A2 = (d1/d2)2 = (50/80)2 = 0.391
Continuity: w2 = w1 · A1/A2 = 2.0 · 0.391 = 0.781 m/s
Resistance coefficient according to Borda-Carnot
Referred to the dynamic pressure of the approach velocity w1:
ζ = (1 − A1/A2)2 = (1 − 0.391)2 = 0.371
Calculate the pressure drop
Δpv = ζ · ρ/2 · w12 = 0.371 · 499 kg/m³ · 4.0 m²/s² ≈ 741 Pa
Check via the shock-loss form: Δpv = ρ/2 · (w1 − w2)2 = 499 · (2.0 − 0.781)2 ≈ 741 Pa — identical.
The loss corresponds to about 37 % of the approach dynamic pressure; a diffuser with the same cross-sections would retain most of it as pressure recovery.
Result
| Resistance coefficient ζ (referred to w1) | 0.371 |
| Pressure drop Δpv | approx. 741 Pa ≈ 7.4 mbar |
All values are illustrative. The applicable standard and project-specific boundary conditions remain authoritative.
Frequently asked questions
To which velocity does the resistance coefficient ζ refer?
This is the most frequent source of error: ζ is only defined together with its reference cross-section. For contractions and orifices, the reference is usually the velocity in the narrower cross-section; for elbows and valves, the pipe velocity. Anyone who adopts a ζ value from a source must check the reference and, if necessary, convert with the square of the area ratio — otherwise errors by whole factors arise.
Why does a sudden expansion generate more loss than a gradual one?
At the sudden expansion, the flow separates at the edge and the jet mixes dissipatively with the surrounding dead-water region (Carnot shock loss); the kinetic energy of the excess velocity is largely dissipated instead of being recovered as pressure. A diffuser with a small opening angle (about 6 to 8 degrees), by contrast, decelerates the flow without separation and recovers a large part of the dynamic pressure as a pressure rise.
What is the difference between differential pressure and permanent pressure loss at a standard orifice?
The differential pressure is the pressure difference between the standardized tappings immediately upstream and downstream of the orifice — the measured quantity of the flow measurement. Downstream, the flow recovers part of it through deceleration; only the remainder is the permanent pressure loss that loads the pump or fan. At the same differential pressure, Venturi nozzles have a much smaller permanent loss than orifices — which is why they are chosen for energetically critical measuring points.
Can I simply add resistance coefficients when fittings lie close behind one another?
Only with reservations. The tabulated ζ values apply to undisturbed approach and discharge flow. If elbows, tees or valves are spaced closer than about 5 to 10 pipe diameters, their flow fields interact; the total loss can turn out higher or lower than the sum. For design purposes, simple addition is usually adequate; for metering runs, however, the standardized upstream straight lengths are mandatory.