Engineering task and calculation objective
The DKEG module evaluates Stodola's steam cone law, the classical relationship between mass flow and pressure drop of a multi-stage turbine blade group. The steam cone law describes how the swallowing capacity of a blading behaves when operation departs from the design point: the mass flow changes approximately with the square root of the difference of the squares of inlet and exit pressure.
In practice, Stodola's law is needed to calculate the part-load and off-design behavior of steam turbines — for example, to determine the new operating point of a blade group after a change in live steam pressure, condenser pressure or an extraction, without having to recalculate the stage in detail. The relationship is also a proven tool for evaluating retrofits and for balancing water-steam cycles.
The implementation is based on the Energietechnische Arbeitsmappe (14th edition, 1995), a well-established German power engineering reference. The module works with the pressures upstream and downstream of the blade group and the mass flows at the design point and at the new operating point, and reports the associated pressure factors.
Standard and calculation basis: Energietechnische Arbeitsmappe 14. Auflage: 1995
Calculation workflow
- Define the design point: For the known reference operation, the pressure upstream of the blade group, the pressure downstream of the blade group and the associated mass flow are entered. This point defines the swallowing capacity of the blading.
- Describe the new operating point: For the off-design condition to be evaluated, the known quantities are specified — depending on the task, these are the new pressures upstream and downstream of the blade group, or the new mass flow.
- Form the pressure factors: From the pressure ratios of both operating points, the pressure factors that enter Stodola's law are formed. They capture the ratio of the differences of the squared pressures between the new and the design condition.
- Solve for the unknown quantity: Stodola's law is solved for the quantity sought: either the new mass flow follows from the new pressures, or, for a given mass flow, the resulting pressure upstream or downstream of the blade group is obtained.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Mass flow | m m* | kg/s |
| Mass flow | m m* | kg/s |
| Pressure at blades entrance | P1 P1* | bar |
| Pressure at blades entrance | P1 P1* | bar |
| Pressure at blades exit | P2 P2* | bar |
| Pressure at blades exit | P2 P2* | bar |
| Factor for pressure | C C* | – |
| Factor for pressure | C C* | – |
| Factor 1 | K1 | – |
| Factor 2 | K2 | – |
Worked example
At the design point, an intermediate-pressure blade group swallows 50 kg/s at a pressure upstream of the group of 100 bar (abs.) and a pressure downstream of the group of 40 bar (abs.). At part load, the pressure upstream of the group drops to 80 bar (abs.); the back pressure remains at 40 bar. Find the new mass flow, assuming an approximately unchanged inlet temperature — a worked example of applying Stodola's steam cone law.
Given values
| Mass flow at design point ṁ₁ | 50 kg/s |
| Pressure upstream of blade group (design) p₁₁ | 100 bar (abs.) |
| Pressure downstream of blade group (design) p₂₁ | 40 bar (abs.) |
| Pressure upstream of blade group (part load) p₁₂ | 80 bar (abs.) |
| Pressure downstream of blade group (part load) p₂₂ | 40 bar (abs.) |
Solution
Form the differences of the squared pressures
Design point: p11² − p21² = 100² − 40² = 10,000 − 1,600 = 8,400 bar²
Part-load point: p12² − p22² = 80² − 40² = 6,400 − 1,600 = 4,800 bar²
Evaluate Stodola's steam cone law
ṁ2 = ṁ1 · √[(p12² − p22²) / (p11² − p21²)]
ṁ2 = 50 kg/s · √(4,800 / 8,400) = 50 kg/s · 0.756 = 37.8 kg/s
Result
| Mass flow ratio ṁ₂/ṁ₁ | 0.756 |
| Mass flow at part-load point | 37.8 kg/s |
All values are illustrative. The applicable standard and project-specific boundary conditions remain authoritative.
Frequently asked questions
Under what conditions does Stodola's law apply?
The relationship applies to multi-stage blade groups with approximately constant flow cross-section and unchanged blading, for moderate departures from the design point. An approximately constant inlet temperature is also assumed; if it changes significantly, an additional temperature correction using the square-root ratio of the inlet temperatures must be applied. For single stages, or when sonic velocity is reached in the narrowest cross-section (critical pressure ratio), the relationship reduces to the simpler linear proportionality between mass flow and inlet pressure.
Why do the pressures enter as squares?
The quadratic relationship follows from integrating the flow relation over the many stages of the group: each stage behaves like a throttle, and stringing together many small expansion steps yields, in total, the ellipse equation of the Stodola cone, in which the difference of the squared pressures upstream and downstream of the group is decisive.
What is a typical source of error when applying the law?
The most common mistakes are using gauge pressures instead of absolute pressures — the law is valid exclusively for absolute pressures — and applying it across extractions: no mass flow may be added or withdrawn between the inlet and exit of the group under consideration. Otherwise, the turbine must be split into several blade groups and each group treated separately.