The purpose is to illustrate that the beta (β) in EN 13480-3 formula 6.4.6-1 and 6.4.7-1 can become negative and leads to invalid results.
Most calculators and software will probably disregard the invalid result, but it could be improved in the EN 13480-3 to prevent misunderstanding.
The proposal is to add after the formulas 6.4.6-1 and 6.4.7-1 the statement:
“or 0.5, which ever is greater.”
Description
The EN 13480-3 clause 6.4 gives the calculation for reducers. Many of the formulas are similar to the pressure vessel code EN 13445-3. The design of a pressure vessel is of different dimension which might be the source of the beta going out of scope.
The formula 6.4.6-1 calculates the beta:
The beta is used to compensate the required wall thickness for the junction between the large end of a cone and a cylinder without a knuckle in formula 6.4.6-2:
The knuckled reducer uses the following formulas
The beta formula is the same as the non-knuckled reducer. The beta is calculated according the following formulas:
The problem is that beta can become negative for certain reducer giving a negative value for the ej wall thickness.
As an example, the DN20 x DN15 S8 and the DN300 x DN250 S8 of EN 12053-2A have the problem of a negative beta value.
Figure 1: EN 10253-2A DN20 x DN15 S8 reducer
Figure 2: EN 10253-2A DN300 x DN250 S8 reducer
The beta calculation for the DN20xDN15 would be, with 1.5 mm corrosion and 12.5% tolerance:
| Before_F 640601 [-1] | divCosAlpha [-] | = 1 / (cos(alpha)^(0.5)) |
| 1.0049 | = 1 / (cos(0.139626)^(0.5)) | |
| Formula 640701 [0] | beta_kna [-] | = 1.0/3.0 * (D_cL / e_ja)^0.5 * tan(alpha) / (1 + divCosAlpha) - 0.15 |
| -0.103909 | = 1.0 / 3.0 * (21.4 / 5.5) ^ 0.5 * tan(0.139626) / (1 + 1.0049) - 0.15 |
The beta calculation for the DN300xDN250 would be, with 1.5 mm corrosion and 12.5% tolerance:
| Before_F 640601 [-1] | divCosAlpha [-] | = 1 / (cos(alpha)^(0.5)) |
| 1.01111 | = 1 / (cos(0.20944)^(0.5)) | |
| Formula 640701 [0] | beta_kna [-] | = 1.0/3.0 * (D_cL / e_ja)^0.5 * tan(alpha) / (1 + divCosAlpha) - 0.15 |
| -0.0319774 | = 1.0 / 3.0 * (297.4 / 26.5) ^ 0.5 * tan(0.20944) / (1 + 1.01111) - 0.15 |
Both beta will result in a negative/invalid wall thickness ej.
It is expected that the beta value would be above 0.5 which could be concluded from figure 6.4.6-1:
The formula 6.4.6-1 and 6.4.7-1 are similar to the straight pipe formula in Clause 6.1 and cone 6.4.4-2. The reducer wall thickness will be checked for these wall thickness also which leads to the conclusion that the factor below 0.5 (or even 1.0) is not realistic.