## Introduction to the stability design of titanium pressure vessels

There are stability problems in the regular design of titanium pressure vessels. We analyzes the main factors that cause the instability of pressure vessels due to internal and external pressure, and puts forward the measures that should be taken to control the instability of titanium pressure vessels.

When the cylinder and head of titanium pressure vessel are subjected to compression load, circumferential instability or axial instability may occur. Circumferential instability can form several axisymmetric waveforms along the circumference of the cylinder or the circumference of the transition zone of the convex head; axial instability can appear along the axial corrugation or wrinkles. There are many factors affecting the stability of the pressure vessel, and its theoretical analysis is relatively complex, in order to facilitate engineering calculations, in the design rules of the pressure vessel, the cylinder introduced a stability factor of safety and the use of graphical algorithms.

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**1. Stability calculation of the external pressure cylinder of titanium pressure vessel**

NaCl titanium pressure vessel design parameters: design pressure Pc = -0.1MPa; design temperature t = 150 ℃; cylinder diameter Di = 2000mm; cylinder length 8600mm; material: TA2; medium: NaCl saturated aqueous solution; corrosion margin: 0mm. pressure vessel sketch is shown in Figure 1.

After the calibration of the external pressure, does not meet the strength requirements, can take two solutions: (1) additional reinforcement ring; (2) increase the cylinder wall thickness. The following are the two programs to calculate and to compare.

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1.1 Program a reinforcement ring

**Calculation of the external pressure of the cylinder**

Figure.1 sketch of pressure vessel

(2) According to the above calculated value, the coefficient A = 0.000413, and then calculated from the coefficient A and the thickness of the external pressure cylinder to obtain B = 28.53 MPa;

(3) Calculate the permissible external pressure:;

(4) Compare Pc and [P], obviously [P] > Pc, the calibration passed.

After adding the reinforcement ring (the equipment is changed from a long cylinder to a short cylinder), the calculated length of external pressure is reduced, and the external pressure calculation meets the requirements.

**Calculation of the moment of inertia of the reinforcement ring**

(1) Select the reinforcement circle of 100 × 8mm flat steel;

(2) strengthening ring and shell combination section moment of inertia Is; Is = I_{1} + A_{S} (45 + 6 – a) ^{2} + I^{2}_{2} + A_{2}a = 1.84 × 10^{6}mm^{4};

(3) Calculate the B value:

We get A=0.0003653;

(4) The required moment of inertia of the combined section of the reinforcement ring and shell I:

(5) Compare I with Is, I < Is, so the choice of 100 × 8 mm flat steel to meet the external pressure cylinder strengthening requirements.

The total weight of the barrel is 2080kg after the reinforcement ring is added.

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1.2 Option 2 increase the cylinder wall thickness

Increase the nominal thickness of the cylinder δn to 17mm (effective thickness δe is 16.1mm):

According to the above calculated values, get A value = 0.0002040, calculate the permissible external pressure:

Compare Pc and [P], then [P] > Pc, meet the strength requirements.

After increasing the wall thickness of the cylinder, the total weight of the cylinder is 4177kg.

The external pressure calculation was carried out separately for the shell according to the two options of adding reinforcement ring and increasing the wall thickness of the shell, and the calculation results of both options can meet the stability requirements. However, the total weight of the shell was 2080kg after the reinforcement ring was added according to scheme 1, and the total weight of the shell was 4177kg after the wall thickness was increased according to scheme 2. The results showed that the amount of material used after the reinforcement ring was added only accounted for 49.7% of the material used to increase the wall thickness of the shell. Obviously, it is more material saving than increasing the cylinder thickness. For TA2, a precious non-ferrous metal, this method is more economical. In addition, the reinforcement ring can reduce the influence of shape defects of large diameter thin-walled pressure vessel and improve the reliability of the structure.

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**2. Convex head stability problem**

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2.1 Elliptical head

For ellipsoidal heads under internal pressure, circumferential compressive stress will be generated at the bottom edge of the head (especially in the transition zone of the head) due to the phenomenon of rounding, which may cause circumferential instability of the head. It is stipulated in the design rules: For head, the minimum effective thickness shall not be less than 0.15%Di; for head, the minimum effective thickness shall not be less than 0.3%Di (K is the shape factor of head).

Under the action of external pressure, ellipsoidal head tends to elongate in the long-axis direction (circumferential elongation) and causes tensile stress in the circumferential direction, so there is no problem of circumferential instability in the transition zone, but in the center part of the head, the short-axis direction tends to flatten and compress, and compressive stress exists in both radial and circumferential directions, so there is a problem of instability in the center part. As mentioned above, the internal pressure instability and external pressure instability of ellipsoidal head are different in that the former is in the transition zone of the head while the latter is in the center of the head. In the calculation of external pressure head, the center part of the head is converted into equivalent external radius and calculated according to the instability of spherical shell, and the equivalent external radius R0=K1D0.

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**3. Conclusion**

In order to effectively prevent the destabilizing deformation of pressure vessel under internal and external pressure, while considering the structural rationality of pressure vessel, the economy of pressure vessel should also be considered, especially the pressure vessel made of precious non-ferrous metals.

For the cylinder under external pressure, the critical pressure of the short cylinder is higher than the long cylinder, so the long cylinder into a short cylinder can improve its critical pressure. The external pressure cylinder is set to strengthen the circle, that is, to change the long cylinder into a short cylinder or shorten the calculated length of the cylinder, in order to improve the stability of the cylinder. By setting the reinforcement circle than directly increase the thickness of the cylinder is more economical.

For the internal pressure of the pressure vessel, the shell as long as there is circumferential film compression stress on the possibility of instability, so in the structural design should try to make the pressure vessel structure simple, to avoid sudden changes in the shape of the structure, to ensure the effective thickness of its shell, it can effectively prevent the pressure vessel instability.

In summary, the pressure vessel only reasonable and targeted design, in order to better ensure its safety, economy and rationality.