Design of packing cup interference fit value of hypercompressors for low density polyethylene production

Abstract

The hypercompressor is one of the core facilities in low density polyethylene production, with a discharge pressure of approximately 300 MPa. A packing cup is the basic unit of cylinder packing, assembled by the interference fit between an inner cup and an outer cup. Because the shrink-fitting prestresses the packing cup, serious design is needed to gain a favorable stress state, for example, a tri-axial compressive stress state. The traditional method of designing the interference fit value for packing cups depends on the shrink-fit theory for thick-walled cylinder subject to internal and external pressure. According to the traditional method, critical points are at the inner radii of the inner and external cup. In this study, the finite element method (FEM) has been implemented to determine a more accurate stress level of packing cups. Different critical points have been found at the edge of lapped sealing surfaces between two adjacent packing cups. The maximum Von Mises equivalent stress in a packing cup increases after a decline with the rise of the interference fit value. The maximum equivalent stress initially occurs at the bore of the inner cup, then at the edge of lapped mating surfaces, and finally at the bore of the outer cup, as the interference radius increases. The traditional method neglects the influence of axial preloading on the interference mating pressure. As a result, it predicts a lower equivalent stress at the bore of the external cup. A higher interference fit value accepted by the traditional method may not be feasible as it might already make packing cups yield at the edge of mating surfaces or the bore of the external cup. Along with fatigue analysis, the feasible range of interference fit value has been modified by utilizing FEM. The modified range tends to be narrower and safer than the one derived from the traditional method, after getting rid of shrink-fit values that could result in yielding in a real packing cup.

Keywords

interference fit value / packing cup / hypercompressor / finite element method (FEM)

Introduction

The reciprocating compressor used in low density polyethylene (LDPE) production is also referred to as hypercompressor, with its discharge pressure reaching over 300 MPa. The cylinders of hypercompressors bear the fluctuation of gas pressure varying by over 100 MPa between suction and discharge. Thus the cylinders have to withstand the high static pressure as well as high-pressure fluctuations under the mentioned working conditions.

The cylinder packing of the hypercompressor is a compound pressure vessel but has no design codes to refer to [¹]. The design of packing cups depends on both the experience of manufacturers and the use of the latest technologies to ensure the absolute safety and reliability of the machine [¹–³]. Shrink-fitting precompresses the packing cups so that they remain in compression during the service cycle [¹,²], and work favorably from the perspective of fatigue analysis [³]. In addition, shrinkage contributes to a more uniform stress distribution throughout the cylinder wall [³]. The interference fit value (IFV) or interference radius is the major parameter that has to be fully considered in designing.

According to Giacomelli et al., the complex geometry of a packing cup should first be simplified. Then, the traditional method based on the theory for heavy-walled cylinder subject to inner and outer pressure should be used to assess the stress level. The aim is to shrink-fit the two parts of the packing cups at a suitable interference radius to obtain compressive circumferential stress at the bore of the inner cup [¹–³].

To design the interference radius as to achieve the same fatigue life for the inner cup and the outer cup is another consideration in designing [⁴]. Despite the fact the traditional method promotes the estimating stress level of packing cups, it is reported that advanced analysis technology, eg. finite element method (FEM), can complement the design. FEM accounts for complex geometries and gives a more complete evaluation [⁵]. Miller and Blanding have analyzed the working data of hypercompressoers and confirmed that unusual events need to be further studied to better understand the complex cylinder design [⁶]. Refurbishment where the modern laser technique is applied on all components ensures the proper quality of cylinder stack, significantly reduces down time [⁷]. Though efforts have been made to complement design of packing cups, no systematic approach is available to design IFV with FEM.

The scope of this study is to determine the stress level of packing cups under service conditions by using FEM and to discuss principles of IFV design. As the traditional method predicts, the critical points appear at the inner radius of the inner and the outer cup, while FEM shows that the critical points are at the edge of lapped mating surfaces. The difference results from the fact that the traditional method can only be used for a simplified packing cup without complicated geometry like lapped sealing surfaces. Fatigue analysis has also been applied to make sure whether the interference fit does any good to reduce the fatigue load of packing cups. As a result, a modified feasible range of IFV has been obtained by using FEM and fatigue analysis. The adjusted range of IFV rips off dangerous values that can lead to yielding at the edge of lapped sealing surfaces or the bore of the outer cup, while these dangerous values may be accepted by the traditional method.

Design requirements of interference fit

Traditional method and proposed method

The traditional method for designing a correct interference fit value is built on the theory for thick-walled cylinders bearing inner and outer pressure, as well as the fatigue analysis done by Smith and Goodman diagram [²]. Given service conditions and the geometry of a packing cup, the interference fit value directly influences the mating pressure P_m, the circumferential stress {\sigma }_{\theta }, and the equivalent stress {\sigma }_{\mathrm{eq}}. Goodman and Smith diagram give the results of equivalent alternating stress or equivalent stress amplitude S_a [¹,²].

Circumferential stress {\sigma }_{\theta }: a high tension circumferential stress may result in the development of cracks at the bore [¹]. The compressive hoop stress at the bore of the inner cup is preferred, since tensile stress makes fatigue failures become easier [³].

Von Mises equivalent stress {\sigma }_{\mathrm{eq}}: the equivalent stress at the inner radius of the outer cup must be lower than yield strength under service conditions [¹]. The geometry of a packing cup is shown in Fig. 1.

As for the proposed method, since complex geometry is taken into consideration with the aid of FEM, hoop stress {\sigma }_{\theta } and equivalent stress {\sigma }_{\mathrm{eq}} can be calculated at places where the traditional method cannot tackle. Critical points with intensified stress other than the bores of cups should also be involved in assessing of stress level. In the proposed method, concerning parameters are listed below.

Circumferential stress {\sigma }_{\theta }: the circumferential stress across the inner cup should be compressive stress, not just at the bore of the inner cup.

Von Mises equivalent stress {\sigma }_{\mathrm{eq}}: Von Mises equivalent stress should be less than yield strength throughout the packing cup, not just at the bore of the outer cup.

Equivalent alternating stress S_a: the equivalent alternating stress at critical positions of a packing cup should be lower than the endurance limit. The interference radius that leads to a lower equivalent alternating stress is preferred in the design process.

The deviation between the traditional method and the proposed method is listed in Table 1.

Calculation of {\sigma }_{\mathrm{eq}},{\sigma }_{\theta } and S_a

The calculation of Von Mises equivalent stress and circumferential stress is based on Lame formula derived from the analysis of a cylinder subject to internal and external pressure. Figure 2 shows a simplified packing cup.

The inner cup bears the gas pressure P_i, mating pressure P_m and axial preloading force P_z. According to Hanlon, we haveP_z=\mathrm{1.2}{P}_i [³]. Without axial contact with other cups, the outer cup is merely subject to the mating pressure P_m. The outer pressure P_o (less than 1 MPa) is much smaller than the gas pressure P_i (over 100 MPa) and can be omitted during the analysis.

The stresses at the bores of both cups can be derived by Lame formula [³,⁸].

(1)\{\begin{array}{l}{\sigma }_{ri}=-P_i,\\ {\sigma }_{\theta i}=P_i{\displaystyle \frac{K_i^2+\mathrm{1}}{K_i^2-\mathrm{1}}}-P_m{\displaystyle \frac{\mathrm{2}{K}_i^2}{K_i^2-\mathrm{1}}},\\ {\sigma }_{zi}=P_i{\displaystyle \frac{\mathrm{1}}{K_i^2-\mathrm{1}}}-P_m{\displaystyle \frac{K_i^2}{K_i^2-\mathrm{1}}}-P_z,\end{array}\hspace{1em}\{\begin{array}{l}{\sigma }_{ro}=-P_m,\\ {\sigma }_{\theta o}=P_m{\displaystyle \frac{K_o^{\mathrm{2}}+\mathrm{1}}{K_o^{\mathrm{2}}-\mathrm{1}}},\\ {\sigma }_{zo}=P_m{\displaystyle \frac{\mathrm{1}}{K_o^{\mathrm{2}}-\mathrm{1}}},\end{array}

where K_i=R_m/R_i, K_o=R_o/R_m. The equivalent stress at the inner radii of both cups can be derived combining the tri-axial stresses.

(2){\sigma }_{\mathrm{eq}}={\displaystyle \frac{\mathrm{1}}{\sqrt{\mathrm{2}}}}\sqrt{{({\sigma }_r-{\sigma }_{\theta })}^{\mathrm{2}}+{({\sigma }_{\theta }-{\sigma }_z)}^{\mathrm{2}}+{({\sigma }_z-{\sigma }_r)}^{\mathrm{2}}}.

Mating pressure P_m can be represented as [²,⁹]

(3)P_m={\displaystyle \frac{(K_o^{\mathrm{2}}-\mathrm{1})(K_i^2-\mathrm{1})}{\mathrm{2}(K_o^{\mathrm{2}}{K}_i^2-\mathrm{1})}}\left({\displaystyle \frac{E\delta }{R_m}}+{\displaystyle \frac{\mathrm{2}{P}_i}{K_i^2-\mathrm{1}}}\right).

Substituting P_m into Eqs. (1) and (2), the equivalent stress at the bores of the inner and the outer cup is

(4)\{\begin{array}{l}{\sigma }_{\mathrm{eq,i}}=\sqrt{{\displaystyle \frac{K_i^4(\mathrm{3}{P}_m^{\mathrm{2}}-\mathrm{6}{P}_i{P}_m+\mathrm{3}{P}_i^2)}{{(K_i^2-\mathrm{1})}^{\mathrm{2}}}}+P_z^{\mathrm{2}}},\\ {\sigma }_{\mathrm{eq,o}}={\displaystyle \frac{\sqrt{\mathrm{3}}{K}_o^{\mathrm{2}}}{K_o^{\mathrm{2}}-\mathrm{1}}}{P}_m.\end{array}

The hoop stress at the inner radius of the inner cup is given by Eq. (1), after substitutingP_m.

(5){\sigma }_{\theta }=-{\displaystyle \frac{(K_o^{\mathrm{2}}-\mathrm{1})K_i^2}{(K_o^2{K}_i^2-\mathrm{1})}}{\displaystyle \frac{E\delta }{R_m}}+{\displaystyle \frac{K_o^{\mathrm{2}}{K}_i^2+\mathrm{1}}{K_o^{\mathrm{2}}{K}_i^2-\mathrm{1}}}{P}_i.

The calculation of equivalent alternating stress does not confine to the simplified model and utilizes three principal stresses {\sigma }_k (k=\mathrm{1},\mathrm{2},\mathrm{3}) [¹⁰]. The equivalent mean stress considering the tri-axial stress is given by

(6){\sigma }_{\mathrm{mean}}={\sigma }_{\mathrm{m1}}+{\sigma }_{\mathrm{m2}}+{\sigma }_{\mathrm{m3}},

where {\sigma }_{mk}=({\sigma }_{sk}+{\sigma }_{dk})/\mathrm{2}, with {\sigma }_{sk} and {\sigma }_{dk} (k=\mathrm{1},\mathrm{2},\mathrm{3}) respectively standing for the principal stress at suction and discharge stroke. The equivalent stress amplitude is given by

(7){\sigma }_{\mathrm{amp}}={\displaystyle \frac{\mathrm{1}}{\sqrt{\mathrm{2}}}}\sqrt{{({\sigma }_{\mathrm{a1}}-{\sigma }_{\mathrm{a2}})}^{\mathrm{2}}+{({\sigma }_{\mathrm{a2}}-{\sigma }_{\mathrm{a3}})}^{\mathrm{2}}+{({\sigma }_{\mathrm{a3}}-{\sigma }_{\mathrm{a1}})}^{\mathrm{2}}},

where {\sigma }_{ak}=\left|{\sigma }_{sk}-{\sigma }_{dk}\right|/\mathrm{2} (k=\mathrm{1},\mathrm{2},\mathrm{3}). Goodman diagram, the fully reversed equivalent alternating stress is

(8)S_a={\displaystyle \frac{{\sigma }_{\mathrm{amp}}{\sigma }_s}{{\sigma }_s-{\sigma }_{\mathrm{mean}}}}.

The structure withstands the fluctuation of pressure when .

FEM of a packing cup

FEM model of a packing cup

Due to the symmetry of a packing cup, a slice of the packing cup is acceptable in FEM. The FEM model is made up of an angular sector of 15°, as demonstrated in Fig. 3.

The FEM model consists of approximately 80 thousand nodes and the analysis is conducted on the ANSYS 15.0 software. The initial load step is to simulate the shrink-fitting. The surface at the external radius of the inner cup is defined as the “contact surface,” and the surface at the bore of the outer cup is defined as the “target surface.” In connection details, an “offset” is added to represent interference radius.

Following the interference step is the autofrettage simulation. The autofrettage of the lube oil hole is divided into two load steps: the increase of autofrettage pressure from 0 up to 1100 MPa and the release of the pressure from 1100 to 0 MPa [³,¹¹,¹²]. Besides, the bilinear elastic plastic behavior is also assumed for the material. When the Von Mises equivalent stress is below the tensile yield strength, the material remains elastic. When the Von Mises equivalent stress is larger than the tensile yield strength but lower than the ultimate tensile strength, the material is assumed to have an elastic behavior but has a different slope [¹¹].

Static structural analysis

When the cylinder works under the discharge condition, the stress level of a packing cup is more severe. The distribution of the Von Mises equivalent stress and the circumferential stress is depicted in Fig. 4, taking interference radius \delta =0.3 mm as an example.

Figure 4 exhibits that stress intensifies at the edge of the lapped sealing surfaces as well as the sealing ring groove. The maximum equivalent stress might occur at the bore of both cups or the edge of the lapped mating surfaces, as interference fit value increases.

Figure 5 illustrates the relationship between the equivalent stress and the interference fit value. When the interference fit value increases, the equivalent stress at the bore of the inner cup tends to decrease while the one at the bore of the outer cup incline. The equivalent stress at the edge of the lapped mating surfaces increases as the interference fit value rises.

Take {\sigma }_s = 1097 MPa [¹³]. According to Fig. 5, the bore of the outer cup yields at \delta = 0.57 mm, while the bore of the inner cup yields when there is no interference fit, and the edge of the lapped sealing surfaces yields at \delta = 0.64 mm.

The circumferential stress in most part of the inner cup declines as the interference radius increases. Viewed from Fig. 4, the maximum circumferential stress of the inner cup occurs at the bore or the sealing ring groove of the inner cup. Figure 6 marks the variation of hoop stress when the interference radius increases. A larger interference radius is needed to gain compressive circumferential stress at the bore of the lube oil hole compared with that in the sealing ring groove. To achieve a compressive hoop stress for across the inner cup, it is necessary to make the bore of the lube oil hole reach the compression state. At an interference radius of \delta \ge0.33 mm, the inner cup gets compressive hoop stress.

It should be noted that the hoop stress at the bore of the outer cup increases all the way as the interference radius rises. It is beneficial for the inner cup to obtain compressive hoop stress. Consequently, the outer cup has to withstand a large tensile hoop stress, which is in combination with the mating pressure, contributes to a large Von Mises equivalent stress.

Fatigue analysis

The principal stresses are obtained by using FEM. The equivalent alternating stress S_a is calculated by using Eqs. (6)–(8). The low-pressure sealing surface of the inner cup, the side surface of the outer cup, and the lube oil hole are examined in the fatigue analysis. Figure 7 displays the relationship between the interference fit value and the maximum equivalent alternating stress in the chosen regions. The endurance limit of the material is {\sigma }_{-\mathrm{1}} = 503 MPa [³].

In the regions of the lube oil hole, the equivalent alternating stress drops as the interference radius rises at \delta<0.3 mm. As the interference fit value goes beyond 0.3 mm, the equivalent alternating stress at the lube oil hole rises. Compared with no-interference, the equivalent alternating stress is still acceptable at \delta>0.3 mm. Shrink-fitting significantly alleviates the fatigue resistance of the lube oil hole when the interference radius is around 0.3 mm.

However, the shrink-fitting does not benefit significantly the low-pressure sealing surface of the inner cup and the side surface of the outer cup. Both of them bear low alternating stress, suggesting that they are safer from fatigue failure compared with the lube oil hole.

Feasible range for IFV

The FEM result indicate that the bore of the outer cup starts to yield at \delta =0.57 mm, while the traditional method starts to yield at \delta =0.66 mm. Considering the fact that the shrink-fitting is not supposed to bring about yield for the whole packing cup, the interference fit value should not exceed \delta =0.57 mm. As shown in Fig. 8, feasible ranges in the demand of and are respectively marked off by two dashed lines.

The interference fit directly influences the mating pressureP_m. Equation (3) does not consider the effect of the axial preloading P_z on the mating pressure P_m. This causes the difference between the FEM result and Eq. (4).

To obtain the compressive circumferential stress, the traditional method requires an interference radius of \delta \ge0.24 mm, different from the proposed method demanding an interference radius of \delta \ge0.33 mm.

In fatigue analysis, the equivalent alternating stresses S_a for chosen regions in Fig. 7 are well below {\sigma }_{-\mathrm{1}}. The interference fit can greatly reduce the fatigue load in the lube oil hole. On the side surface of the outer cup and the low-pressure sealing surface of the inner cup, the interference fit contribute no significant change to the fatigue load that is already very low. The interference fit value can be close to \delta = 0.3 with respect to obtaining minimum fatigue load of the lube oil hole.

According to the analysis above, the determining IFV should prior satisfy no yielding across the packing cup, Moreover, the interference fit satisfies the compressive-hoop-stress state in the inner cup. Furthermore, the lube oil hole should have a comparative low equivalent alternating stress at .

Conclusions

Subjected to pressure fluctuations from 100 MPa to almost 300 MPa, the cylinder of hypercompressors undergoes the most critical service conditions. Careful design is required for packing cups to ensure high reliability and resistance of static or fatigue failures.

In the traditional method, the maximum Von Mises equivalent stress is recognized to occur at the bores of the inner cup and the outer cup. The interference fit should avoid yielding at these places. This paper, resorted to FEM, has proposed that critical points such as the edge of the lapped mating surfaces should be in concern when design the interference fit value. It turns out in this proposed method that the bore of the outer cup yields easier than the traditional method predicts as the interference radius rises. The interference fit value should not exceed the interference radius that brings about yielding in any part of the packing cup.

Furthermore, to obtain a tri-axial compressive state for the inner cup, the interference fit value should be large enough to turn the hoop stress at the bore of the lube oil hole into compressive stress. The fatigue analysis suggests that the shrink-fitting benefits the packing cup for reducing the equivalent alternating stress in the lube oil hole, which is a critical structure regarding fatigue failure.

In contrast to the traditional method, the modified feasible range for the interference fit value has narrowed down. Large IFV leading to yielding at the bores of the outer cup is unpredictable by the traditional method. Therefore, the proposed range for designing the IFV tends to be safer than the traditional method.

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