Specialized in manufacturing compensators, expansion joints, baffle doors

In flue system design, the expansion joint not only needs to absorb thermal displacement, but also produces a significant "pressure thrust" due to internal medium pressure. If this thrust is neglected in the design stage, it may lead to the failure of the fixed bracket, flue deformation and even the instability of the expansion joint itself. Therefore, mastering the calculation formula of flue expansion joint thrust is the key link to ensure the safety of flue structure. This paper will systematically explain the thrust source, calculation formula and engineering application examples of metal expansion joint and non-metal expansion joint.

1. Why do you need to calculate the thrust of the expansion joint

The expansion joint is installed in the flue, and when there is pressure (positive or negative) acting inside, the pressure creates an axial force on the effective area of the bellows or skin. This force will be transmitted to the fixed brackets at both ends and will also act on the expansion joint body.

The core value of the thrust calculation formula of flue expansion joint lies in:

• Determining the structural dimensions and anchoring mode of the fixed bracket
• Check the pressure stability of the expansion joint itself
• Prevent flue interface cracking or expansion joint inversion due to thrust exceeding limit

The consequences of ignoring the thrust calculation are often disastrous: a power plant did not calculate the pressure thrust of the metal expansion joint, which led to the flue fixing bracket at the outlet of the induced draft fan being pushed 30mm away from the foundation, and the flue welds cracked in many places.

2. Thrust calculation of expansion joint of metal bellows

2.1 Sources of Pressure Thrust

Under the action of internal pressure, the effective area of the metal bellows expansion joint will produce an axial expansion force. The magnitude of this force is proportional to the pressure value and the effective area of the bellows.

The basic form of the calculation formula of the thrust of the flue expansion joint (metal bellows) is:

F_p = P × A_eff

Among them:

• F_p — — Pressure thrust, unit: N
• P — — Working pressure in flue (gauge pressure), unit: Pa (Note: the thrust direction is opposite under negative pressure)
• A_eff-Effective area of bellows in m²

2.2 Determination of effective area A_eff

The effective area of the bellows is not equal to the cross-sectional area of the flue because the corrugated structure of the bellows makes its pressure-bearing area between the inner diameter area and the outer diameter area. The following methods are commonly used in engineering to obtain:

Method 1: Check the product sample
The A_eff value is given directly in the technical parameter sheet provided by the manufacturer.

Method 2: Empirical Formula
For standard U-bellows:

A_eff ≈ (π/4) × (D_m) ²

Where D_m is the mean diameter of the bellows = (D_in + D_out) /2, D_in is the inner diameter, and D_out is the peak outer diameter.

Method 3: Inverse calculation by stiffness method
For installed expansion joints, it can be calculated back from the length change under pressure:

A_eff = K × Δ L/P

Where K is the axial stiffness of the bellows (N/mm) and Δ L is the elongation under pressure (mm).

2.3 Corrections in actual operating conditions

Metal bellows expansion joints are usually equipped with tie rods or hinges. The role of the tie rod is to withstand the pressure thrust, thus protecting the bellows. Therefore, the thrust calculation needs to distinguish between two cases:

Key conclusion: For metal expansion joints with tie rods, the pressure thrust is balanced by the tie rods themselves and is not transmitted to the external bracket. However, the design strength of the tie rod must be able to withstand F_p (usually taking 1.5 times the safety factor).

2.4 Calculation Examples

Known:

• Circular flue diameter DN1200mm, metal bellows expansion joint
• Inner diameter D_in =1200mm, crest outer diameter D_out =1320mm
• Operating pressure P = +5000Pa (5kPa positive pressure)
• Try to calculate the pressure and thrust and judge whether the tie rod needs to be installed

Calculation:

1. Average diameter D_m = (1200+1320) /2=1260mm =1.26m
2. Effective area A_eff = π/4× (1.26) ² =1.247 m²
3. Pressure thrust F_p =5000×1.247=6235 N ≈ 636 kgf

Conclusion: If the free expansion joint is used, the fixed bracket at both ends needs to bear a thrust of about 636kgf, which must be included in the design of the bracket. If the type with tie rods is adopted, it can be easily withheld by 4 M16 tie rods (each with a bearing capacity of about 3000kgf).

Thrust calculation of non-metallic fabric expansion joint

A non-metallic expansion joint has a different source of thrust than a metal. Because the fabric skin is so soft that it can barely withstand pressure thrust, the thrust is all borne by the external metal frame and platen.

The calculation formula of flue expansion joint thrust (non-metal) is:

F_p = P × A_duct

That is, the effective area is directly taken as the internal cross-sectional area of the flue (instead of the average area of the bellows).

For rectangular flue:

A_duct = W × H

For circular flues:

A_duct = π/4× D²

3.1 Thrust transmission path of non-metallic expansion joint

The thrust of the non-metallic expansion joint is not borne by the skin, but is transmitted through the following path:

1. The flue gas pressure acts on the end face of the flue
2. The end plate transmits force to the flange connected to the expansion joint
3. The flange is transmitted to the outer metal frame by a platen bolt
4. The frame is then transmit to the flue fixing bracket by a pull rod or bracket

Therefore, for non-metallic expansion joints, installation must ensure that the platen bolts have sufficient strength and pre-tightening force to prevent internal pressure from blowing off the skin.

3.2 Calculation Examples

Known:

• Rectangular flue, width 1500mm, height 1200mm
• Operating pressure P = -8000Pa (8kPa negative pressure, i.e. suction)
• Trial calculation of the thrust to be withstood by the fixed bracket

Calculation:

1. Flue cross-sectional area A_duct =1.5×1.2=1.8 m²
2. Thrust F_p = P × A_duct = (-8000) ×1.8= -14400 N (negative sign indicates directional inward contraction)
3. About 1469 kgf in absolute

Conclusion: The fixed bracket must withstand a tensile force of about 1470kgf (because the negative pressure is inward suction). This value is required for anchor checking during bracket design.

4. Elastic reaction force generated by temperature load

In addition to the pressure thrust, the expansion joint also produces an elastic reaction force when absorbing thermal displacement. This force also needs to be factored into the total load.

Calculation formula of elastic reaction force:

F_e = K × Δ L

Among them:

• K-axial stiffness of the expansion joint (N/mm), supplied by the manufacturer
• Δ L — — Thermal displacement absorbed after actual installation (mm)

For metal bellows expansion joints, the K value is usually 100~500 N/mm; For non-metallic expansion joints, the K value is small (typically

The total thrust (acting on the fixed bracket) is:

F_total = F_p + F_e (When the metal expansion joint has no tie rod)
F_total = F_e (when the metal expansion joint is equipped with a tie rod, the pressure thrust is balanced by the tie rod)
F_total = F_p (non-metallic expansion joint, elastic reaction force can be ignored)

V. Precautions in engineering application

5.1 Thrust direction under negative pressure

When the flue is under negative pressure (such as behind the induced draft fan), the direction of thrust is opposite to the positive pressure, which is the "suction" of inward contraction. At this time, the fixed bracket needs to be subjected to tension instead of pressure. Many engineers only focus on positive pressure thrust and ignore negative pressure suction, resulting in insufficient pull-out ability of the bracket and being pulled out of the foundation.

5.2 Effect of temperature change on thrust

For metal expansion joints, if cold pre-compression/pre-stretching is not performed at the design temperature during installation, the actual Δ L will deviate from the design value, resulting in more than expected elastic reaction force F_e. For example, if the thermal elongation is designed to be 40mm, if it is not pre-compressed during installation, the actual Δ L may reach 2 times the design value and F_e may double, which may lead to overload of the fixed bracket.

5.3 Introduction of safety factors

Regardless of pressure thrust or elastic reaction force, when finalizing the bracket load, the safety factor shall be multiplied by:

• Normal operating load: safety factor 1.5
• Extreme working conditions (e.g. start-stop, failure): Safety factor 1.2 (check according to material yield strength)

That is:

F_design ≥ F_total ×1.5

5.4 Coupling Effect of Multiple Expansion Joints

When multiple expansion joints are arranged in series on the same section of flue, the forces on the fixed bracket are not simply superimposed. A pipe flexibility analysis is required because the stiffness of the expansion joints interacts with each other and the displacement distribution may not be consistent with the initial design. At this point it is recommended to use CAESAR II or AutoPIPE software for simulation.

6. Quick table look-up for thrust calculation

To facilitate engineering site estimation, the following table gives the pressure thrust F_p (non-metallic expansion joint) at ±5kPa pressure for common flue sizes:

Instructions for use: The values in the table are approximate values (converted to 9.8 N/kgf). In practical application, for the expansion joint of metal bellows, the effective area A_eff should be calculated instead of the flue cross-sectional area A_duct, and the value will be slightly lower.

Common Mistakes and Avoidance

VIII. Summary

The calculation formula for flue expansion joint thrust varies depending on the type of expansion joint:

• Metal bellows expansion joint: thrust F_p = P × A_eff (A_eff is the effective area of bellows). When there is no pull rod, F_p is carried by the supports at both ends; When there is a tie rod, it is balanced inside the tie rod, and the bracket only bears the elastic reaction force F_e = K × Δ L.
• Expansion joint of non-metallic fabric: Thrust force F_p = P × A_duct (A_duct is the internal cross-sectional area of flue), and elastic reaction force can be ignored. The thrust force is all transmitted from the external metal frame to the fixed bracket.

Correct calculation of expansion joint thrust is an indispensable step in flue structure design. In engineering practice, the process of "first distinguishing the types of expansion joints, then selecting the correct effective area, and finally counting the elastic reaction force and safety factor" should be strictly followed. For complex pipeline systems, it is recommended to use professional stress analysis software for overall calibration. Through scientific calculation and reasonable type selection, serious accidents such as fixed bracket damage, expansion joint inversion and flue cracking can be effectively avoided.

In flue system design, the core function of the expansion joint is to absorb the thermal expansion displacement caused by the temperature change of the pipe. However, many engineers have little understanding of the calculation of "compensation amount" when designing or selecting models, resulting in insufficient compensation capacity or excessive redundancy of the selected expansion joints. How to calculate the compensation amount of flue expansion joint is directly related to the system safety and investment economy. Based on the principle of thermal expansion, this paper will systematically explain the calculation methods of axial, transverse and angular compensation quantities, and demonstrate the complete calculation process with examples.

I. Basic concept of compensation amount

Compensation amount, also known as displacement compensation ability, refers to the amount of thermal displacement of the pipeline that the expansion joint can absorb, which usually includes three directions:

• Axial compensation amount (Δ L): The amount of elongation or compression along the centerline of the flue.
• Lateral compensation amount (Δ y): horizontal displacement in the direction perpendicular to the flue centerline.
• Angular compensation amount (Δθ): the angle at which the flue axis is deflected.

For most straight flues, the core of how to calculate the compensation amount of flue expansion joint is the calculation of axial thermal expansion. The lateral and angular displacements are not negligible in L, Z or π bend sections.

2. Calculation formula and steps of axial compensation amount

2.1 Basic Formula

The axial thermal expansion Δ L (unit: mm) is calculated as follows:

Δ L = α × L × Δ T

Among them:

• α — — Linear expansion coefficient of flue material, mm/ (m·℃)
  Common values: carbon steel α =0.012 mm/ (m·℃); Stainless steel (304/316L) α =0.017 mm/ (m·℃)
• L-the length of the flue between the two fixed brackets (i.e. the distance to be compensated by the expansion joint), unit: m
• Δ T — — Difference between flue operating temperature and installation temperature, unit: ℃
  Δ T = t_work-t_install

2.2 Key points of parameter values

Temperature difference Δ T:

• The operating temperature T_work shall be the highest possible continuous operating temperature of the flue (excluding instantaneous overtemperature).
• The installation temperature T_install is generally taken as the local annual average temperature or the ambient temperature at the time of actual installation on site, usually calculated at 20℃. If it is installed in winter and no preheating measures are taken, the value should be taken according to the actual lower temperature.

Length L:
Not the full length of the entire flue, but the spacing between the two fixed brackets. The expansion joint should be arranged at one end or in the middle.

Safety Factor:
The calculated Δ L is the theoretical thermal elongation. In actual type selection, the rated compensation amount of the expansion joint should meet the following requirements:

Rated compensation amount ≥ Δ L × K
K is the safety factor, which is generally taken as 1.2~1.5. Take the upper limit for the system with large temperature fluctuation and frequent start-and-stop.

3. Example calculation: axial compensation of straight section flue

Known Conditions:

• Carbon steel round flue, spacing between two fixed brackets L =25 m
• Maximum operating smoke temperature T_work =320℃
• Installation temperature T_install =20℃
• Try to calculate the minimum axial compensation amount of the expansion joint.

Calculation steps:

1. Calculate temperature difference : Δ T =320-20=300 °C
2. Check the linear expansion coefficient of carbon steel: α =0.012 mm/ (m·℃)
3. Substitute into the formula: Δ L =0.012×25×300=90 mm
4. Take the safety factor 1.3: Δ L_required =90×1.3=117 mm

Conclusion: The expansion joint with rated axial compensation not less than 117mm should be selected. If a single expansion joint is adopted, the model with compensation amount of 120mm ~130mm can be selected; If the standard product is only up to 100mm, two expansion joints (each compensating approximately 60mm) need to be arranged within a 25m spacing.

4. Calculation of lateral and angular compensation amount

The expansion joint also needs to absorb the lateral displacement (Δ Y) and the angular displacement (Δ θ) when there is an elbow in the flue, the diameter reduction or the equipment interface is not on the same axis.

4.1 Lateral compensation amount

For L-flues (a 90° elbow), thermal expansion causes lateral displacement on the outside of the elbow. The simplified calculation formula is:

Δ Y = Δ L_vertical × (H/L_horizontal) (approximation)

More accurate calculation requires the force analysis of the fixed bracket and the guide bracket. The following are common experience values:

In practical engineering, the transverse part of how to calculate the compensation amount of flue expansion joint is usually aided by stress analysis software (such as CAESAR II), but the following simplified method can be used for small systems:

Δ y = β × L_vertical × α × Δ t

Where β is the conversion factor, usually taken from 0.5 to 0.8 (depending on elbow stiffness and bracket arrangement).

4.2 Angular compensation amount

When the two expansion joints are arranged in Z-type or π-type, the intermediate pipe segment will generate angular displacement. Calculation formula of angular compensation amount Δθ (unit: degrees or radians):

Δθ = arctan (Δ Y/L_arm)

Where L_arm is the length of the arm that produces the lateral displacement. In actual selection, the angular compensation ability of non-metallic expansion joints is usually ±3° ~ ±6°; Product samples should be checked for metal bellows expansion joints.

V. Comparison of compensation amount between non-metal and metal expansion joints

The compensation ability of different types of expansion joints is significantly different:

After understanding how to calculate the compensation amount of flue expansion joint, it is necessary to choose the appropriate type according to the displacement direction. For example, if the calculation result is 120mm in the axial direction and ±15mm in the transverse direction, priority should be given to non-metallic expansion joints or double metal bellows, which cannot be satisfied by single metal bellows.

VI. Common Calculation Errors and Pit Avoidance Guide

6.1 Error 1: Ignore cold pre-compression

Many installers install the expansion joint directly at the free length without cold pre-compression. Correct practice: For high temperature flues, pre-compressed (or pre-stretched) should be installed ) Δ L_PRE = -0.5× Δ L_CALC. For example, if the thermal elongation is calculated to be 120 mm, the cold compression is installed to be 60 mm, and the other half of the elongation space is reserved.

6.2 Error 2: Confusion of adjacent expansion joint spacing

Some designers mistakenly substitute the length of the whole flue into the formula. In practice, each expansion joint is only responsible for compensating for the pipe segment between the fixed brackets at its two ends. If the total length of the flue is 100m, and four fixed brackets are set to be divided into three sections (30+30+40m), it will be calculated as 30m and 40m respectively instead of 100m.

6.3 Error 3: Ignoring the influence of installation temperature difference

A difference of 45°C in Δ T between winter installation (-10°C) and summer installation (35°C) can result in a difference of approximately 54mm in Δ L for 100m flue. If not corrected to the actual installation temperature, the expansion joint may get stuck in the summer or tear in the winter.

6.4 Error 4: Metal expansion joint does not count pressure thrust

The metal bellows expansion joint generates a pressure thrust (F = P × A_eff) under internal pressure, which needs to be withstood by the bracket. If the influence of pressure thrust on the stability of the bellows is neglected when calculating the compensation amount, the bellows may be unstable and inverted. This problem is particularly critical in the negative pressure condition of desulfurization net flue.

VII. Compensation amount calculation list and verification

Once the calculation is complete, it is recommended to use the following checklist to verify itemized:

1. Is there a clear distinction between axial, lateral and angular displacements?
2. Is the value of material linear expansion coefficient α correct (carbon steel/stainless steel)?
3. Is the length L the spacing between the two fixed brackets and not the total flue length?
4. Does the temperature difference Δ T take into account the minimum installation temperature?
5. Has the safety factor been multiplied (1.2~1.5)?
6. Rated compensation of the selected expansion joint ≥ calculated value?
7. Is the amount of cold pre-compression marked on the installation drawing?
8. Has the pressure thrust been checked for metal bellows?

VIII. Summary

The core of how to calculate the compensation amount of flue expansion joint is the axial thermal expansion formula Δ L = α × L × Δ T, and at the same time, whether the lateral or angular displacement needs to be taken into account according to the flue layout form. It is important to note when calculating that the length L is the spacing between fixed brackets rather than the total length; The temperature difference Δ T shall adopt the highest continuous operation temperature minus the lowest installation temperature; The safety factor of 1.2~1.5 should not be omitted. For non-metallic expansion joints, it is recommended that the axial compensation amount of a single piece be controlled within 80mm, and the number of expansion joints should be increased when it exceeds. The influence of pressure and thrust on the stability of metal bellows also needs to be checked additionally.

After mastering the above calculation method, engineers and technicians can quickly complete the preliminary selection of flue expansion joint. For complex arrangements (multiple elbows, variable sections, limited displacement of equipment interface), it is recommended to use professional stress analysis software for accounting. Correct calculation of compensation quantity can not only avoid flue cracking caused by insufficient compensation, but also prevent cost waste caused by excessive selection, which is the cornerstone of reliability design of flue system.

1. Discussion on the Necessity of High Temperature Flue Expansion Joint

In the design of high-temperature flue systems such as industrial furnaces, boilers and roasters, a frequently asked question is: Does the high-temperature flue need to be equipped with expansion joints? The answer: must be set in the vast majority of cases. During the operation of high-temperature flue, as the temperature rises from normal temperature to hundreds or even thousands of degrees Celsius, the flue material will undergo significant thermal expansion. If the flue is a rigid continuous structure and no expansion joint is set to absorb thermal elongation, the huge thermal stress will lead to flue weld cracking, bracket failure, flange leakage and even overall flue instability deformation. However, whether each high-temperature flue must be provided with expansion joints, how many to set them, and what type to adopt need to be comprehensively judged according to the pipe length, working temperature, direction layout and bracket form. This paper will systematically answer this question from the calculation of thermal expansion, the feasibility evaluation of no expansion joint to the selection of setting scheme.

Calculation of thermal expansion: the basis for judging whether an expansion joint is needed

2.1 Basic Formula of Thermal Expansion

Answer high-temperature flue need to set expansion joint, first of all, calculate the thermal expansion of the flue. The calculation formula is as follows:

Δ L = α × L × (T_WORK-T_INST)

Among them:

• Δ L: thermal elongation (mm)
• α: Coefficient of linear expansion of material (/℃)
• L: Calculate the length of the pipe section (m)
• T_work: Operating temperature (℃)
• T_inst: Installation or initial temperature (℃)

2.2 Thermal expansion coefficient of common materials in high temperature flue

2.3 Calculation Example

A section of carbon steel flue with a length of 20m is set, the working temperature is 450 DEG C and the installation temperature is 20 DEG C, then:

Δ L =12×10⁻⁶ ×20000× (450-20) =103.2mm

This means that a 20m-long flue, after warming up to 450°C, elongates by about 103mm – equivalent to a displacement of 10cm. If both ends of the flue are rigid and fixed, this displacement has nowhere to be released, which will inevitably produce huge internal stress. Therefore, in this case, the answer to whether the expansion joint needs to be set in the high-temperature flue is clear: it must be set.

2.4 The critical length of the expansion joint needs to be set

3. Feasibility conditions of not setting expansion joints

Although the vast majority of high-temperature flues need to be provided with expansion joints, they may not be provided under certain conditions.

3.1 Natural Compensation (Flexible Pipe Design)

The flexibility of the pipe itself can absorb part of the thermal displacement when there are sufficiently long straight sections and elbows in the flue stroke. This is the most common alternative to high-temperature flues. Conditions for natural compensation include:

• The flue is arranged in an L-, Z-or U-shape, with elbows providing flexibility
• The length of the pipe between the two fixed points does not exceed the critical value (see table above)
• The pipe wall thickness is thin (≤6mm) and the stiffness is low

3.2 Short-distance straight pipe sections

For straight pipe sections of very short length (e.g. length ≤3-5m from the outlet of the equipment to the first turn), the amount of thermal elongation is very small (usually ≤10mm), which can be absorbed by the clearance of the connecting flange, the flexibility of the equipment interface or the elastic deformation of the pipe. At this time, no expansion joint can be provided.

Typical example: The connecting section from the roaster outlet to the settling chamber is usually only 2-4m long, and the working temperature is 800-900℃. It can adopt thick-walled tube + large flange structure, and the elasticity of flange bolts is used to absorb a small amount of heat displacement, without separate expansion joints.

3.3 Sliding bracket and elastic connection

For longer flues, if expansion joints are not provided, sliding brackets can be arranged throughout the length, allowing the flue to extend freely, with elastic connections at the ends (e.g. packing box seals). This scheme is commonly used in horizontal directly buried thermal pipes, but it is less used in high temperature flues because its sealing reliability is not as good as that of expansion joints.

4. Typical working conditions where expansion joints need to be set

The answer to whether the expansion joint needs to be set in the high-temperature flue under the following working conditions is yes:

4.1 Long-distance straight pipe sections

When the long straight section of the flue exceeds the critical length (see Section 2.4) and goes straight without turning, axial expansion joints must be provided. Common in:

• Connecting flue from boiler outlet to dust collector
• Straight section of annular flue of roaster
• Original flue at inlet of desulfurization tower

4.2 Connections between High Temperature Equipment

When both devices are fixed independently (such as gas turbine exhaust port and waste heat boiler inlet), there is no common basis between the two, and the relative thermal displacement difference is significant, expansion joints must be set. Typical operating conditions:

• Gas turbine boiler inlet flue: the gas turbine exhaust temperature is 500-650℃, the boiler inlet is about 120℃, and the displacement difference is 30-60mm
• Connecting flue between roaster and cooler

4.3 Where the pipeline changes direction

When the flue changes direction at the elbow, thermal expansion causes lateral displacement of the elbow, creating lateral thrust on adjacent equipment. At this time, hinge type or universal expansion joints should be set on both sides of the elbow.

4.4 Where the flue passes through the wall or floor

When a flue crosses a building structure, wall or floor constraints will limit the axial displacement of the flue, and expansion joints must be provided on both sides of the crossing.

5. Risks and consequences of not setting expansion joints

If the operating conditions that should be set are not set, the following problems will arise:

6. Suggestions on the selection of high-temperature flue expansion joint

6.1 Select Material by Temperature

After confirming whether the expansion joint needs to be set in the high-temperature flue and deciding to set it, the material of the bellows should be selected according to the flue temperature:

6.2 Selecting Structure by Displacement Direction

6.3 Special configuration of high temperature expansion joint

For high temperature flues (≥600℃), the expansion joint requires the following special configuration:

• Guide tube: Prevent high-temperature smoke from directly washing the inner wall of the bellows
• Insulation layer: filled with ceramic fibers to reduce the temperature of the outer wall
• Multi-layer bellows: Reduce single-layer stress and disperse heat load
• Air-cooled or water-cooled structure: External cooling is required under extreme high temperature conditions

VII. Example in which the expansion joint is not required

To give a more comprehensive answer to whether an expansion joint needs to be set in a high-temperature flue, the following example does not need to be set:

1. Short-distance connection section: the distance from the outlet of the equipment to the first fixed point is ≤3m, the working temperature is 500℃, the thermal elongation is ≤12mm, and can be absorbed by the elasticity of the pipeline
2. Fully suspended flexible flue: The flue is suspended by a hanger, and the full length can swing and telescope freely
3. Masonry flue with expansion joints: lined with refractory bricks, with expansion gaps reserved in the brick joints, thermal expansion absorbed by the brick joints, and the metal shell separated in sections
4. Small-diameter pipes with bellows compensator as connectors: such as instrument pipes, sampling pipes and other pipes with diameter ≤100mm

VIII. Summary

The core judgment basis of whether the expansion joint needs to be set in the high-temperature flue is whether the thermal expansion amount of the flue exceeds the bearing capacity of itself and the support. When the amount of thermal elongation exceeds 10 mm or the length of the pipe section exceeds a critical value (20-30 m, depending on the temperature), setting the expansion joint is a necessary and economical solution; For the flue with short distance (≤5m), natural compensation elbow or flexible suspension structure, it may not be set.

After deciding to set the expansion joint, the bellows material should be selected according to the working temperature (321/316L is recommended above 450℃, 310S or Inconel is recommended above 600℃), and the expansion joint structure (axial type, large tie rod type or hinge type) should be selected according to the displacement direction. For the high-temperature flue above 600℃, it is necessary to configure a guide tube, heat insulation layer and multi-layer bellows.

It needs to be emphasized that the pipeline stress damage caused by blindly omitting the expansion joint often appears after a period of operation-it may take months or even years for the crack to expand from microscopic to macroscopic leakage, but once it happens, the repair cost far exceeds the investment of the original configuration of the expansion joint. Therefore, in the design, the calculation results of thermal expansion should be used as the basis, and the scientific decision should be made whether the expansion joint should be set in the high-temperature flue, so as to avoid the long-term potential safety hazard due to saving initial investment.

1. Importance of calculation of expansion joint stent

In pipeline systems, expansion joints are used to absorb thermal displacement and reduce stress, but they themselves cannot withstand internal pressure thrust and pipeline weight. If the bracket is set up incorrectly or calculated incorrectly, the expansion joint cannot compensate properly at least, and the pipeline system will be instable, the bracket will be damaged or even the bellows will be torn at worst. Therefore, the calculation of expansion joint support is a key step after pipeline design and expansion joint selection. Many engineers and technicians either ignore the internal pressure thrust or confuse the force difference between the fixed bracket and the guide bracket when calculating, resulting in serious safety hazards in the design documents. The core of stent calculation is to accurately determine the thrust generated by the expansion joint and rationally distribute it to the fixed stent, guide stent and intermediate stent. This paper will systematically expound the calculation methods of different types of brackets from force analysis, calculation formulas to engineering examples.

2. Types and functions of expansion joint brackets

2.1 Fixed bracket

To understand the calculation of expansion joint stent, first of all, the role of fixed stent should be clarified. A fixing bracket is a rigid structure that completely fixes the pipe and does not allow the pipe to displace or rotate in any direction. In pipes fitted with expansion joints, the fixed brackets bear the following loads:

• Internal pressure thrust generated by expansion joint (blind plate force)
• The frictional or elastic force caused by the thermal expansion of a pipe
• Elastic reaction force generated by bellows stiffness
• Dead weight of pipeline and medium weight
• Wind load, snow load (outdoor pipeline) and earthquake load

Fixed brackets are typically provided at both ends of the expansion joint, at the branch pipe, or where the pipe changes direction.

2.2 Guide Bracket

The guide bracket allows the pipe to move freely in the axial direction, but limits lateral and angular displacement. In the calculation of the expansion joint bracket, the calculation of the guide bracket is essentially different from that of the fixed bracket-the guide bracket does not bear internal pressure thrust, but only bears:

• Vertical load caused by pipeline dead weight
• Axial friction generated by thermal expansion of pipe
• Lateral constraints required to prevent instability

The spacing of the guide brackets is directly related to the stability of the pipeline under the action of compressive force, and excessive spacing will lead to buckling of the pipeline.

2.3 Intermediate bracket (ordinary bracket and hanger)

The intermediate support only bears the pipe's dead weight and vertical load, and does not limit the axial thermal displacement of the pipe. The calculation is relatively simple, and the conventional support and hanger design can be carried out according to the pipeline span and weight.

3. Force calculation of fixed bracket

3.1 Internal pressure thrust (blind plate force)

To answer the core question of the calculation of expansion joint bracket, the calculation of internal pressure thrust is the first task in the design of fixed bracket. Its calculation formula is:

F_p = P × A_eff

Among them:

• F_p: internal pressure thrust (N)
• P: working pressure (Pa)
• A_eff: Effective area of expansion joint (m²)

The effective area A_eff is not a simple internal cross-sectional area of the pipe, but the equivalent area of the bellows that generates thrust under pressure. For standard waveforms, the effective area is usually between the inner and outer cross-sectional areas, which can be approximated as follows:

A_eff = (π/4) × D_m²

Where D_m is the average diameter of the bellows (the average value of the crest diameter and the trough diameter).

Example of calculation:
Assuming that the inner diameter of the pipe is DN500, the average diameter of the bellows is D_m =550mm, and the working pressure is P =0.6MPa, then:

A_eff = π/4×0.55² =0.2376 m²
F_p =0.6×10⁶ ×0.2376=142,560 N ≈ 14.5 tonne force

This means that the internal pressure thrust generated by this expansion joint is up to 14.5 tons, which must be borne by the fixed brackets on both sides.

3.2 Elastic reaction force generated by bellows stiffness

When thermal displacement occurs in the pipe, the expansion joint bellows can be compressed or stretched, creating an elastic reaction force:

F_s = K × Δ L

Among them:

• K: axial stiffness of expansion joint (N/mm)
• Δ L: Actual displacement (mm)

3.3 Pipe Friction and Weight

Friction when the pipe moves on the guide bracket:

F_f = μ × W × L

Among them:

• μ: friction coefficient (0.3 for steel-to-steel sliding and 0.1 for rolling bracket)
• W: Weight of pipe and media per unit length (N/m)
• L: length of pipe from expansion joint to fixed bracket (m)

3.4 Total load of fixed bracket

The final result of the calculation of the expansion joint bracket-the total load of the fixed bracket is the vector sum of the respective partial loads. In the axial arrangement, the loads on both sides of the fixed bracket are in opposite directions, and the total load is the difference between the loads on both sides (take the greater value).

When there are pipes on both sides of the expansion joint, the total axial force on the fixed bracket is:

F_total = max (F_left, F_right) -min (F_left, F_right) The absolute value of, plus the algebraic sum of the co-directional loads on both sides.

4. Arrangement and spacing calculation of guide brackets

4.1 Functional positioning of guide bracket

In the calculation system of the expansion joint bracket, the guide bracket does not bear the internal pressure thrust, and its core functions are:

• Ensure that the expansion joint moves along the axis direction and does not bend laterally
• Prevent column instability of bellows under pressure
• Limit the lateral displacement of the pipe to the specified range

4.2 Determination of spacing between guide brackets

The maximum spacing of the guide brackets is related to the pipe diameter and the rigidity of the bellows. Experience Recommended by EJMA Standards:

The distance between the first guide bracket and the expansion joint port shall not exceed 4 times the nominal diameter of the expansion joint and not exceed 4m.

4.3 Forces on guide brackets

The guide bracket mainly bears the vertical load of the pipeline's own weight and the friction force during axial movement, but does not bear the internal pressure thrust. Its vertical load is calculated by weight distribution within the pipe span.

Note: There should be a gap (usually 2-5mm) between the guide surface of the guide bracket and the pipe to avoid sticking and causing the expansion joint to be unable to expand and contract freely.

5. Calculation characteristics of bracket with tie rod expansion joint

5.1 Function of the tie rod

For expansion joints with large tie rods (such as large tie rod transverse type, hinge type, etc.), the tie rod directly bears internal pressure thrust and is not transmitted to the fixed bracket. This is an important change in the calculation of the expansion joint bracket-the fixed bracket only needs to withstand the elastic reaction force and pipe friction generated by the rigidity of the bellows, and the load is significantly reduced.

5.2 Forces on the fixed brackets on both sides of the tie rod expansion joint

When the expansion joint is set with a tie rod:

F_fixed = F_s + F_f

Where F_p has been internally balanced by the tie rod and no longer acts on the pipe support.

This characteristic makes the tie rod expansion joint particularly suitable for retrofit projects where fixed brackets are difficult to set or carrying capacity is insufficient.

VI. Actual engineering calculation steps

6.1 Collection of underlying data

Before performing the expansion joint stent calculation, the following data need to be prepared:

• Pipe diameter, wall thickness, material
• Operating pressure, operating temperature, installation temperature
• Expansion joint model, effective area, axial stiffness
• Pipe layout drawing, pipe rack position
• Pipes and media weight, insulation weight

6.2 Calculate the total displacement of the expansion joint

Δ L_total = α × L × (T_work-T_inst)

Among them:

• α: Linear expansion coefficient (carbon steel about 12×10⁻⁶/℃)
• L: length of pipe between two fixed brackets (m)
• T_work, T_inst: Operating temperature, installation temperature (℃)

6.3 Calculate the total load of fixed bracket

According to the formula in Section 3, calculate the internal pressure thrust, elastic reaction force, and friction force item by item, and then sum them.

6.4 Check the strength of the stent

The calculated load is taken as the design input, and the strength of the steel structure, anchor bolts and foundation of the fixed bracket is checked. The safety factor is generally 1.5-2.0.

Common Calculation Errors and Correction

VIII. Calculation Examples

Working condition: DN600 steam pipeline, P =0.8MPa, T_work =250℃, installation temperature 20℃, distance between two fixed brackets 30m. An axial expansion joint with effective area of 0.32m² and axial stiffness K =2000N/mm was selected. Friction coefficient μ =0.3, unit weight of pipe and medium is 1200N/m.

Thermal displacement: Δ L =12×10⁻⁶ ×30× (250-20) =0.0828m =82.8mm

Internal pressure thrust : F_p =0.8 x 10⁶ x 0.32=256,000 N

Elastic reaction force : F_s =2000×82.8=165,600 N

frictional force : F_f =0.3×1200×30=10,800 N

total load of fixed bracket: F_total = F_p + F_s + F_f =256,000+165,600+10,800=432,400 N ≈ 44 tonnes force

The fixed bracket should be structurally designed according to the axial thrust of 44 tons, and the safety factor should be 1.5 times.

IX. SUMMARY

The calculation of expansion joint support is a fundamental and critical task in pipeline stress analysis. The core can be summarized as follows: "The internal pressure thrust depends on the effective area, the stiffness reaction force depends on the displacement, the fixed bracket carries the sum, and the tie rod structure can be self-balanced".

In the calculation process, the different functions of the fixed bracket and the guide bracket must be clearly distinguished: the fixed bracket bears the sum of internal pressure thrust, elastic reaction force and friction force, among which the internal pressure thrust often accounts for the largest proportion, which can not be ignored by calculating the effective area formula F_p = P × A_eff; The guide bracket only bears the vertical load and friction of self-weight, and does not bear the internal pressure thrust. The spacing of the guide bracket should be controlled within the specification range to prevent pipe buckling. For expansion joints with tie rods, the internal pressure thrust is self-balanced by the tie rods, and the fixed bracket load is significantly reduced.

In the specific operation, the data of pipeline parameters, working conditions and expansion joint characteristics should be collected first, and the thermal displacement, internal pressure thrust, elastic reaction force and friction force should be calculated in turn. Finally, the safety factor of 1.5-2.0 should be summed and considered. Common mistakes include replacing the effective area with the internal cross-sectional area of the pipe, ignoring the internal pressure thrust, and confusing the functional positioning of the fixed bracket with the guide bracket. Through the correct calculation and design of the expansion joint bracket, the expansion joint can play a safe and reliable compensation role in the pipeline system, and avoid pipeline deformation, expansion joint damage and even safety accidents caused by the failure of the bracket.

1. Harm and Urgency of Boiler Flue Expansion Joint Leakage

In power station boiler and industrial boiler system, flue expansion joint takes on the important functions of absorbing heat displacement, absorbing shock and noise and sealing flue gas. However, long-term exposure to high temperature, corrosive media and alternating stress environment, the expansion joint will inevitably have leakage problems. Whether the leakage treatment of boiler flue expansion joint is timely and appropriate is directly related to the operating efficiency of boiler, environmental protection emission index and the safety of on-site personnel. Flue gas leakage not only leads to the increase of energy consumption of induced draft fans and the aggravation of local flue corrosion, but also may lead to fines for exceeding environmental protection standards. More seriously, high-temperature smoke (up to 150-400℃) leaks to the surrounding areas, resulting in scald risks and fire hazards. Therefore, establishing a set of scientific and rapid leakage treatment scheme is the core skill that boiler operation and maintenance personnel must master. This article will systematically introduce the complete treatment process from leak type diagnosis, temporary plugging to permanent repair.

2. Type and cause diagnosis of boiler flue expansion joint leakage

2.1 Classification by leak location

Before carrying out boiler flue expansion joint leak treatment, it is first necessary to accurately locate the leak location. Common leak sites include:

2.2 Classification by leakage mechanism

Leakage caused by different mechanisms, boiler flue expansion joint leakage treatment methods are also completely different:

• Corrosion leakage: The flue gas contains SO₂, SO₃, Cl⁻¹ and other corrosive media, which forms acidic corrosion under the action of condensate, and the wall thickness of bellows gradually decreases to perforation. It is more common in the expansion joint at the inlet or low temperature section of desulfurization system.
• Fatigue leakage: Under the action of repeated thermal displacement, fine cracks appear at the crest or trough of the bellows, which gradually expand into penetrating cracks. It is common in peak shaving units with frequent start-and-stop.
• Overload leakage: The expansion joint is subjected to displacement beyond the design value due to bracket failure or installation pre-displacement error, resulting in local tearing of the bellows.
• Scour leakage: the guide tube falls off or the guide tube is not set, and the dusty flue gas directly washes the inner wall of the bellows, resulting in abrasion and perforation.

2.3 Methods for rapid diagnosis of leaks

The following methods can be used to quickly locate the leakage point on site:

• Visual inspection: Observe soot traces after shutdown. There is usually white or black soot accumulation around the leakage point
• Soap water leak detection: Spray soap water on the suspected leakage part, and continuous air bubbles can be seen under positive pressure
• Smoke generator: The negative pressure flue can release smoke near the expansion joint, and observe whether the smoke is inhaled
• Infrared thermal imaging: the temperature at the leakage is abnormal, and there are obvious hot or cold spots on the thermal image (positive pressure leakage is a hot spot, and negative pressure leakage of cold air is a cold spot)

3. Temporary treatment measures for boiler flue expansion joint leakage

When a leak is found and the boiler cannot be shut down immediately, the following temporary measures can be taken to control the leak and buy time for the planned shutdown.

3.1 Leak plugging under pressure (positive pressure flue)

For the leakage treatment of boiler flue expansion joint in positive pressure flue, pressure plugging is the most commonly used temporary scheme:

• Small hole leakage (≤3mm): Use tapered wooden plug or metal tapered plug to knock into the leakage hole, and apply high temperature sealant (temperature resistance ≥300℃) on the outside
• Crack leakage: Drill crack stop holes (φ 3-5mm) at both ends of the crack, and then tie them with clamps or steel strips, lined with high temperature resistant gaskets
• Flange leakage: Use flange pressure plugging fixture, inject high temperature sealing injection, or add clamp on the outside

It should be noted that pressure plugging is only used as a temporary emergency measure and should not replace permanent repair.

3.2 Outer Wrap Sealing

For large area leaks that cannot be plugged at a single point:

• High temperature resistant stainless steel sheet (thickness 0.5-1.0mm) is cut into wrapping shape
• Lined with ceramic fiber blanket or high temperature sealing gasket
• Tie and fix on the outside of the expansion joint with stainless steel belt or wire rope
• Apply high-temperature sealant to the interface

This method can significantly reduce the amount of leakage without stopping the furnace, and is suitable for short-term transitions before planned maintenance (generally no more than 1 month).

3.3 Temporary treatment of negative pressure flue

For negative pressure flue, the leakage is manifested by external cold air being sucked in. Although there is no risk of smoke leakage, it will affect the accuracy of smoke monitoring data and increase the load of induced draft fan. Provisional processing may be adopted:

• Apply high temperature resistant aluminum foil tape to the outside of the leak
• Apply high temperature repair mud
• In severe cases, it can be covered with a sealing cloth and vacuum attached

IV. Permanent leakage repair after boiler shutdown

4.1 Local repair welding

For corrosion pits or small-scale cracks, repair welding can be used to repair them. Repair welding in boiler flue expansion joint leakage treatment should strictly follow the following process:

• Clean-up: Thoroughly remove oil, rust and soot around the leakage point and polish to the natural color of metal
• Groove preparation: the crack is polished and grooved, the groove is deep to the bottom of the crack, and the angle is 60-70°
• Welding method: Tungsten Arc Welding (TIG), low current, fast welding
• Welding material selection: Match with base metal, stainless steel expansion joint with the same material welding wire; Transition welding material for dissimilar steel parts
• Post-weld treatment: polishing the weld residual height and performing 100% penetration test

IMPORTANT NOTE: The thickness of corrugated pipe substrate is usually only 1-3mm, which is easy to burn through or cause thermal deformation during repair welding. It is recommended to be operated by welders with thin plate welding qualifications, and the length of single continuous welding does not exceed 10mm, segmental jump welding.

4.2 Local replacement of bellows

When the leakage area is large (greater than 50cm²) or multiple leakage points are concentrated, local repair welding is no longer reliable, and local replacement of bellows should be performed:

• Remove the corrugated pipe from the damaged section, leaving both ends intact
• Prepare a new corrugated pipe section (the wave shape and material should be consistent with the original)
• Butt joint welding was used, and the weld seam was 100% radiographic inspection
• The compensation ability and stiffness of the expansion joint should be re-checked after replacement

4.3 Overall replacement of expansion joints

It is recommended to directly replace the expansion joint in its entirety instead of repairing it locally in the following cases:

• Multiple fatigue cracks in the bellows, approaching the design fatigue life (usually 1000 cycles)
• Improper material selection (if the original 304 is used, the actual need is 316L)
• The guide tube falls off and causes extensive wear of the bellows
• Structural defects of expansion joints (such as wave height and wave pitch not meeting the standard)

When replacing the whole, it is necessary to install in place, butt welding and limit adjustment according to the original design requirements.

V. Liner repair and seal strengthening

5.1 Liner Inspection and Repair

In the treatment of boiler flue expansion joint leakage, lining damage is often the cause of leakage. The repair steps are as follows:

1. Check the guide tube for detachment, deformation and wear
2. Check the heat insulation layer (ceramic fiber blanket) for burn and collapse
3. To repair local damage: fill in ceramic fiber cotton and apply high-temperature cement to the surface
4. If the guide tube falls off, it should be re-welded and fixed, and segment welding should be used to prevent deformation during welding

5.2 Flange Seal Replacement

For flanged connection structure leakage:

• Remove all bolts, clean up old gaskets and corrosion on flange surface
• Check the flatness of the flange surface and smooth the local high points with a grinder
• Select a suitable new gasket: Stainless steel clad graphite pad or high strength graphite composite pad is recommended
• Tighten the bolts 2-3 times in cross-symmetric order to achieve the specified torque
• After tightening, check that the flange clearance is uniform and the deviation is ≤0.3mm

Maintenance strategies to prevent leakage

6.1 Regular inspection system

The establishment of a regular inspection system for expansion joints can greatly reduce the risk of sudden leakage:

• Monthly: Visually check for soot traces and discoloration of the insulation layer
• Quarterly: Infrared thermography examination to compare temperature distribution for abnormalities
• Semi-annually: Internal inspection at shutdown to measure bellows wall thickness (ultrasonic thickness measurement)
• Annually: Check the integrity of deflectors, anchors, lining materials

6.2 Leak warning indicators

Incorporate the following indicators into the boiler operation monitoring system for early warning:

• Induced draft fan current is abnormally increased (flue leakage increases air volume demand)
• Abnormal flue oxygen (high oxygen due to negative pressure leakage and inhaled air)
• Sudden change in surface temperature of expansion joint (abnormal temperature near leakage point)

6.3 Life Management Ledger

Establish a full lifecycle profile for each expansion joint:

• Date of commissioning, design fatigue life
• Time and frequency of each start-and-stop
• Record of previous leaks and repairs
• Analysis of the trend of wall thickness thinning

When the cumulative number of cycles reaches 80% of the design life, it is included in the scheduled replacement list.

VII. Common problems and countermeasures

VIII. Summary

Boiler flue expansion joint leak treatment is a systematic work from rapid diagnosis to standard repair. The key to successfully handling leaks can be summarized as follows: "locate first, classify, plug if urgent, fill if slow, and replace if serious". For the leakage occurring in operation, first of all, the leakage location and mechanism must be accurately judged. Positive pressure flue can be temporarily controlled by pressure plugging, but it must not be relied on for a long time; After the planned furnace shutdown, repair welding, local replacement or overall replacement schemes should be selected according to the damage degree of the bellows, and the lining, guide tube and flange seal should be repaired simultaneously. In the long run, it is more economical and effective to establish a regular inspection system, life management ledger and leak warning indicators than passively handling leaks. For the boiler flue with frequent leakage, it is recommended to fundamentally review whether the expansion joint selection is reasonable, whether the bracket is set correctly, and whether the operating conditions are beyond the design range. Through standardized leakage treatment and perfect preventive measures, the risk of unplanned shutdown caused by expansion joint leakage can be minimized, and the safe, environmental protection and economic operation of boiler system can be ensured.