In fluid dynamics calculation of flue system, the expansion joint is a local resistance element, and its resistance coefficient is directly related to the selection of induced draft fan, the analysis of system energy consumption and the operation economy. However, many designers lack an accurate understanding of the flue resistance coefficient of expansion joint, and often simply take the empirical value or ignore it directly, resulting in small fan selection or large deviation in energy consumption estimation. This paper will systematically introduce the physical meaning, influencing factors, value method and engineering application of the drag coefficient of expansion joint.
1. Why should we pay attention to the flue resistance of the expansion joint
The total resistance of flue system consists of friction resistance along the way and local resistance. Expansion joints are typical local resistance elements-there are deflectors, bellows peaks and valleys, or concave and convex surfaces of fabric skin inside them. These structures can disturb the normal flow of flue gas, create vortices and velocity redistribution, thus causing pressure loss.
The value of accurately determining the flue resistance coefficient of the expansion joint is reflected in:
- Provide accurate total resistance data for induced draft fan selection to avoid insufficient power or excessive margin
- Calculate the operating power consumption of desulfurization and denitrification system and evaluate the effect of energy-saving transformation
- Determine whether the resistance of the existing flue system is abnormally increased due to the aging of the expansion joint (e.g., the inner layer is detached, and the guide tube is damaged)
- Contrast the economy of different expansion joint structures and optimize the design scheme
2. Definition and formula of drag coefficient
The local drag coefficient ζ (pronounced "Zeta") is a dimensionless number characterizing the pressure loss of the local drag element and is defined as follows:
Δ P = ζ × (ρ × v²/2)
Among them:
- Δ P — — Local pressure loss of expansion joint, unit: Pa
- ζ — — local drag coefficient, dimensionless
- ρ — — Flue gas density, unit: kg/m³ (approximately 1.34 kg/m³ under standard conditions, subject to correction according to actual temperature and pressure)
- v — — Average flow velocity in flue, unit: m/s
Therefore, the calculation formula of flue resistance coefficient of expansion joint can be inversely inferred from the measured pressure difference:
ζ =2 Δ P/ (ρ × v²)
For the design phase, the zeta value in the relevant manual or product sample is usually consulted directly.
3. Range of drag coefficients of different types of expansion joints
3.1 Non-metallic fabric expansion joint
Non-metallic expansion joints typically have a small drag coefficient due to relatively smooth inner walls (fabric surfaces) and proper design of the guide tube.
| Structural characteristics | Drag coefficient ζ range | Description |
|---|---|---|
| No guide tube (not recommended) | 1.0~1.5 | The inner layer of the skin directly faces the wind, which has great resistance and is easily damaged |
| Straight type with guide tube | 0.4~0.8 | The inner diameter of the guide tube is consistent with that of the flue, and the transition corner is rounded |
| With guide tube, bell mouth type | 0.2~0.5 | Gradually expanding entrance, minimum resistance, recommended design |
Engineering recommendation: For standard non-metallic expansion joints (with straight guide tubes), ζ =0.6 may be taken. For extended or large displacement expansion joints, take the upper limit of 0.8.
3.2 Metal bellows expansion joint
Because of the existence of corrugation, the inner wall of the expansion joint of metal bellows is wavy, and the guide tube is usually short, and the resistance coefficient is relatively high.
| Bellows Type | Guide tube configuration | Drag coefficient ζ range |
|---|---|---|
| Single wave U-shaped tube | No guide tube | 1.2~2.0 |
| Single wave U-shaped tube | With guide tube (length ≥1/3 of total length of expansion joint) | 0.6~1.0 |
| Multiple waves (3~5 waves) | With guide tube | 0.8~1.2 |
| Duplex (with intermediate connector) | With double guide tube | 1.0~1.5 |
Typical value: Single metal bellows expansion joint (with guide tube) commonly used in industrial flues, ζ =0.9 can be taken.
3.3 Sleeve Expansion Joint
Sleeve type (packing box type) expansion joint has annular gap and packing cavity inside, and the resistance coefficient is the largest.
| Structure | Drag coefficient ζ |
|---|---|
| Single sleeve | 1.5~2.5 |
| Double sleeve | 2.0~3.5 |
Such expansion joints are rarely used in flues with strict environmental protection requirements because of their poor sealing performance and high resistance.
4. Key factors affecting flue resistance coefficient of expansion joint
4.1 Design and installation of guide tube
The guide tube is the most important factor in determining the flue drag coefficient of the expansion joint.
- With guide tube vs without guide tube: the resistance coefficient can be 3~5 times different. When there is no guide tube, the high-speed flue gas directly impacts the bellows trough or the inner layer of the skin, forming a violent vortex.
- Deflector length: The longer the better. When the length of the guide tube reaches 2/3 of the total length of the expansion joint, the drag coefficient can be reduced by 40%.
- Guide inlet shape: right angle inlet ζ =0.8, rounded or bell mouth ζ =0.4. This alone reduces resistance by 50%.
On-site diagnosis: If it is found that the pressure difference on both sides of the expansion joint far exceeds the design value, first check whether the guide tube has fallen off or corroded and perforated.
4.2 Flue gas flow rate
The drag loss is proportional to the square of the flow velocity, but the expansion joint flue drag coefficient ζ itself varies with the flow velocity in the low Reynolds number (Re) region. For the typical operating condition of the flue (Re> 10⁵), ζ can be considered a constant. But it needs to be fixed in the following cases:
- Flow velocity
- Flow velocity> 25 m/s: Compressibility effects should be considered, but this velocity is rarely achieved by flue systems.
4.3 Length-to-diameter ratio of expansion joint
The greater the ratio of length to inner diameter (L/D), the smaller the drag coefficient. Because that guide tube can better direct the airflow. The following is the measured data of a certain type of non-metallic expansion joint (with guide tube, D =1200mm):
| L/D | ζ |
|---|---|
| 0.5 | 0.85 |
| 0.8 | 0.65 |
| 1.0 | 0.55 |
| 1.2 | 0.48 |
Therefore, when the installation space allows, choosing a slightly longer expansion joint is beneficial to reduce the resistance.
4.4 Flue cross-section shape
- Circular flue: the flow field is symmetrical, the drag coefficient is the smallest, and the reference value is taken.
- Rectangular flue: There is secondary flow at four corners, and the drag coefficient is about 15% ~25% more than that of circular flue with the same equivalent diameter.
- Rectangular flue with fillet: When the fillet radius is ≥0.2 times the side length, it can approach the circular resistance level.
V. Engineering estimation and table lookup method
For items lacking detailed product data, the expansion joint flue resistance coefficient can be estimated in the following simplified table:
| Expansion joint type | Description of operating conditions | Recommended zeta value | remark |
|---|---|---|---|
| Non-metallic, rectangular, with guide tube | Desulfurization absorption tower inlet | 0.6~0.7 | Desirable upper limit for high dust content |
| Non-metallic, round, flared guide tube | SCR denitrification outlet | 0.3~0.5 | Minimum resistance |
| Metal bellows, single type, with guide tube | Boiler horizontal flue | 0.8~1.0 | Ripple wave number ≤4 |
| Metal bellows, duplex, with double guide tubes | Long distance high temperature flue | 1.2~1.5 | Including intermediate tube resistance |
| Metal bellows, no guide tube | Out-of-date design | 1.5~2.0 | Not recommended for new projects |
VI. Calculation Example: Influence of Expansion Joint Resistance on Fan Selection
Known:
- Non-metallic fabric expansion joint, rectangular flue 1500×1200mm
- Average flue gas flow rate v =12 m/s
- Flue gas density (working condition) ρ =0.85 kg/m³ (about 150℃)
- The guide tube is straight tube type, take ζ =0.65
- The system has 4 identical expansion joints
Calculate the resistance of each expansion joint:
Δ P = ζ × (ρ × v²/2) =0.65× (0.85×12²/2) =0.65× (0.85×144/2) =0.65× (122.4/2?) Correction:
ρ × v²/2=0.85×144/2=61.2 Pa
Δ P =0.65×61.2=39.8 Pa
Total resistance of 4 expansion joints : 39.8×4=159.2 Pa
Comparative analysis: If the expansion joint without guide tube (ζ =1.5) is used, the single resistance is 1.5×61.2=91.8 Pa, and the total resistance is 367 Pa, which is about 208 Pa more. For the induced draft fan of 300MW unit, the more power consumption is about 25kW, and the more power consumption is about 200,000 kWh per year. It can be seen that attaching importance to the flue resistance coefficient of expansion joint has significant energy saving value.
7. Application of drag coefficient in fault diagnosis
The measured pressure difference between the two sides of the expansion joint can inversely deduce the actual zeta value in operation. If a large increase in ζ is found, it indicates the following problems:
| Abnormal phenomenon | Probable cause | Treatment method |
|---|---|---|
| ζ Increase 20% ~50% | Part of the guide tube falls off and dust accumulates seriously | Stop the machine to clean up the dust accumulation, check and repair the guide tube |
| ζ Increased> 100% | The guide tube is completely detached; Tear and warping of the inner layer of skin | Stop the machine immediately to replace the expansion joint or guide tube |
| Zeta gradually increased (multi-month cycle) | Hardening of dust accumulation in valley of corrugated pipe | Compressed air purge |
| Zeta fluctuations (load independent) | The guide tube is loosened and vibrates with the airflow | Shutdown reinforcement welding |
On-site rapid measurement method: Open a pressure measuring hole 500mm upstream and downstream of the expansion joint, connect it with a U-shaped tube pressure differential gauge or an electronic micro-pressure gauge, record the pressure differential under different loads, and substitute it into the formula to calculate ζ.
VIII. Summary
The flue resistance coefficient of expansion joint is the core parameter of local resistance calculation, which directly affects the energy consumption of fan and the rationality of system design. The main conclusions are as follows:
- Typical zeta values: 0.2~0.8 for non-metallic expansion joints (the lowest is the guide tube with bell mouth), 0.6~1.2 for metal bellows expansion joints, and the guide tube is the most effective component to reduce resistance.
- Key influencing factors: the length of the guide tube and the shape of the inlet, the length-diameter ratio of the expansion joint, and the shape of the flue section (circular is the best, followed by rectangular).
- Engineering estimation: when no detailed data is available, ζ =0.7 (non-metallic) and ζ =0.9 (metallic) can be used for preliminary calculation; For projects with high energy saving sensitivity, the manufacturer should be asked to provide measured drag coefficients or conduct CFD simulations.
- Operation and maintenance value: Regular measurement of the pressure difference of the expansion joint and reverse calculation of ζ can non-invasively diagnose internal faults such as falling off of the guide tube and dust accumulation.
Instead of considering the expansion joint as a "zero resistance element" during the flue system design phase, its local resistance should be included in the total pressure drop calculation. By optimizing the design of the guide tube and selecting the low resistance structure, the resistance coefficient of the flue of the expansion joint can be reduced by more than 50% without affecting the compensation ability, thus significantly reducing the long-term power consumption of the induced draft fan.