Pressure Drop Characteristics of Perforated Pipes ...

15NVC-0224

Pressure Drop Characteristics of Perforated Pipes with particular application to the Concentric Tube Resonator Author, co-author (Do NOT enter this information. It will be pulled from participant tab in MyTechZone) Affiliation (Do NOT enter this information. It will be pulled from participant tab in MyTechZone)

Abstract The bias flow in Concentric Tube Resonator (CTR) is a flow-induced phenomenon in which the pressure gradient along the radial direction is produced by the kinetic energy of the flow. As a result, the flow dynamics in CTR is characterized by bias flow into the annular cavity in the upstream and outflow from the annular cavity in the downstream of the flow. This is due to the change in direction of the radial component of the bias flow at a point called the point of recovery, as a consequence of mass conservation. The pressure drop of CTR is a complex function of the momentum flux and other geometric parameters such as porosity, open area ratio, discharge coefficient of the perforated holes, bias inflow, bias outflow, grazing flow and length. In this study, numerical experiments are conducted to obtain an empirical formula for the friction factor of perforated pipes which are extensively used in automotive mufflers. The finite element based numerical results are validated with test results reported in the literature and are found to be in good agreement.

Introduction The pressure drop, also known as back pressure, associated with flow acoustic elements used for noise control in commercial automotive intake and exhaust systems is an undesirable outcome that needs to be minimized during the design process. In spite of its criticality, the portion of energy that is wasted due to flow restrictions cannot be adequately estimated during the initial phase although the acoustic performance can be satisfactorily evaluated. This is due to the availability of the one dimensional plane wave models such as the transfer matrix method [1] that can be suitably used to predict the acoustic performance up to the plane wave limit, whereas similar models are not available to predict the pressure drop hitherto, despite the fact that such attempts have been made earlier with partial success [2]. Consequently, a full 3-D CFD simulation has to be resorted to, which is tedious and time consuming. The flow mechanism in CTR was first studied by Sullivan [3] by using two pressure probes, one inside the perforated tube and other in the annular cavity. The probes were simultaneously translated along the axial direction and the differential pressure is used to estimate the peak inflow and outflow velocity. The significant outcomes of this experiment are: (1) there is bias inflow (from inner pipe to annular cavity) up to about 60 per cent of the perforated length and bias outflow (from annular cavity to inner pipe) cavity in the remaining 40 per cent. (2) discharge coefficient of the perforated holes varies axially; it is lesser in the inflow zone as compared to the outflow Page 1 of 8

zone due to mass conservation. The back pressure values, however, were not reported.

Figure 1. Illustration of bias flow (bias inflow and bias outflow) by velocity contour plot

The first systematic parametric study of the back pressure of CTR was conducted by Munjal et al. [2], where the open area ratio was used in place of the porosity to express the back pressure. They concluded that, the normalized back pressure is invariant of the mean flow for Mach number ≤ 0.2. The effect of porosity and distribution of holes on the mechanical performance (pressure drop) was investigated by Lee and Ih [4], wherein the focus was on the distribution pattern of the holes. They studied five porosity patterns and observed that the most optimized perforation pattern in terms of Transmission Loss (TL) and back pressure was the one with gradually increasing porosity from the upstream part and gradually decreasing porosity from the middle to the downstream end. Later, Liu and Ji [5] demonstrated that the pressure drop of straight through CTR increases gradually as the porosity increases. However, they did not quantify the dependence of pressure drop on porosity for such acoustic elements. In this work, a parametric study is conducted to quantify the dependence of pressure drop on porosity by introducing friction factor for perforated pipes.

Computational Methods In this study, the CFD module of COMSOL Multiphysics® is used as the computational tool which is based on the finite element method. The geometry is created by using the model builder tool inbuilt in COMSOL as shown in figure 2, wherein a quarter or half geometric model is built depending on whether the number of holes along the circumferential direction is even or odd. This helps to reduce the computation time. Using air as the medium, the flow is considered to

be incompressible and the k-ɛ turbulence model is used for turbulence modeling. The flow in the viscous layer near the wall is approximated by using a wall function formulation. A velocity boundary condition is used at the inlet with turbulent intensity of 0.04 and turbulent length scale of 0.07 times the inner diameter of the perforated pipe. At the outlet, a pressure boundary condition is applied with a normal outflow to atmospheric pressure.

4.

Find the new hole center to center distance along the axial length as C𝑎 ′ =

5. 6.

C2 Cc ′

Select the desired perforated length as 𝑙2 = 2𝑑1 Find the number of holes along the axial length and round it off to the nearest integer Na =

7.

Find the final center to center distance along the axial length as C𝑎 ′′ =

8.

𝑙2

Ca ′

𝑙2 Na

Find the actual porosity from the center to center distances calculated from step 3 and step 7 as 𝜎𝑎 =

πdh 2

4C𝑎 ′′ Cc ′

Mesh independence

Figure 2. COMSOL model of CTR

Steps for controlling porosity and perforated length Due to the inevitable round off errors involved in modeling perforations in pipes during geometry creation, especially while distributing holes along the circumferential direction of the pipe, the actual porosity, 𝜎𝑎 that is achieved deviates from the intended porosity (hereafter nominal porosity, 𝜎𝑛 ) to a significant extent. Since it is not possible to avoid the round off errors, an alternative approach is to use a semi iterative procedure in which the desired perforated length is accurately controlled by adjusting the center to center distance of the perforated holes to obtain the actual porosity. This can be achieved by performing the following steps with reference to figures 3 and 4:

As shown in figure 2, the computational domain is split into several sub-domains to generate mesh individually. A tetrahedral mesh is applied at the perforated holes. For the annular chamber and inner pipe, a physics controlled mesh is applied which is calibrated internally by COMSOL for fluid dynamics applications. Five boundary layers with a stretching factor of 1.2 and thickness adjustment factor of 1.5 is applied to all the walls. A maximum mesh element of 1.5 mm is used in the perforated holes for Mesh 1 and 1 mm for Mesh 2 and Mesh 3. The other mesh parameters are shown in Table 1 below. A nominal porosity of 0.04 and default geometrical parameters given in Table 4 are used for the mesh independence study. Table 1. Mesh size of fluid domains for mesh independence study

Mesh 1 Min size 0.31 mm 1.56 mm

Growth rate

Curvature factor

Resolution

Inner pipe Annular chamber

Max size 2.88 mm 5.22 mm

1.1

0.4

0.9

1.15

0.6

0.7

Inner pipe Annular chamber

2.88 mm 5.22 mm

0.31 mm 1.56 mm

1.1

0.4

0.9

1.15

0.6

0.7

1.1

0.4

0.9

1.13

0.5

0.8

Domain

Mesh 2

Mesh 3 Inner pipe Annular chamber

Figure 3. Definition of porosity and perforated length

1.

Calculate the hole center to center distance for a given nominal porosity as C = Ca = Cc =

2.

2

𝜋

√𝜎

𝑛

Calculated the number of holes along the circumferential direction as Nc =

3.

𝑑ℎ

π(d1 +tw ) C

0.31 mm 0.78 mm

Table 2. Mesh configurations and Normalized pressure drop

Mesh configurations Mesh 1

Normalized pressure drop 0.4694

Mesh 2 Mesh 3

0.4691 0.4692

rounded off to the nearest

integer, tw is the pipe thickness. Calculate the new hole center to center distance along the circumferential direction as Cc ′ =

Page 2 of 8

2.88 mm 4.13 mm

π(d1 +tw ) Nc

Considering the advantage in computation time, Mesh 1 is chosen as the default mesh configuration for subsequent studies.

The results in Table 3 amply validate the COMSOL 3-D predictions on which the following parametric study is based.

Performance parameters The normalized pressure drop introduced by Munjal et al. [2] will be used as the performance parameter for the subsequent analysis. By combining the empirical formula suggested in Ref. [2] with that of the Darcy-Weisbach equation [6], the desired performance parameter can be generalized as follows: The modified Darcy-Weisbach equation for pipe flow in terms of pressure drop is given by ∆𝑝 𝐻

=𝑓

𝑙

(1)

𝑑

where,

Table 3. Pressure drop across CTR Mach number

Experiment (Pa)

Prediction in Ref. [4] (Pa)

Prediction in Ref. [5] (Pa)

Prediction (Pa)

0.085

281.6

250.9

251

232.10

0.17

1074

870.7

878.8

860.23

Table 4. Specific parameters used for validation and default parameters used for parametric study. The underline (____) indicates the default values.

𝑓 is the Moody friction factor

Values

𝑙 is the length of the pipe, 𝑙 = 𝑙2 for the perforated portion

For Validation [4,5]

Default, for parametric study

Total length (l1+l2+l3) mm

400

400

Perforated length (l2) mm Extended inlet and outlet (la = lb) mm Inner diameter of chamber (d2) mm

174 13

2*d1 68

110

110

Inner diameter of perforated pipe (d1,d3) mm Inner pipe wall thickness (tw) mm Perforated hole diameter (dh) mm

32

32

2 4

Nominal Porosity (σn) %

10.3

Air Temperature K

299

1, 2, 3, 4 2, 3, 4, 5, 6 4, 6, 8, 10, 12, 14, 16, 18, 20 299

Inlet flow Mach number

0.085, 0.17 10 x 11.87

0.029

18

5

9

10

Parameters

𝑑 is the inner pipe diameter, 𝑑1 = 𝑑3 = 𝑑 H is the dynamic head denoted by

𝜌𝑣 2 2

The empirical formula suggested in Ref. [2] can be written in terms of the porosity as ∆𝑝 𝐻

= 0.24𝜎

𝑙 𝑑

(2)

By combining equation (1) and (2), we get the friction factor as 𝑓 = 0.24𝜎

(3)

In order to take into account the effect of the inner wall pipe thickness, perforated hole diameter, porosity and perforated length, equation (3) can be generalized as 𝑓 = 𝐹(𝜎, 𝑑ℎ , 𝑡𝑤 )

(4)

where, 𝜎 is the porosity, and

Center to center distance, axial x radial dimensions mm x mm Number of holes in axial direction, Number of holes in circumferential direction

12.8 x 10.68

𝐹 is the function to be obtained here from a parametric study.

Results and Discussion Case Validation Radial bias flow To validate the accuracy of the 3-D simulation, the straight through perforated silencer or CTR used in Refs [4,5] will be used as shown in figure 4. The corresponding dimensions are shown in Table 4.

Figure 4. CTR with two pressure probes at inlet and outlet

Page 3 of 8

The kinetic energy of the grazing flow in the inner pipe induces momentum flux in the radial direction through the openings of the perforated holes. Consequently, the driving pressure (difference between the static pressure of the grazing flow and the stagnation pressure of the gas in the annular cavity) determines the magnitude of bias inflow and bias outflow. The inflexion point of the driving pressure gradient with respect to the axial length is called the point of recovery. As shown in figure 5, it can be seen that the effective bias inflow into the annular cavity of the CTR occurs in about 72 percent of the perforated length and bias outflow from the annular cavity into the perforated pipe in the remaining 28 percent. Moreover, the peak bias outflow velocity is about 7 times greater than the peak bias inflow velocity. This can be attributed to the conservation of mass since the same amount of gas has to leave the annular cavity but with a total opening area of 28 percent only. Therefore, it can be inferred that the discharge coefficient of the perforated holes is not uniform and is lesser in the inflow zone as compared to that of the outflow

zone. This is in agreement with the experimental findings of Ref. [3]. In this case, positive velocity ratio represents bias inflow into the annular cavity and negative velocity ratio indicates bias outflow from the annular cavity back into the perforated pipe.

Effect of porosity By keeping all other parameters at their default values as shown in Table 4, the effect of porosity on the normalized pressure drop of CTR is given in Table 5. The variation of normalized pressure drop with respect to porosity is also plotted in figure 7. Table 5. Effect of porosity. Nominal porosity, 𝝈𝒏

Actual porosity, 𝝈𝒂

∆𝒑⁄𝑯

0.02

0.0221

0.3891

0.04

0.0441

0.3969

0.06

0.0643

0.4000

0.08

0.0827

0.4111

0.12

0.1287

0.4226

0.14

0.1416

0.4255

0.16

0.1545

0.4314

0.18

0.1911

0.4377

Figure 5. Variation of velocity ratio (ratio of radial bias flow velocity to grazing flow velocity) along the length at Mach number 0.17 0.45 Normalized Pressure drop

A graphical visualization of the flow dynamics in the perforated holes is shown in figure 6 by means of the surface bias flow velocity. Due to the inertia of the flow to change the direction [6,7], the effective area of flow through the holes in and out of the annular cavity is relatively small as compared to the grazing flow. Consequently, it can be concluded that the varying flow area is responsible for the nonuniform discharge coefficient.

0.44

data points linear fit

y = 0.3*x + 0.38

0.43 0.42 0.41 0.4 0.39 0.38 0

0.02

0.04

0.06

0.08

0.1 0.12 Porosity

0.14

0.16

0.18

0.2

Figure 7. Variation of normalized pressure drop with respect to porosity across the total length at inlet flow velocity of 10 m/s.

Figure 6. Surface velocity plot of the radial bias flow velocity magnitude for a perforated length of 96 mm at an inlet average velocity of 15 m/s.

Parametric Studies A parametric study is conducted by varying the porosity while retaining the default values for other parameters as given in Table 1. This approach is shown to be justified in the appendix where it is demonstrated that the additional friction factor associated with the bias flow is much lesser than the effect of friction the grazing flow faces while traversing the holes. As a result, parameters like the inner chamber diameter, chamber length, perforated hole diameter and inner pipe thickness are not considered further in the analysis.

By expressing the total pressure drop normalized with respect to the dynamic head as ∆𝑝 𝐻

𝑙1 +𝑙2 +𝑙3 𝑑

+ 𝑎𝜎

𝑙2 𝑑

∆𝑝 𝐻

= 12.5 𝑓0 + 2𝑎𝜎

(6)

By fitting a linear curve in figure 7, the normalized pressure drop can be expressed as a function of the porosity as follows 𝐻

= 0.38 + 0.3𝜎

(7)

By comparing equation (6) and (7), we get f0 = 0.0304 and a = 0.15 Here, f0 is the friction factor of the un-perforated section. The generalized friction factor for the perforated portion can thus be expressed as

Page 4 of 8

(5)

and substituting the default values from Table 1, equation (5) becomes

∆𝑝

Since the normalized pressure drop is found to be more or less independent of the mean flow Mach number [2], an inlet average flow velocity of 10 m/s is used as it gives advantage of lesser computation time. The temperature used for computation is 299 K which is the same as in Refs. [4,5].

= 𝑓0

(8)

𝑓 = 0.0304 + 0.15𝜎

(9)

This is the required empirical formula for the friction factor of perforated pipes.

The empirical formula presented here in equation (9) can now be used to calculate the back pressure of perforated elements with grazing flow.

References Validation of the empirical formula 1. In order to validate the proposed empirical formula, equation (9) is compared with the experimental results published in Refs. [4, 5] which are also given in Table 3. The parameters adopted from Ref. [4] are: air temperature 299 K, inner diameter of perforated pipe 32 mm, nominal porosity 10.3 %, perforated length 191 mm and unperforated length 209 mm, giving a total length of 400 mm. The comparative result is tabulated in Table 6 below. It is observed that the normalized pressure drop predicted by using the empirical formula for the friction factor of perforated pipes given in equation (9) is in good agreement with the test results.

2.

3.

4.

Table 6. Comparison of the normalized pressure drop of CTR

5. Mach number 0.085

Normalized pressure drop CTR Ref. [4]

Equation (9)

0.550

0.573

6.

Munjal, M. L., Acoustics of ducts and mufflers, 2nd edition, Chichester, UK, John Wiley & Sons, 2014. Munjal, M. L., S. Krishnan, and M. M. Reddy. "Flow-acoustic performance of perforated element mufflers with application to design." Noise Control Engineering Journal (NCEJ) 40, no. 1 (1993): 159-167. Sullivan, Joseph W. "Some gas flow and acoustic pressure measurements inside a concentric‐tube resonator." The Journal of the Acoustical Society of America 76, no. 2 (1984): 479-484. Lee, Seong-Hyun, and Jeong-Guon Ih. "Effect of non-uniform perforation in the long concentric resonator on transmission loss and back pressure."Journal of Sound and Vibration 311, no. 1 (2008): 280-296. Liu, Chen, and Zhenlin Ji. "Computational Fluid DynamicsBased Numerical Analysis of Acoustic Attenuation and Flow Resistance Characteristics of Perforated Tube Silencers." Journal of Vibration and Acoustics 136, no. 2 (2014): 021006. Munson, Bruce Roy, Donald F. Young, and Theodore Hisao Okiishi. Fundamentals of fluid mechanics. New York, 1990. Batchelor, George Keith. An introduction to fluid dynamics. Cambridge university press, 2000.

Summary

7.

From the preceding results and analysis, the following observations can be made:

Contact Information

1. Effect of thickness and perforated hole diameter on friction factor is insignificant.

David Neihguk

2. Therefore, it follows that friction factor of the perforated portion is indeed given by equation (9).

email: [email protected] Mahindra & Mahindra Ltd. Mahindra Research Valley, Chennai 603 204. India

It can be noticed that, the friction factor for the perforated portion is considerably more than that of the Moody friction factor of the unperforated portion, which is 0.0283 [6]. Therefore, it can be stated that, the presence of perforations in the pipe significantly increases the friction factor of the perforated pipe.

Conclusions In the literature, the effect of Moody friction factor for nonperforated uniform diameter pipe (f0) is well documented. In the present paper, we have presented a simple expression for the friction factor of a perforated pipe as a function of the porosity. It has been observed that the effect of hole diameter, pipe thickness and the mean flow Mach number on the friction factor is practically negligible.

Page 5 of 8

Acknowledgments The authors gratefully acknowledge the support given to the first author by Mahindra & Mahindra Ltd to carry out this work as part of his Master’s dissertation. In particular the authors wishes to thank Mr. N. V. Karanth and Mr. P. S. Yadav from the Automotive Research Association of India (ARAI), Pune for their moral support. Finally, the authors thank the Internal Technical Review Council of Mahindra & Mahindra Ltd for giving the permission to publish the contents of this work.

Appendix Momentum transfer analysis of Concentric Tube Resonator (CTR) The following simplifying assumptions are used in the momentum transfer analysis of the CTR: 1. 2. 3. 4. 5. 6. 7.

Incompressible Newtonian and single phase fluid. One dimensional steady state flow. Isothermal flow with constant fluid viscosity. No external work and heat transfer between fluid and surroundings. Circular cross section profile for pipe and perforated holes. Uniform perforation throughout the length. Linearly varying bias outflow and bias inflow.

Figure A: Schematic representation of flow dynamics in CTR The momentum equation for the CTR consists of the wall shear stress terms and the momentum flux terms as given below [6] 𝐿

𝐿

𝐴(𝑃𝑖 − 𝑃𝑜 ) = ∫0 𝑟 𝜏𝑖 𝑆(1 − 𝜎)𝑑𝑥 + ∫𝐿 𝜏𝑜 𝑆(1 − 𝜎)𝑑𝑥 + ∯ 𝜌𝑈(𝑈. 𝑛)𝑑𝐴 + ∯ 𝜌𝑈(𝑈. 𝑛)𝑑𝐴 𝑟

(A1)

In the above equation (A1), the first two terms of the right hand side represent the balancing force due to the wall shear stress of the inflow zone (𝜏𝑖 ) and the outflow zone (𝜏𝑜 ) respectively, and the third and fourth terms represent the momentum flux terms. Let 𝑈𝑖 , 𝑈𝑜 𝑎𝑛𝑑 𝑈𝑟 be the average flow velocities at the inlet, outlet and point of recovery, respectively, 𝐴 is the cross sectional area of the inner pipe, 𝑆 = 𝜋𝑑 is the circumference of the inner pipe, 𝑃𝑖 and 𝑃𝑜 are the total stagnation pressure at the inlet and outlet, 𝐿 and 𝐿𝑟 are the total length and length of inflow zone of the CTR respectively. 𝜎 is the porosity of the perforated pipe and 𝑓 is the friction factor of the inner pipe. Assuming a linearly varying grazing flow velocity along the length of the CTR, we can express the grazing flow velocity (𝑈) and bias flow velocity (𝑉) as a function of the axial length in the inflow zone and outflow zone as follows. Inflow (into the annular cavity) region (𝟎 < 𝒙 < 𝑳𝒓 ): In the inflow zone, the grazing flow velocity and bias flow velocity can be written as a linear function of the axial distance as follows: 𝑈=(

𝑈𝑟 −𝑈𝑖 𝐿𝑟

) 𝑥 + 𝑈𝑖 𝑥

𝑉 = 𝑉𝑖 (1 − ) 𝐿𝑟

(A2) (A3)

Outflow (from the annular cavity) region (𝑳𝒓 < 𝒙 < 𝑳): Similarly, in the outflow zone, the grazing flow velocity and bias flow velocity can be written as a linear function of the axial distance as follows: 𝑈=(

𝑈𝑜−𝑈𝑟

𝑉 = 𝑉𝑜

𝐿−𝐿𝑟

) (𝑥 − 𝐿𝑟 ) + 𝑈𝑟

(𝑥−𝐿 𝑟 ) (𝐿−𝐿𝑟 )

Page 6 of 8

(A4) (A5)

Wall shear stress terms [6] Now, the wall shear stress can be written as 𝑓

𝜏𝑜 = 𝜏𝑖 = 𝜌𝑈 2

(A6)

8

Substituting the expression for wall shear stress (A6) into the first two terms of equation (A1) and applying the conservation of mass, i.e., 𝑈𝑖 = 𝑈𝑜 , we get 𝑓 8

𝐿

𝜌𝑆(1 − 𝜎) [∫0 𝑟 (( 1

𝐿𝑆

2

12

( 𝜌𝑈𝑖 2 ) 𝑓

𝑈𝑟 −𝑈𝑖 𝐿𝑟

2

𝐿

𝑈𝑜−𝑈𝑟

𝑟

𝐿−𝐿𝑟

) 𝑥 + 𝑈𝑖 ) 𝑑𝑥 + ∫𝐿 ((

(1 − 𝜎) [1 +

𝑈𝑟 𝑈𝑖

𝑈

2

) (𝑥 − 𝐿𝑟 ) + 𝑈𝑟 ) 𝑑𝑥 ]

(A7)

2

+ ( 𝑟) ]

(A8)

𝑈𝑖

Momentum flux terms [7] Using the momentum correction factor 𝛽 to account for the non-uniform variation of velocity, the momentum flux terms of the momentum conservation equation (A1) can be written as 𝜌𝐴𝛽𝑖 (𝑈𝑟 2 − 𝑈𝑖 2 ) + 𝜌𝐴𝛽𝑜 (𝑈𝑜 2 − 𝑈𝑟 2 ) = 𝜌𝐴𝛽𝑖 (𝑈𝑟 − 𝑈𝑖 )(𝑈𝑟 + 𝑈𝑖 ) + 𝜌𝐴𝛽𝑜 (𝑈𝑜 − 𝑈𝑟 )(𝑈𝑜 + 𝑈𝑟 )

(A9)

The grazing flow velocity and the bias flow velocity at the inflow zone and outflow zone can be related through the continuity equation. By using equations (A3) and (A4) in (A11) and (A14), we get 𝑄𝑖− 𝑄𝑏𝑖 = 𝑄𝑟

(A10) 𝐿

𝐴(𝑈𝑟 − 𝑈𝑖 ) = −𝑆𝜎𝐶𝑑𝑖 ∫0 𝑟 𝑉 𝑑𝑥 𝑈𝑟 − 𝑈𝑖 = −

(A11)

𝑆𝜎𝐶𝑑𝑖 𝑉𝑖 𝐿𝑟

(A12)

2𝐴

𝑄𝑟 + 𝑄𝑏𝑜 = 𝑄𝑜

(A13) 𝐿

𝐴(𝑈𝑜 − 𝑈𝑟 ) = 𝑆𝜎𝐶𝑑𝑜 ∫𝐿 𝑉 𝑑𝑥

(A14)

𝑟

𝑈𝑜 − 𝑈𝑟 =

𝑆𝜎𝐶𝑑𝑜 𝑉𝑜 (𝐿−𝐿𝑟 )

(A15)

2𝐴

where, 𝑄𝑖 , 𝑄𝑏𝑖 , 𝑄𝑟 , 𝑄𝑏𝑜 and 𝑄𝑜 are the volumetric flow rates at the inlet, bias inflow, point of recovery, bias outflow and outlet, respectively. 𝐶𝑑𝑖 and 𝐶𝑑𝑜 are the discharge coefficient of the perforated holes in the inflow zone and outflow zone, respectively. On substitution of equation (A12) into the momentum flux terms, we obtain the inflow momentum flux as 1

𝑉𝑖

2

𝑈𝑖

𝜌𝐴𝛽𝑖 (𝑈𝑟 − 𝑈𝑖 )(𝑈𝑟 + 𝑈𝑖 ) = −𝛽𝑖 ( 𝜌𝑈𝑖 2 ) (𝑆𝜎𝐶𝑑𝑖 𝐿𝑟 ) [2

− 𝐶𝑑𝑖

𝑂𝐴𝑅𝑖 2

𝑉

2

( 𝑖) ]

(A16)

𝑈𝑖

Similarly, on substitution of equation (A15) into the momentum flux terms, we obtain the outflow momentum flux as 1

𝑉𝑜

2

𝑈𝑜

𝜌𝐴𝛽𝑜 (𝑈𝑜 − 𝑈𝑟 )(𝑈𝑜 + 𝑈𝑟 ) = 𝛽𝑜 ( 𝜌𝑈𝑜 2 ) (𝑆𝜎𝐶𝑑𝑜 (𝐿 − 𝐿𝑟 )) [2

+ 𝐶𝑑𝑜

𝑂𝐴𝑅𝑜 2

𝑉

2

( 𝑜) ] 𝑈𝑜

(A17)

where, 𝑂𝐴𝑅𝑖 and 𝑂𝐴𝑅𝑜 are the open area ratio of the inflow zone and outflow zone, respectively. Again, by making use of the conservation of mass, i.e., 𝑈𝑖 = 𝑈𝑜 , and taking the momentum correction factor 𝛽 = 1 for turbulent flow the combined momentum flux associated with the bias inflow and bias outflow can be written as Page 7 of 8

1

𝑉𝑖

2

𝑈𝑖

− ( 𝜌𝑈𝑖 2 ) 𝑆𝜎 [𝐶𝑑𝑖 𝐿1 {2

− 𝐶𝑑𝑖

𝑂𝐴𝑅𝑖 2

𝑉

2

( 𝑖 ) } − 𝐶𝑑𝑜 (𝐿 − 𝐿1 ) {2 𝑈𝑖

𝑉𝑜 𝑈𝑜

+ 𝐶𝑑𝑜

𝑂𝐴𝑅𝑜 2

2

𝑉

( 𝑜 ) }]

(A18)

𝑈𝑜

Finally, the total momentum flux due to the wall shear stress and the bias inflow and bias outflow is given by substituting equations (A8) and (A18) into (A1), the resulting equations yields an expression for the non-dimensionalized pressure drop as given below ∆𝑃

=𝑓

𝐻

𝐿 3𝑑

(1 − 𝜎) [1 +

𝑈𝑟 𝑈𝑖

2

𝑈

𝑉

𝑉

𝑈𝑖

𝑈𝑖

2

+ ( 𝑟) ] − 2 [𝐶𝑑𝑖 𝑂𝐴𝑅𝑖 { 𝑖 − 𝐶𝑑𝑖 ( 𝑖 ) } − 𝐶𝑑𝑜 𝑂𝐴𝑅𝑜 { 𝑈𝑖

𝑉𝑜

𝑈𝑜

𝑉

2

+ 𝐶𝑑𝑜 ( 𝑜 ) }] 𝑈𝑜

(A19)

Order of magnitude analysis For practical applications, the following simplification can be applied to equation (A19), the justification for this can be seen in figure 5. 𝑉𝑖

≪

𝑈𝑖 𝑈𝑟 𝑈𝑖

𝑉𝑜 𝑈𝑜

and

𝑉𝑖 𝑉𝑜

,

𝑈𝑖 𝑈𝑜

≪1

(A20)

~1

(A21)

Due to mass conservation, the same amount of gas entering the annular chamber from the inflow zone must escape in the outflow zone but within a much shorter length. Therefore, the discharge coefficient of the outflow zone (𝐶𝑑𝑜 ) is greater than that of the inflow zone (𝐶𝑑𝑖 ). 𝐶𝑑𝑜 > 𝐶𝑑𝑖

(A22)

Equation (A19) now becomes ∆𝑃 𝐻

𝐿

𝑉𝑜

𝑑

𝑈𝑜

= 𝑓 (1 − 𝜎) + 2 [𝐶𝑑𝑜 𝑂𝐴𝑅𝑜

]

(A23)

Using the above simplifications and the data from Ref. [3], it can be seen from Table A that the momentum flux due to the radial bias flow is about one third of that contributed by the wall shear stress. Thereby, the effect of chamber diameter, perforated length, perforated hole diameter and the pipe thickness are not significant for practical applications. Table A: Comparison of momentum flux due to wall shear stress and radial bias flow

Page 8 of 8

Parameters

Ref [3]

f [equation (9)]

0.0359

L (Length of CTR)

257.0 mm

Lo (Length of bias outflow zone)

102.8 mm

D

50.8 mm

𝜎

0.037

𝐶𝑑𝑜

0.500

𝑂𝐴𝑅𝑜

0.299

𝑉𝑜 ⁄𝑈𝑜 (Peak bias outflow velocity ratio)

0.144

Momentum flux due to shear stress

0.175

Momentum flux due to radial bias outflow

0.043