### 1. Introduction

*k-ε*model and compared it with that of the ASHRAE DB; they also proposed a formula to estimate dynamic loss coefficients according to the Reynolds number.

### 2. Methodology

### 2.1 Numerical Method

*ϕ*is numerically calculated using the following equation (Ansys, 2009).

_{k}*is the diffusion coefficient,*

_{k}*S*is the source term supplied for each of the

_{ϕk}*N*scalar equations,

*ρ*is the density of the fluid,

*u*is the velocity component in the

_{i}*i*direction, and

*x*is the coordinate in the

_{i}*i*direction.

*k-ε*model and Menter-Lechner wall treatment, which is known to be less sensitive to y+, are selected. The numerical method for flow analysis is shown in Table 1.

### 2.2 Dynamic Loss Coefficient

*C*is the dimensionless dynamic loss coefficient,

*∆P*is the total pressure loss,

_{t}*∆P*is the friction loss, and

_{f}*V*is the average flow velocity of the pipe section.

### 2.3 Verification

*D*, the inlet flow velocity is

*V*,, the elbow radius is

_{0}*r*, and the angle is

*θ*. Each 10

*D*long straight pipe is connected to the upstream and downstream of a round elbow with an angle

*θ*=90°. Inlet velocity and outlet pressure are used as boundary conditions, and the analysis conditions are shown in Table 2. The flow analysis is performed under conditions of

*Re*= 1.0E+04 ~ 1.0E+06 for comparison, and a grid is selected by setting the first spacing such that

*y*+ becomes closer to 30 in each condition according to the Ansys Fluent User's Guide (Ansys, 2009).

*Re*= 1.0E+05 is shown in Fig. 2, and a summary of grids under the conditions of

*Re*= 1.04E+04, 1.04E+05, and 1.04E+06 is provided in Table 3. Fig. 3 shows the results of this study and Zmrhal and Schwarzer (2009). The velocity of the ship exhaust gas is approximately 30~40 m/s, and the Reynolds number corresponds to approximately 2.0E+05. In this study, the verification was performed with Reynolds numbers ranging from 1.0E+04 to 1.0E+06. In the range of

*Re*= 1.0E+05~1.0E+06, which is the operating range of the exhaust gas, this study produced results consistent with Zmrhal and Schwarzer (2009), but with a difference of approximately 10% at

*Re*= 1.0E+04. However, the numerical analysis of non-standard fittings in Section 3 is performed at approximately

*Re*= 2.0E+05.

### 3. Analysis of EGP Fittings

*k-ε*model and Menter-Lechner wall treatment are used for flow analyses. Constant velocity is applied as an inlet boundary condition, and for simplicity, the pressure as the outlet boundary condition is assumed to be a gauge pressure measurement.

### 3.1 Case 1: Double elbows

*D*, the elbow angles are each

*θ*, and the distance between two elbows is

*L*. The flow analyses are performed with a variable distance

*L*and a variable angle

*θ*, and a grid is selected separately for each analysis case. The grid of the condition

*θ*= 135° and

*L*= 2

*D*is shown in Fig. 6.

*D*= 0.8 m) and the distance between the two elbows (

*L*= 2

*D*) are fixed to predict the dynamic loss coefficient using the angle

*θ*as a variable. The analysis is performed by increasing the angle

*θ*from 115° to 155° in 10° increments. The numerical results of the dynamic loss coefficient are shown in Fig. 7. It can be observed that as the elbow angle

*θ*increases, the pipe becomes closer to being a straight pipe, thus reducing the pressure loss.

*D*= 0.8 m) and the elbow angle (

*θ*= 135° and 145°) are fixed to predict the dynamic loss coefficient using the distance

*L*between two elbows as a variable. The distance

*L*is increased from 1

*D*to 4

*D*in increments of 0.5D. The analysis results of the dynamic loss coefficient are shown in Fig. 8. In the condition of elbow angles

*θ*=135° and 145°, it appears that the influence of the distance

*L*on pressure loss is minimized at

*L*= 2

*D*or above. If a double elbow is used in EGP for space allocation, it is recommended to design the elbow with a large angle or with the distance between two elbows greater than 2

*D*if possible.

### 3.2 Case 2: Double inlets

*D*,, the diameters of the two inlet pipes are each

_{o}*D*,, and the distance between the inlet pipes is

_{i}*L*. The flow analysis is performed with the distance

*L*as a variable, and a grid is selected separately for each analysis case. The diameter

*D*of the outlet pipe is 0.8 m, and the diameters

_{o}*D*of the two inlet pipes are each 0.8/2

_{i}^{0.5}m. The grid for the case of

*L*= 2

*D*is shown in Fig. 9.

*L*from 2

*D*to 5

*D*in 1

*D*increments. The numerical results of the dynamic loss coefficient are shown in Fig. 10. The dynamic loss coefficient of the inlet located furthest from the outlet is denoted as C1 and the dynamic loss coefficient of the inlet located closest to the outlet is denoted by C2. In the range of the distance

*L*= 2

*D*~ 5

*D*, the pressure loss of the inlet furthest from the outlet slightly decreased as the distance

*L*increased. The pressure loss of the inlet closest to the outlet gradually increased as the distance

*L*increased.

### 3.3 Case 3: Y joint

*D*, the diameters of the two inlet pipes are each

_{o}*D*, and the angle between the inlet pipes is

_{i}*θ*. The flow analysis is performed using the angle

*θ*as a variable and a grid is separately selected for each analysis case. The diameter

*D*of the outlet pipe is 0.8 m, and the diameters

_{o}*D*of the two inlet pipes are each 0.8/2

_{i}^{0.5}m. In order to prevent pressure loss due to reduction of the flow area, the inlet pipe extends its diameter to

*D*immediately before connection, and the length of the expansion pipe is

_{o}*D*/2. The two inlet pipes are symmetric about the center axis of the outlet pipe. The grid with the angle

_{o}*θ*= 80° is shown in Fig. 11.

*θ*from 60° to 100° in increments of 10°. The results of the dynamic loss coefficient are illustrated in Fig. 12. In the range of the angle

*θ*= 60° to 100°, as the angle

*θ*increases, the pressure loss increases almost linearly except for the case of

*θ*= 80°. In general, numerical errors arise from various sources, including selection of grid and numerical methods, which can be cumulative (Mumma et al., 1998). However, even though the same method was applied to the analysis cases in this study, the case of

*θ*= 80° indicates a slightly different result. Further experiments are required.

### 3.4 Case 4: Capped wye (drain pot)

*L*is connected downward to be used as a drain pot. In Fig. 5(d), the diameter of the outlet pipe is

*D*, the diameter of the inlet pipe is

_{o}*D*, and the angle formed between the inlet pipe and the outlet pipe is

_{i}*θ*. The flow analysis is performed with angle

*θ*and the ratio of diameter

*D*as the main variables, and a grid is selected separately for each analysis case. Outlet piping diameter

_{i}/D_{o}*D*=0.8 m is applied to the analysis cases. The grid with the angle

_{o}*θ*= 45° and the ratio

*D*= 1 is shown in Fig. 13.

_{i}/D_{o}*θ*increases from 30° to 60° in increments of 15° and the ratio

*D*of the diameter increases from 0.5 to 1 in increments of 0.25. The numerical results of the dynamic loss coefficient are shown in Fig. 14. It can be inferred that the pressure loss decreases with diameter ratio

_{i}/D_{o}*D*increasing. Furthermore, in the range of angle

_{i}/D_{o}*θ*= 30°~ 60°, pressure loss generally increases as the angle

*θ*increases. The larger the ratio of

*D*, the greater the effect of angle

_{i}/D_{o}*θ*on the dynamic loss coefficient. However, in the case of the ratio

*D*= 0.5, the dynamic loss coefficients at angles

_{i}/D_{o}*θ*= 30° and 45° are similar.

*D*= 1) is fixed, and the flow analysis is performed for the inlet flow velocities of 20 m/s, 30 m/s, 35 m/s, 40.7 m/s, and 48.8 m/s. The inlet flow velocities are selected in consideration of safety margin based on the ships delivered from SHI Geoje shipyard. The analysis results of the dynamic loss coefficient with varying Reynolds number are shown in Fig. 15. The influence of the Reynolds number on the dynamic loss coefficient is not observed in the range of Reynolds numbers studied.

_{i}/D_{o}### 3.5 Case 5: Rupture disk

*D*, the diameter of the branch pipe is denoted by

_{m}*D*, and the length of the branch pipe is denoted by

_{b}*L*. The flow analysis is performed with the ratio

*D*and length

_{b}/D_{m}*L*as variables, and a grid is selected separately for each analysis case. The diameter of main pipe

*D*= 0.8 m is applied to the analysis cases. A grid with a diameter

_{m}*D*= 0.6 m and a length

_{b}*L*= 0.8 m is shown in Fig. 16.

*L*increases from 0.2 m to 1.6 m, and the ratio

*D*increases from 0.5 to 1 at intervals of 0.25. The analysis results of the dynamic loss coefficient are described in Fig. 17. As the ratio of

_{b}/D_{m}*D*increases, so does the dynamic loss coefficient. Larger dynamic loss coefficients are evident in length

_{b}/D_{m}*L*= 0.2 m compared to

*L*= 0.4 m ~ 1.6 m. The ‘rupture disk’ fitting is one of the major non-standard fittings of exhaust gas pipe. However, the fitting has the shape of clogged pipe, so it does not seem to play a significant role in pressure loss. In this case, the dynamic loss coefficients of the rupture disk fitting are calculated under 0.06, and it appears that the pressure loss due to this shape is not significant.