Roughness Length (Input Form 3B)

The roughness length (E) of a tunnel is the average height of the uniform protuberances from the tunnel wall. The roughness length is a function of the type of construction and finishing techniques used within the tunnels of the system. Table 4.1 gives typical ranges of values for common types of materials used in subways. Roughness lengths are measured in feet. The roughness lengths are entered in Form 3B. The SES program distinguishes between tunnels with uniform roughness and tunnels with ribs. A sketch of a tunnel with uniform roughness is given in Figure 4.1 and a sketch of a tunnel with ribs is given in Figure 4.2. These two types of tunnels are discussed below.

Table 4.1 Ranges of Values for Common Types of Materials Used in Subways

Material Roughness Length E, ft.

Concrete Rough Forms 0.0055 - 0.01

Concrete Smooth Forms 0.001 - 0.0055

Track Bed with 2” Ballast See Text

Tile Walls 0.0008

Ribbed Tunnel See Text

4-3 E

F ig u re 4 .1 S c h e m a tic D ia g ra m o f T u n n e l w ith U n ifo rm R o u g h n e s s L e n g th s

T u n n e l W a lls

E = A ve ra g e R o u g h n e s s L e n g th D = D ia m e te r o f th e T u n n e l

D

b h

F igure 4.2 S che m atic D iagram of R ibbed T unnel T u nnel W alls

h = he igh t of ribs b = w idth of ribs

λ = c enter to c enter s pacing betw een ribs Do = diam e ter of tunne l

D_o R ibs

λ

Tunnels With Uniform Roughness

The Reynolds number is defined as follows for a given flow throughout the system:

N VD Q

RE = ρ = p

µ ν

4

where NRE = Reynolds number

ρ = mass density of air, slugs/ft³ V = velocity of the air, ft/sec

D = hydraulic diameter of the tunnel, ft µ = absolute viscosity of air, slugs/ft-sec

Q = volumetric rate of airflow in the tunnel, ft³/sec p = perimeter of the tunnel, ft

ν = kinematic viscosity of air, ft²/sec

The relative roughness of the walls in a system is defined as the ratio of the roughness length and the hydraulic diameter of the tunnel. This is expressed symbolically as E/D where E is the roughness length and D is the hydraulic diameter of the tunnel.

The Darcy-Weisbach friction factor of the walls within a system depends on the relative roughness of the walls and the Reynolds number for the airflow through the system. This Darcy-Weisbach friction factor is defined as follows:

∆P fLV

= ρ D² 2

where ∆P = pressure drop over the given length, lb/ft² ρ = mass density of the air, slugs/ft³

L = length of the tunnel, ft

V = velocity of the air in the tunnel, ft/sec D = hydraulic diameter of the tunnel, ft f = Darcy-Weisbach friction factor

The user must enter the roughness length for each line segment in a system. The program utilizes the user-entered roughness lengths to calculate the friction factor for the walls within each line segment. The SES calculates and prints the hydraulic diameter, relative roughness, and fully turbulent friction factor in the input verification for each line segment in a system.

The roughness length of a wall is obtainable from many different sources. Figure 4.3 is a Moody diagram showing the relationships between the friction factor, Reynolds number, and relative roughness for various types of tunnels with uniform roughness. Since the airflow in subway tunnels is almost always fully developed turbulent flow, the SES assumes fully turbulent flow if the Reynolds number is greater than 2000. Therefore, the user may determine the relative roughness, and thereby the roughness length, for a tunnel where the only known parameter for the walls is the friction factor due to the fact that the friction factor is no longer Reynolds number-dependent for fully turbulent flow.

Table A in Appendix B provides the Darcy-Weisbach friction factor as a function of relative roughness for fully developed turbulent flow. The user may use this table to determine the relative roughness, and thereby the roughness length, for a tunnel instead of using a Moody diagram.

Table B in Appendix B provides the Darcy-Weisbach friction factor as a function of Reynolds number and relative roughness. This table is simply a Moody diagram in tabular form.

7 89 2 3 4 5 6 7 89 2 3 4 5 6 7 89 2 3 4 5 6 7 89 2 3 4 5 6 7 89 2 3 4 5 6 7 89

Table C in Appendix B provides the relative roughness as a function of the Darcy-Weisbach friction factor for fully developed turbulent flow. The user may also use this table to determine the relative roughness, and thereby the roughness length, for a tunnel instead of using a Moody diagram.

Ribbed Tunnels

If the protuberances in a tunnel are spaced widely enough so that the roughness of the tunnel can no longer be considered uniform, the tunnel is considered a ribbed tunnel. A sketch of a ribbed tunnel is given in Figure 4.2. The fully turbulent friction factor for a ribbed tunnel can be determined from Figures 4.4 and 4.5. Once the friction factor for a ribbed tunnel has been determined from Figure 4.4, the user must use Tables A through C in Appendix B to determine the appropriate equivalent roughness length for the tunnel. The user cannot enter the height of the ribs as the roughness length — the user must first determine the friction factor and then work backwards from Tables A through C in Appendix B as previously explained to obtain the roughness length.

0.050

Figure 4.4 Effect of Ribbing on Pipe Flow Friction Factor (Ref.6) D_o

λ b h

10^-2

4-7 f D

L p

U

o L

= ∆

1 2

ρ

10^-2 10^-1 10⁰ 10¹

10^-1 10⁰ f

λD_o

Figure 4.5 Effect of Shape of Internal Ribbing on Pipe Flow Friction Factor (f based on D_o) (Ref.6)

10^-2 10¹

hD_o=0 070.

b/h = 2 b/h = 1

Varying Roughness Length Along the Tunnel Perimeter

The walls of a subway system will generally have varying roughness lengths along the different portions of the tunnel boundaries. For instance, the trackbed will have a different roughness length than the ceiling and the walls. The perimeter of a line segment can be entered in separate lengths to account for the various surfaces comprising the total inner surface of the tunnel segment. A roughness length must be entered for each of the separate fragmented perimeters. Up to eight perimeter “segments” and

corresponding roughness lengths may be entered for each line segment in the system. An example of how segmented perimeters and corresponding roughness lengths may be used is given in the following example. The SES calculates a weighted average roughness length when more than one segmented perimeter and corresponding roughness length are entered for a line segment. The weighted average roughness length is printed in the input verification for each line segment (See Example 4.1).

Example 4.1 Consider the tunnel line segment cross-section given in Figure 4.6. The walls of this tunnel line segment have various items attached to them such as pipes and walkways. The entire perimeter of this tunnel line segment can be broken into the

segments outlined in Table 4.2. Table 4.2 provides the approximate theoretical roughness lengths and resulting friction factors for each of the segmented perimeters. The weighted average friction factor is also provided.

Third Rail

Figure 4.6 Example of Segmented Perimeters Section not to scale

Rails

Drainage W alkway

Conduit

4-9 Table 4.2 Theoretical Tunnel Friction Factors for Segmented Perimeters

(described in example 4.1)

Weighted tunnel friction factor: 0.028

* For use with Moody diagram to evaluate ft

** Roughness characteristic of ribbed tunnels (h = height, λ = spacing); ft evaluated using Figure 4.4

The user does not have to perform all the calculations shown in Table 4.2. The SES computes the friction factors, the user only has to enter the roughness lengths for each perimeter segment.

The user may use a particular friction factor for a line segment by entering the total perimeter of the line segment and the corresponding weighted average roughness length required for the tunnel friction factor using the tables in Appendix B.