Reach of a Flanged Line Sink in a Streaming Flow A line sink of strength (Q/L) equal to 30.0 m²/min (129. SCFM/ft) lies in a horizontal plane over which air passes from right to left, as in Figure 7.6. The sink is located at the origin. Far upwind of the line sink the air has a uniform velocity

Question

Reach of a Flanged Line Sink in a Streaming Flow
A line sink of strength (Q/L) equal to 30.0 m²/min (129. SCFM/ft) lies in a horizontal plane over which air passes from right to left, as in Figure 7.6. The sink is located at the origin. Far upwind of the line sink the air has a uniform velocity [latex](U_0)[/latex] of 30.5 m/min (100. FPM or 0.508 m/s).
To do:
(a) Compute and plot the shape and location of the dividing streamline.
(b)Compute and plot the values of [latex]U_x[/latex] and [latex]U_y[/latex] at points along the dividing streamline.
(c) Describe the region within which the air speed is 200 FPM or greater. Repeat for air speeds of 150 and 120 FPM.

Accepted Answer

(a) A plot of the dividing streamline can be generated from Eq. (7-46) by setting [latex]ψ = Q/L[/latex] and computing a series of radii (r) for a range of 0 from a small value (such as 1°, 0.0174 radians) to 180° (π radians). The coordinates (x,y) are computed from Eq. (7-27). The authors used Excel to do the calculations; the file is available on the web site. Figure E7.2a shows the dividing streamline ([latex]ψ = ψ_d[/latex]).
The reach is the region between the dividing streamline and the x-axis. Using Eqs. (7-51), (7-53), and (7-54), the following distances are computed:

[latex]ψ=U_0 r sin heta +(\frac{Q}{\pi L} ) heta[/latex] (7-46)

[latex]x=r\cos heta y=r\sin heta r=\sqrt{x^2+y^2}[/latex] (7-27)

[latex]r_0=\frac{Q}{\pi LU_0}[/latex] (7-51)

[latex]h_\infty =\frac{Q}{LU_0}[/latex] (7-53)

[latex]h_0 =\frac{Q}{2LU_0}[/latex] (7-54)

[latex]r_0=\frac{\frac{Q}{L} }{\pi U_0} =\frac{30.0\frac{m^2}{min} }{\pi 30.5\frac{m}{min} } =0.313 m[/latex] [latex]h_\infty =\frac{\frac{Q}{L} }{ U_0} =\frac{30.0\frac{m^2}{min} }{ 30.5\frac{m}{min} } =0.984 m[/latex]

and

[latex]h_0 =\frac{\frac{Q}{L} }{ 2U_0} =\frac{30.0\frac{m^2}{min} }{ 2(30.5\frac{m}{min} ) } =0.492 m[/latex]

(b) At the coordinates (x,y) along the dividing streamline obtained in Part (a), [latex]U_x[/latex] and [latex]U_y[/latex] are computed using Eqs. (7-49) and (7-50). Figure E7.2b was created with the spreadsheet, and shows

[latex]U_x=-U_0-\frac{Q}{\pi L}\frac{x}{x^2+y^2}[/latex] (7-49)

[latex]U_y=-\frac{Q}{\pi L}\frac{y}{x^2+y^2}[/latex] (7-50)

these velocity components as functions of x. The speed of the air along the dividing streamline varies from [latex]U_0[/latex] at x » 0 to zero at the stagnation point. [latex]U_x[/latex] and [latex]U_y[/latex] are both negative along the dividing streamline, indicating flow downward and to the left. The magnitude of the air velocity, i.e. air speed (U) along the dividing streamline is also plotted in Figure E7.2b. The maximum air speed along the dividing streamline is about 0.64 m/s (126. FPM), which occurs at x ≈ 0.30 m. The region near the maximum air speed along the dividing streamline is sometimes called the "shoulder".

(c) Suppose an industrial operation is conducted for which the capture velocity (see Chapter 6) is 200 FPM (1.02 m/s). The ventilation engineer wants to locate the region within which the air velocity is equal to or larger than this value, which will be called [latex]U_{min}[/latex]. The capture region is the locus of points (r, θ) within which the air speed exceeds [latex]U_{min}[/latex] . From Eqs. (7-47) and (7-48),

[latex]U_r=-\frac{∂\phi }{∂r} =-[U_0 cos heta +\frac{Q}{\pi Lr} ][/latex] (7-47)

[latex]U_ heta =-\frac{1}{r} \frac{∂\phi }{∂ heta } =U_0 sin heta[/latex] (7-48)

[latex]\mid \vec{U} \mid ^2=U^2_{min} =U_r^2+U_ heta ^2=[-(U_0 cos heta +\frac{Q}{\pi Lr} )]^2+(U_0 sin heta )^2[/latex]

which can be manipulated into quadratic form for unknown r as follows:

[latex](U_{min}^2-U_0^2)r^2-\frac{2U_0 cos( heta )}{\pi} \frac{Q}{L} r -(\frac{Q}{L} )^2\frac{1}{\pi^2} =0[/latex]

In the spreadsheet, r is calculated from the above equation using the quadratic formula for various values of θ. Because of the quadratic nature of the equation, r is double-valued. Fortunately, it turns out that one of the r values is negative and unphysical; that value of r is rejected. The values of r versus θ are converted to y versus x, and are plotted as an isopleth of value [latex]U_{min}[/latex] = 200 FPM in Figure E7.2a.
The air speed below this isopleth is greater than 200 FPM, and that above the isopleth is smaller than 200 FPM. The capture region is therefore the area enclosed by the isopleth and the x-axis. The calculations are repeated for [latex]U_{min}[/latex] =150 and 120 FPM; these isopleths are also plotted in Figure E7.2a.

Discussion: The capture region grows as [latex]U_{min}[/latex] decreases, as expected. The isopleths at [latex]U_{min}[/latex] = 200 and 150 FPM lie entirely within the dividing streamline, indicating that the air speed along the dividing streamline is everywhere less than 150 FPM (0.762 m/s), which agrees with the results shown in Figure E7.2b. Particles under these isopleths are predicted to get removed by the suction system, according to the concept of capture velocity. The isopleth at [latex]U_{min}[/latex] =120 FPM crosses the dividing streamline, indicating that there is a region (shaded in Figure E7.2a) above the dividing streamline in which the air speed exceeds 120 FPM. If the capture velocity of the process is 120 FPM, particles in this region are predicted to be removed by the suction. However, this is unphysical since even the air itself in the shaded region does not get sucked into the slot! This illustrates a fundamental flaw in the concept of capture velocity - particles in the shaded region are unlikely to be removed, even though the air speed in that region is greater than the capture velocity. Furthermore, capture or non-capture of a particle is dependent on the initial direction of the particle, as well as its location. The trajectory of a particle in a flow field is calculated in a more sophisticated manner in Chapter 8.

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