Flanged Slot of Finite Width in Quiescent Air Consider a slot in an infinite plane as shown in Figure 7.10. The width of the slot is 2w (w = 0.020 m) and the aspect ratio of the slot is very large, L/w » 1. Air is withdrawn through the slot at a volumetric flow rate (Q/L) of 12.0 m²/min, which

Question

Flanged Slot of Finite Width in Quiescent Air
Consider a slot in an infinite plane as shown in Figure 7.10. The width of the slot is 2w (w = 0.020 m) and the aspect ratio of the slot is very large, L/w » 1. Air is withdrawn through the slot at a volumetric flow rate (Q/L) of 12.0 m²/min, which produces an average face velocity, [latex]U_{face}[/latex] = 300. m/min. This corresponds favorably with typical slot velocities recommended by the ACGIH for different types of industrial ventilation hoods.

To do:
(a) Predict and plot several streamlines for this flow.
(b) Predict the normalized velocity components ([latex]U_\mathrm{x}/U_{face}[/latex]and [latex]U_\mathrm{y}[/latex]/[latex]U_{face}[/latex]) and the normalized air speed,

[latex]\frac{U}{U_{face}} =\frac{\sqrt{U_\mathrm{x}^2+U_\mathrm{y}^2} }{U_{face}}[/latex]

at points along the streamline ψ= Q/(2L). This is the streamline along the y-axis (x = 0), i.e. along the axis of the slot (axis of symmetry), normal to the slot face. Plot U/[latex]U_{face}[/latex] as a function of the normalized radial distance to the center of the slot (r/w) along this streamline,

[latex]\frac{r}{\mathrm{w}} =\frac{\sqrt{x^2+y^2} }{\mathrm{w}}[/latex]

Compare with the air speed that would exist if the slot were a line sink of strength Q/L.
(c) Predict the velocity components ([latex]U_x[/latex] and [latex]U_y[/latex]) at an arbitrary point (x,y). Find the velocity components at the particular point x = 0.63 m, y = 0.025 m.

Accepted Answer

(a) Eq. (7-73) is solved for x/w,

[latex]1=\left[\frac{\frac{\mathrm{x}}{\mathrm{w}} }{cos(\frac{ψ}{K} )} \right] ^2-\left[\frac{\frac{\mathrm{y}}{\mathrm{w}} }{sin(\frac{ψ}{k} )} \right] ^2[/latex] (7-73)

[latex]\frac{\mathrm{x}}{\mathrm{w}} =cos(\frac{ψ}{K} )\sqrt{1+\left[\frac{\left(\frac{\mathrm{y}}{\mathrm{w}} \right) }{sin\left(\frac{ψ}{K}\right) } \right] ^2}[/latex]

A value of the stream function ψ is selected. An range of [latex]\mathrm{y}/\mathrm{w}[/latex] values is generated (from 0 to 1.4), corresponding to the flanged surface. A value of x/w is computed for each y/w value using the above equation. The computation is repeated for several values of ψ, and the results (the streamlines) are plotted in Figure E7.3a. The authors used Excel, and the spreadsheet file is available on the web site.

(b) Along the axis of symmetry of the flow, x = 0 and ψ = Q/(2L) since half of the total volumetric flow rate enters from the right half of the slot. An array of y/w values from 0 to 5.0 is generated. Using the second part of Eq. (7-70), the value of Φ at each y value is computed along this streamline. Using Eqs. (7-81) and (7-80), the velocity components, [latex]U_\mathrm{x}[/latex] and [latex]U_\mathrm{y}[/latex], are computed at each value of y. ([latex]U_\mathrm{x}[/latex] is verified to be zero along the axis of symmetry.) The normalized air speed (U/[latex]U_{face}[/latex]) is then computed at each y value. The normalized radius (r/w) is computed from the center of the slot upward; in this case, r/w = y/w since x = 0. The result is plotted in Figure E7.3b, which also shows how the speed varies if the slot were to be replaced by a line sink, as given by Eq. (7-34). It is seen that for both cases the air speed decreases away from the inlet. At radial distances less than about three slot half-widths, the air speed for a slot of width 2w is lower than that predicted by a line sink with the same volumetric flow rate. Beyond about three slot half-widths, the air speeds of the two flows are virtually the same. Thus for practical purposes, at radial distances much greater than the slot width (2w), a slot of finite width and a line sink yield similar results. Indeed, if the streamlines in Figure E7.3a were viewed from "far away", they would be indistinguishable from those of a line sink, i.e. they would look like rays to the origin. Finally, the empirical expression for U/[latex]U_{face}[/latex] along the symmetry axis of a rectangular flanged inlet, Eq. (6-5), is also plotted in Figure E7.3b. An aspect ratio of 10:1 is assumed, and y is used in place of x in Eq. (6-5) because of the coordinate system used here. The results are between the line sink and the finite width flanged slot calculations. All three curves merge beyond about three slot halfwidths.

[latex]\frac{\mathrm{x}}{\mathrm{w}} =cos h(\frac{\phi }{k} )cos(\frac{ψ}{k} ) \frac{y}{\mathrm{w}} =sin h(\frac{\phi }{k} )sin(\frac{ψ}{k} )[/latex] (7-70)

[latex]U_\mathrm{x}=-\frac{∂ψ}{∂\mathrm{x}} =-\frac{k}{\mathrm{w}} \frac{sin h\frac{\phi }{k} cos\frac{ψ}{k} }{(sin h \frac{\phi}{k} cos \frac{ψ}{k} )^2+(cos h \frac{\phi}{k} sin \frac{ψ}{k} )^2}[/latex] (7-80)

[latex]U_\mathrm{y}=\frac{∂ψ}{∂\mathrm{x}} =-\frac{k}{\mathrm{w}} \frac{cos h\frac{\phi }{k} sin\frac{ψ}{K} }{(sin h \frac{\phi}{k} cos \frac{ψ}{k} )^2+(cos h \frac{\phi}{K} sin \frac{ψ}{k} )^2}[/latex] (7-81)

[latex]U_r=-\frac{Q}{2\pi L}\frac{1}{r}[/latex] (7-34)

[latex]\frac{U(\mathrm{x},0)}{U_{face}} =1-\frac{2}{\pi} arctan\left\{\frac{2\mathrm{x}}{a}\left[\left(\frac{2x}{b} \right)^2+\left(\frac{a}{b} \right)^2 +1 \right]^{1/2} \right\}[/latex] (6-5)

(c) To find the velocity components [latex]U_\mathrm{x}[/latex] and [latex]U_\mathrm{y}[/latex] at an arbitrary point (x,y) it is first necessary to know the values of the potential functions Φ and ψ at this point. Equations (7-70) must be solved simultaneously for Φ and ψ at (x,y). There are several ways to do this - trial and error, Newton's method, etc.; the authors chose to use Mathcad's Solve Block feature. The following results are obtained at x = 0.063 m and y = 0.025 m:

ψ(0.063,0.025)=1.503 [latex]\frac{m^2}{min}[/latex] and Φ(0.063,0.025)=7.249 [latex]\frac{m^2}{min}[/latex]

From these results, Eqs. (7-80) and (7-81) are used to find the velocity components:

[latex]U_\mathrm{x}[/latex](0.063,0.025)=-555.4 [latex]\frac{m}{min}[/latex] and [latex]U_\mathrm{y}[/latex](0.063,0.025)=-23.2 [latex]\frac{m}{min}[/latex]

The Mathcad program can be found on the web site for this book.
Discussion: The potential flow equations used here for a flanged slot do a better job than those for a simple line sink at predicting the velocity field near the inlet face, and the infinite velocity at the origin is removed. However, both potential flow approximations do an excellent job far from the inlet.

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