Unflanged Slot of Finite Width in Quiescent Air Air enters an unflanged rectangular inlet that has a large aspect ratio, L/w » 1 (Figure 7.13). In the far-field the air is motionless. The width of the inlet is 2w (w = 0.020 m) and the volumetric flow rate per unit length entering the inlet is Q/L

Question

Unflanged Slot of Finite Width in Quiescent Air
Air enters an unflanged rectangular inlet that has a large aspect ratio, L/w » 1 (Figure 7.13).
In the far-field the air is motionless. The width of the inlet is 2w (w = 0.020 m) and the volumetric flow rate per unit length entering the inlet is Q/L = 12.0 m²/min. The face velocity, [latex]U_{face}[/latex] = Q/(2Lw), is 300 m/min which is typical of the slot velocities recommended by the ACGIH for industrial ventilation systems. The numerical values used in this example problem are the same as those used in the previous problem; the only difference here is that the inlet is unflanged.
To do:
(a) Select several values of the stream function (ψ) and compute and plot streamlines near the inlet face.
(b)Compute and plot the normalized air speed, U/[latex]U_{face}[/latex] , along the line of symmetry, i.e. the xaxis. Compare to the air speed predicted for a line sink and from the empirical results of Chapter 6 for an unflanged slot.
(c) Compute and plot several isopleths along which the normalized air speed (U/[latex]U_{face}[/latex]) is constant.

Accepted Answer

(a) The locus of points (x,y) of a streamline can be found as follows: A particular value of ψ is selected, and a range of y/w values is defined (here, 0 < y/w < 2). The value of Φ is calculated at this ψ and at each y by solving Eq. (7-91) for 2Φ/k,

[latex]\frac{y}{W} =\frac{1}{\pi} [2\frac{ψ}{K}+exp(2\frac{\Phi}{K} )sin(2\frac{ψ}{K} ) ][/latex] (7-91)

[latex]\frac{2\Phi }{K} =\ln \left(\frac{\frac{\pi y}{w}-\frac{2ψ}{K} }{sin (\frac{2ψ}{K} )} \right)[/latex]

Eq. (7-90) is then used to solve for x/w for each value of y/w. The process is repeated for several values of ψ. The authors used Excel to generate the streamlines; the file is available on the book's web site. Figure E7.4a shows ten streamlines in the upper half of the flow field. Unlike the flanged slot example above in which the axis of symmetry was the y-axis, the x-axis is the axis of symmetry here.

[latex]\frac{x}{W} =\frac{1}{\pi} [2\frac{\Phi }{K}+exp(2\frac{\Phi}{K} )cos(2\frac{ψ}{K} ) ][/latex] (7-90)

(b) Along the axis of symmetry of the flow (x-axis), y = 0 and ψ = 0. Using Eq. (7-90), the value of x/w is computed for several values of Φ along this streamline. Using Eqs. (7-93) and (7-94), the velocity components, [latex]U_x[/latex] and [latex]U_y[/latex], are computed at each value of x/w. ([latex]U_y[/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 x/w value. The normalized radius (r/w) is computed from the center of the slot forward; in this case, r/w = (x + l/π)/w since the face is located to the left of the origin at normalized distance l/π, and y = 0 along the axis of symmetry. The result is plotted in Figure E7.4b, which also shows how the speed would vary if the slot were to be replaced by a line sink, Eq. (7-34).

[latex]U_x=-\frac{K \pi}{2w} \frac{1+exp(2\Phi /K)cos(2ψ/K)}{[exp(2\Phi /K)sin(2ψ/K)]^2+[1+exp(2\Phi /K)cos(2ψ/K)]^2}[/latex] (7-93)

[latex]U_y=-\frac{K \pi}{2w} \frac{exp(2\Phi /K)cos(2ψ/K)}{[exp(2\Phi /K)sin(2ψ/K)]^2+[1+exp(2\Phi /K)cos(2ψ/K)]^2}[/latex] (7-94)

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

The empirical expression for U/[latex]U_{face}[/latex] along the symmetry axis of a rectangular unflanged inlet, Eq. (6- 4), is also plotted in Figure E7.4b. An aspect ratio of 10:1 is assumed. The results show much disagreement among the three curves for radial distances less than about four slot half-widths. Beyond this, the three curves show reasonably good agreement.

(c) Equations (7-93) and (7-94) can be used to compute normalized velocity components [latex]U_x[/latex] and [latex]U_y[/latex], from which air speed (U) can be found. Unfortunately the values of Φ and ψ for coordinates (x,y) are not known explicitly. To compute these values, implicit Eqs. (7-90) and (7-91) must be solved numerically. The authors used Mathcad's Solve Block feature. Once computed, the velocity components [latex]U_x[/latex] and [latex]U_y[/latex] are calculated at (x,y) using Eqs. (7-93) and (7-94), and normalized air speed (U/[latex]U_{face}[/latex]) is calculated for an array of points (x,y) upwind of the inlet. Mathcad's contour plotting routine is used to generate the contour plot; Figure E7.4c shows the results of this computation.

Discussion: The shape of the isopleths in Figure E7.4c compare favorably with those of Figure 6.8, although the magnitudes differ. This is not surprising since the centerline air speed also differs significantly, as seen in Figure E7.4b, especially near the inlet face. The reader should note the factor of two, however; x and y are normalized by the slot half-width in Figure E7.4c, while they are normalized by the full slot width (s = 2w) in Figure 6.8.

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