Consider the two dimensional duct shown in the figure. With the axial distance from the...
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Consider the two dimensional duct shown in the figure. With the axial distance à from the inlet and the half width h(x) given in m, the duct consists of the following three sections: a) a contraction for 0 < x < 2, where h(x) = 1/x; b) a straight section, for 2 < x < 6, where h(x) 0.5; and c) a straight diffuser with a half angle a, for 6 < x < 10, where h(x) ≈ [0.5+ a(x − 6)] (it has been assumed that tan a ≈a). h(x). 0 2 = 6 10 m a (a) Assume that the flow through the duct is potential and one-dimensional with an approximately uniform velocity ue(x) that is nearly parallel to the x axis. Further assume that at x 4 m/s. Use continuity (u₂(x)h(x) 4) to calculate 1 m, ue = ue(x) along the duct. (b) Calculate Im at x = 6; calculate C₂ = (ue/um)² - 1 and dCp/dx for x > 6 m. (c) Use the laminar Stratford criterion to estimate (i) the location x, of the laminar separation point for a = : 0.02 rad (ii) the value of a for the separation point to move to îs 10 m (iii) the value of a for x, = 10 m, if the boundary layer is removed at x = = 6 m. = = (d) Calculate 8* = 1.72√√/(v/um), where ♬ is the "equivalent flat plate length” found in (b) above. Does the "effective channel width" (h − 8*)/h justify the assumption of one-dimensional flow? Consider the two dimensional duct shown in the figure. With the axial distance à from the inlet and the half width h(x) given in m, the duct consists of the following three sections: a) a contraction for 0 < x < 2, where h(x) = 1/x; b) a straight section, for 2 < x < 6, where h(x) 0.5; and c) a straight diffuser with a half angle a, for 6 < x < 10, where h(x) ≈ [0.5+ a(x − 6)] (it has been assumed that tan a ≈a). h(x). 0 2 = 6 10 m a (a) Assume that the flow through the duct is potential and one-dimensional with an approximately uniform velocity ue(x) that is nearly parallel to the x axis. Further assume that at x 4 m/s. Use continuity (u₂(x)h(x) 4) to calculate 1 m, ue = ue(x) along the duct. (b) Calculate Im at x = 6; calculate C₂ = (ue/um)² - 1 and dCp/dx for x > 6 m. (c) Use the laminar Stratford criterion to estimate (i) the location x, of the laminar separation point for a = : 0.02 rad (ii) the value of a for the separation point to move to îs 10 m (iii) the value of a for x, = 10 m, if the boundary layer is removed at x = = 6 m. = = (d) Calculate 8* = 1.72√√/(v/um), where ♬ is the "equivalent flat plate length” found in (b) above. Does the "effective channel width" (h − 8*)/h justify the assumption of one-dimensional flow?
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Related Book For
Physics
ISBN: 9781119539636
11th Edition
Authors: John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler
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