MODIFICATION EQUATIONS OF AIR BUBBLES DISTRIBUTIONS AT SELF-AIR ENTRAINMENT CONDITION

Abstract

The Chanson’s equation for distribution of air bubbles in vertical direction in the developing zone at self-air entrainment condition is used when the air bubbles concentration at the flow surface is 90%. Otherwise, if this condition is not satisfying, then the equations of Straub and Anderson can be used. The results of these two equations are not similar with experimental results. Therefore, these two equations need to be modified accordingly. These modification equations can also be used to predict the air bubbles distributions in vertical direction. Hence, the main objective of this study is to modify these equations for vertical air concentration distribution in the developing zone and validate them with experimental results. The steep channel in the form of flume with 10 m long, 0.2 m wide and 0.4 m high with slopes varies between 20° and 25° were used in this experimental work. The discharge water was 9 l/s, 12 l/s and 21 l/s with Froude numbers between 6.9 to 8.0. The Thomson weir (V Notch) was used to calibrate the discharge flow of water. A set of video cameras was used to record the motion pictures of the air bubbles. The air bubble was analyzed using Ulead Video Studio 11 software program equipped with Imagej software. The results of this study indicates that the modifications of equations of Straub and Anderson were the equation air concentration distribution (C) in the underlying zone value was 0.647 m at 20° slope of channel bed, the equation air concentration distribution (C) in the underlying zone the value was 0.542 m at 25° slope and the equation in the mixing zone remained the same. The original Chanson equation was modified mainly in terms of the hyperbolic tangent (tanh) equation which originally had a power of 2 while the modification was 0.8; and the C_e was 0.9 sin α, whereas at the modified Chanson’s equation, C_e was converted into C_e= 0.6 sin α.