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Equations for Flow Rise

The fabric barrier is implemented primarily to create a pressure differential across the barrier which will be used to create the necessary head across the turbine. When the barrier is stretched across a flow, it will limit the cross sectional area at a section of the river. Conservation of mass predicts that this will result in an increased flow velocity around the barrier, and an increase in kinetic energy in the flow. The specific energy (energy per unit mass) in a flow can be represented by Bernoulli’s equation seen below

From this equation, if it is assumed that there is no change in elevation, a increase in velocity would correspond with a reduction in pressure. This reduction in pressure would result in a pressure differential across the barrier, a key component in any hydraulic turbine.

The equations that accurately predicted the flow behavior over a barrier involved calculation of the total energy in the flow. This was calculated by integrating Bernoulli’s equation, which gives an expression for energy per unit mass. If Bernoulli's equation is integrated over the cross sectional area of the flow, and multiplied by the mass flow rate per unit area, then an expression for the total power in any cross section of the river can be generated. These calculations were analyzed before the addition of any turbine, so the assumption of constant power in the flow is valid. Therefore, the power in the flow before the barrier will be equivalent to the power in the flow over the barrier.

            This equation is an expression for the power in any unit area in the flow. Note that the substitution  is made. Because the flow over the barrier is discharged airborne to the atmosphere, it is assumed that the pressure in this section is zero. It is also assumed that the pressure in the portion of the flow before the barrier is linearly increasing with depth, with a slope of the specific weight of water. The velocities in any one cross section of the flow is assumed to be independent of the position, however velocities at different cross sections are different. The flow was assumed to be rectangular. With these assumptions in place, it is important to define the variables that will be used in the solution equations. The figure below shows which variables are used.

H0 is the height of the barrier, v0 is the velocity of the flow over the barrier, H is the height of the flow after it has been blocked, and v is the velocity of the flow after it has been blocked. Note that H and v will be different then the height and velocity of the flow before the barrier is in place. Because the velocity is assumed to be constant through out the cross sectional area, it was not necessary to integrate the term for kinetic energy in the flow. The below equations show how the solution was derived from the assumptions. A temporary variable h is introduced as the depth from the surface.

           This relates H, the height of the flow before the barrier, with H0 the height of the barrier and the velocities of the flow. Conservation of mass allows us to relate the velocity of the flow with the height of the flow, so Q is introduced as the volume flow rate and w is introduced as the width of the flow. This leaves us with the following final equation for the flow.

In this equation, the only unknown is the raised height of the river, H. This equation can be solved numerically for the value of H. In theory, this equation balances the pressure forces pushing the water over the barrier with the resistance forces of the barrier. The higher the raised height, the more the pressure forces will push the water over the dam, so the water level will rise until this force is significant enough to drive a flow rate of Q over the barrier.
Physical experiments to verify these calculations can be found here