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I'm in the process of using Excel to automate the design of sediment basins used for erosion control purposes. My goal is to have Excel automatically calculate basin parameters based on user-entered data, without the use of macros, Excel Solver, or external programs. This will keep the worksheets distributable and allow for easy design optimization.

One option for areas in tight spaces is to allow the emergency spillway to carry part of the design discharge, while the principal spillway carries the majority. Both can be modeled as weirs, for which the following relationship exists:

Q = CLH^{1.5}

Where:

Q = design discharge (known)

C = weir coefficient (known)

L = weir length (known)

H = head or depth of water above weir bottom (unknown).

When both spillways are modeled as weirs with bottom elevations separated by an [known] elevation E, the result is:

Q = C_{1}L_{1}H^{1.5}+C_{2}L_{2}(H-E)^{1.5}

or, since the C and L are known for each weir, simplified as:

Q = AH^{1.5}+ B(H-E)^{1.5}

Is there any way to solve for a value of H, given that the other variables are all known, in a single equation? Since I'm modeling a real system, I'm only concerned with the real, positive solutions. If it's not possible, I'm willing to use Excel to guess-test-iterate to a solution, but that's a non-ideal scenario.

Thanks in advance,

-Nick

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# Solving Weir Flow Equations Algebraically

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