Solve the same problem as given above as Example Problem 6.17, except that, the rate dq/dt at which the flow rate past the gate is increased linearly with time until 250 s, and held constant thereafter. In other words, for 250 s dq/dt = at + b. The rate of linear increase of dq/dt is such that a =

Question

Solve the same problem as given above as Example Problem 6.17, except that, the rate dq/dt at which the flow rate past the gate is increased linearly with time until 250 s, and held constant thereafter. In other words, for 250 s dq/dt = at + b. The rate of linear increase of dq/dt is such that a = 1.0 × [latex]10^{−4}[/latex] and b = 1.0 × [latex]10^{−3}[/latex] (ft/s)². Also predict the position downstream from the gate where a secondary surge will likely develop, and the time at which this surge will first appear. In predicting [latex]x_{s}[/latex], note that since the rate dq/dt is not constant but increases with time, the position may not, and very likely is not, at the leading edge of the unsteady flow, or at the zone of quiet. Therefore, [latex]x_{s}[/latex] should be computed for each time, and the value that is the smallest will represent the location of the formation of the surge.

Accepted Answer

Integration of dq/dt with respect to time gives q as a function of time, or q = [latex]q_{o}[/latex] + (at + b)t, in which a = 1.0 × [latex]10^{−4}[/latex] and b = 1.0 × [latex]10^{−3}[/latex]. The computer programs above can be modified slightly to solve this problem. These modifications might consist of: First, instead of reading dq/dt, the values of a and b above need to be read, and from these, the flow rate per unit width q is computed from the above integrated equation, (Note that since the q given in these programs first applies for the flow upstream from the gate, a and b will be read as negative values.) second, a statement needs to be added to compute the position of the secondary surge from the equation:

[latex]x_{s}= \frac{\left(V+c\right)^{2}}{1.5\left({g}/{Y}\right)^{{1}/{2}}\left({\Delta Y}/{\Delta t}\right)}[/latex]

and third, the variable for [latex]x_{s}[/latex] needs to be added to the written statement. Using the following line of input data:

Question 6.19: Solve the same problem as given above as Example Problem 6.1...

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