Is this a new basin?  If so, do you have to match the existing outflow discharge?  If so there is your answer.  

If this is an existing basin that you are evaluating then route your inflow hyd through under existing condition and determine the overtopping flow by using a rating curve for a weir and culvert(if there is a culvert).  The flow rate that you determine to overtop is approximately the increase in flow rate that you will need to add to your culvert capacity.

I am guessing at your situation so more information would be helpful!

These are sediment basins for cattle feed yards.  This one happens to be an existing basin.  However, there could be some new basins designed in the future.  New basins aren't really a problem, because we can design them at the required size.  However for an existing basin, the only thing we can do to keep it from overtopping during the design storm is change the culvert size.  However, I don,t want the culvert so large that the runoff isn't slowed down enough to settle out the solids.  It's cheaper to clean a sediment basin than it is to clean a primary lagoon.  We have to design off the 25 yr 24 hr rainfall event for the given Kansas county we are working in.  The sed basin has to provide at least 30 minutes of detention.  More detention time is allowed, as long as the basin is dry within 24 hours.  We used to install concrete overflows large enough to control the peak discharge.  Consequently, smaller rains wouldn't make it to the lagoon.  Now, the state doesn't want water standing in the sed basins unless we install a clay liner and prevent seepage in excess of .25 inches per day.  Furthermore, if the client qulified for equip, the NRCS requires no water stand in the sediment basin for longer than 24 hours.  This is the case for my current client.  I suppose we could install the concrete overflow for the larger storms and put in a perforated riser to control the smaller rains.

Example:

Existing sediment basin volume is 142,000 cuft, and peak discharge is calculated at 349 cfs.  Is there an equation to calculate the required discharge to provide a minimum 30 minutes detention time without the basin overtopping?

as indicated by gbam, you need to route the hydrograph through the basin.  you will need a stage - storage - discharge relationship for the basin and the outlet to perform the routing.  You can use HEC-1 (or another program) to perform the routing.  

You could construct an overtopping spillway and leave the culvert small.  This would allow de-silting and also handle larger flows.  

If I understand the question correctly you are looking for the discharge rate needed to allow a storage time of 30 min.  

In the past I have had to calculate detention times for basins, so this should work:

Vol / time = flow

142000 ft^3  x 1/30 min x 1 min/60 sec = 78.8 cfs

Hope that helps.

nvy047
Thanks for the input.  However, would that 78.8 cfs discharge be sufficient to prevent my basin from overtopping during a 25 yr 24 hr storm, generating a peak discharge of 349 cfs?

it is a little more complicated than this, because the culvert will not flow at a constant rate.  The discharge through the outlet culvert will vary with the depth of water in the basin.  Additionally, if the basin doesn't contain the entire storm runoff volume, the routing must still be done to size the outlet and spillway to prevent overtopping.

The simplied equation mentioned above has been used to determine storage times in extended detention basin, where the discharge is typically through a small orifice.  Therefore I would use this as a starting point for the discharge rate needed and model/adjust the basin accordingly.  

As far as overtopping, you would have to adequate storage in the basin to compensate for the additional inflow (78 vs 349).  Tr-55 estimate approx. 8.7 Ac-ft of total storage would be require in the basin to maintain a release rate of 78 cfs.  This too is only a estimate, but it is another starting point to work from.