Question #3: Coastdown Recovery With Trickle (easy).

Given a machine with stable running at L=L_Peak loses the ability to continue trickle injection. Luminosity decays at some rate corresponding to some luminosity lifetime. Presume that this coastdown is of sufficiently long duration that exclusively trickle recovery is not worthwhile (as in question #2 above).

Consider the picture to the right: At time t=0 ability to inject is lost. At time t=1 luminosity has decayed to a level L=Lmin but injection is again available. Data logging is abandoned and injection proceeds under filling conditions. At time t=2 a luminosity LDeliver is achieved. Injection is switched to trickle and data logging begins. At time t=3 the luminosity has again achieved the value Lpeak and the interruption is over. The dashed yellow line shows the integrated luminosity from this course.

Can you show that in this case the optimum luminosity level at which to switch from filling injection to trickle injection is again L_Deliver=L_peak(1-R_trickle/R_fill)?

Solution to Question #3

This problem is very similar to the problem of filling from scratch after an abort.

In that problem we took an approach which formed an expression for the luminosity lost as a result of the interruption. We then minimized the luminosity lost by finding a value L_deliver such that the derivative of luminosity lost w.r.t. L_deliver was zero.

As in the abort problem, the luminosity is all lost during three intervals. In tallying up that lost luminosity in the abort problem we came up with an expression:

(Eq I):Luminosity Lost = ∫^t=1_t=0 L_Peakdt + ∫^t=2_t=1 L_Peakdt + ∫^t=3_t=2 L_Peakdt - ½(L_Peak+L_Deliver) (t₃-t₂)

A similar expression can be written for the coastdown case. In doing so one finds a luminosity lost expression which is different in two ways:

• The lost luminosity in the interval between t₀ and t₁ is reduced by a term related to the luminosity decay rate (still integrating what is still there).
• The lost luminosity in the interval between t₂ and t₁ is also reduced since the filling time interval is less (there is less filling to do).

Both of these additional terms involve L_min (the first also includes the luminosity lifetime) and not L_Deliver. So in differentiating this new luminosity lost relationship w.r.t. L_Deliver all of the new terms drop out, and the result is the same is in the abort recovery case.