Make particle emission number vary in time.

Hello.

I have a scene where a lot of pit balls fall in slow motion from inside of what i think will be a dress (not made wet). These particles are emmited in 1 frame and contained in a box until they are let out by a valve made from a shape keyed circle.

The problem is that I want the amount of balls falling to be increasingly higher, as if it would be starting to rain, drop by drop, and then the rain came down.

I know emission numbers cant be animated; and i guess I could try doing it manually, animating the valve to fit 1 ball, 2, 3, and so on, increasing the flow; but I wanted to know if there is a simpler “standard” workaround for this.

Tried attaching the blend file but it doesn’t upload or something.