The T-3 motor is almost a clone of the T-2 motor. I wanted a similar motor to the T-2, but I wanted to add a smoke tracking grain without losing propellant weight. So I decided to make a new casing 1/2" longer than the T-2. I also used a welded forward closure, rather than a bolt on closure, the welded closure gained about 1/2" in extra length. With the added length I now had room in the casing for a 1" smoke grain. I was going to use the nozzle from the T-2 motor, but I had some extra 1 5/8" steel bar stock, so I made a new nozzle as well. The nozzle has a .42" diameter throat, the same as the T-2. I reduced the nozzle length in two areas, the throat region is shorter, and the exit cone is smaller. The throat area has shown little or no erosion after numerous firings, so I decided a long throat area was not needed. The exit cone was reduced to .94", making for an expansion ratio of 1:5. The T-3 nozzle came in at 186 grams, 20 grams lighter than the T-2 nozzle, while actually making the major regions of the nozzle thicker and stronger.
For my first try at a smoke tracking grain, I am going to use a 1" thick slug of KNO3/epoxy/Fe2O3 composite propellant. It's the same formula I use to start my hybrid engine. It may burn a little faster than I want, but it will be a starting point.

Here is a diagram of the T-3 dimensions. The forward bulkhead is welded from the outside and is .41" thick. The inside chamber length is 11.605". The nozzle is retained by (8) 10-32x3/8" machine screws. The nozzle is sealed by applying a bead of high temp RTV around the casing, then inserting the nozzle. The 10-32 screws are inserted and the nozzle is pulled back against the screws and the RTV allowed to cure.

Above is a picture of the T-3 motor after its first firing.

Here is a close up of the nozzle.

Above is the thrust time curve for the test. This test used standard 65/35 sugar propellant with the KNO3 in the straight from the bag (not powdered) form .
Burn Time: Propellant .795 seconds
Burn Time: Smoke 11.5 seconds
Total Impulse: 129.769 lb. seconds
Maximum Thrust: 332.23 lbs.
Average Thrust: 163.231 lbs.
Propellant Weight: 1.0268 lbs.
Grain Geometry: 2) 1.5" diameter x 5.25" Clyindrical. Outside Inhibited, core & ends burning.
Core Diameter: .5"
Throat Diameter: .42"
Isp: 126.38
A couple of points on this test.
First of all, the burn time was only .795 seconds. That's a lot faster than the T-2 motor using the same propellant. The reason? I believe, and I hadn't considered this before, the smoke grain at the top of the motor was unrestricted and burned some (or all) of the paper inhibitor off the grains. I think using an inhibitor on the outside of the smoke grain would eliminate this.
Second, the Isp was about 5 seconds better than prior testing. That's probably due to two things, 1) faster burn created higher chamber pressure resulting in higher Isp. 2) The smoke grain would no doubt, add to a small extent the thrust produced. So the slightly higher Isp is no surprise.