propeller mixer design

Sounds like homework. Think of the propeller as a screw, turn it one revolution, calculate the distance it moves forward and multiply this by the diameter. Now you have a volume, multiply this by a speed and g and density and the head you need to overcome and you get hydraulic power. You choose the units, you choose the efficiency. If it is not homework you are in more trouble as there are many factors such as viscosity, shear etc etc to take into account which would require some actual research (work).

Any how, the hard part is designing the shaft and bearings so that the thing is balanced and the bearings don't get hammered into submission.
You wouldn't believe the nubers of knackered strirrers I see.

The starting point is the agitation power requirments, which is a function of the vessel size and shape, the number of baffles in the vessel, the shape of the agitator, its speed, and the density and viscosity of the fluid being agitated, etc. The calculation is mildly irritating iterative. Details can be found in good Process Engineering textbooks. Try Perry: "The Chemical Engineer's Handbook", any edition.

As already mentioned the problem is much more complex than it appears at first view. I would suggest a simple approach which leads to a maximal torque and thus to a maximal power.

Consider that you have the blades at 90° with respect to the plane they move in. Calculate the maximal force if the water in the tank is not moving (this means with highest relative velocity) and forget the other effects. This force is proportional to the area of the blade, its speed (r*ω) and a coefficient which you can find in any book dealing with aerodynamics or hydrodynamics (it depends on the ratio L/B of the blade). The power will be the product of the ( force/blade* number of blades) with the speed. Choose a motor with a serious power and torque reserve and you are so far. You will not have the exact value but most probably the result will cover the actual requirements.

Hope it will help. It is a pragmatic approach since you only expect a power indication. if you want more precise values you should base all computations on already available research results. The reynolds number and the relative speed play an important role.