# How can I calculate the optimum loiter airspeed for an aircraft?

I am in the conceptual phase of designing a 10 seater amphibian aircraft. I have done the initial sizing using Raymer and Roskam's methods. I have got my design point. To do a constraint analysis as well as to find the fuel fraction for loiter phase I need to calculate the optimum loiter velocity for a loiter time of 60 minutes at 7000ft. Breguet's range equation doesn't help since I don't know the fuel fraction of loiter phase. How can I calculate the loiter airspeed?

The optimum cruise speed is the speed at which fuel consumption is minimized per unit of velocity. So, you need to compute the derivative (C/v)' where C is rate of fuel consumption and v is rate of travel (airspeed). (I assume you want this by "loiter". If you want to minimize fuel consumption, that is called "maximum endurance" and that is explained below on the chart.)

Since fuel consumption is proportional to power, and power divided by velocity is thrust, we can graphically determine this point by the thrust required curve. The diagram below shows the relevant relationships:

So, to compute the optimum cruise speed, first draw the thrust-required curve, as shown in the lower part of the diagram above, then you draw a line from the origin tangential to the thrust required curve. The point of tangency lies above the optimum cruise speed on the x-axis.

Maximum Endurance

If by "loiter" your goal is to minimize the fuel flow while maintaining a particular altitude, then you use the power required chart (the upper chart in the figure above). The fuel flow is proportional to power, so the maximum endurance speed is found by locating the lowest point on the power required curve. In the example above, that point is located at 80 CAS. Mathematically, the maximum endurance is found by solving the equation P'=0, where P is the power required function.

• To loiter is to stand or wait around without apparent purpose. The loiter speed or holding speed is the same as the maximum endurance speed. So, why not turn it around and make the main answer about the maximum endurance speed and the "bonus" about maximum range. I'd vote you up for it :-) – DeltaLima Oct 9 '17 at 20:02

Would be great if you add your existing approach. Roskam has already an example towards an amphibious aircraft if I am not mistaken.

If you don't have other concerns and if your design is not already constrained by some other points (take off, max speed, etc) here's the first approach:

Several certification regulations enforce a minimum loiter speed that is 1.3 times the stall speed.

Remember that there's a trade off between max speed and loiter speed.

• Thanks for your help but I don't have stall speed for my aircraft. To calculate stall speed I will need Clmax and wing area, which I don't know because I haven't reached the wing design phase. To do wing design I need to do constraint analysis so that I can get W/S and hence S i.e wing area. So I am stuck in a loop. What I want to know is if there is any rule of thumb that can give me loiter airspeed when you know empty weight and max take off weight. – Anirudh Prabhu Oct 10 '17 at 9:02