Steampunk Airship Alternate Propulsion - Sails

In a previous post I looked at, and rejected, the idea of steam powered airships.  Simply put steam engines are too heavy to practically lift, which is why the airplane and airship had to wait for more weight efficient diesel engines.  But why stop looking at the idea there?  After all steam power isn't the only means of locomotion used by 19th century sea going vessels, why should it be the only method used by airships of a society with similar technology?  Ships of that time, and today, used the wind to move, by sail.

Before we look at getting a ship into the air, let's first just look at making it move using the wind.  Early box sails operated by catching higher wind pressure behind a ship.  The limit of course being that one can only move downwind and no faster than the ambient wind.  Triangular sails, it was eventually found, allowed a vessel to sail almost directly into the wind at greater speeds.  Although it wasn't known at the time these sails operate on exactly the same principles as wings, and are simply airfoils.  Common figures for sail performance:
Coefficient of Lift = 0.8
Lift-to-Drag Ratio = 5
Sails generate propulsive force based on their angle relative to the apparent wind, beta.  Apparent wind factors in both the ambient wind and the effect of the ship's motion.
Sails generate force by precisely the same mechanism as wings do, so the propulsive force generated is just:

Note that the sail area, A, is a total.  A ship can have move than one sail.  Indeed it will be necessary, especially for an airship which cannot have the stabilizing effect of a conventional keel or the steering power of a conventional rudder.  Sails will produce both forward and lateral thrust.  On most sailing ships this lateral thrust is counteracted by an opposing force generated by the motion of the keel through water.  An airship does not receive this benefit since it has not surface submerged in much denser water.  This means additional sails or other stabilizing mechanisms will need to counteract this force.
Another well known mechanism for both stability and additional buoyancy is outriggers.  Since we cannot suspend an envelops above the ship, where the sails now are, this seems an excellent place to put gas bags.  These outrigger bags will improve ship stability and provide the necessary lift.  Now let's see if they can be made to anything resembling a reasonable size.
The mass of air displaced by hydrogen in the gas bags must be greater than the mass of the airship.  We'll assume the drag optimal bag configuration used by early 20th century airships, with length to radius ratio of 10.  Let's first consider just the hull, ignoring mass of sails, rigging, and the bags themselves.
How big will the hull be?  There have been many designs through history, and we won't try to find an optimal hull configuration for an airship here.  A few reasonable numbers for a sailing ship, however, in terms of length:
Width, T = 0.1L
Height, T = 0.1L
Surface Area = 1.7L + Vol/T or about 0.27L^2
Thickness t = (0.03L + 3.8) mm (for L in meters)
Setting the combined volume of the gas bags times the difference in density equal to the mass of the airship and rearranging:

For a wood airship this be approximated into a gasbag length to hull length relation:

Which does tend to underestimate by about 4 - 9% for lengths over about 10m (which it overestimates by less than 1%).  These numbers are not totally unreasonable.  It suggests that for a 10m long airship two outriggers approximately the same size as it would be necessary to keep it aloft.  For a 100m airship the outriggers would be half the size of the ship.  Consider an increase in gasbag volume of 1.5 - 2 times, so the outrigger length would be increased by a factor of about 1.25.  This, of course, will work better for smaller ships where more of the final loaded mass is structure.  The gasbag volume for large ships could increase drastically to account for cargo.  Internal cargo capacity for a sailing ship can be estimated by the Builder's Old Measurement.  A minor oversimplification will let us say that the mass of cargo in terms of the average internal volume should be about:

Which gives additional required length of

Another estimation that is often used for civilian ships is the Dead Weight Tonnage, or DWT, coefficient, the ratio load weight to lightweight.  This averages about 2.3 for cargo ships and 1.6 for passenger ships, and would be much smaller for light vehicles.  Adding this factor in, along with a reasonable safety factor finally suggests that the ratio of lengths should be about:

With this we can finally return to the question of propulsion.  Let's have a look at the total drag acting on the airship:

Hulls are designed to be fairly streamlined, and  a reasonable value for its coefficient of drag is about 0.03.  Inserting everything we've worked out and found so far:

Which highlights the important difference between apparent speed, seen by the sails, and true speed, seen by the hull.  The beta term will have a maximum of about 1.02, inserting the estimate for an airship on the order of 10m long, and assuming that the true and apparent speed are approximately equal finally gives a relation between sail area and length:

Astounding as it may appear the sail propelled airship does seem initially viable.  There's no reason that it couldn't be constructed and used, except that by the time the technology necessary to build and lift it was developed much faster and more reliable methods of moving by land and sea than sails had been invented.  The idea is, however, undeniably cool.