Csdrhrt

Yes, there are flaws. You don't get enough power.

Why?

Rain terminal velocity is about 10 m/s.

10 cm/day of rainfall means you will get on average 1.1 μm/s, or 0.0011 $l/scdot m^2 $.

The average power delivered by that rain is given by its weight times its velocity, thus, remembering that 1 liter of water weight about 10 N, we have that the average power is 0.1 $W/m^2$.

To move horizontally just 60 kg overcoming the rolling resistance of the wheels and the bearings, you would need to provide about 1.5 W (assuming wooden wheels with radius 1 meter), corresponding to about 15 square meter in the ideal case of 100% yield, more than the 9 you have available.

In the extreme case where you want to move 60 kg at 1.5 m/s of constant speed up you would need 900 W, meaning that you would need, a collecting surface of at least 9000 $m^2$, which is way larger than the 9 $m^2$ that you have.

And, mind, this is the estimate just for moving 60 kg. If we add to the calculation the mass of the vehicle and its payload the required surface further increases.

As has already been described, you can't get enough power from using the rain directly on an individual vehicle.

Hence the only way to proceed must be to use the rain indirectly.

We're going to build rain gathering header tanks all around the village each one will drive a waterwheel linked to a continuous belt along a stretch of road. A vehicle wanting to pass along that stretch of road will hook itself to the belt and ring the bell for the tank to be opened.

This approach removes the size limitation from the vehicle power supply.

You're still going to have limited power making climbing hills difficult. It's going to be well worth hooking vehicles into the system on the decent as well as ascent. That way you get the funicular effect of the descending vehicle helping to pull the ascending vehicle up the hill, while also controlling the decent speed and reducing the risk of runaway vehicles.

Just skip Wheeled stuff. [EDIT: apparently that is against OPs requirements, but I like my answer so it'll stay]

Replace your roads with perfectly level canals and "power" them by only having one end supplied with water at any time.

You need two between each city (back and forth) and each city needs a very large area to power its canal (unless you use small canoes only), but its viable, and once the canals are set up pretty cheap.

Put the cart in a river and let the flow of the river carry the cart. If the river is shallow enough the cart wheels will touch the riverbed.

The river is fed by the rain, and concentrates water from a comparatively large area into a small stream to give you the watts per square meter you need.

This is unidirectional, but being able to go anywhere was not in the question requirements.

This answer has been invalidated by requirements added to or clarified in the question after posting, but is being left for future web-searchers to whom it may be useful.

Use a system based on a Funicular to store water in raised reservoirs and convert it to mechanical energy. You can use this to drive carts along rails, like a cross between a cable-car and a tram/railway.

Since you rely on the reservoirs refilling between carts, this is more likely to be a scheduled system for transporting goods at scheduled times, rather than a public ad-hoc transportation.

It won't work, no way

[This is a repost of a previous, wrong and therefore deleted answer. Orders of magnitude are hard. The deleted answer had accumulated a number of upvotes, showing that orders of magnitude are difficult for others too.]

To fix ideas, let's assume that the vehicle is supposed to work by catching water on its roof and directing it to fall on a waterwheel. Let's compute available power.

1. 10 cm of rain per day means 1.16 µm per second.




2. Suppose the catchment area is 3 m by 2 m = 6 m².




3. This means that 6 × 1.16E-6 = 6.94 grams of water will be caught per second.




4. Say that this water falls 2 meters: since energy is weight times distance, it results in 9.6E-3 × 9.81 × 2 = 136.25 milliwatts.

If 136 mW are quite obviously not enough to power the vehicle; powering the vehicle from rain directly won't work.

Note: L.Dutch had the idea of using the kinetic energy of water droplets to power the waterwheel; see their answer for the computation. It may give more power than trying to catch water and make it fall on a waterwheel, it still is a pitiful amount of power per square meter.

• 
  

  But wait! We can charge the reservoir!

  
  
  

  Let's borrow an idea from Mr. Musk's electric cars. We don't have to power the vehicle all the time: we can charge a reservoir over a period of time and use the accumulated energy at a higher power.

  
  



• Let's assume that we accumulate water for 10 hours; the total accumulated energy will be 136.25 mW * 36000 = 4905 J.




• This is enough to give us 1 horsepower for about 6.6 seconds. Still too little.




• Let's make the charging area larger. We can catch water over a larger area, and then pour it into the vehicle's reservoir. Let's say that we want the vehicle to have 4 horsepower for 1 hour; this comes to about 11 MW of energy.




• To get that amount of energy from water falling 2 meters we need some 546 tons of water. This is a ridiculous amount; there is no way to make it work: the required catchment area would be enormous, the reservoir on the vehicle would be a hundred meters tall, and the vehicle won't be able to move under its own power.

Two ways to make it work:

1. A wide catchment area at either end of the road collects rainwater and forces it through a Pelton wheel. The wheel drives a chain of belts from one end of the highway to the other. So, you now have two belts on either side of a road, rotating in opposite directions. Your vehicles will latch on to the belt in the direction it wishes to travel and be carried along. The size if the catchment and the flow rate determines the maximum load that can be moved, and the diameter of the turbine wheel determines the speed.


2. Same Pelton Wheel as before, but this time, it is used to charge a battery, which can then be used to drive a motor, moving the car.

Can you collect the rainwater in a cylinder and light a fire under it turning it into steam? If the rainfall is constant, you could potentially have a steam powered vehicle which doesn't need to be refilled from water towers as it could simply be left in the rain to refill it's water tender. Of course you would have to make some of the components from metal and others could potentially be non-combustible material such as stone. By 1800, the likes of James Watt and Richard Trevithick had created rudimentary steam engines which were just beginning to be commercially useful.

Another option, if you have a world where there are many mills powered by waterwheels, you could use these are stationary winding engines so you'd attach a rope to your cart and have the action of the turning wheel pull the cart to the mill. Once it arrives there, disconnect it and attach it to the next mill. Assuming you are delivering to locations which have proximate mills, this could be an option.

The only other thing that springs to mind would be a rain powered impeller so rain falling on the vehicle is channelled to a drain point, this drain point contains an impeller driving some sort of motor to charge a battery. If it's raining all the time, this would be constantly charging. The cart left in the rain for a while would increase it's battery charge. Problem is, for this to work you'd need metal for the windings in the motor and the charger and it isn't exactly pre-industrial.

I'm going to assume other posters got the value of 900W correct.

As a comparison:

• An untrained adult can put out about 100W of power when riding a bike.


• A competent commuter could be doing 150-200W


• A race sprinter can do well over 700W but only for short periods.


• Team Sky riders can do 400W for the duration of their race.

At an estimate, your cart would need 4-6 people pedalling through a shared geared axle to drive a roadwheel. Getting up to speed would be the most effort, but once its rolling then speed maintenacne of 5 km/h or 3 mph will have a lower power input requirement.

Something vaguely like this but with less beer.

Downsides

• significant climbs are going to be very slow, because of the load on the cart.




• Your cart won't be particularly aerodynamic, so more than 5 km/h will be exponentially harder due to wind resistance.




• Metal requirements - ideally you'd have metal bearings and races to decrease friction, and a metal chain and cogs and axles for strength. However hardwood wooden chain/gears and bearings certainly have existed.




• 6 people on a 3 metre long cart will use a lot of load space carrying the people around. I'd guess less than half of the deck will be available for load.

Another variation on this is to use the people as draught animals, and have them tow the cart on traces and shoulder harnesses. Perhaps a punishment for convicted criminals?

Other posters already showed that it seems to be impossible with the given requirements.

However, when able to alter those requirements, those would be my proposals :

• Altering the streets to decrease friction - have your vehicles "glide" on a water film(similar to a phenomenon known as aquaplaning).
  For that purpose, you would probably have to slightly adapt the streets, however.




• Besides, if it was possible, I would suggest your character to add a hot air balloon to it's vehicle; it's quite simply to build & attach, and you can travel by wind - as your salesman doesn't really care where he goes next, that should be fine - and if the wind is moving you the wrong way, just make your balloon stick to the ground and wait until wind direction changes ;)

Would storing the energy temporarily be acceptable? Since there is plentiful wood, you could harvest the rain and use a steam engine to convert the energy of the rain in to tanks of compressed air. Which can be used to power a vehicle, similar to the one you pictured.

Edit to answer your question: There seems to be no obvious flaw in your idea, if you consider that your story takes place on a planet different from earth. This does not mean you can disregard the laws of physics, it means you can change constants that are specific to our planet in your math.

To be more specific:

After finding the equations you find most suitable, alter the constants that you want, so that enough energy can be sufficiently be harnessed.

Your story is obviously not situated on our planet, which means that you are free to modify:

1. gravity


2. liquid that the rain consists off


3. density of materials for construction


4. mass of the creatures, and the materials they are interested in carrying.

Dont let our planet restrict you, unless you are set on the story playing out there.

If you do want the planet to be as earth-like as possible, play around with changing the constants minimally. Some should give you more likely vehicles, with less modification then others.

Okay, so water from above wont give you enough energy to get your vessel moving? Than combine it with something, that might offer a more lasting impression, like... Sodium (Na).

While that might collide with your "no much metal" rule and maybe with your expected level of technology, you still could put (handwave) your inventor or your society into a situation where the availability of an element that reacts in the same (or even more exothermic) manner with water would be possible.

But I have to admit that I cannot offer any calculations about how much thrust that could apply, nor what it would do to your vessel... at least, there would be enough water around to put out the fires....

This is actually a very similar reason as to why the sparks from a welder don't hurt. Individually they carry very little energy, just like a rain drop.

So you can't use the power of a falling raindrop to power anything beyond erosion.

As for a reservoir, now your fuel weighs more than the power it generates. So no matter what, the plan doesn't work. Water simply doesn't carry enough energy to do much of anything. Your cart would be better off with pedals and having the water collect in a small jug to quench the thirst of whoever is doing the peddling.

This is a bad idea:

Imagine the government had too much money and built small rivers, charged with water. And your carts has at the sides water wheels. The water wheel powers the cart (it can't have a battle racing mode)

When turning directions you just disassemble the wheel and push the cart on the other side of the street, assemble the wheel, with an inverter (since the water is flowing in another direction) and your good to go.

Or you just make a small nuclear fusion power plant on your cart.

Hope this helps :)