posted 12-07-2012 05:53 PM ET (US)        

--a passage of a particular distance (in D miles) is to be made;

--the fuel available is limited (to G gallons);

--there are two possible speeds available for travel during the passage;

--there is a speed (S1), but at this speed the fuel burn (of F1 gallons-per-hour) would consume all the fuel available before reaching destination;

--there is a speed (S2) with better fuel efficiency (F2), which can be used to conserve fuel; you could make the whole passage at this speed but we assume that S2 is a slower speed and we wish to minimize the total time to make the passage; and,

--by combining the two speeds the passage can be made in the least time if we run as long as we can at S1 until we have just enough fuel left to continue at S2 to the destination.

There are some situations where this method cannot be used. Let me state the obvious:

--if the passage cannot be made entirely at the more efficient speed, there is no solution possible

--if the passage can be made entirely at the less efficient speed (and assumed to be faster speed), there is no solution necessary

The algorithm computes the amount of time (T1,T2) at each speed (S1,S2) that will satisfy the conditions above. The times and speeds do not have to be run in order or in continuous segments. You could mix them up, as long as the total time at each speed is correct.

When using the algorithm you should hold in reserve some fuel. The algorithm uses all the fuel in its calculation. You must provide a reserve by entering less than the total amount of fuel you actually have. A ten percent reserve is the minimum I would hold back.

Now I explain the algorithm's details:

A boat has a fuel capacity of G (gallons) and wishes to make a passage of D (miles) in distance.

The boat operates at two speeds. At S1 (mph) the boat goes a certain speed and burns fuel at a rate of F1 (gph).

At S2 (mph) the boat goes a slower speed and burns fuel at a rate of F2 (gph), a more economical rate

The total time to make the passage will consists of the time spent at S1 called T1 (hours) and the time spent at S2 called T2 (hours).

We can then say that the total distance travelled will be

(1) D = (S1*T1) + (S2*T2)

and the total fuel burned will be

(2) G = (F1*T1) + (F2*T2)

Whatever fuel we don't burn at S1 will be burned at S2, so we can say

(3) F2*T2 = G - (F1*T1)

and the distance travelled at S2 is whatever we did not cover at the other speed, so we can say

(4) S2*T2 = D -(S1*T1)

The amount of time we can run at S2 is determined by how much fuel is left after running at S1 for time T1 at burn rate F1. We can then define T2 as

(5) T2 = [G -(T1*F1)] / F2

Now we substitute the equivalence (5) for the variable T2 in equation (1) and solve for T1

(6) T1 = [ D - (S2*G/F2) ] / [ S1 - (S2*F1/F2)]

Now we evaluate to find a solution for a particular set of conditions as follows

D = 250 miles
S1 = 25 mph
F1 = 9 gph (this is about 2.75-MPG)
S2 = 6 mph
F2 = 1 gph
G = 75 gallons (the amount of fuel we have to burn)

We find that we must run at S1 for 6.89-hours and at S2 for 13-hours

CHECKS:

25-mph for 6.89-hours = 172-miles
6-mph for 13-hours = 78-miles
TOTAL = 250-miles

9-gph for 6.89-hours = 62-gallons
1-gph for 13-hours = 13-gallons
TOTAL = 75-gallons

Here is another check, this time for 300-miles and the same speeds, rates, and fuel available

S1 = 5.172-hours
S2 = 28.448-hours

LEG ONE was 5.172-hours at 25-mph and 9-gph for 129.31-miles and 46.55.16-gallons
LEG TWO was 28.4484-hours at 6-mph and 1-gph for 170.69-miles and 28.4485-gallons

posted 12-08-2012 09:01 AM ET (US)            

Here is another method to solve the problem of mixing two rates of fuel burn.

We have a distance of D (miles) to cover with only G (gallons) of fuel available. Therefore we must make good a fuel economy of D/G miles per gallon for the passage. We have two speeds available. At S1 (mph) we get M1 (mpg) which is not enough to make the passage, but at S2 (mph) we get a better fuel economy of M2 (mpg) which is more than enough to make the passage. We want to blend these rates to get an average of D/G (mpg) so we can make the passage on our limited fuel. We need to know how much time at each speed we can run.

The average fuel economy of the trip will be a blend of the two economies as follows:

(1) D/G = a*M1 + b*M2, where a,b are coefficients such that a + b = 1

The coefficients tell us the weighting of each fuel economy to be used in the averaging. We will know D, G, M1 and M2, so we can solve for a and b. With some algebra, we find

(2) a = [(D/G)-M2] / (M1-M2)

Now we check with the same parameters we used before and compute a:
D = 250-miles
G = 75-gallons
M1 = 2.77-miles per gallon
M2 = 6-miles per gallon

a = 0.8275
b = 0.1725 (from the relationship b= 1 - a)

First we check to see if this will blend to our target fuel economy:

D/G = (0.8275 * 2.77) + (0.1725 * 6)
D/G = 2.2922 + 1.035
D/G = 3.327
which agrees closely with
D/G = 250/75 = 3.33

Now we know the weighting coefficients, but this also tells us how much fuel we have to burn at each speed. (This may not be obvious but it is true. See my earlier article, hyperlink at bottom of this article)

At S1 we need to burn 75 * 0.8275 = 62.0635-gallons
At S2 we need to burn 75 * 0.1725 = 12.9375-gallons
TOTAL FUEL = 75-gallons

From this we can figure the times and distances:

To burn 62.0635-gallons at 9-gph needs 6.896-hours covering 172.4-miles
To burn 12.9375-gallons at 1-gph needs 12.9375-hours covering 77.6-miles
TOTAL DISTANCE = 250-miles

For more on weighting factors for averaging fuel economy see my article in the REFERENCE section:

posted 12-08-2012 02:04 PM ET (US)            

In actually running a passage this way, you would get on plane and do your best to maximize fuel economy. The fuel remaining (FR) would be closely monitored along with the distance to go (DTG). At any moment the needed fuel economy to complete the passage will be DTG/FR. The moment DTG/FR began to approach your best fuel economy at the lower speed, you should get off plane and transition to the more fuel efficient speed. If you didn't you would run out of fuel before reaching destination.

posted 12-12-2012 11:04 PM ET (US)            

It was quite explicitly mentioned that the algorithm does not allow for any reserve. The algorithm burns 100-percent of the fuel you input as being available. It was also explicitly mentioned that a reserve allowance should be provided when making a passage using the speeds and rates calculated by the algorithm.

Apparently you agree with my recommendation, but the way in which you have replied makes me think you are suggesting something that I have not already recommended. Perhaps you need to re-read the article to find my clear recommendations about the fuel necessary.

It is prudent to carry extra fuel in any operation of a boat (or other transportation device) that runs of fuel. Usually an allowance of one-third fuel reserve is often suggested. I don't think experienced boaters who attempt passage of more than 200 miles between fuel docks will have much confusion about fuel reserves.

There is a further problem with the recommendation as made--we are to "just carry the extra fuel." The problem is how much extra fuel?

posted 12-14-2012 12:02 AM ET (US)            

In the passage that was used in the example, the distance is easily covered with the available fuel in just the main tank. We have 70-gallons and can expect 6-MPH. This gives a range of 420-miles. Derating by holding one-third in reserve still gives us 277-mile range. The passage is only 250-miles.

The purpose of the algorithm--as stated at least twice in the initial article--is to explore how to blend a faster speed with a slower speed in order to reduce the time of the passage as much as possible within the constraint of a fixed amount of fuel to be burned. This has nothing at all to do with the notion that we must provide an allowance for things to go wrong.

We always have to provide an allowance for things to go wrong or to go less than perfectly. But before we can allocate an allowance for fuel needed we need to know just now much fuel we need if things go right. That is basis from which we calculate an allowance. Exactly how much to allow for a reserve is up to the individual vessel master, the nature of the passage, and the circumstances under which it is to be made. I don't know how you put that into an algorithm.

Regarding the recommendation to be "more worried about the weather." There is no input or variable in the algorithm for weather. The algorithm tries to find the fastest way to cover a distance with a limited amount of fuel.

Concern for the weather is a constant and ineluctable element in operation of a vessel. I do not think that it is necessary to build into my algorithm any consideration for the weather.

In order to introduce a variable into an algorithm and employ it in a calculation there is the presumption that there is some quantitative value that can be employed. If you can develop an algorithm for introducing weather into the calculation, please let me know.

posted 12-14-2012 07:46 AM ET (US)            

Do your algebra before you take your trip
Make sure you check your math!
If you run out of fuel out there
Your subject to jimh's mighty rath!!!!!!

posted 12-17-2012 06:20 PM ET (US)            

quote:

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6-mph for 13-hours = 78-miles

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posted 12-17-2012 10:44 PM ET (US)            

The smallest I have seen are 25 gallons, but they take up quite a bit of room when full, something like 4'x4'x.5'. When they are empty, they can be rolled or folded then stowed, which is an advantage over jerry cans.

Unfortunately, a few that I looked at on the web were around $500.00.

posted 12-20-2012 09:27 AM ET (US)            

After many trips with my boat and motor, I have found that I can save fuel by moving at displacement speed for some portion of a long leg. On several of our trips we have tried going along at displacement speed for an hour or two. When moving at displacement speed we can typically go along at 6-MPH. If you are in a scenic area, it is not a big problem to just relax and enjoy the scenery. We call this the Whaler-as-trawler mode of travel.

What becomes tedious is having to steer for hours while moving at 6-MPH. If our boat had an auto-pilot it would be much more relaxing. We'd let the auto-pilot steer on some of these legs. The problem with the auto-pilot is the cost of installing one. It would be several thousand dollars for me to set up an auto-pilot.