When is it More Cost-Effective to Add a Staircase vs. More Floorspace?

[H2Bro]

Thought experiment:When is it "cost-effective" to add a staircase Vs more floorspace

This is a thought experiment I, erm, thought of.

Imagine a cylindrical (upright) office building with a central staircase at the focus. Workers exiting the staircase walk to their desks, and eventually walk back to the staircase. At what point does the energy required to walk from the staircase to the outermost desk, i.e. building circumference, exceed the energy required to walk up an additional flight of stairs?

I posed this while thinking about whether there are "optimal" proportions to human habitats. I lack any schooling in physics, so after buggering about in wikipedia I came up with the following calculations. Do please look it over/offer revisions because the value I found surprised me.

Assumptions
- office worker is 70kg
- 70kg person walking @ 4km/hour burns 216 cal/hour, or 54 cal/km (taken from http://www.brianmac.co.uk/energyexp.htm)
- This translates into walking efficiency of .226 Watts per meter (
- horizontal length of the staircase, i.e. length of stairs added together, is 10m
- floor-ceiling height is 2m with no discernible floor thickness
- People travel 1/3rd as fast when going up stairs (as viewed from above)

So! Let it begin.

Energy to go up the stairs
F(stairs)= energy to ascend 2m + energy to traverse 10m
Lifting a 700N person up 2M requires 1400J of energy.
A person approaching the staircase at 4km/h would drop down to 4/3km/h, meaning it takes 0,0075hours or 27 seconds to travel the 10m (seems about right).
1400J over 27 seconds = 51 Watts
Plus the energy required to traverse 10m, as defined above = .226 Watts X 10m = 2.26 Watts

Energy to go up staircase = 51 W + 2.26 W = 53.26 W

Energy to travel to desk

Now, it would be most efficient if a person only walked up the flight of stairs if it took more energy to walk to their desk and back (a distance of 2r), in other words when:

F(desk) - F(stairs) \geq 0

Since we know the value of F(stairs) = 53.26W, and that it takes .226W to walk 1m at 4km/h, finding how far one could walk for 53.26W of energy is:

53.26W / .226W = 235 meters

Because this distance is a round trip, our ideal worker would not want to have a desk further than 117 meters from the stairs.

I did a similar calculation for a person walking at 7km/h, which burns 411 calories/hour, or 58cal/hour = 245 W/km = .245W/m. Using this value I find a maximum distance of 397/2 or 198m (!).

Now, this value seemed quite high (and an immensely large office building, at that), and I have a suspicion I didn't correctly translate/convert the energy units all the way through. Also, perhaps you can see places to refine the calculation, i.e. energy cost of standing up/sitting down, stopping/starting, or what-have-you.

 

[Michael C]

An interesting question. I haven't checked all your calculations, but the end result doesn't surprise me: walking up stairs is a lot more tiring than walking on a flat surface.

In order to calculate the optimal proportions for the building, however, you'd also need to take into account the time it takes to walk to the desk: walking 117 metres on a flat surface will take much longer than walking up a flight of stairs.

 

[256bits]

H2Bro
Did you take into account the energy and time needed to walk back down the stairs to the person's starting position.

 

[H2Bro]

Thank you Michal C and 256bits for your feedback.

@Michael:
The time taken to walk to the desk would be 3.5 minutes at 4 km/h, giving an energy input of 15.2 Watt minutes. Because I made the total energy required by both walking to the desk and up the stairs equal, going up the stairs uses the same amount of energy in 30 seconds giving 106.25 Watt minutes of input energy.

Now, I have these two figures: how do I go about using them to find the optimal proportions? Would you suggest taking the staircase energy input and finding how far 118.3 Watt minutes would take a worker on flat ground?

@256bits:
This is a good suggestion. The original energy cost of ascending the stairs was
E(ascend) = E(up) + E(over)
E(ascend) = 51W + 2.26W = 53.26W

I will take a crude estimate that it requires 1/3rd as much energy to lower oneself down each step than it does to lift oneself up, but of course the horizontal distance stays the same.

E(descend) = 51W/3 + 2.26W = 19.26W
E(descend) + E(ascend)= 72.52 Watts, and seeing as it takes 27 seconds to go up and for the hell of it let's round this to a 1 minute round trip,
E= 72.52 Watt minutes of input energy. Perhaps you can just help with the conceptual part of this. Would it make sense to use Watt minutes as my yardstick, so I try to measure the energy to go up stairs and energy to walk across flat ground in Watt-minutes and not just Watts?

At first I thought this might result in a pyramid shaped building, because a worker would walk much further on the first floor rather than walk up and down, say, 3 flights of stairs. But, the marginal energy cost remains constant, so if the first floor desks are far away enough to justify going to the 2nd floor, and the 2nd floor desks are filled to the same amount as the 1st floor, it will still be "rational" to ascend an additional flight of stairs. So the building have a constant thickness, I believe.

Edit: adjusted original staircase watt-minutes to reflect 30 second travel time. Now I must have made an error, because here I have 106 Watt minutes to go up the stairs and 72.5 Watt minutes to go up AND down.
2nd Edit: saw what I did wrong there. Roundtrip cost of going up and down the stairs is 145 Watt minutes. Any ideas how to use this number to go about finding optimal building size?