Cost Estimation capacity exponent formula

[LT Judd]

TL;DR Summary

How to modify the capacity exponent formula for cost per unit of capacity

To estimate the cost of an item of equipment or plant from another of a known cost and size or capacity , we use a well known formula Cost B = Cost A * ( CapacityB/Capacity A)^n, where n is a factor usually around 0.5-0.8 depending on the plant or equipment involved. These exponents are published in various books and publications,How do you modify this formula to get the cost per unit of production. For example ,the $/MW of a 3MW Boiler vs a known 100 MW Boiler. I can convert it to total cost , perform the calculation then turn it back into cost per unit, but I thought there may be a more "elegant" way to do it. I have a feeling it has something to do with logarithms , but just can't get my head around it.

 

[anorlunda]

Cost B = Cost A * ( CapacityB/Capacity A)^n,

What are the units in that formula? Cost B, is that $/day for the plant or $/item made?
Capacity B, is that units per day? Units per dollar?

 

[LT Judd]

What are the units in that formula? Cost B, is that $/day for the plant or $/item made?
Capacity B, is that units per day? Units per dollar?

No, For the first formula, it's total capital cost so for example it might cost 200k to purchase an 11 tonne per hour boiler how much does it cost to buy a 25 tonne per hour boiler if the exponent is 0.7. I understand that part.
What I want know is what's the formula if instead of $ for cost, you use $/tph.

 

[ChemAir]

What I want know is what's the formula if instead of $ for cost, you use $/tph.

For cost, the units are currency. $/currency capacity (edit) is a different unit. Can you explain what you mean by this, or clarify your question? This sounds like you may be trying to optimize capacity, rather than calculate cost for just a larger widget.

The units on the upscale calculation are pretty self-evident.

If a 10 ton unit costs $1,000, then using this formula, a 20 ton unit costs $1,625, using a 0.7 exponent if my arithmetic is right.

You can attempt to change the units of capacity to incorporate a time component, but the correlation won't necessarily be correct. I am not sure I am answering the question you are asking.

 

[LT Judd]

Sorry didnt explain myself very well, Its really a maths problem rather than an engineering one.
I am trying to do a cost comparison of different types of power generating plant. One standard way to do that is to find a known plant of a certain size at a certain time in a certain place and then convert it to the present time and the desired capacity and the desired location via a number of indices which are available in various textbooks and publications , with the formula from my first post.
In my case I am talking about power plant, but the same idea applies to chemical plant and indeed individual items of major equipment in such plants , and I am only talking about the capacity index , not the time or place.
For some one who is not a professional estimator in business like myself , The "known" values are hard to come by as they are usually the IP of large firms and consultants. What is publically available is often quoted in ($/kw ) or dollars per some other parameter for equipment of a certain size .
For example a industry guide may say;
for a theoretical plant of 380 MW , major equipment cost may be xx/MW, grid connection cost may be YY/ MW and Balance of plant may be ZZ/kw. exponents for each may be different.
In my case I have the "known figure" in $/MW and the out put in $/MW, I can convert the known figure to total dollars , do the calculation and then convert back again, but because I am trying to model a large number of case in a large excel spreadsheet , that introduces more chances of error and is a bit unwieldy. Something tells me there must be a simpler way but can't quite think of it.
See attached diagram , the right hand one is what I imagine the relationship between cost per unit and capacity must look like.
Is one the integral of the other or something like that?