# Consumption modeling of wood in a boiler

I know very little about thermodynamics, and I need it to do my master's dissertation in Operations Research. I don't know if what I did is correct, and if it is, I don't know how to model the amount of wood I would need to setup the machine, nor how to model the loss of heat to the environment. Any help would be very welcome.

I'm trying to model an industrial boiler's behavior, specifically how much wood $$w_i$$ I would need to spend to produce the product $$i$$ and the amount of wood $$s_{ij}$$ the setup from $$i$$ to $$j$$ would cost - I have the setup time, processing time and processing temperature of each product $$i$$ for now. My objective is to build a capacity constraint of a mathematical programming problem, as explained below.

Params:

• $$w_i$$: the amount of wood that the product $$i$$ uses.
• $$s_{ij}$$: the amount of wood that the machine needs to use to change from production of product $$i$$ to product $$j$$.
• $$W$$: the amount of wood available.

The Decision variables:

• $$x_i$$: amount of product $$i$$ produced.
• $$z_{ij}$$: represents the exchange of production from product $$i$$ to product $$j$$.

The constraint would be like this:

• $$w_i x_i + s_{ij} z_{ij} \leq W$$

From some basic books of physics, I've seen something that I think I could use.

$$Q = c m \Delta T$$

With that, I could model the amount of energy that the boiler, product, and water inside would need to change from a temperature to the temperature the product would need to be processed. But I don't know how to model the loss of heat to the environment, that I think I would need to discover $$s_{ij}$$ from the time of setup or temperature variation (each product has a specific temperature).

Also, with the Net Calorific Value (NCV) of the wood used I could relate the amount of wood that I would need to generate the heat used as in $$Q_{combustion}=NCV w_i,$$ By the end, to discover $$w_i$$, considering the boiler and water is stabilized at the specific temperature needed to process the product $$i$$, I think I could use the efficiency $$e$$ of the process using the wood and have something like this: $$Q = c m \Delta T = e NCV w_i$$ $$w_i = \frac{c m \Delta T}{e NCV}$$

I don't know if the efficiency $$e$$ just depends on the wood used and the boiler, as at least I understood in Morissette et al. (2013) study, or if there is a variation with time.

I'm trying to set this data as real as possible, but for now, I don't have access to the company's data. If the company that I'm aiming to apply my model sees interest, I would like to have almost everything done, to just get their numerical data and test my model.

PS 1: Is there any way to know the efficiency of a type of wood in a boiler without specific experimentation (no input/output energy)? What kind of data would I need to have to do it?

PS 2: The kind of process here is of dye shoelace. There is a set of products kind, each representing a specific color of shoelace.

• What about the moisture content of the wood? Anyone burning wood knows dry wood burns better and is also better for the stove. Commented Jan 14 at 16:44

Wood has an energy content (you'll need to assume this, wood you get in a rainstorm makes less steam than wood that sat in a desert for a year). That times boiler efficiency equals the energy in steam you can produce. No one besides Babcock & Wilcox (boiler makers) is modeling heat loss to the environment. Everyone else is taking an assumption of efficiency. Go find some boilers and see what the manufacturers rate them at.

I assume whatever product you are making uses steam energy. So work backwards and figure out how much energy you need to feed the boiler.

Now as to your modeling project, I'm not sure why you need to model a boiler. You buy a boiler big enough to provide the steam you need, and it will produce it. Chipped wood feeds a wood boiler using variable speed augurs that change according to steam pressure. Use more steam, pressure drops and the augurs speed up. Modeling exactly how fast that happens is beyond the scope of pretty much everyone not working for our friends at B&W (or a competitor). The energy content of steam doesn't really change a lot at the normal pressures a saturated steam boiler will run at, so the pressure doesn't have to be super-exact, if it did you wouldn't burn wood.

I'm not sure if or how this helps you. I have run a wood fired boiler before, no one cared how to model it other than the tuning constants for the pressure control loop.

Certainly given the amount of steam you need, you can determine the wood required and the resulting carbon and particulate loads.

• Thanks for the answer. I want to model the boiler because I would like to know in this fake scenario the possible environmental impact of the wood combustion and also to generate a solution to a lot sizing problem that is viable for the amount of wood available in the company. Commented Jan 15 at 13:51
• But for what I understood, at least in the company that you worked for, this would not worth it. Still, is there anything you suggest to calculate the setup wood cost $s_{ij}$ or the only way you see not worth it? Commented Jan 15 at 14:10
• @JosaFerreira wood cost is legit, you need a source, and these days you will be competing against pellet manufacturers, for grills Commented Jan 15 at 19:20