# Heat Loss for a Manufacturing Plant

## Homework Statement

I have a manufacturing plant, and I know the material/thickness of every wall and the roof. I want to estimate the plant as an empty box, where I know the surface area of each material surrounding the box (ex. 4000 sq ft concrete, 1000 sq ft of windows, 1000 sq ft of xyz roofing, etc.). I'm trying to determine how long this plant would take to go from 70F to 32F internally if the outside air is 0F and the heating to the building were discontinued.

I'm going to assume that no convection is taking place and only conduction and radiation are taking place. Would a building radiate a significant amout of heat, or should I only concern myself with conduction across the walls/roof?

## Homework Equations

Unsure, but heat transfer equations.
Possibly:
ΔT=RQ
ΔT=Q(rate)(R)/(A)

## The Attempt at a Solution

Not really sure to start with this. I'm planning on determining the rate of heat that is going to escape the plant, then looking at the specific heat of the internal air and determining how long it would take to drop the inside air to 32F. I'm looking for a lot of insight though. I'm not too sure of what factors I should be considering and what heat transfer principles/equations would apply.

For reference, I'm an upcoming senior Mechanical Engineer in undergraduate. I've taken thermo, however have not yet taken heat transfer.

Thanks for any help.

Chestermiller
Mentor
Would you be satisfied with getting a lower bound to the amount of time it would take?

Chester,

Yes a lower bound would be fine. I realize that the equipment inside will have heat interia (?) and will continue to heat the building a little longer than my estimate will say. I'm not sure the extent of this effect, and may explore that once I figure out the basic problem.

So yes, I'm looking for a semi-rough estimate, in a worst case senario (minus convection) situation.

Chestermiller
Mentor
Chester,

Yes a lower bound would be fine. I realize that the equipment inside will have heat interia (?) and will continue to heat the building a little longer than my estimate will say. I'm not sure the extent of this effect, and may explore that once I figure out the basic problem.

So yes, I'm looking for a semi-rough estimate, in a worst case senario (minus convection) situation.
OK. To get this lower bound, you're going to assume that all the equipment and all the air within the building are at the same exact temperature at any given time. So you are going to include the equipment in this approximation. Figure out the weight of all the air in the building, and multiply it by the heat capacity of air. Figure out the weight of all the equipment in the building, and multiply it by a representative heat capacity for the metal. Add the two together. This is the thermal inertia of the building contents. You are going to assume that the temperature on the inside surfaces of the walls is equal to the temperature of the building contents. You are also going to assume that the temperature at the outside surfaces of the walls is equal to the outside temperature (0 F). So, if you know the temperature of the building contents, you can calculate the total rate of heat loss through the walls. Set the thermal inertia of the building contents times the rate of change in inside temperature equal to minus the rate of heat loss through the walls. This will give you a differential equation for determining how the inside temperature varies with time. The solution to this equation will tell you the fastest that the inside temperature can cool down.

Chet