Optimizing Heat Loss in a Rectangular Building

In summary: So are you saying instead of just 20x I should have 20x^2?No, 20x^2 is only if the walls are square, which they do not have to be. Let the height be z, the length be y and the width be x. Volume = xyz = 4000.
  • #1
fk378
367
0

Homework Statement


A rectangular building is being designed to minimize heat loss. The east and west walls lose heat at a rate of 10 units/m^2 per day, the north and south walls at a rate of 8 units/m^2 per day, the floor at a rate of 1 unit/m^2 per day, and the roof at a rate of 5 units/m^2 per day. Each wall must be at least 30 m long, the height must be at least 4 m, and the volume must be exactly 4000 m^3.

Find the dimensions that minimize heat loss. (check the critical pts and boundary pts)




The Attempt at a Solution


Since Length x Width=120, the height must be 33 1/3 m.

I'm having trouble starting with the equation for heat loss. I so far I have 2(10)x + 2(8)y + (5+1)z.

I'm not sure about the z part. The floor loses by a factor of 1, and the roof by a factor of 5. Since they both must have equal area, I figured they must lose at the same rate--that's why I added the 2 numbers together. Can anyone explain it better?
 
Physics news on Phys.org
  • #2
Well for some reason you're not using the notation x,y,z to represent the dimensions directly, instead in your formulation they represent surface areas of the walls. Why not just use them directly?

And yes it's ok to group the floor and roof areas together since their areas are the same. Just find the critical pts and boundary values as the question said. Alternatively you could use the method of Lagrange multipliers if you've learned it already.
 
  • #3
Defennder said:
Well for some reason you're not using the notation x,y,z to represent the dimensions directly, instead in your formulation they represent surface areas of the walls. Why not just use them directly?

So are you saying instead of just 20x I should have 20x^2?

The way my equation is now, I just get real numbers for my partial derivatives, so I can't even find any critical pts or plug in the boundary points...
 
  • #4
No, 20x^2 is only if the walls are square, which they do not have to be. Let the height be z, the length be y and the width be x. Volume = xyz = 4000.

To set up the total heat loss equation you have to decide which walls correspond to the compass bearings east, west, north, south and their respective surface areas. It's arbitrary but you must be consistent in this choice throughout the question.

And as for your earlier attempt, where does it say that length x width = 120? Remember that you are given the minimum dimensions but not the maximum ones.
 
  • #5
fk378 said:

Homework Statement


A rectangular building is being designed to minimize heat loss. The east and west walls lose heat at a rate of 10 units/m^2 per day, the north and south walls at a rate of 8 units/m^2 per day, the floor at a rate of 1 unit/m^2 per day, and the roof at a rate of 5 units/m^2 per day. Each wall must be at least 30 m long, the height must be at least 4 m, and the volume must be exactly 4000 m^3.

Find the dimensions that minimize heat loss. (check the critical pts and boundary pts)




The Attempt at a Solution


Since Length x Width=120, the height must be 33 1/3 m.
What tells you that length times width= 120?

I'm having trouble starting with the equation for heat loss. I so far I have 2(10)x + 2(8)y + (5+1)z.
I have no idea if this is correct because you haven't said what x, y, and z are! Are they areas? If you find x, y, and z, how will that help you answer the question?

I'm not sure about the z part. The floor loses by a factor of 1, and the roof by a factor of 5. Since they both must have equal area, I figured they must lose at the same rate--that's why I added the 2 numbers together. Can anyone explain it better?
 

Related to Optimizing Heat Loss in a Rectangular Building

1. How does heat loss occur in a rectangular building?

Heat loss in a rectangular building can occur through conduction, convection, and radiation. Conduction is the transfer of heat through materials, such as walls, floors, and windows. Convection is the movement of heat through air currents, such as drafts or air leaks. Radiation is the transfer of heat through infrared waves.

2. What factors affect heat loss in a rectangular building?

The factors that affect heat loss in a rectangular building include insulation levels, air tightness, window quality, and building orientation. Insulation helps to reduce heat loss through conduction, while air tightness prevents heat loss through convection. High-quality windows with low U-values can also reduce heat loss. Building orientation can also impact heat loss, as buildings facing south typically receive more direct sunlight and therefore may have lower heat loss.

3. How can I optimize heat loss in a rectangular building?

To optimize heat loss in a rectangular building, you can take several steps. First, ensure that the building is well-insulated, with proper levels of insulation in walls, floors, and ceilings. Next, seal any air leaks or drafts to prevent heat loss through convection. Consider upgrading to high-quality windows with low U-values to reduce heat loss. Finally, adjust the building's orientation if possible to take advantage of natural sunlight and reduce heat loss.

4. What are some common mistakes to avoid when trying to optimize heat loss in a rectangular building?

One common mistake is using too much or too little insulation. Too much insulation can lead to moisture problems and reduced air circulation, while too little can result in high energy bills and discomfort. Another mistake is not addressing air leaks or drafts, which can significantly impact heat loss. It's also important to consider the building's orientation and how it affects heat loss, as well as the quality of windows and their U-values.

5. Are there any new technologies that can help optimize heat loss in a rectangular building?

Yes, there are several emerging technologies that can help optimize heat loss in a rectangular building. These include smart thermostats that can adjust heating and cooling based on occupancy and weather conditions, as well as energy-efficient building materials such as aerogel insulation. Building automation systems can also help to regulate heating and cooling systems more efficiently, reducing heat loss in the process.

Similar threads

{Calculus} Minimize the heat loss
    • Calculus and Beyond Homework Help
Replies
9
Views
3K
Heat loss from the temperature difference
    • General Engineering
Replies
16
Views
2K
Engineering Tubular Reactor with catalytic walls
    • Engineering and Comp Sci Homework Help
Replies
6
Views
804
Understanding calculation of 2nd order LTI DE response to step input
    • Calculus and Beyond Homework Help
Replies
8
Views
277
Optimizing Air Flow for CO2 and Heat Loss in Vans
    • Biology and Medical
Replies
7
Views
2K
Engineering Heat transfer from composite pipe -- Is my answer right?
    • Engineering and Comp Sci Homework Help
Replies
1
Views
963
Heat Transfer and Combustion: Estimating Heat Loss in Insulated Pipes
    • Engineering and Comp Sci Homework Help
Replies
1
Views
2K
Calculate the Heat Loss in a Hot Water Tank from a Shower
    • Introductory Physics Homework Help
Replies
2
Views
2K
Heat Transfer -- Rate of heat loss from a pipe with superheated steam
    • Engineering and Comp Sci Homework Help
Replies
1
Views
2K
A Average temperature in a greenhouse
    • Thermodynamics
Replies
2
Views
1K

Hot Threads

Recent Insights

Back
Top