U R A ΔT, and Other Building Science Blandishments

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heat flow by conduction spray foam insulation in wall

Remember that science class where you learned about the three forms of heat flow: conduction, convection, and radiation? Well, class, let's make it real today. You insulate your walls, floors, and ceilings to limit the amount of heat your home loses on cold days and gains on hot days. That tells us that the amount of heat flow depends on the stuff in the surfaces that make up the building envelope, but what else affects it?

This is where we have to do a little thinking. Without too much difficulty, you might figure out that size matters, right? A bigger house will lose more heat on a cold day than a smaller house, all else being equal. The third factor is the difference in temperature between inside and out. These three factors are tied together nicely in a simple, little equation:

heat flow by conduction equation

U is the quantity that accounts for the insulation and how well the materials conduct or resist heat. A is the area of the surface that heat is flowing through. ΔT is the temperature difference. Multiply them all together and you know how much heat flows through a wall, floor, or ceiling. Pretty simple, right?

To find the total amount of heat flow through your home, you just do this calculation for all the surfaces of the building envelope, the boundary between conditioned and unconditioned space. The answer will be in BTU per hour, where BTU stands for British Thermal Unit (an antiquated unit that the British no longer use). When we do a Manual J load calculation, our software does this calculation for all the surfaces in the house we model. The same thing happens in a home energy rating.

If U seems unfamiliar to you, maybe you've heard the term R-value. Those two are intimately related, and if you know one, you know the other. R tells you how much something resists heat flow, and U tells you how much it helps it. When you're trying to reduce heat flow, you want high R-values and low U-values. They're inversely related to each other:

heat flow by conduction r to u equation

heat flow by conduction u to r equation

The other really important bit of building science here is that heat flows from higher to lower temperatures. If it's warm inside and cold outside, your house loses heat to the outside.

heat flow by conduction wall diagram

It seems obvious to us, but that's only because our experience has conditioned us to take it as a fundamental truth. The universe didn't have to be this way, though. There's nothing but our observation that heat actually does behave this way that tells us that it must be so. In other words, heat from the cooler air outside won't start piling up on its own inside your house. That's a good thing because if it did, your house might just burst into flames for no reason. This observation that heat flows from warmer to cooler is one part of the second law of thermodynamics.

If you want to think about how weird this really is, meditate on the fact that only the second law of thermodynamics stands between you and suffocation. That's right. For the same reason that heat won't pile up in your house, all the air molecules in the room you're sitting in now won't pile up in one corner, leaving you gasping for air. According to statistical mechanics, though, it's possible.


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Nov 14 2011 - 8:46am

Mr. Bailes, 
Your continue to look at the Manual J as Doctrine when as a home modeling tool it is simple and crude at best. On old houses you don't know the inputs.Insulation values are SWAG at best, Infiltration rates vary with wind speed, a Homes color,temperature differences, whether the house has a fence around it, the size and position of the house next door, what is going on inside and many other variables. Duct losses/gains are substantial and again quite difficult to determine without testing. Infiltration rates and system loss/gain values are not static, they are constantly changing, why not focus on this fact. 
As I am sure you are aware, on old houses, contractors learn quickly how to manipulate the manual J to give them the answers they want, they work it backwards. Easy to do.You have said you like to see 1000 sq. ft/ton on new houses, no problem. 
Even if manual J is perfect, you have selected a design condition that will oversize your equipment 98% of the Time. In your equation above, the temperature difference is constantly changing from 0 to a lot.. Why not look a models that focus our attention on the fact that loads are constantly changing, and therefor look a equipment that modulates capacity. Why are we still using the same tools and approach used by Fred Flintstone? I am also sure you know Manual J cannot pass the BESTEST. The happiest and most comfortable homeowners I know are the ones with Modulating Heat and Staged AC, equipment that changes capacity to changing conditions, yet these systems are less than 5% of all those sold. I suggest we focus on how loads constantly change rather than what they are at some arbitrary point. 
Keep up the good work, 

John Poole
Nov 14 2011 - 9:53am

I don't see any partial differential equations here, but this is a good start. Deltas are good. Little deltas even better! :-D

Allison Bailes
Nov 14 2011 - 10:42am

Phil: You make a lot of good points, of course. Yes, there's a lot of uncertainty in modeling existing homes. Yes, equipment that can change its capacity can perform better. It's also more expensive to install and repair. As with anything, there are pros and cons. Don't get me wrong. I like variable capacity, but it's not always the best solution. 
John P.: No, no. That comes next week, when we solve the partial differential diffusion equation.

George Reynolds
Nov 14 2011 - 5:01pm

The picture of the spray foam insulation with the "nominal" thickness variation (which exceeds the Version 3 grade 1 spec, raises a more difficult question than balancing equations. An IR analysis will normally yield quantitative variation of 25%+ unless the cavity is full and flush. Is there any guidance from EPA or Resnet that excuses spray foam from having voids in cavities? 

Allison Bailes
Nov 14 2011 - 6:05pm

George R.: Good eye! In this case the walls were undersprayed intentionally because they're not your standard 2x4 walls. I'm pretty sure the one in that photo is a 2x8 wall, so it can still be Grade I as long as the rater doesn't overstate the R-value.

John Mattson
Nov 15 2011 - 1:57pm

Good article overall. Small point, all else being equal (which never is) a house with larger SURFACE AREA will loose/gain more. "Size" can mean volume as well as surface area so be careful using "size". Your equation comes out in heat per unit of surface area. Second, and a really small point, of interest only to physics types, it is Grad-T, the temperature gradient, not simple difference. They are subtly different.

Nov 15 2011 - 2:14pm

Mr. Bailes, 
1. When would modulating equipment not be the "best" solution? 
2. Why use a crude approach like manual J with all the flaws when better tools are available?

David Butler
Nov 15 2011 - 9:16pm

Phil: While I agree with most of your points (although I would hardly refer to Manual J as being a crude tool for modeling design loads), it is not necessary to have modulating equipment to achieve comfort. I've heard that argument many times before, mostly from the very folks who sell this (very expensive) equipment.  
The reason most homes are uncomfortable is *not* because the equipment is oversized during part-load conditions, but rather because of poor design and installation practice (especially duct systems) and egregious oversizing, along with a litany common envelope flaws. The fact is, a poorly designed and installed modulating system in a home with a faulty envelope will not deliver comfort.  
Let's get the basics right before we start talking about higher performance equipment, lest we waste the client's hard-earned money on the small potatoes. 
BTW, my current home and previous home (both with single-stage heat pumps) are the most comfortable and economical I've ever lived in.

Nov 15 2011 - 10:46pm

Thank You for your response. 
You are the first person I have ever met, ever, claim single speed AC and Heat is as comfortable as modulating equipment. 
Since loads are constantly changing why would a single speed AC/Heat make any sense? Would you buy a car with an on/off switch for a gas pedal? Full speed or stopped? 
In many towns, Boston for instance, carrier will state that their 2 stage AC will remove 30x the moisture, run 84% of the time on first stage, and provide far superior comfort. IAQ devices only work while units are running. Your manual J sized single stage furnace will short cycle all the time except when it is -10 outside.  
I do not know anyone with 2 stage AC units who would ever go back to a single stage unit.They would certainly not agree their money was "wasted" 
Is someone buying a geo system wasting their money? What if I don't like unsightly outside units, or noise.What if I'm a Greenfreak? How can you make this call for someone else. 
Fact:It is impossible to know just how comfortable you can be until you are comfortable. Most folks with single stage units think that is all comfort is. 
Since your last 2 Homes had singe stage units this would seam to apply in your case.You may not have experienced comfort in your home yet, A tragedy. 
I am sure I can find a contractor buddy that would be happy to get you a nice price on a 2stage HP, $50 says you change your mind. 

David Butler
Nov 16 2011 - 12:04am

I never said single speed is "as comfortable", but when compared to what most folks are used to, I doubt many people would be able to tell the difference.  
You said... 
>I do not know anyone with 2 stage AC units who would ever go back to a single stage unit. 
I would argue that this is because their dealer did a great job brainwashing them (sorry, couldn't resist). Seriously, it's because their point of reference was likely so bad. After spending the money, and if their new system is designed and installed properly, then of course they're going to feel that way. Multi-stage and variable capacity equipment is good stuff. Really. It's just too expensive for what it provides. And has some long term service issues.  
I have literally hundreds of clients throughout the country for whom I've designed and specified HVAC who are thrilled at the comfort they have (check the testimonials on my website about-us page). My clients too often come from bad homes with bad HVAC systems. I wonder if they could really appreciate the subtle difference. It's like going from 1950's TV via rabbit ears to HD-720P (single stage done right), then to fully modulating (1080P).  
And BTW, I never specify ground source heat pumps. I've yet to find a situation where they're cost justified, even with 30% tax credit. In my own home, my 15 SEER Lennox air source HP only consumes around 900 kWh in winter, and 2200 kWh in summer. A ground source (assuming it would even work here) might save $150 (40%) off what I spent for hvac before I went net-zero energy. 
I believe the most important prerequisite for comfort is a good envelope. Perhaps it is you that hasn't experienced how comfortable one can be with single stage HVAC.

Nov 16 2011 - 7:47am

OK,I think I've got it. 
To avoid the horrors of oversized equipment we use math and science and our manual J spreadsheet to select on/off equipment that is oversized 99% of the time.  
To those folks that elect to choose a heating source that delivers just the amount of capacity they need we tell them their perceived benefits are due to a post hypnotic suggestion. 
And then we call ourselves building scientists. 
You gotta love it, What a business!