Insulation Contractor Derby KS
Once upon a time, people thought the cost of a house was the price they paid for it, and the cost of insulating a house was the amount they paid the lumberyard or the insulation contractor. In the late 1970s, and periodically since, we learned that the cost of owning a house includes the cost of heating and cooling it in the years following purchase. The long-term cost associated with insulating a house is the life-cycle cost, consisting of the initial cost of the insulation plus the cost of energy lost or gained through the insulation over the remaining life of the house.
Illustration 1 shows the life-cycle cost of insulating and heating 1 square foot of an exterior wall of a house. The horizontal scale is thickness of insulation; the vertical scale is dollars per square foot. Curve A represents the cost of insulating 1 square foot of wall. We assume that there is a minimum fixed setup and overhead cost, and thereafter an additional cost per inch of material installed. Curve A is therefore a straight line. Curve B represents the cost of energy lost (and/or gained in a cooling climate) through the same surface over the lifetime of the insulation. It starts at the cost of energy lost through an uninsulated wall but drops rapidly as insulation is added. In fact, it drops by nearly half with the addition of the very first inch. But then the law of diminishing returns sets in. Since total energy loss can never drop below zero, curve B gradually approaches zero at very high insulation thicknesses.
The curve we are most interested in is C, the life-cycle cost. Curve C is plotted by adding the values of curves A and B: the installation cost of 1 inch of insulation plus the cost of energy loss through the wall with 1 inch of insulation, and so on, inch by inch. It turns out that curve C always has a minimum: i.e., there is always a unique thickness of insulation resulting in the lowest combined cost of insulating and conditioning the house over its life. We call this precise thickness of insulation the optimum thickness.
The obvious way to discover this thickness is to calculate curves A and B, inch by inch, and add their values together. But that is a lot of work! Economists use a trick they call marginal analysis. Heres how it works. The marginal cost of doing something is the incremental cost of doing it just a bit more. In our case, it is the incremental cost of adding just one more inch of insulation. The marginal benefit is the incremental benefit realized by doing that last bit more -- in this case, the additional lifetime energy savings due to the addition of that last inch of insulation. Illustration 2 shows the marginal analysis for wall insulation. Curve A is once again the cost of insulating. This time curve B is the value of energy saved due to insulating (lifetime energy savings). It generally costs the same amount to increase insulation thickness from 9 to 10 inches as to increase it from 1 to 2 inches. The marginal cost in this case, therefore, has a constant value, as shown by curve C. The energy savings realized by increasing the insulation thickness from 9 to 10 inches is a lot less than that realized by going from 1 to 2 inches. Therefore, the marginal savings is given by the continually decreasing curve D. The genius of marginal analysis lies in the fact that the optimum thickness occurs at that precise point where marginal cost equals marginal saving. At any thickness less than optimal, the marginal savings would exceed the marginal cost. On the other hand, at any thickness greater than optimal, the incremental cost of the additional inch of insulation would exceed the resulting incremental savings.
Each of these curves can be expressed mathematically, so we can define the optimum thickness by equating the expressions for marginal cost and marginal saving.
Cost of Insulating
Except for the special case of the open attic floor, a strict cost accounting of insulating a square foot of an exterior surface of a house must include: 1) the cost of the insulation, 2) the cost of any additional framing required to house the insulation and 3) the value of interior space taken up by the insulation that otherwise could be used as living space. See Cost of Insulating for the equation.
The marginal cost (MC) of insulating is the additional cost of one more inch. Compute the cost of any two thicknesses (5 and 6 inches, for example), and you will see that MC = a1 + a2 + a3.
Recall from the previous installment that the quantity of heat energy lost through a surface may be calculated by the Heat-Loss Equation.
What we require here is an equation for the total number of BTUs lost through the surface over an entire heating season (keep in mind that an equivalent equation can be found for heat gained during an entire cooling season). See Season BTU Loss.
At this point, if you dont remember (or didnt take, or faked your way through) differential calculus, youll have to either calculate the fuel costs at two different thicknesses (say 9 and 10 inches) or accept on faith that the marginal savings are given by the differential of FClife with regard to T, (Ã³FClife/Ã³T). See Savings.
According to marginal analysis, the optimum thickness, Topt occurs at the precise point where MC = MS. See Optimum Thickness.
One small correction remains to be made. We have assumed that the R-value of the construction is due only to the insulation. In fact, the uninsulated construction has a small R-value, which we will call R0. Therefore, the optimum thickness of insulation will be R0/Rin inches less than given above. See Optimum Thickness Corrected.
The formulas are straightforward and require only a common calculator with the square-root function. You will find, however, that getting the values for some of the terms -- a2 and PWF in particular -- will require some heavy mental lifting. A very thorough discussion can be found in my earlier book, From the Walls In, Atlantic Monthly Press. The book is now out of print but is available in many libraries and for sale at http://www.half.com .
If you are interested in increasing insulation levels in an existing home, you will find an excellent article, Estimating the Payback Period of Additional Insulation, at http://www.eere.energy.gov/consumerinfo/refbriefs/ea3.html.
Current DOE Recommendations
The U.S. Department of Energy periodically issues recommended R-values for residential buildings. These values are often shown in the form of simple R-value maps in insulation manufacturers sales literature. Unfortunately these simplified maps do not account for different fuel types and prices, so the values shown may not apply to your home. Fortunately the DOE also issues a map (Illustration 3) and chart (see page 58) that do account for fuel source. To find the recommended total R-values for the surfaces of your home, just identify your geographic zone and heating and/or cooling type (whichever accounts for your primary energy bill). For example, a home in Chicago heated by natural gas would find its recommended R-values in the row (shaded) defined by Zone 2 and Natural Gas.
In the next issue we will see how to choose the best insulation for each area of your home and -- even more important -- how to (and how not to) install it.