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Determining the optimum thickness of insulation for heated buildings
Beach, R. K.
NATIONAT RESEARCH COUNCIL CANADA
DIVIsION OF' BUILDING RESEARCH
DETERMINING THE OPTIMUM THICKNESS OF INSULATION FOR HEATED BUILDINGS
by R. K. Beach
A N A L Y Z E D
Technical Paper No. f 87 of the
Division of Building Research
OTTAWA M a y 1 9 6 5
DETERMINING THE OPTIMUM THICKNESS OF INSULATION FOR HEATED BUILDINGS
by R . K . B e a c h
It is cornrnon Practice to improve the thermal characteristics of a building by adding insulation. lThe rnain advantagee of this practice are the irnprbvement in the cornfort conditions within the building, realized rnainly by the increase in the inside surface temperature, and the re-duction of the annual heating costs due to the reduced rate of heat loss. Other benefits such as improved fire resistance, cond.ensation control and sound control may also be achieved.
Experience has shown that reasonably comfortable conditions in buildings can be obtained by the u.se of relatively srnall arnountg of insu-Iation. The use of more insulation than is necessary to rneet minirnum comfort requirernents will reduce the annual heating coets, and hence, can be justified on economic grounds. Each additional inch of insulation added, however, achieves a srnaller saving than the previous inch of insu-lation until a point is reached where the saving just corresponds to the cost of installing the last inch of insulation. This is the economic lirnit, and the total thickness of the insulation corresponding to this tirnit is called the optirnurn thickness of insulation.
Figure I is a graphical representation of the annual cost of insu-Iating and the annual cost of heating. The curve representing the total 'cost of insulation and fuel has a definite rninirnurn value which occurs at the point corresPonding to the optirnurn thickness of ineulation as
previously defined.
A report published by the National Research council in 1940, rrThe optirnum Thickness of rnsulation for canadian Hornes!r by J.D. Babbitt (NRC No. 8?4) and a subsequent paper issued by the Canadian Mineral Wool Manufacturerst Association, 'rThe Econornic Thickness of Insulation for Ganadian Hom€s,tt both of which are based on a graphical presentation, have contributed rnuch to the understanding of the econornic advantages of insulation. They have become less useful in recent years because of changing costs and they do not include the use of natural gas and electricity as heating fuels. This present paper u6es the sarne rnathernatical basis as the earlier ones, but extends it to include these and additional factors that can be significant in many cases.
Although there are a large number of factors that rnay have sorrre effect on the optirnurn thickness of insulation, experience has shown that the significant ones can be grouped into four rnajor iterns as follows:
( I ) The cost of the fuel consurned (Zl The cost of the heating system
(3) The cost of supplying and installing the tl insulation
2
-It can be shown that each of these iterns may be repre sented by an equation and, by adding the four equations together, an equation representing the total cost is obtained. From this, an equation for to, the optimum thickness of insulation, can be derived. As the unit coste and savings are independent of the total area involved, w e n e e d o n l y c o n s i d e r a u n i t a r e a w h e n c o n s i d e r i n g t h e g e n e r a l c a s e . For a specific case, the total costs can be obtained by multiplying the unit costs by the area under consicieration.
The annual cost of supplying the heat lost through one square foot of ineulated area (c1) maf be erqrreesed by the equation:
z 4 D Z
(r)
where D e g r e e d a y s ,
Cost of fuel in cents per unit of fuel, Efficiency of utilization of the heating
system over the whole heating season, Heat content of the fuel in Btu per unit of fuel,
Thermal conductivity of the insulation i n B t u p e r ( s q f t ) ( h r ) ( ' f ' p e r i n . o f thickne s s),
O v e r - a l l c o e f f i c i e n t o f h e a t t r a n s m i s s i o n of the uninsulated structure in Btu per ( s q f t ) ( h r ) ( ' F ) ,
Thickness of insulation in inches.
I n s o r n e c a s e s , h o w e v e r , f u e l c o s t s a r e a t a x - d e d u c t i b l e item. Thus, the actual cost to the owner is reduced by the amount t h a t i s s a v e d i n t a x e s . I f r i s t h e a p p l i c a b l e t a x r a t e e x p r e s s e d a s a
decimal, then the net annual cost of heating, obtained by rnultiplyrttg E q . ( t ) b y ( l - r ) , i s c f E F D
z
E FU
2 4 D z ( l - r ) (
k
)
rc
The initial or capital cost of o f i n s u l a t e d a r e a ( C " ) m a V b e e x p r e s s e d
( z l
the heating system per sq ft by the following equation: t fw h e r e 3
-c
b ( T . - T ) s l 0T .
1T
oIncremental unit capital cost of the complete heating system in cents per Btu per hour of capacity,
I n s i d e d e s i g n t e m p e r a t u r e i n o F , O u t s i d e d e s i g n t e m p e r a t u r e i n o F .
( 3 )
( 5 )
As this is a capital cost item it cannot be added to or com-pared to the annual cost of heating until it has been converted to an equivalent annual cost. This can be done by dividing it by the
appropriate Present Worth Factor (W) which rnay be referred to as the rrpresent value of an annuityrr. By applying this factor to Eq. (3), the equation for the annual cost of the heating system (c*) becomes
b " ( T i - T o ) ( t )
vir
-
(klu+Tj
In general, the cost of supplying and installing insulation is made up of the cost of labour and material which is dependent on t h e t h i c k n e s s u s e d , a n d o t h e r expenses which are not. Therefore, the annual cost of insulating (ci) may be e)<pressed by the equation
( 4 )
c . 1
where Initial or capital base cost of insulating i n c e n t s p e r s q f t ,
Initial or capital incremental cost of insulating in cents per sq ft per in. o f t h i c k n e s s ,
P r e s e n t W o r t h F a c t o r . b .
1
In order to install the insulation, space must be available. If there is not sufficient space, then it must be provided by installing e t r a p p i n g , b y i n c r e a s i n g t h e s i z e o f t h e studs used, or by some other method. The annual cost of this additional work (c*) follows the same pattern as that of the insulation. Hence, the equation rnay be e>rpressed a s a i b i t
w w
a . = 1w
4
-w h e r e Initial or capital base cost of additional w o r k i n c e n t s p e r s q f t ,
Initial or capital incremental cost of additional work in cents per sq ft per i n . o f t h i c k n e s s .
By adding Eq. (21, (4), (5) and (6) we obtain an equation for the total annual cost (c1):
z 4 D z (l - r )
In
J k / U + t
" T = E F .
Once to is deterrnined, the various annual or capital costs rnay be found by substituting the value for to in the appropriate equations. The total cost for a particular building can be obtained by rnultiplying the unit cost by the area involved. The costs rnay be expressed as capital costs sirnply by rnultiplying the annual costs b y t h e P r e s e n t W o r t h F a c t o r ( W ) .
The following is an example of the use of Eq. (8) to determine the optimum thickness of fibreboard roof insulation for a public school to be built in the Ottawa area. The building is to be h e a t e d b y a n o i l - f i r e d h o t w a t e r s y s t e r n ; o t h e r f a c t o r s a r e a s s u m e d a s f o l l o w s :
a w b * t
" *
=
T t
T
+.t+] ,.I
b " ( T i - T o )Those persons acquainted with calculus will know that an equation for to, the optimum thickness of insulation, can be obtained by taking the first d'erivative of Eq. (71, equating it to zero and solving for to. The resulting equation is
+ _ /fu"(ri - ro) r zupz(r-rtl :*
l _ k/v
(s)
'o=
VL *
'F-liryq
Incrernental cost of insulating B a s e c o s t o f i n s u l a t i n g
Thermal conductivity of insulation k
b i = l l . 0 c e n t s p e r a i = 0 c e n t s P e r s q- (included with sq ft per inch of thickne s s ft roofing) = 0 . 36 Btu per (sq ft) (hr) ( " F p e r i n . )
P r e s e n t W o r t h F a c t o r
( I n t e r e s t rate of 5j per cent and I n c r e m e n t a l c o s t o f heating s y s t e m I n s i d e d e s i g n t e m p e r a t u r e O u t s i d e d e s i g n temperature 1 r . 9 5 of. 20 years) 1 . 8 c e n t s p e r B t u p e r h r o f capacity 7 0 ' F . h e a t t r a n s -etructur e 5
-w
u
D
z
F E a b T t e f m eT
= - 1 3 ' F
oOver -all coefficient of m i s s i o n o f u n i n e u l a t e d D e g r e e d a y s
C o s t o f f u e l , N o . 5 oil Heat content of fuel
0 . 3 6 B t u p e r ( s q f t ) ( h r ) ( ' F . ) 8 7 4 0
I 0 c e n t s p e r g a l 1 8 2 0 0 0 B t u p e r g a l 0 . 7 0 .
Efficiency of utili zation
The remaining factor s -t h e -t a x r a -t e ( r ) - d o n o -t a p p l y i n -this S u b s t i t u t i n g i n E q . ( 8 )
cost of other work, a* and b*, and c a s e , a n d h e n c e h a v e z e t o v a l u e . 1 . 8
g9_
- l 3 ) 1 . 9 5It
0 0 xn
40 t 8 8 7 x x 0 0 . 3 5 x I I . 9 5 1 l 0 . 3 5 - 0:%-2 . 4 i r . . , s a y Z l i n . n o m i n a l thickness.In a like manner the optirnum thickness of insulation for the w a l l s c a n a l s o b e d e t e r m i n e d . A s s u m i n g t h a t t h e w a l l s a r e c o n s t r u c t e d a s a m a s o n r y c u r t a i n w a l l ( U = 0 . 2 5 ) , a n d t h e c o s t o f f o a m e d p o l y s t y r e n e i n s u l a t i o n ( k = 0 " 2 7 ) i n p l a c e i s I0 cents per board ft, the calculated o p t i m u m t h i c k n e s s w i l l b e f o u n d t o be 2.0 in.
I f t h e b u i l d i n g i n q u e s t i o n w a s a c o m m e r c i a l b u i l d i n g rather than a public building, then the effect of the tax rate (r) on the fuel c o s t s w o u l d b e i n c l u d e d . W i t h a t a x r a t e o f 4 7 p e r c e n t t h e c a l c u l a t e d o p t i m u m t h i c k n e s s e s w o u l d b e 2 . 0 i n . f o r t h e r o o f i n s u l a t i o n a n d 1 . 5 i n . f o r t h e w a l l i n s u l a t i o n . R o u n d i n g t h e s e o f f t o t h e n e a r e s t c o m m e r c i a l s r z e w o u l d g i v e 2 i n . f o r t h e c e i l i n g and l* in. for the walls.
6
-This equation may also be used to determine the optirnum thickness of insulation for houses. For example, determine the optimum thickness of insulation for the walls and ceiling of a solid m a s o n r y h o u s e i n t h e T o r o n t o s u b u r b s h e a t e d b y e l e c t r i c i t y .
M i n e r a l w o o l b a t t s a r e t o b e u s e d i n t h e w a l l s a n d l o o s e - f i l l m i n e r a l w o o l i n t h e c e i l i n g . T h e w a l l s a r e t o r e c e i v e a d r y w a l l
finish over strapping. There will be a first mortgage on the house at 6 j p e r c e n t f o r ? 5 y e a r s . O t h e r a s s u m e d f a c t o r s a r e a s f o l l o w s : W = l 2 . Z D = 7 1 0 0 Z = l. 25 cents per kwh E = 1 . 0 F = 3415 Btu per kwh r = 0 ( n o t a p p l i c a b l e ) k = 0 . 2 5 B t u p e r ( s q f t ) ( h r ) ( " F ) per in. of thickness
, Incremental cost of insulation b. = 2 cents per sq ft per in. of t
thickne s s
Incremental cost of strapping
b* =
:J..ff:J""r
sq ft per in. of
Over all coefficient of heat trans
-m i s s i o n o f u n i n s u l a t e d structure U = 0 . 2 4 B t u p e r ( s q f t ) ( h r ) ( " F ) . Substituting in Eq. (8) the optimum thickness of insulation f o r t h e c e i l i n g w e g e t :
I n c r e m e n t a l c o s t o f heating system b" = 1.5 cents per Btu ($51. OO/tcw)
I n s i d e d e s i g n t e m p e r a t u r e T . = 7 0 ' F 1 O u t e i d e d e s i g n t e m p e r a t u r e T o = - 3 " F P r e s e n t W o r t h F a c t o r D e g r e e d a y s C o s t o f e l e c t r i c i t y
E fficiency of utili zation
Heat content of fuel
T a x r a t e
Therrnal conductivity of insulation
7
-to ceiling = l - 1 . 5 x ( 7 0 - - 3 ) . ? 4 x ? 1 0 0 x t . ? 5
t T l . u * i 4 r 5
9.4 in. , 6oy 9 in. nominally,
1r
]L
0 . 2 5
I
7,L 2 . Zz.z
x+
z 5
2
I
Il
0 . 2 5
-ffi
0 , 2 5
-ffi
t walle o l . 5 x ( 7 0 - - 3 )-tz. z
!
I0 .
z
l
z 5
l . 5n
4 l 1 0 0 x 3 x t. o
I
3
5. 6 in. , edy 6 in. nominally.
In thie case it was assutrred that the two tlpes of mineral wool insulation have the same incremental cost and insulating value, w h e r e a s i n p r a c t i c e t h e r e may be slight differences. I f t h e h o u s e w a e heated by either gas or oil the optimum thickness of insulation would be 6 in. for the ceiling and 3 in. for the walls.
I n t h e p r e c e d i n g e x a m p l e s , t h e v a r i o u s cost itema are consistent over the complete range of insulation thicknesses and the answer can be found by straight substitution in the formula. There are cases, however, when this may not be true such as, for example, a frame building with z x 4 stud walls. A nominal four in. of
insulation can be installed without incurring any cost for additional work, but if more insulation is required then either strapping or l a r g e r s t u d s m u s t b e used. In locations where the climate is cold or the cost of fuel high, considerably more insulation than four in. of insulation can be economically justified even when the additional costs are considered. To solve this type of problem, calculate the
optimum thickness without considering the additional expense. If
the answer is greater than the space available, recalculate the optimum thickness including the cost of additional work. If this result is grearer than the space available, then it is the true optimum thickness. If it is less than the space available, however, then the true optimum thick-neas is the actual space available.
Probably the most difficult part about using Eq. (g) is
obtaining the correct values to substitute in the equation. The following c o m m e n t s h a v e b e e n i n c l u d e d to assist in this problem.
The tax rate depends entirely on the ownerts financial status, and he must provide the figure to be used. The owner rnust also make the final decision on the value to be used for the Present
Worth Factor. This factor, which is an elq)ression of the cost of money, d e p e n d s o n t w o o t h e r f a c t o r s : the rate of interest and the period of y e a r s c o n c e r n e d . T h e s e d e p e n d o n t h e o w n e r r s a b i l i t y t o b o r r o w o r i n v e s t m o n e y a n d o n t h e service life of the item concerned. o n t h e other hand, the owner may arbitrarily decide that the additional cost
8
-must be recovered in a specific number of years and at a specific interest rate which may not bear any relationehip to the service life or investment terms. Although different values for w may apply to each different item, in practice, good results are obtained by the use of a eingle representative value. The equations have been
derived on this baeie. Having selected the rate of interest and number of yeare, the valuee of the Preeent worth Factor may be obtained
from the standard interest tables for the rrpreeent value of an annuity, tl Sufficiently accurate values may aleo be obtained by interpolation from F i g u r e 2 .
Although the owner must specify the ineide deeign
temperature that he wisheg to be maintained in the building, the out-side design temperature and the number of degree days depend on the location of the building. Normal deeign values for theee items are available in Supplement No. I to the National Building Code (NRC 6483), Heat from electric lights, occupants and equipment in the buitding and other sources, however, provide a portion of the heat required to maintain the inside temperature. If the amount of heat from theee
eources is great, then the use of the normal degree day values will result in the estimated savings in fuel being greater than could be achieved in practice, and hence the calculated optimum thickness of insulation would also be greater than it should be. Ae the optimum thicknese is proportional to the square root of the number of degree days, the error in calculating the optimurn thickness will be much less than the reduction in degree days. In rnany cases this error witl b e o f f s e t b y o t h e r f a c t o r s s o i t i s o n l y n e c e s s a r y to reduce the normal degree-day figure when the amount of heat from other sources is v e r y l a r g e .
Chapter ?3 of. the 1963 Guide and Data Book of the American society of Heating, Refrigerating and Air-conditioning Engineers*, contains tabulated k values for various building materials and the U values of many representative wall and ceiling constructions. The publication also explains how to determine the U value from basic principles and much other useful information. The k value of the insulation may be obtained from the manufacturer as well and
should be used in preference to values given in the publication which a r e a v e r a g e v a l u e s .
There are four variables that are related directly to the heating system. These are the incremental cost of the system, the
Available as reprint from the Division of Buitding Research, o r d e r n o . N R C 7 7 8 8 .
9
-actual cost of fuel, the heat content of the fuel and the efficiency of utilization over the heating season. This information should be readily available from mechanical contractors and from local fuel suppliers. Although the incremental cost of the heating system is probably the moet difficult to determine, a reasonably accurate value can be obtained by dividing the estimated cost of the heating system by the maximum rate of heat loss or the capacity of the s y s t e m .
The remaining items are the cost of supplying and installing the insulation and the cost of any additional work required to provide space for the insulation. Although these items may be estimated, probably the best thing to do is to obtain estimates from contractors for a range of insulation thicknesses. The incremental costs can then be determined by dividing the difference in cost for two different thicknesse6 by the difference in thickness and the total area to be insulated. The cost of additional work rnust include such things as the cost of adjusting the door and window frames to suit the thicker walls as well as the cost of extra strapping or larger studs.
The accuracy of the final result will, of course, depend on the accuracy of the values that are substituted for the many vari-ables in Eq. (8), as well as the accuracy with which the cost equations represent the actual costs. Although the possible error is large, it is found that the errors in the various items tend to offset one another and that, in practice, the results are quite accurate. Also, the
curve of total costs is such that at the point of optimum thickness it i s r e l a t i v e l y f l a t ( s e e F i g u r e l ) and hence, the costs are nearly constant. The effect then is to minimize the effect of errors on the optirnum thickness and to permit some latitude in selecting the actual amount of insulation to be used without significantly increasing the total annual cost.
The final choice as to how much insulation to use rnay also be affected by the type of building involved. In the commercial building field there is a very strong tendency to keep the capital
costs as low as possible without considering the possible effect on the operating costs. Although an allowance has been made for the effect of taxes on the net operating costs, taxes do change, and therefore there may be aome justification for giving major consideration to the capital costs rather than equal consideration in the case of commercial buildings. on the other hand, this type of thinking should not extend t o p u b l i c b u i l d i n g s o r o w n e r - o c c u p i e d h o u s i n g . In these cases, the operating costs are not tax-deductible but they are just as important as the capital costs in determining the total cost to the public, who in
1 0
-the long run must bear -the cost of -the public butlding and to -the houee owner. The final decision on the thickneee of ineulation to be uaed s h o u l d b e b a s e d o n e o u n d b u s i n e s e and engineering practice, and it is hoped that the rnethod outlined in thle paper will aaeiet in making t h e c o r r e c t d e c i e i o n .
atl
