Converting heating degree-days from below 65 degrees F to below 18 degrees C

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CONVERTING HEATING DEGREE-DAYS FROM BELOW 65 O F TO BELOW 18 OC

Donald

W.

Boyd

C u r r e n t values of heating degree-days published by the

meteosolsgical services in North America will probably b e expressed as

degree days below 18 "C starting sometime in 1975 and continuing f o ~ a considerable time before tables of normal degree-days below I8 O C are

available. To compare cument values with t h e normal it will be

necessary t o convert t h e published, normal degree-days below 65 OF t o

degree days below 18

"c.

Because 65 "F and 18 "C are slightly different temperatures the degree-day conversion is not simply a matter of

multiplying by a constant. In a d d i t i o n to t h e multiplication t h e r e is an adjustment, which in a 30-day month may mount to 18 F a r e n h e i t

degree-days or 10 Celsius degree-days. For any month in which t h e

d a i l y mean temperature exceeds 65 O F for one or more days the conversion

cannot be exact but the method o u t l i n e d i n t h i s Note w i l l give

adequately accurate r e s u l t s .

The heating degree-days f o r a single day are d e f i n e d as the number of degrees t h a t the mean temperature f o r t h e day i s below the base temperature.

PE

t h e mean temperature for the day is above the base temperature then the degree days are zero. The h e a t i n g degree- days for a month are t h e sum _{of t h e daily values. The normal h e a t i n g} degree-days for a particulas month are the average of t h e values for t h a t month over some standard period, usually 30 years.

From these definitions it follows t h a t t h e only method af

computing exact normal values is t o go through the records day by day for 30 years. Degree days below 45 "E have been computed for a number of weather stations in t h i s way. In Canada, however, most of t h e published normal values are based on afz inexact b u t s u f f i c i e n t l y

accurate computation method devised by H . C. S

.

Thorn,

'

He showed t h a t normal monthly degree-days below 65 "F could be computed from t h e normal monthly mem temperatures, t, and t h e i r standard deviations, s. To do this h e used two equations:

where N is t h e number o f days in t h e month and D is t h e average number of _{degree days p e r day. The value of h i s defined by Eq. [I). The} v a r i a b l e E i s a f u n c t i o n of h defined in Thorn's paper by a graph and a l s o by a t a b l e . Knowing t h e monthly mean temperature and its standard d e v i a t i o n one can compute h from E q . ( I ) and hence o b t a i n R from the

t a b l e . D is then computed from Eq. (21 and t h e degree days below 65

"F

f o r t h e month are given by ND.

It seems, therefore, that t h e ~ e a r e three obvious methods of o b t a i n i n g heating degree-days below 18 'C:

1. To convert d a i l y temperatures to C for the past 30 years and compute C degree days exactly,

2. To use Thorn's method to compute degree days below 64.4 "F from

monthly mean temperatures and standard deviations and multiply

t h e r e s u l t b y 519, and

3 . To use Thorn's t a b l e to establish t h e relationship between degree days below 65 O F and below 6 4 . 4 "F, and multiply by

5/9 to convert to degree days below 18 OC.

The f i r s t method is exact but would require a great deal of computation. The second is reasonably accurate but would be tedious except on a

d i g i t a l computer. The t h i r d should equal the second i n accuracy and should be simple enough f o r hand calculation. The remainder of this note is concerned o n l y with the third method in the hope that it will be useful until complete tables can be computed using e i t h e r method one or t w o .

0

A is t h e normal number of degree days below 6 5 F in a month, but it is mere convenient to work w i t h D, t h e average number of degree days p e r day, where:

As 18 "C is exactly equal to 64.4 O F , the average C degree days per day below 18 C' will be e x a c t l y equal to 5/9 of the average

F

degree days

p e r day below 64.4 O F (Dl)

.

What is required, therefore, is the conversion from D to Dq.

Equations (I] and (23 can be combined to give:

Knowing D, s and N , one can compute h + R and from Thorn's t a b l e obtain

t h e corresponding value of h

.

Equation (11 gives (65 - t'g

,

and t

.

Thorn's method can then be used in t h e usual order to give D', the

average degree days per day below 64.4 O F .

These calculations weTe done f o r odd values o f D from I to 21 and f o r five values of s ~ ranging from ~ / ~ 5 to 25. The mean temper- atures in Cmadian Normals, Volume 1 and t h e standard deviations given by Kendall and Anderson3 were searched and only four station-

months of concern were found t h a t were not covered by these ranges.

They were Carmi, British Columbia, in May, Holman and P o r t Radium, Northwest Territories, in August and Embarras, Alberta, in April.

Thorn's values for h and R are quoted to two figures and is

the r e l a t i o n s h i p between D and D'. The values of R were therefore smoothed and t a b u l a t e d to four figures. For values of h from 0 . 7 8 to

zero and of R from zero to 0.19 the R-values themselves were smoothed. Were there were

n

identical values of II, say

L,

they were replaced by

the series:

where m takes only odd i n t e g r a l positive values less than 2n. For

values of h from -0.70 to zero and of E from 0.70 to 0 . 1 9 t h e same smoothing was applied to the sum h + R.

Using these smoothed values the calculations of D l were repeated. The results, expressed as values o f the difference D - ! I t are shown in Table I and are plotted in Figure 1.

The plotted values obviously needed some smoothing and after

t r y i n g several methods the following equation was finally selected:

log

(a

-

y) = log d

-

C D / S N ~ / ~

₁₅₁

where :

d = 0.6, the difference in degrees

F

between the base temperatures 65 O F and 18 "C or 6 4 . 4 O F .

y = D - D l , the required adjustment in

F

degree days per day.

c = 1.5, a dimensionless constant.

This equation is shown in F i g u r e I for each of the f i v e values of

S N I P

in Table I. No attempt has been made t a j u s t i f y t h e form of t h e equation on theoretical grounds; it is purely empirical. Na attempt has been made to use any mathematical analysis t o obtain the b e s t value f o r the

constant c , it is s i m p l y a convenient. number which g i v e s a sufficiently close fit to t h e computed values of y. "Suf.EicientLy close" in t h i s case means t h a t most o f the plotted p o i n t s a r e within about 0.05 F

degrees [on the y-scale) of the corresponding curve, This corresponds to about one degree day per month and hence is more accurate t h a n the

degree days computed from monthly mean temperatures quoted only to

0 . 1 O F , corresponding to about 3 degree days p e r manth.

The conversion from A (the F degree-days below 65 "F) to

B

( t h e C degree-days below I 8

'c)

is therefore accomplished by using t h e two e q u a t i o n s :

These equations may be u s e f u l when s is known and when t h e conversion is being done by a d i g i t a l computer.

In many cases, however, t h e value of s will n o t be readily available and t h e precision of t h e s e equations will n o t be necessary. It is evident from Figure 1 t h a t the central curve f o r = 15 is

corresponds to about 3 degree days per month. If this is sufffciently accurate then Eq. ( 5 ) reduces to:

log (d

-

Y ) = log

a

-

0.1

D.

171

The values of Ny need not be calculated any more accurately than one degree day. Equation (7) was heref fore used to calculate t h e range o f D , or of A =

ND,

f o r each integral value of Nyformonths of28 1/4, 30 and

31 days. The three t a b l e s d i f f e r only slightly. If the central one f o r N = 30 were used for all months t h e m a x i m u m e m o r in Ny would be one degree day per month.

Table I1 w a s calculated u s i n g N = 30 b u t it could be used for any month. Knowing only A ( t h e F degree-days below 65 O F ] , one can obtain a value of B (the C degree-days below 18 "C) in two steps:

obtain the adjustment Ny from Table TI, and compute £3 from the equation:

The _{inverse conversion from Celsius to Fahrenheit can be}

accomplished with equal accuracy and simplicity. For N = 30, Eqs. 16) and (73 can be written:

where :

z = 5yS9 (or y = 1.823

Combining these gives:

Equation (8) was used to calculate the range of B for each i n t e g r a l

value of 302. The results are given in Table 111. Although it is

based on N = 30 and S N I - J ~ = 1 5 , it could be used for any month. bowing only I3 (che C degree-days below 18

'c) ,

one can obtain a value of A

0

[the F degree-days below 6 5 F), in two steps: obtain the adjustment Nz from Table 111, and compute A from the equation:

CONC LUS ION

Monthly h e a t i n g degree-days can be converted exactly from below 65 O F to below 18 " C only by recalculation en a day-by-day b a s i s .

Most o f t h e normal monthly degree-days below 65 *F in Canada were

computed from normal monthly mean ternperarures using Thorn's method. It is assumed t h a t h i s method yields adequately accurate r e s u l t s .

Monthly degree-days computed by Thorn" method can be

converted without any significant l o s s of accuracy by using Eqs. (5)

and (6) provided the standard deviation of the monthly mean temperatures

A more practical conversion which may involve errors of about 3 F o r 2 C degree-days per month can be accomplished without considering the standard deviation or the number af days in t h e month. Table I1 and Eq. ( 6 ) can be used to convert from monthly degree-days below 65 O F to

below 18 "C. _Table

_III

_{and Eq. (9) can be used for the inverse} conversion.

REFERENCES

( I ) Thorn, H.C.S. The r a t i o n a l felationship between h e a t i n g degree

days and temperature. Monthly Weather Review, V o l . 82, No. 1, p . 1-6, Jan. 1954.

(2) Canadian Nomals, Vol. 1, Temperature 1941-1970. Atmospheric Environment Service, Downsview, 1973.

(3) Rendall, G.R. and S.R. Anderson. Standard deviationsofmonthly and annual mean temperatures. Canada, Dept. of Transport, Meteorological Branch, Toronto, Climatological Studies Number 4 . 1966.

TABLE I

Values of y = D

-

D' in F degree-days p e r day

TABLE I 1

Adjustments Ny in F degree-days to be used i n equation: B = 5 (A - Ny)/9

A , in E degree-days

h'y, in

TABLE I I I

Adjustments Nz in C degree-days to be used in equation: A = 9

(B

+ N z ) / 5 B, in C degree-days Nz, in C degree-days TABLE IV PRINCIPAL SYMBOLS F : F a h r e n h e i t C : Celsius A : F degree-days below 65 O F B E C degree-days below 18 OC

!I : AJN, average F degree-days p e r day below 6 5 "F D t : Average F degree-days per day below 64.4 "F N : Number of days in t h e month

c : 1.5, a dimensionless c o n s t a n t

d : 0.6, the difference between 65 and 6 4 . 4 O F

h : Factor in Thorn" t a b l e R : Factor in Thorn's t a b l e

t : Normal monthly mean temperature

s : Standard d e v i a t i o n of monthly mean temperatures y : D-Df, t h e adjustment in F degree-days per day

0 0 2 4 6 8 1 0 12 24 1 6 18 20 2 2 24 D I N F D E G R E E - D A Y S PER D A Y F I G U R E 1 1 V A L U E S OF y FOR V A R I O U S V A L U E S O F D AND s ~ '