FOREST PRODUCTS LIBRARY
FOREST RESEARCH LABORATOR Y
OREGON STATE UNIVERSIT Y
RATE OF TEMPERATURE CHANGE IN LAMINATE D
!MUMS HEATED IN AIR UNDER CONTROLLE D
RELATIVE HUMIDITY CONDITION S
Reviewed and Reaffirmed
July 1954
No. 11434
UNITED STATES DEPARTMENT OE AGRICULTUR E
FOREST SERVIC E
FOREST PRODUCTS LABORATOR Y
Madison 5, Wisconsi n
in Cooperation with the University o! Wisconsin
PA C
'
:1W OF PET ERA.TIIRE
.
,,• 4
'nArI' r7 LAl' I?A'Y t 7t tBERS HEATED?
1 .p IN AIR UNDER CONTROLLED RELLTIVE IRIDIIDITY CONDITIONSz.
1
r ri
Iv-
_..^
../ }
.
!
~.,
w
t-I
By
J . D . MaoLEAN, SENIOR ENGINEER
In gluing laminated timber with resin glues that must be heated t o
obtain the required setting temperatures, it is desirable to have a mea~a s
of estinet?ng the time needed to reach a given temperature in the inner '
d,lue line, where temperature changes take place most slowly . Informatitn
on she rate of temperature change in laminated timbers is of interest in
the fabrication of boat keels, ribs, and similar construction .
A large amount of experimental work has been done at the Fores t
Products Laboratory in studies of both heat conductivity and the rate o f
temperature change in mood heated in different mediums . Results of thes e
investigations have been discussed in a number of publications (1 to o),L .
The purpose of this paper is to discuss the rate of temperatur e
change in laminated timbers heated in air under controlled humidity con . ditors . In p revious studies of the rate of temperature change in wood ,
it has been found that the heating medium, as well as the temperature conditions employed, affects the rate at which the temperature increases (6) .
To obtain. data, therefore, on the rate of heating under dry-kil n
conditions when the humidity is controlled to maintain a definite moistur e
equilibrium, a series of experiments were made on laminated blocks of whit e
oak, red . oak, maple, hickory, southern pine, and coast Douglas-fir . 'Thes e
blocs were constructed by gluing 6 to 14 laminations 3/4-inch thick, 4 an d
inches wide, and 4 feet long .& This length was sufficient to avoid th e
ei ect of end heating at the 3T id length, where the temperature reading s
were taken . Dry-bulb temperatures employed were approximately 110°, 160° ,
and 200° to 205° F . The humidity was regulated to obtain an equilibriu m
ucisture content of about 12 percent . Resin glues of the low-temperatur e
pbe : colic ty p e were used in gluing the laminations .
Results of these experiments showed that when the dry-bulb temper a .,
tare was 110° F . and 160 0 F . and the relative humidity about 70 and 80 per cent, res p ectively, the diffusivity was about the same as when panels o f
-This mimeograph is one of a series of progress reports prepared by th e
sorest Products .Laboratory to further the Nation's war effort . Result s
here reported are preliminary and may be revised as additional dat a
b ecome available .
==nnerels in parentheses refer to publicatl ens named in the list of refer ences at the end of this paper .
. .imeo . Nee L_1 434
fri
the same species were heated in a hot press . The rate of heating as indi cated by the diffusivity, which is discussed later, was between 10 and 1 5
percent faster, however, when the wood was heated at a temperature o f
approximately 200° F . and the relative humidity was about 85 percent . This
might be expected considering the greater amount of water vapor in-the air .
For example, at 200° F . and 85 percent relative humidity, the air would hav e
nearly 2-1/2 times as much water as at 160° F . and 80 percent relative
humidity.
Three laminated test blocks, two of white oak and one of Douglas-fir ,
were heated at 160° F . without humidity control, to compare the results wit h
those obtained under controlled conditions . It was found that the rate o f
heating was somewhat slower under these conditions and some checking of th e
glue joints occurred . All blocks heated in a dry kiln with the humidity
controlled to give a moisture equilibrium of about 12 percent were in good
condition after heating .
Factors Affecting the Rate of Temperatur e
Change in Woo d
Important factors affecting the rate of temperature change in wood
are ; the heating medium, the direction of heat travel, the density of th e
wood and the moisture content of the wood .
Heating Medium
Experiments show, when wood is heated, that steam heats faster than
liquids, liquids heat faster than hot plates, and hot plates heat faste r
than dry air (6) .
Direction of Heat Trave l
The rate of temperature change in wood heated in the direction o f
the fibers is two to three times that of wood heated across the fibers (2) .
While there may be a small difference in the rate of temperature chang e
across the annual rings (radial direction) in comparison with the rate o f
change parallel with the annual rings (tangential direction) experiment s
indicate that whatever difference exists is too small to justify consider ing these two directions separately (2) . Except in short lengths wit h
large cross-sectional areas, the effect of end heating does not need to b e
taken into consideration as a factor in the temperature rise at the cente r
of the block . For this reason, the present discussion will be confined t o
heating in the transverse direction .
Mimeo . No . 81434
-2-
Density or Specific Gravity
In ge rer-al, heavier 'w o 's, heat metee slo'Wly than lighter woods .
Although , t: e heat Cohductivi .t of wood increases with an' ine ,rease i n
density, the rate of temperature- change, on the c .oitrary ., deeireases wi'th '
an increase in density . That is, the ligh'ter ' woods ; although more resistant to heat flow, will change temperature mere rapidly,, un'de'r any
given heatig conditions, than the heavier woods . that are 'better hea t
conductors . , For a given temperature more heat units are absorbed pe r
unit volume by the heavier woods because of their greater amount of woo d
substance ; this will naturally.'tend'to decrease the rate of-heating as th e
density increases .
The effect of density on the rate of temperature change depends t o
a considerable extent on the heating 'medium, For example, its effect i s
more cons-pi-eTTus with' steam as the heating medium than when 7 :1a ds ar e
employed . Likewise, wood'density has less effect on the rate of heatin g
when hot plates are used than when .wood is heated in liquids .
This variation appearb'to be the result' of'differences, in surface resietaiO. e t o
heat transmission from the, hea ++ing medium to the 'Wood in' contact''wra .th it .
When liquids or gases are used ; the specific heat, rate of circulation ,
and similar variables 'probably have an important bearing on the result s
obtained .
1
1
'
1.
Tloisture Content
Previous experiments . on green timbers heated in steam and in liquid s
slimed that green material hes .ted. faster than seasoned wood (6) . Above th e
fiber-saturation pei :nt (about 30 percent voistuta eonten~t)• the spate of heating under any given temperature conditions was not appreciably affected b y
further increases in moisture content (2, 6),,
For practical purposes it is sufficiently accurate to assume tha t
the rate of heating under equilibrium moisture content conditions does no t
vary to an important extent' as a result .ofmoisture content variation, ove r
the range of about 10 ' to 20 percent . The difference in the rate of heatin g
Within these moisture content limits is .therefore not important in makin g
temperature determinations .
Determination of Temperature Change s
Lt.
Tlie rem of he :t tranend ssidn ap y spec'ifieal .y 4-6 isotropic sv,b,stances, that is, substances having the same physical characteristics an d
the same properties in all di regtions : Although wood does not cio.me under '
this class ' .fication, . it does hesre- sufficient uniformity in at•u,c. x°e s o
that .mathemaetical calculations ear, be applied to ddetermi
the rate. o f
temperature charge after experimental data have been obtained . uncles 'variou s
heating conditions . 'chile certain variations from the average ' will occu r
in individua l timbers, because of differences in growth conditions and othe r
r:
'himeo . No . P1434
-3 -
variable factors, the computed data will be withie,e' nits o f c w t
needed for practical purposes . Formulas employed in d .ltulating the #rate
of temperature change in rectangular seeti=s of solid . wood, will als o
apply i computing temperature changes in laminated timibe`rs . of the salhtcposs-s:eetion dimensions .
In the-heating of any solid, the rate of +emp,era .tre t am.ge' at a
given point-depends on the diffusivityL of thee . subs znce, ,a factor that . _
nay be considered a constant over normal ranges of temperatu•re . .Diffii~t y
h2 repr oentme /
may be expressed by the following equation in which
.
the
specific
heat,
and
iZ
the
conduc.kileey
fusivity, P the density,- e
,
h
o In
Since the diffusivity required in making computations rs dependent ,
on the heating medium used, it is necessary to dete-rmin .e the diffusivity _
for the particular heating conditions employed . The value determined fro m
such experimental tests may, for this reason, be considered an . appa.Fent
diffusivity rather than the true diffusivity of the wood .
Any important influence of the glue on the rate of heating als o
affects the diffusivity, Tests made on some laminated sections heated i n
the kiln without glue indicated that the difference in re,suits obtaine d
with and without glue was negligible for the thickness of laminations used .
Diffusivity was computed for each laminated block used in the series .
of experiments made in the dry kiln and the results weiaveraged for eac h
species . The average diffusivity for woods heated in the dry kiln wit h
dry bulb temperatures of 110° F . and 160° F, are included in table 1- ..
While all the species -listed in this table were not used in thes e
experiments, extensive tests made with various heating mediums (6) hav e
shown that woods of similar density heat at about the same rate under th s
• same conditions . It is, therefore, possible to determine the approximat e
- effect of density wi_th . any particular heating medium, when experiment s
have been made on woods of several different densities . This relation wa s
taken into consideration . in preparing table 1 from the experimental dat a
obtained .
Since the rate of temperature change is approximately the same fo r
both the radial and tangential directions, the average diffusivity value s
given in table 1 for the respective species a pply generally to heating i n
either the radial or tangential direction . The diffusivity factor fo r
heating in the longitudinal d`iretti- on: is higher since the rate of heatin g
'Diffusivity measures the change of temperature that would be produced i n
a unit volume of the substance by the amount of heat that flows in uni t
time through unit area of a layer of unit thickness and having uni t
difference of temperature between the faces .
h'iimeo . 1 o . R1434
-4-
n th%rats
th
across tO -1110espse
is considera4 .Y faster along the fibers
the length of ,the tim1wgrv isoo-sstuncidt .6k0twesCi4ft4 tefttomr4mOr4he effett
of end heat jiOg ew,tendifteo talae.jib, agettreon whe_0poMb *nimum ratt of temperature char tp- . occur*- longkAba.nal Oefusivitmewbinvolved ' i the
'• 0
temperat4rt 0QUriPeAkans tor this MEW* .
. 2 It 1
P
AV
'NspettelApan increrseiin theArtOte .INt.heat"Ete amount 4f water
1
o
The
amount
oftAetitpr
.
Mapor
required
t
vapor
is itiAmMi4ei
obtain a Mien rektive hogidity increases rapidjrloel the higher tempeaPs.
tares and4ppargifty for is reasonl 41p 4AgIftsOnOWtvdetermined in itt@el ,
exlNiArivea4sigiisralkst tn- the higher tempwstttgr.,e (:;T.OgSt. Th.m-4ke" . period rreelde*AboRa Omen tsaperaturv
At'otfzarr Pra'ssigtof:4 to .
the diffutsivitIN liteviodiv when .6heo,lelVitsivity as_ .increasedi :k
taAreggettP -1
age ' it
r@dmotwi grgt r hooting, pe'iod 'bp take sale.
L.
.
. 'UsisleW a
Equatio
(anndiNI)t :m$ used . to-AIPWridWItile tt .iZigiftitpl
_
for the timbers used in the experiments .
*d
Cqputatiqp f. Temperature
's!-- :
tr .
.1
If computations of temperature are 2NIAt ,VeNT-JeUrkssurdfailfl,itMeArtkYlo :,
heating temperature, initial wood temperature, and parti e
a timber of given dimensions, simple proportional timoviter,oeratAum r-ela£
tions can be used that make it possible to apply the data for finding the
temperature at
atm.
(9q
ilaftt-‘P-t 't varegaes mentioned
,-*rM
given
IstUr!
are different . - This 7:73.11 be MTr-ated -15TZ
b The following factors are involved in reputing the -terdie'raturt s
attained .in heating . roct'angl:ar timbers in a: dry kilzi under controlle d
relative) humidity .
4 .
Initial 'weed temperature .
2, Temperature of the heating medium, in this case ; the dr. ibla b
temperature of the_
:-iln,.
.
6
3.
Diffusivity-of the wNio ,
L .
Hedtg period .
5 . Cross-sectional dimensions (w4eteAhe eff,eet of .en1d, hettixg is
be -eona4deped .).,,, neglected the length do,Q. n® t 6.
Positi:dn df the paint with reapegt to the-four -ice surfaces .
All of these Ddctors ore included in equakton 4 .lp,the appendix .
A largo amount of tedious computations d* .ngegssaryt to .use; eqa_aat.:,On 4 for .
temperature calculations . For this reason, the time-tgmpwrature curves i n
figures 1 to 9 have been prepared to-make it . pQs4iblp to compute the temperature easily'-ithout using the basic equation ,
Mimeo . No . R1! 34
-5 -
These computations are based on the assumption that the temperatur e
of the mood surface is constant throughout the heating period and tha t
goad, circulation of the air is maintained . It is not usually possible to
maintain a constant temperature from the time heat is first applied, since
some of the early part of the heating period may be required to bring th e
kiln to the desired maximum temperature . Under such conditions, the heat 'period should be extended to compensate for the gradual increase i n
surface temperature or, in other words, for the lower average temperatur e
before the maximum is reached . When the rate of increase in the kiln tem perature is reasonably uniform it would probably be satisfactory to increas e
the heating period about one-half the time needed to reach the maximum tem perature . This is assuming tha t) during the period the temperature is bein g
raised, the temperature change in the wood is about the same as would b e
attained in one-half the time if the maximum temperature had been applie d
at the start . For example, if 1 hour is taken in bringing the temperatur e
up to the max t, the heating period should be increased about 1/2 hour ove r
that determined from the charts .
If the temperature conditions are not fairly uniform on all su•face s
and at all parts of the timbers being heated, the length of the heatin g
period should be increased a sufficient amount to compensate for the tem perature difference . The kiln operator must depend on his judgment in
estimating how meeh di
e s=hould be taken to compere : .te fa ng the
variable conditions .
Charts and Formulas for Temperature-Computation s
Definitions of the symbols used and a list of the auxiliary formula s
derived from equation 4
en in the appendix . Auxiliary formulas are
indicated by letters for purpose of reference . Figures 1 to 8 sham com puted temperature curves based on an assumed initial wood temperature o f
60 ° F ., a heating temperature of 200 ° F . aRdr ll di fti.s
ty of 0 .00025 .
These curves are shown for timbers having actual thicknesses of 1, 2, 3 ,
L, 6, 8, 10, 12, and 14 inches and actual widths varying from the squar e
dimensions to 16 inches . Temperatures are plotted for the eenter of al l
sizes given and are also shown for 1-1/2 inches from the surface for a
thickness of 6 inches (fig . 4) and for each inc h aitnterval from the surfac e
to a depth of 4 inches in timbers 8, 10, and 12 inches thick with width s
up to 15 inches (figs . 5, 6, and 7) . Figure 9 shows . the computed tempera ture at the center of square timber dimensions ranging from 2 by 2 inche s
to 18 by-18 inches in cross section . Examples illu t•atT g the use of the
figures show how temperature computations can be made fow :intermediat e
thicknesses . In determining the timber dimensions it is necessary to ad d
the thickness of the wood cauls as both the width and thiehAess 1ff%,,a+t th e
rate of heating . While the cauls may not always be of the same specie s
and hence may 'not have quite the same diffusivity as the matewi]y Wf, .g
glued usually the effect on the rate-of heating :does net require . arz
special consideration . If the cauls are thick In proportion to the, electio n
being glued the lamer diffusivity of the two woods should be used iz com puting the required heating period to insure an ample time allowance .
•,
Mimeo . No . R10I
-6-
When the width and thickness of a timber are not the same, the temperature will always change more slowly along the transverse axis perpendicula r
to the wide faces than along the axis connecting the narrow faces . For example, in a timber 6 by 12 inches in cross-section, the temperature chang e
takes place more slowly at the mid section along the 6-inch axis connectin g
the faces 12 inches wide, than it does along the 12-inch axis connecting th e
faces 6 inches wide . The temperatures given in the figures for points betwee n
the center and surface are therefore temperatures at distances measured alon g
the shorter transverse axis and apply for the portion of the timber wher e
end heating does not affect the results .
Under commercial conditions, desired information relating to tempera ture changes in rectangular timbers will be either :
t
(1) The time Tb required to obtain a given temperature Ux at some
particular point or ,
(2) The temperature U x when the heating period Tb is given .
Only a very simple computation is necessary to use the data plotte d
in figures 1 to 8 for any particular heating conditioA and for any
species having a diffusivity different from 0 .00025, that assumed for
making the temperature computations . To avoid the arithmetical calcula tions for finding values of Ux or U (as indicated in equations U1 and Cl
appendix) figure 10 was prepared to make it possible to determine thes e
temperatures directly after connecting the initial wood temperatures U a ,
and the heating medium temperature Ub , with a straight edge .
Method of Using Figure 10
In this figure the initial wood temperature Ua and temperature Ux ar e
read on the vertical left-hand scale . Values of Ub are read on the vertica l
right-hand scale, and values of U are given on the bottom scale . To use the
chart, place a straight edge connecting the given initial wood temperature U a
on left-hand scale, with the given heating temperature Ub on the right-han d
scale . This is illustrated in figure 10 by the full line connecting th e
initial wood temperature 60° F . and the heating medium temperature 220° F .
and also by the dotted line connecting the initial wood temperature 75° F.
and the heating medium temperature 200° F . -.assume, for example, that the
initial wood temperature and heating temperature are those connected by th e
dotted line and that Ux = 180° F . To find the corresponding temperature U ,
(any temperature obtained from the figure for the timber in question) pas s
horizontally from Ux = 180° F . on the left-hand scale to the intersectio n
with the dotted line . Directly below this point, the temperature U is found
on the bottom scale to be about 177° F . If U is known and it is desired t o
find Ux read upward to the line connecting the two temperatures and find Ux
on the left-hand scale opposite the point of intersection . For example, again
using the dotted line assume U = 155° F . Opposite the point of intersectio n
directly above 155° F., Ux is found to be about 160° F .
Mimeo . No . RJ L alt
-~7 -
.1
desired at a part .c l r poet ,
a timb p.-r- err-h n theg
is Ua and the hating ~nedYt- temperature is . .Ub, the f ow'
be used . Determine temper .ti e U by• 'm &ns . oaf equatim-(C 1)
ie
using figure 10 . From the prig er fie- (f
1 tea )
ttrlktl
gtestion ; find the hating pgp''
:r°'1r'ta-ch~ hemp
rn
tie T, fcl'
} . required poet. Determiner bit 41figure, by the factor P' taker Mee tabs. 1, $g.y, .the s
considerat0,
To find w per
twh
kvsl.tirajg%
Tb by the factor F, taken from table 1 for the
_
_
.
iod !I
_
de
_is g
si., l orc
T (
find the corresponding heating period T . In this case T
Tb .
'
F
s .
c)~~~~,~'
.
Find, from the proper figure, (figs .. 1 to 9) the temperature U
after heat u. for T ho w, ..peterm e the t er-at'gx
g
(C1) appendix, or figure 10 .
v.
-ll
Exanip1-t
Data give n
Dry-bulb temperature - - 175° = Ub
Initial wood temperature r• 70°
Cr
tt
al . nensi
8 by .11 .5 it hw,s _
'Sprc ies ' --
. white
Ua
r
cl•tIcl
, ino ,udg
.
.eki't
. .~ .
.
oak
-t_ Y
Require d
Find heating period needed to, obta
17:0,° F . at .the _cente-- ,
n
-a
tLemy.O±Ett ure
-
Solutio n
U., .
From equation (C) or figure lcL f old the c,
using equation (C 1)
r
U
9BO T
r
170) (l40)
,
_
200 - L
7 ,or about . 113 a r Ji L1
Zf figure 10 is used place the straight edge on%VO° .,' 1-eft-hay ~
on 175° F . right-hand scale . From 170°, the requi0od roper u
Mimeo . No . Rl434
,
,
,
1. '
left-hand scale pass horizontally to the line connecting the tempertb.le s
of 70° and 175° F. and read the temperature U on the bottom scal e
found to be but 193° F ., the
r. ate+ obtained rasing equa
:t
a
tem
ratur•e",
,
_
FirIA from figure 5',4
d to red
-Vbilroe.i
;'of
a
t.
fiber
~
193° , F'os'flit 4
nter `lf' ' die's ' i om the surface "
a' $ X «'
usivity
,inehes'f•
't sound to be about 19 .5 hours when t1d " •
("OM
is 0 .00025 .0' . -,= :
r:+=+-:_ w
1
-1
'
1
Sine.E'"table 1 shows that F = 1 .25 for oak, the time T found from th e
chart-must be multiplied by 1 .25 . 19,5 x 1,25 z T b 24 .4 or about 24 . 5
hours, Tip Yieat~i
p4rio'd• nee l,ed'' to obtain a temperature of 170° F . at th e
center cf a timbe 8-by 14 .5 inc'hes'.
Example 1 b
Assuming the . same timber and same, heating colditi~ rs as gen in.
cl- from one , of the .
example l a , find t`fie -temperature . at a distance of 1
surfaces 8 inches apart after the timber has been heated for' 14 hours .
Since 14 hours = Tb , the corresponding time T, to be used in finding th e
temperature U from the chap' ). 14 +I .25. 11 .g .zhours
'
.t• .-Tige,te pegeatur e
'
J obtained,At, .1 my from thrf_e he,
from f gure t to te ' aoou;'' ~.d V. ~ e,
.•
equation {B1) or from figu?e 10 .
Using '{ e uation (Bl ~ Ux
l7~
= 16 6
This can 'be' fo.pIrITI - bm f't'gu tk* ' by placin ►klt
188° F. oi'
in exa i l-e a
Te din u r -tr
70° and 175° F . ' i'e-gM` !r y''i b tip
across from the point of .i
€,01oni.;. .
the temperatures PC. by using' equation TB") . • and
Exa mpie. 2
~,
D-ter gi ve•'x . ~~
:`
•.~ .
©m scale t
i -hand scale_i
ocnetig
it 16'60'.
T 1:1' tit*
-aft
I
'mt . '
, q ;' • - '; k t, .
i ~ti I T . 11 . : ,:r .11.
u" •-
+
- :''t" -T
Dry-Ibul'b+ tempe'ratu .ne
a' PM "
F.
`l ilI-i
,. i
•• +
n
F . =' Ua .,r ) .:` -auk e: '} ~7 e .
.. .
. t ,,o
p
diiYhes includin g
CI no
.vifi p!6 7
to. al dim
thickness of cauls .
• 't • l r1 r.fmi -. .
~~ • .••
In+ati~l wood temis,~ '
r .sll ,
Speel,rs fShoast UQ r .~ ter,
-1 .
I - ,
.! imeo . Trio . 'R143 4
I
C
L
-9 :,.
Require d
Find ho
g period necessary to c tAtn
190° F . = (Ux ) at the center.
to .erat a of
.
As previously explained, whew, the dxxyw u.lb temperature is betwee n
180° and 200° F . the .dif
iv`ii - g
n• ix-table should be Increased eimat
10 percent, since the rate of heating as indicated by the diffusi v=ity, .i a
somewhat faster in this higher temperature rage, apparently because of the
greater-amount of water vapor in the air . From table 1 the diffusivit y
for Douglas-fir, for temperatures below 180° F . is given as 0 .00022 . Tao
creasying. this by 10 percent gives 0 .000242 . The factor F (last colum n
table 1) would then be 0 .00025 + 0,000242 = 1 .03 ap proximately . This faqi r
is the same as would be obtained by dividing F = 1 .14, taken from table 1 ,
by 1 .1,
In this example it is necessary to interpolate for the thickness o f
6-1/2 inches which is between two thicknesses for which temperature compu tations have been made (figures 4 and 5) . This can be done by first find ing the heating periods required for timbers 6 by 8-1/2 and 8 by 8-1/ 2
inches .
As in example la , the first step is to determine U by using eithe r
equation (C1) or figure 10 ap illustrated in the preceding examples, I n
this caste U is found to be about 191 0 F .
The second step is to find from figures 4 and 5 the time T require d
to obtain a temperature of 1WJ° F . at the center of a 6- by 8-1/2- and o f
an 8- by 3-1/2-inch timber . From figure 4 the time T2 required to reach
a temperature of 191° F . . at the center (5 inch from the surface) of a
6- by 8-1/2-inch timber- is fend to be about 8 .75 hours and from figure 5
the time T1 required to reach this te:rterature'at the center of an 3- b y
8-1/2-inch timber is found to be about 12 .5 hours . The difference in
thickness of these two timbers is 8 - 6 = 2 inches and the difference in
the heating period required to obtain the same timperature at the cente r
is 12 .5 - 8 .75 = 3 .75 hours, The timber 6-1/2 by 8-1/2 inches in cross section is 6-1/2 - 6 or 1/2 inch thicker than the one 6 by 8-1/2 inche s
for which the required heating period was found to be 8 .75 hours . Dividing
1/2 by 2 gives 1/4 . The heating period required for the timber 6-1/2 b y
8-1/2 inches in cross-section would then be computed as 8 .75 plus 1/4
(12 .5 - 8 .75) = 9 .69 or about 9 .7 hours when the diffusivity is 0 .00025 .
Since the time 9 .7 must be multiplied by 0 .00025 + 0,000242 s 1.03
the computed--heating period Tm needed to obtain a temperature of 190° a t
the center of the 6 .1/2 by 8-1/2 inch timber is, for the conditions given ,
(9 .7) (1 .03) = 10 hours, approximately ,
If in the foregoing example the dimensions were 7-1/2 by 8-1/2 inche s
instead of 6-1/2 by 8-1/2 inches the required heating period would b e
Mimeo . No . 81434
-10 .
computed as 8 :75 plus (?~
_
(12 .5 - 8030 * 11 .5f. Sours. MultipT
ing this by 1 .03 gives 11 .9 hours required te reach - . #ernperatKTe of• 190°F . i n
the 7-1/2 by 8-1/2 inch coast Douglas-fir ;ri .bens I44ermarple 2•-it is - •
assumed that the difference in heating wefiod required f'or•the•intermedia E :_ .
in th i•
thickness p prop_ortioKj.1,~to the diffe r
ence in hmerng periods reed for't
. i
Expressed in terms o
ness of a timberlint.ermedi
another havin { a ste: lie r •
W
a
have the
t. sam- wv
.` iWI
r
temperature
at a given
.r .
of the timber with a thickness rm'wou
i.e~l f'i '
Dm - D2
T2 + ( DDm D ) (T1 - T2) v litre T1 is the . time fegnz1'e 4
1
2
temperature Ux al :-+e sane-'proportional 4,0xeVAOrlknes s
DI and T 2 Is the tome required to obtain IMILiAirnIgIVAMSTilOtel'Milme
Proportional distance in the timber of I ti''t?
g oZ
~►
~:~, 0 `
This is illustrated in example 2 where Dm = 6-1/q ; 40 .
"°1t„ '
D 2 ^ 6", T= 12 .5 hrs . and T 2 = 8.75 hrs . The width of all three timbers
is the sayte, 8-1/2" . Similarly for the timber 7-1/2 x 8-1/2 inches i n
cross-section Dm = 7-1/2 inches while Dl, D 2 , T1 and T2 are the same as in
the first computation .
Humidity Curves
iT
,s
R
they el tive
.Figures 11 and, .12 are added fat mtveni ppice , in,i;i-acl
humidity required to obtain any give r 1 .equiflibriu4a moist ;e •F ontel.: ori t o
find the required wet bulb temperature for a given relative humi1 7 aridr
dry-bulb temperature . Figuur
sho is the i*elati'dn sof' :' 'qu-i~lib S i i Sri®e~~4tu' ®
content, dry bulb tem p erature and relat4T'• hui irdi j4•ari figur di jsbiowff ., .
aa1btt-, eratu sJ4 ., ' '
the relative humidity for different wet ! aVaddl
APPENDIX
_a
,4
-
~
,
iiFormula Used For Computing the Rath of Te-3 ayture Ch i
.
tl
r
Rectangular Timbers Surrounded by a Heating Medium at'C tenstant 'Te 'n erature
•
.
If h2 repre 'tints the diffusivit,y i, the ,rajSal direction .'.o the
diffusivity in the' tangential direction and q', . time diffusil- ty in the
a<'ime o . No . R1434
-11-
longitudinal direction, the change of temperature in a timber woeId t e
expressed by a partial differential equation in the general form ,
5U
6t -
52U
2
5x2 + k
h
2
62 U
5y2 +
q
2
5 2U
(1,
z
r
In this expression U represents the temperature at any point thin th e
timber and t represents the time heat is applied .
If the length,of the timber is assumed to be sufficient to pre Ven t
end heating affecting the tefl1 tature at ' the mid length, where temperatur e
changes take place most slowly, and when for practical purposes the tan ^-ential dif fusivity = 2 may be assumed to be the same as the radia l
diffusivity h2 , equation (1) can be written ,
U
5U = h 2 I52
`
rb'
5x2
62 U
77d
(2 )
In order to compute the diffusivity or the temperature changes i n
timbers of various cross-sectional dimensions equation (2) must be solve d
subject to the follauing boundary conditions :
LetZ=(U--U I )
U=U0 whent = 0
Z = (Uo - Ul ) when t = 0
Z = 0 when x = 0, y =0
and t
0
x=a, y=b,i
In this case U is the temperature at any point (x, y) in a trans verse section of the timber ; Uo is the temperature of the wood before hea t
is applied ; U1 is the temperature of the heating medium ; x and y are dis tances from the coordinate axes at x = 0 and y = 0 ; a and b are the cross-sectional dimensions of the timber and t is the time in seconds .
Substituting Z in equation (2 )
FZ
6t
h2
2Z
-6 : x 2
&2Z
8 y2
(3 )
A solution of this equation to meet the boundary condition$ gives,
cc
Z = Z 2
n 1 n=1
Am, n
.Mimeo . No . 81434
sin m
sin
..
b
e
-12 -
TI2
h2
t
(a2 + ~2) .
where Am,n is the definite integra l
ab
a .fb f(
00
X ,fL)
sin
k
ma sin
b µ• d~ d u
when f(k,p.) is a constant and equal to (U 0 - U1 ) a.s in the heating c
tions assumed,
4(Uo - .U l )
iri
ab
4 (U o -
m:
sin nb N• dk dp.
.n
TJ
'-&a cos riTfr
[1 - cos m Tr
mn7r2
16(U o - U
TI 2
MI
when .rrj Apt", yie,•ztefg i1 e h odd, ant
0 w.
we
7.
+ 1 sin
a
sin
sin 3bY e
sin
5a
+ 1 sin
7T
sin
be
by
e
+ 1 sin l x s in
be
a
in 7rxa
+ 71 sin
±' imeo . ho . 81434
sin 77rv
e
b
7T
'o r rhrs est
.Th~. !
st few te'nv
the„ ,
ternpe-,a-tuxe.•II at any Pgr{,x,Y)E Ptn r
be written,
-
J
., , .
tiny; flat , .` . . .
ape? - t may th
.
7T
9 )
2 h2 t
2h2
7f•t
b
2
25
2_ + 'b
(a
b2
- -T2 h2 t (a2+'
25
b
7T2 h2 t (ag +
7r h2 t (12+
a
-13 -
2
1
)
)
49 )
b
+•
•
•
(4 )
geimaddion (
covert b
ze
that 'we mom
verg r-
secttea l c
r' J
-t i
esw
A1* valves
tee
,
a and
s>
se' sand okly a
ns as n led
heating periods . It may be noted
t is large amd also $I
the cro gsr
since t
le e1
eI (
* _.
a
dr tions are Different frtse
Data for Time-tem
I
~;
r
: 1
ting
G
--
.
I
.qua .®
The equations (A) to (C) are relations derived)* - +
show
the
relation
of
time
and
diffusivity
and
methods
ofg
thwltmg
the
These
weed
b
e
tove
wee
.
t
tke
lieat;
r
s•
t
orPesPoaA*ng
*'
from tVtWe wetted ice' dORPWM the data for IOW temp
in the fig es . '
Syg*p Used ' vO
strat3
LqEdggrCalfirotAwt
-
@Np at'
I
la
'~
91ying to comput9 data (figures 1 to 9, inclusive )
-e .
U0 = Initial wood temperature assumed as 60° F .
Ul = Heating medium temperature as,su ed .
h2 = Diffusivity ab
U
ed a
ZOO' F .
Q0 O9025 (s quire inch per second) .
t at acmes :o . toter
distance from the su&faee . Values of U are temperature s
Temperature obtained in a givcm ti
within the wood at ttld, ; astance from the surface shows by
the figure .
. "
T = Time of heating (im, hours) nece ary to obtain the .tern ate *
U . This period of heating is determine d , frem the fiiguie Se
the timber under consideration*
Mimeo . No t R143)4
-14,
S, bo1s Appiring to 'T'ece!'-)e'rathiee CVnIputatibi-isl''.1hen Diffusivit ,
.
, .
Feating
end', or'anth"
''
are Different from Those Used n.
Computing Data Plotted in Figures 1 to 9, Inclusive
a-a- -
-
w6od 't-enperatu p i ; ,r
Ub = Any heating medium tem.perature i.
-
r. l*#I
,
ado, .voiOlie' .of
. ..
,
di ffafS. ivity'astaumerl fioe'a 41ee wood , ,( sepla,r-piitt a
per second) .
Temperature obtained at. a clefiNe .te point in the timber whe n
. the
WO..Qd temr2taturd .ta, the heating temperature Ub ,
or both are different , from U o .= 60 0 and. U l
200° .F ., th e
Ux
initial wood and' iaelItt'Lug .meditlm temperatures used in eemputin g the- curve - ehO Wff -iri. - the figure's& '
Tb
.7;
Time of heating (in hours) required to obtain the temperatur e
Uwhet: either thedi.ffusivItT. h , the initial woods femp .epa e
tare Ua , the heating medium. ter,7)erature Fboner all til.'ree
variables are different frc
60 0 , U l
200° F ., and
0 .00025, the valuer used in computing data for the curves .
The relation of the heating period T and the heating period T b i s
readily determined from the actuation,
T, (h2 )
= T ( 0 . : 0 0 2 5. )
.
'
---
----
-1,1,,
(A )
In -LI-Le expression T is the computed heating period found from the . figur e
for the timber in question, and 0 .00025 is the diffusivity ces,sume .d in
ma'-inc the cDoputa.tions ..
Tb - is the heating period. to be determiaerf,eo r
-vhich, is assumed when the wood has a diffusivity
Values of h2x
for different species are given in
,
'
ean be readily shown that the relation of the tempe,ratures U and ;
U j assumed in compalting the- values ;. of U pl .otted in . the figures, anti =an y
other
wood' arLd heating med.im temperature U a and . U b is expresse d
by the equation ,
-1 .
U = Ux b
I I (U b e U 9. )
( TT ' - ' o )
-where Ux is the temperatu r e obtained at a particular distance fronii th e
sua . 'ace when either ;U.o: i Ub , or -both are differerA -far
and U I ‘
Yi1neo . -Ko . R14-34
7hen Ux is assumed, it is necessary to salve for U to dete .rmin e
the heating period T from the-proper f-igure . . -Solving for U tn equation
( B) gives,
(Ub - Ux) (Ul "( U'b ' - Us )
(C )
- -
.
Since in computing the time-temperature curves shown'in figure s
I to 9, inclusive, U0 was assumed as 6 00 ar'd U1 'al.-20 00 '. ;` subs:titlittin•
these numerical values in equations (B) and (C) gives ,
Ux
U
U,~ 20
.-
(Ub
- Ua ) (200
- U) I
f
,
140
-T(Ub .- .- ux) .:,(140).l .
T
..
•--
(B 1 )
-
(Ub - Ua)
Values •e' U and U can also be determined by means of figure 10 a s
x
explained in the text .
1 !
a
Reference s
(1) MaeLEMI, J . D .
1930 . "Studies of heat conduction in wood -- results of ; steamin g
' green round southern pine timbers , " Proc . A ."T .1i. ,
pp . 197-217 .
(2)
(3 )
19'32 . "Studies of heat conduction in wood -- Par 'II ---Result s
of steaming green sawed southern pine timbers," Pzoc .
pp . 303-329 .
1934 . "Temperatures in green southern pine timbers after variou s
steaming periods," Proc .. A .W.P .A., pp . 355-373 .
( 4)
1935 .
(5)
Mimeo .
" Temperature and moisture changes in coast Douglas-fir, "
Proc . laseST .P .A ., pp . 77-103 . -
'
1936 . "Average temperature and moisture .reduction- calculations
for steamed round southern pine timbers , " Proc . A .W.P .A . ,
pp . 256-279 . .
R1434
. ;i
(6 )
( 7)
Mac LEAN? , J . D .
1940 . "Rela t. ion of wood density to rate of temperature change i n
wood in different heating mediums,'4 Proc . A .W .P,A . ,
pp . 220-248 .
1941 . "Thermal conductivity of wood," Heating, Piping and Ai r
Conditioning 13(6) :380-391 .
-
(8 )
7
- :_
--,
.,-
-
1942, "The rate of temperature change in wood panels heate d
between hot plates, " Mimeograph No . R1299 of the
Forest Products Laboratory .
(9)
1943 . "Method of computing the rate of temperature change i n
wood and plywood panels when the two opposite face s
are maintained at different temperatures," Mimeograp h
r
No . 81406 of the Forest Pro4.uats Laboratory .
w
,r .
-*-.
-- +
1•
~.. .
w
- -~.. -
.I
nA
Mimeo, No . R1434
_17_
r
Table 1 .--Average diffusivity factors determined from experiments in th e
dry-kiln with the humidity controlled to dive equilibrium
moisture content conditions of about 12 percent, and fromother data .
(For moisture content range of about 10 to 20 percent and dry-bul b
temperatures up to about 180° F . )
Specific
gravity
range
(air dry
wood)
0 .4 0
to
.5 0
Woods tested
t
Other woods of
Average
?t similar specific : di.ffu s; gravrity
sivity
facto r
•
hx
Facto r
F ?.
0 .0002 5
:Port Orford `white- : 0 .000220 :
cedar, eastern
: redceciar, _hemlock, :
:mahogan y
:Douglas-fir
hx
1 .1 4
.50
to
.60
:Southern yellow
:pine
Blackgum, black
walnu t
.000210 :
1 .1 9
.60
to
.70
:Red oak, white
:oak, suga r
:maple, hickory
:Ash, yellow birch
.000200 :
1 .2 5
.
-The factor F will be found convenient in finding T b where T b (hx)
=
T(0 .00025) . This is shown as equation (A) in the appendix . From
this equation T b -
C .0002 5
25
(T) =FT or solving for T, give s
T =
Tb
The diffusivity values. hX should be increased about 10 percent fo r
dry-bulb temperatures between 180 0 and 200° F . and about 15 percent
for dry-bulb temperatures over 200° F . The factor F should be divide d
by 1 .1 when the diffusivity is increased 10 percent and should b e
divided by 1 .15 when it is increased 15 percent .
Pimeo . No . 8143 4
cd
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m O
H 0
0
co 4
o
•
a
'
44
b
m
0
0 0
4%o
O
m
4-1
+)
0D
0
o
'
4,
o
4-1
I
&I
r.
43
,-I
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Pignre 4 .--Temperatures at center and halfway between the center and surfaces of timbers 6 inches
thick, after various heating periods . Temperature of heating medium, 200° r . ; initial
wood temperature, 60° 7 . ; diffusivity, 0 .00025 .
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Figure 11 .--8elation of dry-bulb temperature, relative humidity, and equilibrium moistur e
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