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DE-ICING ANALYSES - THE THERMAL PROPERTIES OF ACCRETED SALINE ICE
I. Horjen
To cite this version:
I. Horjen. DE-ICING ANALYSES - THE THERMAL PROPERTIES OF ACCRETED SALINE ICE. Journal de Physique Colloques, 1987, 48 (C1), pp.C1-397-C1-403. �10.1051/jphyscol:1987155�.
�jpa-00226301�
JOURNAL DE PHYSIQUE
Colloque C1, supplement au n o 3, Tome 48, mars 1987
DE-ICING ANALYSES
-
THE THERMAL PROPERTIES OF ACCRETED SALINE ICE I. HORJENNorwegian Hydrotechnical Laboratory, Sintef Gr., Klaebuveien 153, N-7034 Irondheim, Norway
&sum& Une analyse thkorique du d6givrage est proposke ; elle est bade sur l86quation transitoire de la chaleur. Cette hation contient trois coefficients qui sont fonction de la salinitk de la glace et de la temp6rature : wnductivitk thermique, chaleur sgcifique et densitk de la glace. ~'6quation de la chaleur a 6t6 r6solue avec l1hypoth&se selon laquelle la glace initiale ne contient pas de bulles et pss&de une salinitg constante. Des tests de sensibilit6 aux pardtres ont kt6 faits pour trouver la variation de 1'Qnergie de d6givrage ( ~ / m ~ ) avec la tmp6rature de l'air, 1'6paisseur de glace et la pxlissance @orifique apport6e. Ce travail presente aussi une nouvelle thkorie sur le rejet d'eau salke durant le refroidissement B la surface libre de la glace.
ABSTRACT
In this paper a theoretical analysis of de-icing is given, based on the transient heat conduction equation. This equation contains three coefficients which are functions of ice salinity and temperature:
thermal conductivity, specific heat capacity and density of the ice.
The heat conduction equation was solved assuming the ice initially does not contain air bubbles and the salinity is constant.
Sensitivity tfsts have been done to find the variation of energy of de-icing (J/m ) with air temperature, ice thickness and supplied power of heat.
I . T H E O R Y O F D E - I C I N G
We consider an ice layer of uniform thickness formed by spray icing on a vertical surface.
In order to simplify the analyses we assume that icing has terminated when de-icing sets in. Further we assume that the ice falls off as soon as the temperature at the ice/structure interface has reached the melt- ing point of the ice.
Assuming that heat transport takes place only in the horizontal direction (x), the heat conduction equation reads:
(the symbols are defined in the nomenclature) Boundarv conditions
Initially we assume that the ice temperature is uniform and equal to the air temperature:
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987155
Cl-398 JOURNAL DE PHYSIQUE
At the ice/structure interface a constant power of heat Q ( w / m 2 ) is supplied from t = 0. (Fig. 1 ) : 0
-k ae/ax = Qo for x = 0, t >
o
( 3 ) At the ice/air interface we have the boundary conditionk aslax = Q~ for x = h, t
, o
( 4 )Q is the net heat flux at the ice surface and consists of several terms (heat loss by convection, evaporation, sublimation and S
radiation).
For the de-icing analyses we are only interested in knowing the temperature history of the ice close to the underlying structure. A
simpler outer boundary condition is then normally sufficient:
In this paper we assume Eq. 5 to be valid.
W A R M PLATE
Fig. 1 . Sketch of an ice layer with heat fluxes during de-icing.
The solution of Eq. 1 with the boundary conditions Eq. 2, 3 and 5 will be a function of the following parameters:
The necessary de-icing time t = t is then the solution of m
where B m is the melting temperature which is a function of the ice salinity S:
e = -(as)/(? - I O - ~ S ) m
where a = 5.4113
.
O C (Assur /I/)The energy of de-icing per unit area is defined by
2 . THE THERMAL PROPERTIES OF ACCRETED SALINE I C E
The temperature history of a solid body depends on the boundary condi- tions and the followinq "thermal properties" of the body: thermal con- ductivity, specific heat capacity and density. Formulas of these para- meters for saline ice will be given below.
2.1. Thermal conductivity
This term depends on the distribution and orientation of brine channels and air pockets in the ice.
In order to find a mathematical expression of the thermal conducti- vity we must consider an idealized model of the ice. This model consists of horizontal and vertical brine channels.
Neglecting the effect of air pockets the thermal conductivity in the horizontal direction becomes
ki and kb are the thermal conductivity of pure solid ice and brine respectively. E~ and E~ are fractions of the relative brine volume vb in horizontal and vertical channels respectively. (Note that
Eh + E = I . ) v
2 Ssecific heat capacity
Neglecting the heat of crystallization (or dilution) of salts in the brine the specific heat capacity of saline ice may be written
where
y is the mass ratio of precipitate (including its water of crystalli- zation) and brine and
r
is the mass ratio of salts in the precipitate(hydrate) and salts in the brine. Eq. (11) may be deduced by remem- bering the definition of specific heat capacity of a medium and using relations given in Cox and Weeks ( / 2 / ) , later called CW. (Note that in their paper C is used for y and k is used for r.)
When 0 > -8.2 the mass of solid salts in the brine is insigificant
0 0
and we put y = 0. Between -8.2 C and -23 C the precipitated salt Na2S04
.
10H20 (sodium sulphate decahydrate) affects slightly the thermal properties of the ice. From the results CW the mass ratio yJOURNAL DE PHYSIQUE
may be approximated by
(Correlation coefficient r2 = 0.9882. )
At -23'~ another hydrate, NaCl
.
2H20, starts to precipitate and y is no longer given by Eq. (12).Defining now a as the mass ratio of crvstalline water and salts in the precipitate the following relation is easily deduced:
where Sb is given in per mille.
0 0
In the temperature interval -8.2 C to -23 C a is almost conztant /2/, and w e will here use the mean value a = 1.30. (For 0 > -8.2 C n is about 2.7.)
When 8 > -8.2 C we have 0 y = 0 and Eq. (11) reduces to the equation of Zubov (/3/) (note that a sign error occurs in the English trans- lation)
.
2.3. Ice density
From CW the sea ice density is given by
where v a is the relative volume of air and F(0) is defined by
This function may be approximated by a ratio of polynomials of third order in 0 /2/.
The density of pure ice is from Pounder (/4/) given by
where Q . LO = 9 17 kg/m3 is the ice density at 0'c and oi = 1.53
.
is the coefficient of volumetric expansion of pure ice.
When heating of the ice layer starts, we assume there is no void volume in the ice (v = 0). The ice densicy may then be expressed by
a
No brine is expelled during the heating process. If gravity drainage is negligible, it may be shown that the coefficient of volume expan- sion of saline ice and pure ice is theoretically equal, i.e. u = a
(Cox /5/). i
3 . NUMERICAL RESULTS
The de-icinq theory specified in chapter 1 and 2 is implimented in a computer model developed at NHL. In this model we assume the ice salinity is constant (gravity drainage of brine is neglected).
From icing theory (Horjen and Vefsnmo / 6 / ) it may be shown that during continuous spray icing the ice salinity near the phase inter- face is given by
where n is the freezing fraction of impinging spray. If brine drainage processes are negligible, this equation does also approxi- mate the ice salinity during de-icing. In the examples follgwing we have put S = 30 /oo (corresponding for example to Sw = 33 /oo and n = 0.9 in Eq. 17).
The fraction of total volume of brine channels in the ice sample, which is orientated horizontally ( E ~ ) , is a parameter in the model.
This parameter has a value between 0 and 1 . Indirectly a value of E~
may be found by calibrating the model against experimental data. By using E = 0.8, the calculated and the measured results agree quite well.
his
value will hence be used in the following calculations.In order to examine the relative effects of the different parameters in the de-icing model, several sensitivity tests have been carried out.
The following standard condition has been chosen:
Air temperature: Ba = -15' c
Ice thickness : h = 5 cm Supplied power : Qo = 800 w/m2
The de-icing process will continue until the interface structure/ice has reached the melting temperature.
Figure 2 shows the temperature distribution in the ice layer at about 10minutes time intervals. For curve 3 (t = 33 min) we have 8(0) = -1.6 C , which is approximately the "melting temperature" of the ice
containing 3 0 '/oo of salts. The de-icing time tm will hence be about 33 minutes.
In Fig. 3 the supplied de-icing energy per unit area, Eq. ( 9 1 , is plotted as a function of air temperature, ice thickness and supplied power. The point where the curves intersect gives the de-icing energy for the standard condition.
JOURNAL DE PHYSIQUE
I C E THICKNESS I r m l
Fig. 2. Temperature distri- bution in an ice layer of thickness 5 cm at various times after the de-icing power input (800 w/m2).
- 10 - I5 - 20
I
A I R I E M P E R A I U R E I ' C I
0
O l 2 3 L S
ICE I H I C K N E S S I c n l
Fig. 3. Supplied de-icing energy per w n t area as a function of air temp- rature, ice thickness and supplied power.
4 . CONCLUSIONS
Assuming an ice layer has been formed from imp'nging sea spray droplets and the ice salinity is constant (30 '/oo), the following conclusions may be drawn:
1 . The increase in temperature per unit time at the
ice/structure interface is largest at the beginning of the de-icing period. For the standard condition defined in section 3 the de-icing time (tm) is about half an hour (Fig. 2).
2. The energy of de-icing per unit area Em increases with decreasing air temperature, ice thickness or power supply
(Fig. 3). For the standard condition we have Em
--
1.5 MJ/~'.
- 1From Fig. 3 we also see that Em
-
Q o (approximately),.
- 2hence from Eq. ( 9 ) tm
-
Q o.
NOMENCLATURE
C = specific heat capacity (J/kg C) 0 k = thermal conductivity ( W / m 0 C)
Q = unit heat flux (W/m 2 )
h = ice thickness normal to the phase interface (m) t = t i m e ( s )
S = salinity (per mille)
2
Em = heat energy of de-icing per unit area (J/m )
1 = specific latent heat of fusion (J/kg) B = temperature ( 0 C )
p = density (kg/m 3 )
v = volume fraction
a = air
b = brine occluded in the ice layer i = pure ice
ss = precipitated salts w = spray
m = melting
o = ice structure interface s = ice/air interface
REFERENCES
/ 1 / Assur, A. (1958): Arctic Sea Ice, U.S. National Academy
Sciences - National Research Council Pub. 598, 1958, 106-138.
/2/ Cox, G.F.N. and Weeks, W.F.: "Equations for determining the gas and brine volumes of sea-ice samples", J. Glaciology, Vol 29, NO 102, 1983, 306-316.
/3/ Zubov, N.N.: "Arctic Ice", 1943, Translated to English by US Navy Oceanographic Office and the American Meteorological Society.
/4/ Pounder, E.R.: "The physics of ice", Pergamon Press, 1965, 151 pp.
/5/ Cox, G.F.N.: "Thermal expansion of saline ice", J. Glacio- logy, Vol 29, No 103, 1983, 425-32.
/6/ Horjen, I. and Vefsnmo, S.: "Computer modelling of sea spray icing on marine structures", Symposium on Automation for Safety in Offshore Operation (ASSOPOI, Trondheim, 1985, 315 -323.
