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Analytical model of heat transfer in porous insulation around cold pipes

Tom Guldbrandsen a,, Per W. Karlsson b, Vagn Korsgaard a

a Department of Civil Engineering, Technical University of Denmark, DK2800 Lyngby, Denmarkb Department of Mathematics, Technical University of Denmark, DK2800 Lyngby, Denmark

a r t i c l e i n f o

Article history:

Received 1 July 2010

Received in revised form14 September2010

Accepted 14 September 2010

Keywords:

Insulation

Condensation

Piping

Moisture

Aircon

Wicking

a b s t r a c t

A thermal insulation system is analysed that consists of a cold tube insulated with a porous material

faced with a vapour retarding foil.Water vapour will diffuse through the vapour retarding foil and condense on the cold tube. To avoid

build-up of water in the insulation a hydrophilic wicking cloth is wrapped around the cold tube and

extended through a slit in the tubular insulation and a slot in the facing to the ambient so that condensed

water can evaporate into the air. Some of the moisture in that part of the wicking cloth situated in the slit

in the tubular insulation will diffuse backwards to the cold pipe and contribute to the heat uptake of the

cold tube. This part is calculated for the stationary case and compared with the sensible heat transfer

through the tubular shaped insulation material, using measured dry k values and measured fictitious

moist k values.

2010 Elsevier Ltd. All rights reserved.

1. Introduction

Air conditioning installations in warm humid climates include

piping carrying chilled water from the refrigeration plant to the

chillers in the rooms to be air-conditioned.

Usually the temperature of the chilled water and hence the sur-

face temperature of the piping will be below the dew point of the

ambient air which means that the water vapour in the air will con-

dense on the piping. To prevent dripping, which of course is unac-

ceptable, the piping must be thermally insulated. The thickness of

the insulation must be sufficient to increase the surface tempera-

ture of the insulation to be equal to or higher than the dew point

of the ambient air.

As insulation materials are not completely water vapour tight,

measures must be taken to minimize water vapour from diffusing

into the insulation. This flow of water vapour leads to interstitial

condensation in the insulation layer and dew formation on the sur-

face of the pipe itself. Interstitial condensation will reduce the

insulation value and may cause deterioration of the insulation

material, and dew formation on a metal pipe may cause corrosion.

This results in a significant waste of energy and loss of process con-

trol from the original design. Wet insulation can also contribute to

indoor environmental problems by either promoting mould

growth on exterior surfaces of the insulation or dripping water

onto other building materials.

Besides, by diffusion moisture can enter into the insulation with

moist air through slits and flaws in the vapour retarder facing due

to barometric pressure fluctuations and, for vertical piping, due to

the inverse chimney effect as well. The amount due to barometric

fluctuations will be insignificant compared to the other

contributions.

The amount due to the inverse chimney effect can be significant

and should be prevented by an airtight but diffusion open tape

over the wick slot in the vapour retarder facing.

Different measures are available to control water vapour trans-

fer into thermally insulated piping and hence reduce the amount of

condensation. (ISO 15758).

(a) Installing a vapour retarder on the surface of diffusion open

insulation materials.

(b) Use of insulation materials with a high water vapour resis-

tance factor (low permeability).

(c) Use of a capillary active fabric to continuously remove con-

densed water from the pipe surface to the environment. (The

wicking concept) [1].

The last measure was developed as a consequence of the fact that in

practice the two first measures have not shown to be able to pre-

vent substantial moisture ingress into cold pipe insulation over

time.

0017-9310/$ - see front matter 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijheatmasstransfer.2010.09.044

Corresponding author.

E-mail addresses: birtheogtom@mail.dk (T. Guldbrandsen), p.w.karlsson@mat.

dtu.dk (P.W. Karlsson), v.korsgaard@mail.tele.dk (V. Korsgaard).

International Journal of Heat and Mass Transfer 54 (2011) 288292

Contents lists available at ScienceDirect

International Journal of Heat and Mass Transfer

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j h m t

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2010.09.044mailto:birtheogtom@mail.dkmailto:p.w.karlsson@mat.%20dtu.dkmailto:p.w.karlsson@mat.%20dtu.dkmailto:v.korsgaard@mail.tele.dkhttp://dx.doi.org/10.1016/j.ijheatmasstransfer.2010.09.044http://www.sciencedirect.com/science/journal/00179310http://www.elsevier.com/locate/ijhmthttp://www.elsevier.com/locate/ijhmthttp://www.sciencedirect.com/science/journal/00179310http://dx.doi.org/10.1016/j.ijheatmasstransfer.2010.09.044mailto:v.korsgaard@mail.tele.dkmailto:p.w.karlsson@mat.%20dtu.dkmailto:p.w.karlsson@mat.%20dtu.dkmailto:birtheogtom@mail.dkhttp://dx.doi.org/10.1016/j.ijheatmasstransfer.2010.09.044

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2. The wicking concept

A hydrophilic wicking cloth is wrapped fully or partly around

the cold pipe and extended through a slit in the tubular insulation

to the outer surface of the insulation. The water vapour condensingin the wick around the cold pipe will by capillary action be sucked

to the surface of the insulation from where it can evaporate into

the ambient air [2,4].

When the condensed water is sucked out through the slit in the

insulation, its temperature will increase, and hence also its vapour

pressure. When reaching the outer surface it will be close to the

saturation pressure, at the ambient air temperature.

In practice the porous tubular insulation is covered with a va-

pour retarder facing or jacket to reduce the diffusion of water va-

pour into the insulation. The wicking cloth is extended through a

slot in the facing/jacket and adhered to the outer surface, Fig. 1.

For insulation materials with a very high permeability (e.g. min-

eral wool) the vapour retarding facing/jacket will cause the water

vapour to diffuse directly to the surface of the cold pipe and notcondense in the fibre matrix, and hence the insulation will stay

dry. This has been verified by X-ray scanning of the insulated pipe

as shown in Fig. 2.

In contrast to the very vapour permeable mineral wool, mois-

ture will condense in closed cell insulation where the temperature

is below the dew point of the ambient air.

The moisture content in the wick is found by experiment to be

less than 50% of the saturation value. Even though the heat

conductivity of water is approximately 15 times larger than that

of mineral wool, the water content in the thin wick aroundthe pipe

will only have a marginal influence on the insulation value.

3. Backward diffusion

However, one could expect that water evaporating from the

moist wick in the slit will diffuse backwards to the tube and con-

dense and hence increase the heat flow significantly and conse-

quently decrease the insulation value.

As it is difficult to measure this diffusion flow directly the pur-

pose of this paper is to set up a mathematical solution to the

problem.

4. The physical system

The physical system to be mathematically investigated is

shown on Fig. 1.

Nomenclature

a real constant (dimensionless)D diffusion coefficient (of water vapour in air) (m2/s)h latent heat of vaporization (J/kg)H heat flow density per length (W/m)i imaginary unit (dimensionless)

k real constant (dimensionless)m molecular weight (kg/kmol)n whole number (dimensionless)P water vapour pressure (Pa)r radius (m)R gas constant 8.32 103 (J/(kmol K))p permeance (kg/(m2 s Pa))T temperature (K)

Greek symbolsu angle (radians)k heat conductivity (W/(m K))

Subscripts

amb ambientb backward diffusiondry dry insulationmoist moist insulationo outside (at vapour retarder)r vapour retarder facingi inside (at cold tube)

Alufoil vapour retarder

Insulation

Wick fabric

Cold pipe Condensation

Ti

ro

Tamb r i

ToBackward

diffusion

Fig. 1. Cross section of cold pipe insulated according to the Hygrowick concept.

Fig. 2. The picture is constructed from a computer calculation based upon scanned

X-ray transmission data. Because the variations in density are small the uncertainty

is considerable. Nevertheless a qualitatively correct variation versus radius and

along thewet wick is clearly seen. Colours refer to theinternetversion of thearticle.

Gray scale (in brackets) refer to the paper version. The wick around the cold pipe is

moist: yellow (gray). The insulation close to the wick and the wick in the slit is

hygroscopic moist: fuzzy yellow (fuzzy light gray). Most of the insulation is dry:

green (dark gray). The steel pipe and the fixture are red (black). The air in the pipe

and the surroundings is transparent to x-rays. The colour is blue (black). CompareFig. 1.

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Choudhary et al. [2] have set up a mathematical model of the

system shown and by numerical calculations demonstrated that

the pipe insulation system incorporating the wicking concept is

effective in getting the condensed moisture out of the insulation

system.

In the steady state situation the amount of water vapour diffus-

ing through the vapour retarder facing equals the amount of con-

densed water evaporating from the protruding part of the wick.

As mentioned above some of the moisture in the wick in the slit

will evaporate and diffuse backwards to the surface of the cold

tube and condense. This moisture flow also represents a heat flux

that will decrease the insulation value of the system. The decrease

is difficult to measure directly. In the present paper an analytical

model is developed to calculate this heat flux. The results of the

model have been verified using laboratory experiments.

5. Mathematical model

5.1. Derivation of the analytical solution for the stationary distribution

of water vapour pressure P in the insulation

The analytical solution is found by solving the Laplace equation

with the known boundary conditions. One could object that the

water vapour pressure does not exactly fulfil the Laplace equation

since the diffusion coefficient is varying with temperature. The rel-

ative change in diffusion coefficient is, nevertheless, neglected

since it is small compared to the relative change in water vapour

pressure. We consider this approximation as reasonable in view

of the modest requirement for precision.

We are looking for a solution to the Laplace equation in polar

coordinates with radius r and angle /.

The traditionally used partial solutions are of the form

Pp1r;/ rk expiku 1

where i is the imaginary unit, k is a real constant and all four com-

binations of signs are possible [3].

The possible values of k are determined by the boundary

conditions.

Because of the special geometry caused by the wick and by the

retarding foil (see Fig. 1) it is preferable to use the slightly different

form:

Pp2r;u rik expku 2

where k is still real.

From (2) we can create the linear combination:

Pp3r;u sin k lnr

ri

coshk u 3

where ri is radius of the cold tube and u = p at the wick in the slit.Expression (3) fulfils the boundary condition that requires mir-

ror symmetry around the slit.

Let

kn 2n 1 p

2 ln rori

4where nP 0 is a whole number and ro is the radius of the retarding

foil. Then for k = kn, n = 0, 1,2 . . ., Eq. (3) also fulfils the boundary

condition that the diffusion flow density through the retarding foil

must be zero (to a first approximation). In order to fulfil the bound-

ary condition at the cold tube we must finally add the known va-

pour pressure Pi at the cold tube.

After insertion of (4) into (3) and summation over all n we can

express the complete solution to the Laplace equation for the water

vapour as:

Pr;u Pi Po Pi

X1n0

ancoshkn p

sin kn lnr

ri

coshkn u

5

where Po is the vapour pressure P(ro, p). If we insert this in (5) wefind the constraint:

X10

an 1n

1 6

Po can be calculated from the known temperature at that position.

The coefficients an can be determined if the water vapour pres-

sure along the wick in the slit is known.

5.2. Calculation of heat transfer caused by backward diffusion

The heat transfer Hb per length of the pipe due to backward dif-

fusion is approximately given by:

Hb 2hH2O DH2O mH2OR T

Zrori

1

r

@

@uPr;u

up

dr 7

where the factor 2 is due to the diffusion to both sides from thewick. hH2O is the latent heat of vaporization, DH2O is the diffusion

coefficient and mH2O is the molar weight of water vapour. R is the

gas constant and T is the absolute temperature.

As mentioned before we have neglected the temperature

dependence of the diffusion coefficient. Similarly we now neglect

the temperature dependence of the latent heat.

In (7) we then insert the mean temperature and the latent heat

and diffusion coefficient at the mean temperature.

By inserting (5) in (7) and applying the substitution u = knln(r/ri)

the integral in (7) can be calculated as

IX1n0

an kn tanhp kn 1

kn

Z2n1p2

0

sinudu

" #

X1n0

an tanhp kn %X1n0

an 8

where the approximation in (8) is justified because the argument of

the tanh function is large for all realistic values of ri and ro and for

all n.

Insertion of (8) in (7) then yields:

Hb 2hH2O DH2O mH2OR T

Po Pi X1

0

an 9

Example based upon case 2 in [2]:

With an ambient temperature of 305 K (32 C) and a cold tube

temperature of 275 K (2 C) the temperature at (r, p) is determinednumerically to be%285 K (12 C) so the average temperature along

the wick is 280 K (7 C) and we get: hH2O = 2.5106J/kg,DH2O = 2.510

-5 m2/s, mH2O = 18 kg/kmol, Pi = 700 Pa, Po = 1400Pa

(at 285 K) [5].

It seems reasonable to assume that partial solutions with low

values of an are dominating.

As an example, let a0 = 1 and all other coefficients an = 0.

After insertion in (9) we then find: Hb = 0.68 W/m.

As another example consider the case a0 0 and a1 = 0 and

an = 0 for all other values ofn:

It seems obvious that P(r, p) must vary monotonously from ri toro. It can be shown that this requirement leads to the limits:

1/8P a1P 1/4 corresponding to 9/8P a0P 3/4 and thus to

5/4P IP 1/2 and, finally, to 0.85 W/m > Hb > 0.34 W/m.

For comparison the heat flow density of the same tube with dry

insulation and without a wick is given by:

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Hdry 2p k To Ti

ln rori

10With k = 0,04 W/(m K) (typical for porous insulation) and ro/ri = 2.5

we find: Hdry = 8.2 W/m. Hence, in this example, the backwards dif-

fusion represents 11%, respectively 5.6% of the dry heat flow.

5.3. Calculation of heat transfer due to diffusion of water vapourthrough a non-ideal vapour retarder and directly to the cold tube

Only in the start up period water vapour in the ambient air will

be able to diffuse through the slot in the facing. The start up period

will last until the water vapour pressure in the wick in the slot has

reached a value equal to the water vapour pressure in the ambient

air and equilibrium is established.

If the vapour retarder facing is not completely diffusion tight

the vapour pressure in the wick will increase to the saturation

pressure at a temperature close to the ambient temperature. A lar-

ger part of the protruding wick will then be moist enabling evapo-

ration of the intruding vapour. This vapour will in turn condense

on the cold pipe and be sucked backwards by the wick in the slit

to the protruding part of the wick.

With ro = 40 mm, an ambient temperature of 305 K and a rela-

tive humidity of 90% the ambient water vapour pressure is

4270 Pa. From (5) it appears that for all angles not too close to pthe vapour pressure just inside the vapour retarder is close to the

vapour pressure Pi = 705 Pa at the cold tube. With a permeance

of 1 perm= 5.7 1011 kg/(m2 s Pa) we then approximately find:

Hmoist Pamb Pi 2p ro hH2O 5:7 1011 kg=m2sPa

0:13 W=m $ 1:6% of Hdry 11

6. Experimental verification

It is difficult, as already mentioned, to measure the backward

diffusion directly. Indirectly, however, it should be possible to ver-ify the backward diffusion as calculated by the above-developed

analytical solution. This is done by first measuring the k-value of

the dry insulation kdry. Thenkdry is compared with thek-value mea-

sured when the state of equilibrium for the same test tube has

been established after a certain period of time in a moist

environment.

Such tests have been carried out at the testing laboratory FIW

(Forschungs Institut fr Wrmeschutz) in Munich, Germany.

The test tubes used consisted of a steel pipe with an outer

diameter of 54 mm insulated with a 30 mm of glass wool faced

with a glass fibre reinforced Al-foil and with the wicking system

integrated as shown in Fig. 1. So for the test tube ro = 57 mm,

ri = 27 mm.

The test tube was first placed in a dry environment and the heatuptake by the cold pipe was calculated by measuring the water

flow through the pipe and the temperature difference between

the inlet and outlet. The surface temperature of the Al-foil facing

was kept constant at 22 C and the water in the pipe at 6 C. From

these measured data the dry k-value of the glass fibre material was

calculated to be 0.0311 W/(K m). The same test tube was then

placed in a climate chamber with a constant temperature of app.

23 C and a relative humidity of 80%. During the initial period of

a few weeks, water vapour from the air in the climate chamber dif-

fuses through the wick-slot in the Al-foil and condenses in the wick

around the cold test pipe. After this period, the weight of the

test-tube stabilised with a moisture content of 0.3% by volume that

remained unchanged during the subsequent 6 months period.

Practically all the condensed moisture was situated in the wickaround the tube and in the slit in the tubular glass wool shell.

The 0.3 volume% represents 0.3 mm of water layer around the cold

tube and will have no significant influence on the insulation value

of the tubular glass wool shell.

As the reinforced Al-foil facing was undamaged and the end

caps sealed, no further water vapour could enter the system, as

the vapour pressure in the wick in the slot in the Al-foil facing

was equal to the vapour pressure of the ambient air.

After a period of 6 months under stable conditions, the fictitious

k-value of the moist insulation material was measured to be

0.0330 W/(K m). This corresponds to a 6% increase of the dry k-

value.

7. Comparison between mathematical models and

experiments

The heat transfer from the ambient air to the cold tube can be

calculated from Eq. (10).

With To Ti = 226 = 16 K, ro/ri = 57/27 and kdry = 0.0311

W/(K m) we find Hdry = 4.18 W/m.

With kmoist = 0.033 W/(K m) we find Hmoist= 4.44 W/m.

The increase in Hdry from 4.18 to 4.44, e.g. 0.26 W/m, corre-

sponds to approximately 6% of Hdry and is assumed to be due to

the backward diffusion.

In comparison, Hb is calculated using Eq. (9) with a0 = 1, where

Pi is the saturation vapour pressure at the cold tube temperature of

6 C viz., 940 Pa.

Po is the vapour pressure at the point where the wick protrudes

through the vapour retarder. Since (in contrast to case 2 in [2]) the

vapour retarder in this case is almost tight the net flow out of the

wick is small. This in turn means that the vapour pressure Po is only

slightly lower than the ambient vapour pressure Pa.

With Pi = 940Pa and Po % Pa = 2270Pa (23, 80% rh) we get

Po Pi % 1330 Pa and we find the conservative value Hb = 0.62

W/m corresponding to a 15% increase of Hdry.

Another identical test tube was situated in a climate chamber

with the same temperature of 23 C but with a lower relative

humidity of 50%.The dry k-value was measured to bekdry = 0.0315 W/(K m). After

13.5 months the moist k-value was measured to be

kmoist= 0.0318 W/(K m) and the moisture content to be 0.2% by

volume.

Using Eq. (9), Hb is calculated to be 0.223 W/m corresponding to

2.7% of Hdry.

Using Eq. (10) we calculate Hdry = 4.238 W/m with kdry =

0.0315W/(K m), and with kmoist= 0.0318 W/(K m), Hmoist=

4.278 W/m and hence Hmoist Hdry = 0.040 W/m corresponding to

app. 1% increase ofHdry.

8. Concluding remarks

A mathematical model was developed to calculate the back-ward diffusion in a cold pipe insulation systemincorporating

the wicking concept. The wick fabric wrapped around the pipe

extends through a slit in the tubular insulation. When hygrother-

mal equilibrium after some time has been established the mois-

ture in the wick fabric situated in the slit will evaporate and

diffuse backwards to the wick fabric wrapped around the cold

pipe and condense, and by capillary action be sucked back again.

This moisture evaporation/condensation circuit represents a heat

flow that will add to the heat uptake of the fluid in the cold

pipe. This heat flow can be calculated by means of the mathe-

matical model developed in this paper. Laboratory experiments

measuring fictitious k values appear to confirm the mathematical

model.

Depending on the ambient conditions the backward diffusionwill increase the heat uptake by a few percent only.

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References

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outer water-absorbing layers connect through slot in intermediate heat-

insulating layer, US Patent No. 5441083 (1995).

[2] M.K. Choudary, K.C. Karki, S.V. Patankar, Mathematical modelling of heat

transfer, condensation, and capillary flow in porous insulation on a cold pipe,

Int. J. Heat Mass Transfer 47 (2004) 56295638.

[3] H. Margenau, G.M. Murphy, The Mathematics of Physics and Chemistry, D. Van

Nostrand Company, Inc., New York, 1943 (chapter 7).

[4] G.C.P. Crall, The use of wicking technology to manage moisture in below-

ambient insulation systems, in: A.O. Desjarlais, R.R. Zarr (Eds.), Insulation

Materials: Testing and Applications, ASTM STP 1426, vol. 4, ASTM International,

West Conshohocken, PA, 2002.

[5] Physical property values applied are taken from Handbook of Chemistry and

Physics, Chemical Rubber Publishing CO.

292 T. Guldbrandsen et al. / International Journal of Heat and Mass Transfer 54 (2011) 288292