Methods of determination of liquid transfer coefficient in building materials

International Journal of Heat and Mass Transfer 55 (2012) 4318–4322

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Methods of determination of liquid transfer coefficient in building materials Vadzim I. Nikitsin a, Beata Backiel-Brzozowska b,⇑ a b

Pope John Paul II State School of Higher Education in Biala Podlaska, 95/97 Sidorska Str., 21-500 Biala Podlaska, Poland Bialystok University of Technology, Faculty of Civil Engineering and Environmental Engineering, 45 Wiejska Str., 15-351 Bialystok, Poland

a r t i c l e

i n f o

Article history: Received 7 November 2011 Accepted 30 March 2012 Available online 21 April 2012 Keywords: Liquid transfer coefficient Capillary water uptake Building materials

a b s t r a c t At present there is no unified, widely recognized method of estimation of the dependency between the liquid transfer coefficient and the moisture content in material. It is possible to determine such relation by several methods requiring long-term and laborious experimental research, expensive apparatus and complex calculating procedures. The original method of determination of the liquid transfer coefficient in building materials is presented in this work. This method is limited to obtaining experimental data of the kinetics of one-way capillary water uptake by specimens and calculations based on the suggested model. Accuracy of the given method was confirmed using experimental results of capillary absorption process of lightweight concrete and ceramic material. Since the suggested method is theoretically grounded and its accuracy is comparable with other known methods and at the same time it is less laborious and cost-consuming, it can be recommended for practical application. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Coupled, non-stationary heat and moisture transfer processes occur under influence of the surrounding environment in capillary-porous materials of building envelopes. To describe this processes numerous models of different complexity were formulated. However, comparison of experimental results of exploited exterior walls with calculation results achieved using these models shows that the actual moisture in structural materials repeatedly exceeds the calculated values. Increased moisture of building envelopes decreases their thermal protection and lowers sanitary-hygienic inside conditions. Moreover, corrosion processes of materials intensifies and this lowers service life of buildings. According to the authors’ opinion, the significant difference between the theoretical and experimental results is caused by the fact, that an influence of the surrounding environment is not wholly taken into account in calculation procedures. Above all it refers to precipitations which wash exterior surfaces and water penetrates deep into the material structure by capillary pulling up from the surface. It is possible to formulate the equation of density of one-dimensional, non-stationary moisture flow in the form:

@w : j ¼ Dw @x

⇑ Corresponding author. Tel.: +48 85 746 96 22; fax: +48 85 746 95 59. E-mail address: [email protected] (B. Backiel-Brzozowska). 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.03.080

w ¼ u  q ¼ w  qW :

ð3Þ

Since it is possible to express the density of moisture flow by the following formula:

J ¼ Dw

@w @u @w ¼ Du ¼ Dw ; @x @x @x

ð4Þ

after taking the relation (3) into account we obtain:

Dw ¼

Du

q

¼

Dw

qw

:

ð5Þ

ð1Þ

It is possible to write the moisture balance formula:

  @w @ @ @w : ¼  ðjÞ ¼ Dw @s @x @x @x

The value of the coefficient Dw in a general case depends on the pore structure of material, its moisture content and properties of liquid. This coefficient is called differently in literature. According to the international standard ISO 9346:2007 the parameter Dw is called the moisture diffusivity and it depends on moisture content w. Two other coefficients are also used to describe the liquid transfer process: coefficient Du dependent on moisture u and coefficient Dw dependent on moisture w. It should be emphasized, that indicators w, u, w are connected through the following relation:

ð2Þ

It is obvious, that in order to obtain the accurate quantitative description of the process of the moisture transfer in capillary-porous building materials, the accurate values of the capillary transport coefficients should be used. Solving this very complex issue using the known methods usually leads to significantly different results. Therefore the aim of this work is to elaborate scientifically justified, simplified, and yet enough accurate method of determination of the coefficient of liquid transfer in building materials.

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Nomenclature sorption coefficient (kg/(m2 s0.5)) liquid transfer coefficient dependent on w (moisture diffusivity) (m2/s) liquid transfer coefficient dependent on u (kg/(m s)) liquid transfer coefficient dependent on w (kg/(m s)) height of the specimen subjected to the process of water capillary uptake (m) density of moisture flow along X axis (kg/(m2 s)) volume of pores filled with water in capillary uptake process per material unit volume (m3/m3) relative volume of mobile liquid (m3/m3) mean capillary radius (m) mean equivalent capillary radius (m) degree of pore saturation moisture content mass by mass (kg/kg) capillary moisture content (kg/kg)

A Dw Du Dw h j P P(u) rm re s u uk

2. Selected methods of determination of the liquid transfer coefficient At the present there is no unified, widely recognized method of estimation of the dependency between the liquid transfer coefficient and the moisture content in material. It is possible to determine such relation by several methods requiring long-term and laborious experimental research, expensive apparatus and complex calculating procedures. The coefficient of the liquid transfer (moisture diffusivity) can be evaluated using moisture profiles w(x, s), measured in tested specimens after different time elapsed after start of water capillary uptake process. Direct and indirect methods of measurement can be applied to determine these profiles (e.g. slice-dry-weight method, electrical methods, gamma-ray attenuation, neutron radiography, magnetic resonance, etc.). They are described among others in [1,3]. The liquid transfer coefficient Dw can be determined based on the moisture profiles w(x, s) by solving Eq. (2) using the Boltzmann transformation method, the Profile method, etc. [1,2]. According to the Boltzmann method quoted among others in [4,5], it is possible to state, that for one-way capillary water uptake in specimens of infinitive length made of homogeneous, isotropic porous material, having constant initial moisture content wo (w = wo for x > 0 and s = 0) and constant moisture content on the surface which is in contact with water wk (w = wk for x = 0 and s > 0), differential equation (2) has only one solution. This solution is described by characteristic curve called the profile w–k, where k is a Boltzmann variable determined from k = x/s0,5. If material is homogeneous and isotropic, experimentally outlined moisture profiles in the different moments of time reduce themselves to only one curve w–k. Then the coefficient Dw is calculated from the following model:

Dw ðwi Þ ¼ 

1 2 @w @k

Z

us w wk

sorption moisture content (kg/kg) moisture content mass by volume (kg/m3) capillary moisture content (kg/m3)

Greek symbols g dynamic viscosity (Pa s) h contact angle (rad) k Boltzmann variable n(u) tortuosity of capillaries filled with mobile liquid q apparent (bulk) density of material in air-dry-state (kg/m3) qw density of liquid (water) (kg/m3) r surface tension (N/m) s time (s) sk duration of capillary absorption process (s) w moisture content volume by volume (m3/m3)

In the Profile method an additional profile obtained through averaging of two profiles measured for s and s + ds is used to determine the Dw coefficient. For the illustration we will demonstrate calculation of the Dw coefficient by the Profile method with application of the model [1]:

1 Dw ¼   dt

Rx

x¼xo

ðwðx; t þ dt  wðx; tÞÞdx @wo @x0

;

ð7Þ

which explains Fig. 2. The constant Profile method described in [1,14,18] is also wellknown. According to this method only one profile is determined after density of moisture flow through the specimen is stabilized. This moisture profile does not depend on time. The density of moisture flow resulting from steam diffusion through the upper surface of the specimen is minimum. It is obvious that to apply these methods some experimental and computational difficulties should be overcome, as well as errors associated with random character of experimental data or relations which approximate these data should be expected. How it is shown in [17] the accuracy of determining the Dw coefficient at low and high moisture level is very low. Another procedure of determination of the moisture transfer coefficient called ‘‘moment method’’ should also be mentioned [7,8]. Its realization does not require the knowledge of moisture profiles, however it requires from the researcher a great experience

wi

kdw:

ð6Þ

w0

Interpretation of the model (6) is shown in Fig. 1. If the assumptions of the Boltzmann method are complied and at time s = s0 we have a moisture profile u(x) than according to the Matano method quoted in [6] the moisture transfer coefficient can be describe based on the equation:

Du ðux0 Þ ¼

1 Þ 2s0 ðdu dx x0

Z

1

x0

x

du dx; dx

where ux0 is moisture content on the moisture profile for x = x0 and s = s0.

Fig. 1. Solution of the differential equation (2) using the Boltzmann variable.

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Dimensionless function b(s)

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

Fig. 2. Graphical interpretation of the Profile method.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

The degree of pore saturation s for correct realization of experiment. There are data approximation formulas for some materials and proposed coefficients in [19]. The authors of this work have also developed the method of determination of the liquid transfer coefficient [9]. Experimental part of this method requires only getting data referring to kinetics of one-way capillary uptake in building material specimen Dm(t)exp. According to this method function Du in any form should be initially given. Then Dm(t) can be calculated by solving Eq. (2). Du function for which the calculated values Dm(t) fit to the experimental Dm(t)exp is taken. Thanks to the original computational procedure not only the moisture transfer coefficient, but also the moisture profile from the start to the end of the capillary uptake process, as well as function of the pore size distribution were developed. However, mathematical complexity of the model and the necessity to apply computational technique for processing the experimental data can be perceived as weak points of this method.

Fig. 3. Dimensionless function b(s).

The tortuosity can be evaluated from the model shown in [10]

nðuÞ ¼ 

1 2

þ sin



PðuÞ

arcsinð2PðuÞ1Þ 3

2  ;

PðuÞ ¼

q ðu  us Þ: qw

ð10Þ

The mean capillary radius rm can be evaluated using the formula shown in [11],

rm ¼ r e n2 ðuk Þ;

ð11Þ

where re is mean equivalent capillary radius for which the value of the water penetration resistance is equal to the resistance of the examined capillary-porous material. The value of can be calculated from 2

2gh : sk r cosðhÞ

3. Suggested method

re ¼

To eliminate experimental and computational difficulties in described methods the authors suggest application of analytical equation introduced in [10] for determining the moisture transfer coefficient in capillary-porous material Du. The calculation requires data referring to radius of capillaries containing liquid menisci, sorption membrane thickness, specific surface of contact between mobile liquid and sorption membrane, function of pore size distribution, wetting angle, liquid viscosity and other factors. The function of pore size distribution can be evaluated by mercury porosimetry or using model for this function based on the principle of maximum entropy [10]. At low-level moisture content function of the liquid transfer coefficient Du is [10]:

The sk time corresponds to the maximum relative volume of mobile liquid P(uk) which is also called in literature the active porosity available for capillary transport Ua (P(uk) = Ua). It is possible to use Ua as the lower value of P in Eq. (9). To calculate the degree of pore saturation (see Eq. (9)) it is recommended to assume P = 1.05Ua based on numerous experimental data. From [11] results, that

Du ¼

q  r  cosðhÞ  rm  bðsÞ: g  nðuÞ

ð8Þ



2

h

sk

¼

A wk

ð12Þ

2 ð13Þ

:

Then, after considering models (11), (12), (13), and (8) we receive the formula for the liquid transfer coefficient

Du ¼

2qA2 n2 ðuk ÞbðsÞ ; w2k nðuÞ

Dw ¼

Du

q

ð14Þ

;

The degree of pore saturation can be calculated from the following formula:

qðu  us Þ : qw P  qus

800

ð9Þ

The P value can be determined basing on absorbability N evaluated by standard methods according to the formula P = qN/qw. Determining the lower value of P will be shown farther in this paper. Sorption moisture us, kept by Van der Vaals forces on walls of capillaries and pores in the form of thin membranes regarded as motionless, depends on temperature and relative humidity of air in pore space. How it was shown in [20] the heat front associated with sorption processes moves right before the moisture front during capillary saturation of initially dry porous material. The value of material sorption moisture at the relative humidity of air equal 80% can be used as the us. With the known value of the function s based on the graph shown in Fig. 3 it is possible to determine the dimensionless function b(s).

equaon (14) equaon (15) staonary method [14] method of the work [9]

700

Liquid transfer coefficient Du·104, kg/(m·s)

s¼

600 500 400 300 200 100 0 0

2

4

6

8

10

12

14

16

18

20

Moisture content u, % Fig. 4. Dependence between liquid transfer coefficient and moisture content in ceramic material determined by various methods.

V.I. Nikitsin, B. Backiel-Brzozowska / International Journal of Heat and Mass Transfer 55 (2012) 4318–4322

4321

Fig. 5. Dependence between the liquid transfer coefficient and the moisture content in lightweight concrete determined by various methods.

where wk is the mass of liquid absorbed by the unit volume of the specimen in the moment of time sk (wk = q(uk  us)). Methods for determining A, wk and sk are given among others in [1,12]. It should be emphasized that A/wk ratio is decisive in the known empirical equation [13]

Dw ¼ 3:8

w 1 A2 1000wk : 2 wk

ð15Þ

As it can be seen from Eq. (14) this ratio is sufficiently theoretically grounded. For comparison, calculations of the liquid transfer coefficient Du were performed based on models (14) and (15) and on the experimental method [9] using data obtained for specimens of ceramic material characterized by the following parameters: q = 1750 kg/ m3, A = 0.132 kg/(m2 s0.5), Ua = 0.288, N = 0.307, us = 0.0024 kg/kg. The results of these calculations are presented in Fig. 4. Additionally, the results from [14] received with the Stationary method (Bryling method) for specimens of ceramic brick with density 1700 kg/m3 is presented in Fig. 4. Fig. 4 shows that calculation results from the model (14) and results from the work [9] are similar. This means that Eq. (14) can be applied in the method from the work [9] for the first approximation. Values of the coefficient Du determined with the Stationary method turned out to be dozen times lower than values determined from the model (14). It is possible to explain such a considerable difference by the fact that the Stationary method evaluates the moisture state of material after completion of the process of capillary uptake, when the moisture profile is constant in time and the stationary moisture flow is associated with steam diffusion through top surface of the specimen. Additional test of accuracy of the given method was performed using experimental results of capillary absorption process of various density lightweight concrete obtained by different methods:  method of moments (Gafner method) (q = 600 kg/m3) [8],  method of gamma rays and solving Eq. (2) with Boltzmann method (q = 520 kg/m3) [15],  stationary method, (polyconcrete q = 400 kg/m3) [16],  electrical method and the transient Matano method (q = 407 kg/m3 and q = 363 kg/m3) [6] Calculations according to the suggested method and Eq. (14) were performed based on experimental data obtained by the authors using specimens of cellular concrete with q = 565 kg/m3. The outcome of this confrontation is shown in Fig. 5. It can be seen from Fig. 5 that considered methods gave results differing both qualitatively and quantitatively. Although it should be emphasized that suggested quantitative method corresponds

with all compared methods except to the Stationary method. The explanation of this difference was given earlier in this paper. Fig. 5 shows that accuracy of Dw evaluation at low and high level of the moisture content differs considerably. It is associated with the random character of experimental data and with the accuracy of the approximation of these data. Inaccuracies of this type were precisely described in [17]. 4. Conclusion The method of determination of the liquid transfer coefficient in building materials is presented in this work. This method is limited to obtaining experimental data of the kinetics of one-way capillary water uptake by specimens and calculations based on the model (14). Since the suggested method is theoretically grounded and its accuracy is comparable with other known methods and at the same time it is less laborious and cost-consuming, it can be recommended for practical application. References [1] M. Janz, Methods of measuring the moisture diffusivity AT high moisture levels Lund: division of building materials, Report TVBM-3076, 1997. [2] V.P. De Freitag, M. Krus, H. Künzel, D. Quenard, Determination of water diffusivity of porous materials by gamma-ray attenuation, in: Proceeding of the International symposium on Moisture Problems in Building Walls, Porto, 11-13.09.1995, Porto, 1995, pp. 445–460. [3] Bezpalko Natalia Zastosowanie techniki TDR do badania procesów przenoszenia masy i energii przez wybrane przegrody budowlane (praca doktorska), Politechnika Lubelska, Lublin (in Polish). [4] S. Roels, J. Carmeliet, Analysis moisture flow in porous materials using microfocus X-ray radiography, Int. J. Heat Mass Transfer 49 (2006) 4762–4772. [5] J. Crank, The Mathematics of Diffusion, Oxford University Press, United Kingdom, 1989. ˇ erny´, Moisture and heat transport and [6] M. Jerman, M. Keppert, J. Vy´borny´, R. C storage characteristics of two commercial autoclaved aerated concretes, Cement-Wapno-Beton (Cement-Lime-Concrete) 2011/1, pp. 18–29. [7] D. Gaffner, The moment Method for measuring moisture diffusivity of porous building materials, in: Thermal Performance of the Exterior Envelopes of Whole Buildings International Conference 2011, . [8] D. Gaffner, Determination of moisture flow coefficients for porous materials by using the ‘‘moment method’’ Building Physics in the Nordic Countries, Swedish Council for Building Research, DW13. Stockholm, 1988, pp. 423–427. [9] V. Nikitsin, B. Backiel-Brzozowska, M. Bołtryk, Experimental-analytical method of evaluation the moisture migration coefficients and parameters of ceramic material porous structure Papers of the Commission on Ceramic Science, Polish Academy of Science, Polish Ceramic Bulletin Ceramika/Ceramics, vol. 84, 2004, pp. 371–376 (in Polish). [10] A.V. Afonin, V. Nikitsin, Vycˇislenie koéfficientov perenosa zˇidkoj vlagi v kapilljarno-poristych stroitel’nych materiałach Vest, BGTU Stroit. i Arhit. 1 (2002) 10–15 (in Russian). [11] V. Nikitsin, B. Backiel-Brzozowska, Evaluation of water penetration resistance in ceramic material Papers of the Commission on Ceramic Science, Polish Academy of Science, Polish Ceramic Bulletin Ceramika/Ceramics, vol. 103/2, 2008, pp. 1031–1036 (in Polish).

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