Simulation and Verification of Coupled Heat and Moisture Modeling

Transcription

1 Simulation and Verification of Couled Heat and Moisture Modeling N. Williams Portal 1, M.A.P. van Aarle 2 and A.W.M. van Schijndel *,3 1 Deartment of Civil and Environmental Engineering, Chalmers University of Technology, 2 Deartment of Architecture, Building and Planning, Eindhoven University of Technology 3 Deartment of Architecture, Building and Planning, Eindhoven University of Technology *Corresonding author: P.O. Box 513, 5600 MB Eindhoven, Netherlands, a.w.m.v.schijndel@bwk.tue.nl Abstract: The modeling of couled heat and moisture transort is found to be a valuable method in which ossible damage-related rocesses in building materials and comonents can be redicted. This aer includes the imlementation and comarison of two tyes of moisture otentials used in the modeling of couled heat and moisture transort: the natural logarithmic of the suction ressure (LPc) and the relative humidity (Rh). Two finite element models evaluating the couled thermal and hygric transort have been develoed using COMSOL Multihysics. The two develoed models were verified with the normative benchmark test of Euroean Provisional Standard ren These models aear to be valid redictive tools to investigate the imact of a change in climatic conditions on building materials and comonents. eywords: Heat, moisture, verification, redictive tool 1. Introduction The driving force of this aer is the Climate for Culture roject found within the Euroean Union s 7 th framework rogram. This roject attemts to face the challenges of climate change, while acknowledging the need for the reservation of cultural heritage. It is imortant to consider that a change in climatic conditions can cause not only damage, but can also destroy the basic structure of these sites, as well as their associated interior artefacts. As such, the modeling of couled heat and moisture transort is found to be a valuable method in which ossible damage-related rocesses in building materials and comonents can be redicted. A main issue that has been encountered when modeling couled heat and moisture transort concerns the choice of the moisture otential from a numerical oint of view [3]. The following moisture otentials are numerically not suitable: (1) Moisture content [kg/m3]. The main roblem with this otential is the discontinuity in moisture content at the interface of two materials. (2) Partial vaor ressure [Pa]. While using this otential, a roblem can occur if local temerature dros raidly. Furthermore, the saturation vaor ressure dros simultaneously and the relative humidity could rise above 100% (anomaly). (3) Suction ressure Pc [Pa]. The suction ressure ranges from 1 to ~10 9 Pa if liquid water transort is included. Esecially when liquid water (e.g. rain) is enetrating the material at a boundary that is dry initially, a sudden change from ~10 9 Pa to 1 Pa is exected. Such a change easily causes numerical instabilities and may roduce Pc < 1 Pa (anomaly). (4) Relative humidity Rh [%]. Rh encounters a similar roblem as described for Pc. However, if there is no liquid water enetration at the boundaries, the Rh is a suitable moisture otential. The latter will be illustrated in this aer. (5) The natural logarithmic of suction ressure (LPc) [Pa]. This otential seems to be best suitable for extreme conditions as will also be discussed in this aer. This aer resents the imlementation of various moisture otentials namely, the natural logarithmic of the suction ressure (LPc) and the relative humidity (Rh), into the governing equations for couled heat and moisture modeling. The COMSOL Multihysics modeling rocedures used for the two case scenarios are discussed; followed by, the verification of the models using a normative benchmark test. Lastly, the numerical simulated results obtained from the two models are comared. 2. Couled Heat and Moisture Transort The models are based on one-dimensional conductive heat transfer, q cd, according to Fourier s Law: q = q cd (1)

2 q cd = λ T = λ x where T is the temerature [ C, ] and λ is the moisture deendent thermal conductivity [W/m]. The total moisture transfer, g, includes both one-dimensional vaour, g v, and liquid flow, g l. Moisture transort can be characterized by various otentials as is demonstrated in the two develoed models, which include natural logarithmic of the suction ressure (LPc) and relative humidity (Rh). The following Equation (2) however describes a general form of the total moisture transfer using artial vaour ressure as the otential: g = g v + g l g v = δ = δ x (2) ξ ξ g l = D w = D w sat sat x where is the artial vaour ressure [Pa]; sat is the saturation vaour ressure [Pa]; δ is the vaour ermeability [kg/mspa] ; Dw is the liquid transort coefficient [m 2 /s]; and lastly, ξ is the moisture caacity [kg/m 3 ]. Furthermore, PDEs for energy and moisture balance are used to exress dynamic heat and moisture transort mechanisms. These balance equations can be exressed by: c ρ = ( λ T) (3) w = ( δ ξ D w ) sat (4) g v +g l where c is the secific heat caacity [J/kg]; ρ is the density [kg/m 3 ]; t is time [s]; and lastly, w is moisture content [kg/m 3 ]. The transformation of Equations (3) and (4) using LPc and Rh as moisture otentials is described resectively in Sections 3 and LPc model The heat and moisture transort can be described by the following PDEs using LPc as otential for moisture transfer [1]. C C T LPc = ( 11 T + 12 LPc) = ( 21 T + 22 LPc) (5) With: 10 LPc= log( Pc) CT = ρ c 11 = λ M w 12 = llv δ φ Psat, ρ art w (6) C LPc = M w 22 = δ φ Psat ρ art Psat 21 = δ φ, where t is time [s]; T is temerature [ C, ]; P c is caillary ressure [Pa]; ρ is material density [kg/m 3 ]; c is secific heat caacity [J/kg]; λ is thermal conductivity [W/m]; l lv is secific latent heat of evaoration [J/kg]; δ vaour ermeability [s]; φ is relative humidity [-]; Psat is saturation ressure [Pa]; M w = [kg/mol]; R = [J/mol]; ρ a is air density [kg/m 3 ]; w is moisture content [kg/m 3 ]; and is liquid water ermeability [s]. 4. Rh Model In the Rh model, the heat transort is described by the energy balance PDE resented in Equation (3). The moisture transort is formulated to include relative humidity, ϕ, as the moisture otential. Accordingly, the total moisture transfer shown in Equation (2) is exressed in the following [2]: g v = δ sat φ = δ sat φ x (7) g l = D w ξ φ = D w ξ φ x Furthermore, the left-hand side of the PDE describing the moisture balance in Equation (4) is simlified in order to be a function of relative humidity. The simlification is accomlished by exanding the artial derivative of moisture content and introducing general definitions for moisture caacity and relative humidity:

3 w = w = w φ φ φ = sat w φ = ξ φ = 1 sat (8) These above relationshis are subsequently introduced to yield the final form of the moisture balance equation for this model: ξ φ = ( δ sat D w ξ) φ (9) As observed in Equation (9), the moisture balance is now exressed solely as a function of relative humidity. Lastly, the boundary value roblem is formulated with the inclusion of two Neumann boundary conditions, namely the convective thermal and hygric fluxes: q c = α c (T s T a ) g φ = β φ (φ s φ a ) (10) where α c is the convective heat transfer coefficient [W/m 2 ]; β ϕ is the moisture [kg/m 2 s]; subscrit s denotes the surface boundary condition; and lastly, subscrit a denotes the surrounding air boundary condition. 5. Use of COMSOL Multihysics The finite element method was used to evaluate couled one-dimensional thermal and hygric transort by means of COMSOL. The Coefficient Form PDE Interface (c) multihysics modeling tool was used to describe the combined transort mechanisms for both LPc and Rh models. 5.1 Modeling equations The energy and moisture balance equations described for both models were simulated in COMSOL using the multihysics modeling tool entitled Coefficient Form PDE Interface (c). This tool contains the feature of a scalar coefficient form equation that is described by a balance equation and boundary conditions. The simlified PDE roblem neglecting convective heat transfer inside the material is described by the following: u d a = ( c u) in Ω n (c u) = g h T µ on Ω (11) u = r on Ω where u is a deendent variable on the comutational domain Ω; d a is a daming/mass coefficient; c is a diffusion coefficient; g is a boundary source term; h is a boundary coefficient; µ is a Lagrange multilier; n is an outward unit normal vector on the domain boundary Ω; and lastly, r is a known vector. The deendent variable, u, can be exanded into a vector including two deendent variables describing both heat and moisture transorts [2]: u = T LPc or u = T (12) φ Furthermore, the daming and diffusion coefficients are accordingly exanded into vector form. For examle, the following matrices describe the coefficients used in the Rh model: da u = d a_t 0 0 d a_φ φ With: d at = ρc d a_φ = ξ ( c u) = c T 0 0 c φ T φ With: c T = λ c φ = δ sat + D w ξ (13) (14) Introducing the above coefficient matrices into the simlified PDE yields the following formulation: ρc 0 0 ξ φ + λ 0 0 δ sat + D w ξ 2 T 2 φ (15)

4 The methodology resented above is also alied to the LPc model. This formulation also holds for two- or three-dimensional cases, as well as non-isotroic materials. The directional roerties can be described accordingly in the relevant coefficients. 6. Model Verification The verification of the LPc and Rh models was comleted by means of the normative benchmark test of Euroean Provisional Standard ren15026 [4]. This benchmark test is based on an analytical solution for onedimensional couled thermal and hygric transort in a homogeneous semi-infinite domain. Figure 1 shows the 20 m domain that is used for verification. that the internal boundary conditions may have on the domain. Accordingly, the changes in the temerature and moisture distributions taking lace from the exterior to the interior can be catured adequately in the numerical simulation rofiles. Temerature and moisture rofiles after 7, 30 and 365 days are to be calculated by the model. These rofile results are required to fall within +/-2.5% of the analytical solution. The hygrothermal material roerties and general data used in the verification are secified by the ren15026 benchmark test. 6.1 LPc model verification MatLab is used for the imlementation of the material functions. These functions are used to convert measurable material roerties such as, φ, δ and λ. These material roerties are deendent on the moisture content into PDE coefficients and also deendent on the LPc and T. This is schematically shown in Figure 2. Figure 1. Overview of the domain used in the verification. The exterior boundary of the domain is defined at 0 m, and as such the interior boundary is located at 20 m. As er the benchmark, the domain is in equilibrium with constant surrounding conditions of T=20 C and Rh=50%. The domain is thereafter exosed to a ste change to T=30 C and Rh=95% at the exterior boundary. The boundary conditions are introduced using Neumann boundary conditions, similar to those resented in Equation (10). Boundary resistances and moisture sources (rain) are to be neglected according to the benchmark. Nonetheless, it is necessary to include heat and moisture s in the models to solve the PDE boundary value roblem. The s were selected to ensure that constant boundary conditions are maintained during the numerical simulations. For examle, low surface transfer coefficients were alied at the interior boundary in order to eliminate any influence Figure 2. The conversion from measurable material roerties into PDE coefficients. The results of the conversion from material roerty into PDE coefficients are resented in Figure 3. Figure 3. PDE coefficients CT, CLPc, ij as functions of LPc and T calculated

5 At each oint in the material the vaour ressure can be calculated using similar corresonding functions. The governing PDE Equations (5) are imlemented using the coefficients from Figure 3. Table 1 rovides the secific boundary values, which are defined as Neumann tye boundary conditions. Table 1: Boundary conditions No Boundary condition 2 4 {0;0} 1 {100*(30*flc2hs(t,10)-T); 2e-8*(4037.5*flc2hs(t,10)- Pv(LPc,T))} In the boundary No. 1 condition, which corresonds to the interior boundary, the command flc2hs is imlemented to create a Heaviside function. The vaour ressure v at the surface can be exressed as a function of LPc and T similar to the coefficients of Figure 2. The numerical simulation included a calculation eriod of 1 year and time stes of 1 hour. A mesh containing 1639 triangular elements with at the exterior boundary a maximum element size of and an element growth rate of 1.2 was observed to be quite sufficient. The heat and moisture rofile results ertaining to the LPc model were lotted for 7, 30, and 365 days against the required ermissible error range secified by the benchmark. The numerical simulation results were found to meet the requirements stated by the benchmark as is deicted below in Figure 4 and Figure 5. Figure 4. The numerically simulated temerature distribution calculated using the LPc model. Figure 5. The numerically simulated moisture distribution calculated using the LPc model. The LPc model is shown to be a valid redictive tool to investigate the imact of variable thermal and hygric conditions on building materials according to the verification results. 6.2 Rh model verification In the Rh model, the initial and boundary conditions secified by the benchmark are introduced into COMSOL using global definitions. The heat and moisture transfer coefficients shown in Table 2 were selected to ensure constant boundary conditions in the calculations. Table 2: Heat and moisture s used for the verification of the Rh model. Name Exression Descrition α e 1.00E4 [W/m 2 ] Exterior heat α i 1.11E-10 [W/m 2 ] Interior heat (adiabatic) β e 1.00E2 Exterior moisture β i [kg/m 2 s] 7.78E-5 [kg/m 2 s] Interior moisture (adiabatic) The material roerties are tabulated and inutted into the Rh model by means of material interolation functions defined in COMSOL. The numerical simulation included a calculation eriod of 365 days and time stes of 3600 s. A mesh containing 1000 elements distributed with a geometric sequence and an element ratio of 10 was found to adequately

6 cature the raidly decreasing moisture rofile at the exterior boundary. The heat and moisture rofile results ertaining to the Rh model were lotted for 7, 30, and 365 days against the required ermissible error range secified by the benchmark. In accordance with Figure 6 and Figure 7, the numerical simulation results meet the requirements stated by the benchmark. Figure 6. The numerically simulated temerature distribution calculated using the Rh model. Figure 7. The numerically simulated moisture distribution calculated using the Rh model. As such, this model aears to be a valid redictive tool to investigate the imact of variable thermal and hygric conditions on building materials. 7. Model Comarison Although both models roduce similar results, they differ in moisture otentials and sace dimensions. A small comarison of the models using COMSOL : The LPc model consists of 290 elements, 1742 degrees of freedom and a solution time of 19 s. The Rh model consists of 1000 elements, 4002 degrees of freedom solved and a solution time of 11 s. 8. Conclusions Two tyes of moisture otentials used in the modeling of couled heat and moisture transort: the natural logarithmic of the suction ressure (LPc) and the relative humidity (Rh) develoed models were verified with the normative benchmark test of Euroean Provisional Standard ren These models aear to be valid redictive tools to investigate the imact of a change in climatic conditions on building materials and comonents. The Rh based model has the advantage that the measured material roerties can be directly imlemented as functions in COMSOL. The disadvantage is that this model is numerical not suitable for liquid water fluctuations at the boundaries. The LPc model is the best suitable for extreme conditions at the boundaries including liquid water fluctuations. The main disadvantage is the PDE coefficients are calculated from the measured material roerties as using MatLab as a re-rocessor. If an error occurs in this re-rocessing hase, it could be challenging to notice ossible errors from the PDE coefficient. References 1. A.W.M. van Schijndel, Heat and Moisture Modeling Benchmark using COMSOL, Proceedings of the COMSOL Users Conference 2008 Hannover, (2008) 2. N. L. Williams Portal, Evaluation of heat and moisture induced stress and strain of historic building materials and artefacts, Master s Thesis. Chalmers University of Technology, Gothenburg (2011) 3. A.W.M. van Schijndel, Multihysics modeling of building hysical constructions. Building Simulation: An International Journal, 4(1), (2011) 4. Euroean Provisional Standard ren 15026, Hygrothermal Performance of Building Comonents and Building Elements Assessment of Moisture Transfer by Numerical Simulation (2005)