In this work the simplification by Milly [99] of the model originally developed by Mualem [98] is used to model the effects of hysteresis. The model is based on
92 CHAPTER3
(a) adsorption (b) desorption
Figure 3.5: Ink bottle effect: different modes of filling the pore for the same capillary pressure
the ink bottle effect and only requires the main adsorption and main desorption isotherm as material input. The model was originally developed for soil science where ink bottle effects are the main reason for hysteresis. However, in the hygro- scopic region the other phenomena mentioned in the previous paragraph also play a major role in the occurrence of hysteresis. Despite this fact the Mualem model is used by other authors [100] to model hysteresis for wood in the hygroscopic region. Although the model is not as accurate in this region as more advanced models, it has the advantage that it has a physical basis and that it only requires a limited set of input data.
Graphically the relation between relative humidity and moisture content can be represented by this model in a two dimensional space: the Mualem space (Fig. 3.6). In this space the abscissa (f ) represents the sorption locations as function of relative humidity, the ordinate (a) represents the accessibility of these locations and the gray surface bounded by abscissa and ordinate represents the moisture content. This graphical representation of the model gives a better insight when deriving the model equations.
In Fig. 3.6 four situations are depicted that result from a different sorption history: main adsorption, main desorption, primary adsorption and primary de-
HEAT AND MOISTURE TRANSFER 93 a f 1 1 RH RH a RH a b 1 1 1 ψ f 1 RH a f 1 1 RH RH c a f 1 1 RH RH d ψ ψ ψ ψ0 ψ0 ψ0 ψ ψ 1 1 2 2 2 0 0 0 0
Figure 3.6: Visualization of hysteresis in a Mualem space. a) Main adsorption b) Main desorption c) Primary adsorption d) Primary desorption
sorption. Main adsorption is the sorption process which starts from an initially dry material. During adsorption all pores are accessible. This is visible in (Fig. (3.6a)) as the accessibility a= 1. Hence if the main adsorption isotherm is known (w = w(0ψ)1= wa(ψ)), the f-function is also known. Main desorption is the
sorption process which starts from an initially saturated material. During desorp- tion the inkbottle pores are not accessible as liquid has to pass the small neck of the pore. The accessibility function a is thus different from1. The moisture content w= w(ψmax
ψ)2is then calculated as follows:
w= w(ψmax
ψ) = wd(ψ) = wa(ψ) + (wmax− wa(ψ))A(ψ) (3.25)
In this equation the second term of the right hand side represents the moisture blocked in large pores by smaller pores (necks of the ink bottle pores) containing liquid at a relative humidity of ψ. Both terms of the second hand side are also visible in Fig. (3.6b) as respectively1and2. If the moisture content during main desorption (wd) and during main adsorption (wa) is known the accessibility (A(ψ))
can be determined: A(ψ) =wd(ψ) − wa(ψ) wmax− wa(ψ) (3.26) 1RH rises from 0 to ψ 2RH drops from ψ maxto ψ
94 CHAPTER3
When the accessibility is known the Mualem model can be used for the calculation of primary adsorption and primary desorption. Instead of starting from initially dry conditions the primary adsorption process starts from a value on the main desorption curve at relative humidity ψ0. The moisture content is calculated as:
w ψmax
ψ0
ψ= w
a(ψ) + (wmax− wa(ψ))A(ψ0) (3.27)
The second term on the right hand side of Eq. (3.27) now represents the part of the moisture blocked at a relative humidity ψ0, which is not present at relative
humidity ψ during main adsorption (Fig. (3.6c)). Primary desorption, being the desorption starting from a point on the main adsorption curve at relative humidity ψ0, can be calculated as:
w 0ψ0ψ= wa(ψ) + (wa(ψ0) − wa(ψ))A(ψ) (3.28)
where the second term on the right hand side represents the part of the moisture content wa(ψ0) blocked at a relative humidity ψ (Fig. (3.6d)).
Primary adsorption and desorption are also referred to as first order scanning curves. Using the Mualem model it is possible to obtain higher order scanning curves (i.e. the switch point between adsorption and desorption does not lie on one of the main sorption isotherms), yet this requires a different expression for the different orders of scanning curves. To avoid the complexity of keeping track of the sorption history Milly [99] suggested a simplification of the Mualem model in which all scanning curves are considered as first order. To model the scanning curves as first order a virtual switch point on the main adsorption or desorption curve is necessary. This virtual switch point is chosen in such a way that the real switch point lies on the new first order scanning curve. If w1is the moisture
content at the real switch point ψ1then the virtual switch point ψ00can be obtained
by filling in w = w1 and ψ = ψ1in Eq. (3.27) and Eq. (3.28). This results in
respectively Eq. (3.29) and Eq. (3.30).
wa(ψ00) = wa(ψ1) + (wmax− wa(ψ1)) w1− wa(ψ1) wd(ψ1) − wa(ψ1) (3.29) wd(ψ00) − wa(ψ00) wmax− wa(ψ00) = w1− wa(ψ1) wmax− wa(ψ1) (3.30) Substituation of Eq. (3.29) and Eq. (3.30) in respectively Eq. (3.27) and Eq. (3.28) results in an adsorption and desorption model for the moisture content independent of the order of the scanning curve and only requiring the real switch point (ψ1, w1)
4
Numerical implementation and
validation
In the previous chapter the transport equations were given which need to be solved to simulate heat and moisture transport in air and porous materials. In the first part of this chapter the numerical implementation and coupling of these transport equa- tions are described. This way a coupled CFD-HAM model is developed applicable for 2D and 3D problems. The second part of the chapter consists of a verification and validation study of the new coupled model.
4.1
Integration of HAM and CFD codes
Before looking into the numerical implementation of the different transport equa- tions, the coupling strategy of the models for air (CFD) and porous media (HAM) is elaborated.