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With this assumption, a two-equation version of Huang s three-equation model (which included ltration ow Huang 1979) can be expressed as the following differential permeability-based model: 2 pv m m pv e l pv e l T + rl = mm + rl pv t T t x x x 2 e pv e T 2 T f T + (rc p ) - hfg r l l = fB 2 + B - hfg r l l B pv t T t x x x where: r = the density (kg m-3) rl = the liquid (water) density (kg m-3) el = the volume fraction of liquid (water) in porous medium pv = the partial pressure of water vapour (N m-2) T = the temperature (K) mm = the differential permeability (s) x = the spatial dimension in the direction of transfer (m) hfg = the latent heat of vapourisation (of water) (kJ kg-1) f = the thermal conductivity of porous medium (W m-1K -1) cp = the speci c heat capacity (kJ kg-1 K-1) (subscript B refers to the bulk porous material).

338 339 340 340 341 341 342 344 345 346 347

Manganese is classi ed as an essential nutrient that is required in trace amounts COMA (1991) felt that there was insuf cient data to set rm dietary standards for manganese but they suggested a safe intake for adults of at least 14 mg/day (18 mg/day in the USA) There are several important manganese-containing enzymes including pyruvate carboxylase which is important in gluconeogenesis and mitochondrial superoxide dismutase which is important in removing the damaging superoxide free radical (see 5) Large numbers of enzymes are activated by manganese although many, but not all, of these are also activated by other metals especially magnesium.

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Equation 31 describes mass conservation of liquid water (the mass of water vapour in the pores is assumed to be negligible due to the low vapour density compared with that of water) Equation 32 describes heat energy conservation The local volume fraction of liquid in porous material (el) is governed by the vapour pressure (pv) and temperature (T) through the material sorption isotherm The sorption isotherm of a material relates the local volume fraction of liquid in material to the local relative humidity of air at constant temperature Relationships for a variety of construction materials can be found in the literature (eg Hansen 1996) Partial discretisation and solution of equations for heat and mass transfer In many applications, reducing Equations 31 and 32 to a pair of ordinary differential equations (ODEs) by discretising the spatial terms only can have advantages over full discretisation.

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Manganese de ciency has been experimentally induced in several animal species but de ciency symptoms in humans that can unequivocally be attributed to manganese de ciency are extremely rare and even the few potential examples are restricted to individuals consuming manganese de cient semi-puri ed diets or receiving TPN containing no manganese Average intakes in the UK are estimated to be around 5 mg/day with as much as half of this coming from tea; the highest UK consumers ingest over 8 mg/day from their diet Given the lack of any spontaneous examples of manganese de ciency and given that average intakes are more than three times the UK safe intake there seems to be no need for supplemental manganese to prevent de ciency .

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SECTION 13.6. Trigonometric and Hyperbolic Functions, page 626 Purpose. Discussion of basic properties of trigonometric and hyperbolic functions, with emphasis on the relations between these two classes of functions as well as between them and the exponential function; here we see on an elementary level that investigation of special functions in complex can add substantially to their understanding. Main Content Definitions of cos z and sin z (1) Euler s formula in complex (5) Definitions of cosh z and sinh z (11) Relations between trigonometric and hyperbolic functions Real and imaginary parts (6) and Prob. 3

First, the equations can be reduced to a form that is consistent with any other equations present in the model (for instance, if plant were to be included see 5) Second, the equations can then be solved using one of a large number of standard ODE solvers, some of which are especially suited to problems involving mixed dynamics so called stiff problems (see Section 54) The spatial terms, for example, the vapour pressure terms on the right hand side of Equation 31, can be dealt with as follows: 2 pv x 2 @ pv (n + 1) - 2 pv (n) + pv (n - 1) Dx 2 (31).

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