HEAT STORAGE CAPABILITY ON ASYMMETRICALLY COOLED WALL BY TRANSIENT HEAT FLOW

Transcription

1 HEAT STORAGE CAPABILITY ON ASYMMETRICALLY COOLED WALL BY TRANSIENT HEAT FLOW L. Hach 1, Y.Katoh 2 1 Intitute of Applied Phyic and Mathematic, Faculty of Chemical Engineering, Univerity of Pardubice, Pardubice, Czech Rep. 2 Department of Mechanical Engineering, Faculty of Engineering, Yamaguchi Univerity, Ube, Japan Abtract A trong rationale for energy efficiency and it aving in building aim to further develop ne concept a ell a more accurate calculation and imulation technique. In the paper i decribed efficient implementation of baic relation governing the thermal energy tranfer proce ithin the vertical all into a tate-variable model of all egment ith time-dependent propertie. 1 Introduction The heat tranfer occurring at building envelope i one of primary caue of overall heat loe and there are everal paive a ell a active arrangement to reduce them ignificantly. While the heat tranfer phenomena are alay acting in three different kind of energy tranport (conduction, convection and radiation), the energy tored ithin the envelope ma i in fact dynamical in nature. Thi ork dicue the ay of automated meaurement and calculation of main quantitie characterizing the time-dependent heat torage amount in building envelope part, i.e. on indo plate and outide all. On the outide-faced all urface area are meaured temperature via urface temperature enor, imilarly like on the inide-faced all urface. The meaurement data et i part of input vector onto a mathematical model hich include beide the envelope egment adacent air layer. With the model ere invetigated to thermal tate: in normalized tate (50 % of air of relative humidity), and ith real data meaured. Both tate ere imulated a one-dimenional heat tranmiion through a concrete all and a reult ere obtained dynamic characteritic: overall time contant, n c ρ V 1 h A 1,2,... n thermal damping (Fig.1) and heat capacity of the all egment, repectively: n 1 () -1 ( J.K ) m c 1,2,... n (2) Both adacent egment (gypum) of the all, i.e. the one facing exterior, rep. interior, ould atify Bi > 0.1, a reult of a dramatic change of temperature gradient on the air-urface boundary. To obtain the heat capacity (2) a ell a time contant (1) of the -ma egment the lumpedcapacity type analyi a ued. The aement of thermal ma of the all urrounding an encloure room of an adminitrative building, i neceary in order to evaluate time-depending thermal comfort of the encloure [1], [2]. The thermal torage in the encloure all influence indoor environment through to apect: (1) qualitative, by expoing a much of internal urface toard indoor air due radiant and convective load and, (2) quantitative, by increaing (decreaing) the air flo along the urface, thu by increaing (decreaing) it heat and ma tranfer coefficient. Hoveer, there a ome limitation on the laminar airflo regime on it bordering to the tranient one determined by indoor comfort parameter. Mathematically, Gr.Pr> 10 9 [3], i.e. excluding the turbulent flo. (1)

2 2 Thermal Ma Function There are three dominant function of the thermal ma : 1. energy torage, 2. temperature, rep. heat flux dumping, and 3. ater vapor aborbent feature. Yet, conidering only enible heat change, the thermal ma i accepted a term for the overall effect of the amount of material ith it thermal capacity. Within the ma can be tored a certain amount of heat, reduced heat and moiture (ma) tranmiion a ell a their fluctuation, reulting in more acceptable thermal condition. An energy balance q in qout qtored (W) of an external all egment ho Fig.1: T e T i ka T, e x T e x 0, ie m c dt, d m& i ka T i T, i x x 0, - (exterior) all thickne (m), ds - urface area element expoed to olar radiation (m 2 ), Q - olar radiant heat rate aborbed by urface element ds (W). Fig.1 External all egment - an energy balance of the all region. ( ) ( ) dt, x, m c W The um term d include a heat amount temporarily tored in the egment all, o the problem of determining it apparently reduce on obtaining: - pecific eight (denity) of the egment ith it dimenion, and - local pecific heat capacity. The ma in our tudy, the concrete lab vertical all ith double gypum, refer to claical material that have the capacity to tore thermal energy for extended period of time ithout phae change under hich i tored and releaed latent heat. The enible heat could oin in very exceptional condition the latent heat from ater vapor change (ice crytal melting on the outide-facing all urface and partially penetrating the gypum). Depending on the material, the pecific heat varie ith it ater content. With moiture laden air penetrating the material due a preure difference and condening in the all cavity a the air temperature decreae, and it occur alongide the temperature profile acro the egment thickne of the torageable egment. Becaue of time-variable T ( ) f T ( ), 1,2,... K [ ] ( ) temperature on it urface, ome form of initial condition, mut be knon. To repect vertical temperature profile on inner urface of the all, adequate nodal equence in vertical direction ould form a rectangular net of nod on hich the value ere calculated. Further, both egment (plater) of the all, i.e. the one facing exterior, rep. interior, ould atify Bi > 0.1,,, n x x 0, a reult of a dramatic change of temperature gradient on the air-urface boundary. That allo utilize the lumped-capacity type of analyi in order to obtain the heat capacity (1) a ell a time contant (2) of the -ma egment. The ame doe not apply for aniotropic part of the -all and the all egment a partitioned accordingly ith patial tep x of the finite difference method (FDM) ued in thi ork. Applying Laplace equation onto an iotropic homogenou part of uch all egment ith different urface temperature T e, T i and boundary condition yielded temperature profile acro the region, and completed ith vapor content aement, led to the heat capacity evaluation. T, e

3 3 Baic Equation and State-Variable Model Baic relation of phyical phenomena that govern the thermal energy tranfer proce ithin the lab ere employed: the la of ma conervation and the firt la of thermodynamic. The adacent air layer to the all urface here are placed the ytem boundarie i conidered a the elltirred ytem. 3.1 Heat and Ma Tranfer Coefficient Heat tranfer coefficient are calculated imultaneouly ith ampling time and are baed on relation decribed through everal dimenionle number (Nu, Pr, Re) in literature. What follo i common practice in determining the convective heat tranfer coefficient through group of thee number, here the dimenional analyi a employed to etimate heat tranfer from vertical urface on the urface area x h x b y. x h i the height of the all egment and in the horizontal direction it extent to the velocity, rep. temperature profile change adacent to the all urface, Fig. 2: b y ka T, i x x 0, ~ 0.4(T Ti) vx,max T i Velocity profile Temperature profile T e T i,e - indoor (outdoor) air temperature T - internal urface all temperature T e - external urface all temperature v x,max - maximal velocity difference (m. -1 ) - -egment all thickne (m). T Fig.2 Velocity and temperature profile of convective airflo adacent to innermot all egment (T < T i ). Auming a teady air motion ithout vicou diipation, baed on π-theorem, yield for 2- dimenional motion (L x, L y ) ith thee relation [3], [4]: 1/ 2 Gr by Nu. Re f Pr,, Gr 5 / 2 Re xh *1/ 5 * 1/ 5 Re by NuGr. f Pr,, Gr *2/ 5 Gr xh *1/ 5, (3) The equation are valid ithin a laminar mixed convection motion over a vertical plate, valid in a more or le ide region of either Nuelt or Raileigh number, excluding the turbulent cae (Gr.Pr > 10 9 ), [5]. On the urface are unknon heat and ma (moiture) fluxe q (), q m () and the velocity profile in the area cloe to the all, both reulting from complex of quantitie forming Grahof, Nuelt and Prandtl number and can be grouped a hon in [4]. Particularly for the calculation of free convection the equation (4) yield the formula: b * 1/ 5 y NuGr. f Pr, Gr xh *1/ 5 Free convection motion characterize the dimenionle group (4) (-) (5)

4 Gr.Pr 10 8 x 3 ( T T ) h i (-) (6) and by a colder all urface (aumed T e < T i ) dra air donard along the coordinate b y. On the vertical urface all, uch flo i decribed through a ytem of equation [5] ith the boundary T q ( ) k 0 condition u 0, v 0 and y x P at b y 0 and u u 0, T T 0 at b y. Ma tranfer coefficient With the vapour permeability propertie of the material deal Fick la in ame manner a Fourier la incorporate the heat tranfer coefficient of heat conduction problem: m& A D dc dx (kg.m ) (7) here m& ma flux (ater content) per time unit, kg. -1 A expoed urface area, m 2 D diffuion coefficient, m C apor concentration, kg.m -3 dx ditance element perpendicular to ma tranfer, m. With introducing the ma-tranfer coefficient h D in imilar manner to the heat tranfer coefficient: D ( C C ) m& h A 2 1 here C i apor concentration in to different location (kg.m -3 ), yield Wherea for convection i or for laminar (buoancy dominated) air flo h D h x k h x k D x f ( Re,Pr ) f ( Gr,Pr) there the dimenionle group complete the Schmidt number (Gilliland [6]): hd x f D ( Re, Sc) (kg. -1 ) (8), (10), (11) ith to unknon variable, namely D and Schmidt number Sc f (D ). Thee equation ere et up into a model of non-tranparent part of exterior all, Fig. 3 belo. The model ue familiar thermalelectric analogy ith dc-circuit (direct current) electric reitance and capacitance, equivalent to the thermal and permeance reitance, repectively: 1 R cv ha p RD m vt (12) (kg -1.m K) (14) (m. -1 ) (15)

5 They are parameter in mathematical model hich repreent an equivalent thermal circuit coniting of active and paive component: thermal ource, thermal reitor and thermal capacitor. The 2-dimenional model include the boundary condition on both all urface. Auming the uniform vapor content ditribution acro the ufficiently thin liced all egment, the lumped parameter model incorporating both heat and ma tranfer coefficient of the vertical all illutrate Fig. 3: h D,e ds h cv,e ds T,e (k/ ) ds T,i h D ds h cv ds T o T i Q - all thickne (m), ds - urface area element expoed to olar radiation (m 2 ), Q - olar radiant heat rate aborbed by urface element ds (W). Fig. 3 Analog cheme of heat tranfer through exterior all expoed to olar (hort-ave) radiation. 3.2 State-Variable Model of Wall Segment The n-egment all model a et up according to the analog cheme in Fig. 3 in the MATLAB/Simulink environment. It comprie the (n+2)-nodal net ith the innermot egment expoed to the indoor air temperature T (), rep. enthalpy i (), and on the external urface outdoor air temperature T e () and enthalpy i e (), Fig. 4: Fig. 4 Analog of exterior all temperature ditribution ith convective boundary condition (Matlab/Simulink environment).

6 T n+1, n 0,1,2,,m, refer to the node lined up evenly in perpendicular direction to the egment all urface here the temperature, a ell a other qualitie, are pinned. It correpond to the value of Fourier equation in central difference cheme. The heat rate refer to thermal equilibrium erved a the initial condition in the tranition procee through the all part ith urface area A expoed to variou thermal excitation, Figure 5-7. The overall time repone, referring effectively to the 36.8 % of the initial temperature difference T e, yield from: td n 1 () (16) here denote the overall time contant in econd, td i time lag (uually td 0 ) and n ytem order. In the 6-egment compoed model a etimated the value ~ , and the temperature indicated quite monotonou profile, Fig. 5: 295 T Boundary condition: h 5.7 W.m -2 K -1 ; k 1.43 W.m -1 K -1 x 40 mm; 6nod T i K; T i 0; 0 ( 180, the 3 rd -order Runge-Kutta method) 280 T 0 () Input tep at 0 T e K < 0 T e K Time (econd) x 10 4 Fig. 5 Exterior all temperature ditribution profile ith convective boundary condition; h cv 5,7 W.m -2 K -1 on both ide. The material relative humidity a kept contant (diffued air of 50 % relative humidity), hile in the econd cae a ued the real meaured data, curve b in Fig. 6: a) 1 T o, 285 T i T o T a mm 200 mm b) Time (econd) x 10 4 Fig. 6 Tranient temperature repone (a) acro 200 mm-thick plain brick exterior all uddenly expoed to ambient air temperature (b) at initial all temperature T i 293 K.

7 For the approximation of the ambient air humidity and temperature daily ing (curve b at Fig.6) ere found hortened time contant (Eq. (2)) over 15 % for upper end of human comfort area (30 70 % r.h.) correponding to the air enthalpy content of 48 kj.kg -1. An adequate thermal capacity, -1 m c 386 kj.k Eq.(1) of 20 cm thick brick all ith ratio A /V 0.2 m -1 ould be kept at value 1 (c J.kg -1.K -1, cae in Fig.5 and 6), and the convective heat tranfer coefficient roughly folloed the air enthalpy ratio: i2 i ( ) 1 h cv 1 hc cv 2 () 6 (J.kg -1.K -1 ) (17) i.e. the value of 5.7 W.m -2.K -1 ould reach 6.7 W.m 2.K -1. That repreent the difference 17 % equivalent to the increae of R cv + R D Eq.(14-15), it may releae the heat accumulated in the fabric proportionally hortly, under otherie ame condition. Thi demontrate Fig. 7 in hich the releae of accumulated amount of heat lag behind the ame cae in Fig. 5. The difference in the later ituation caued expoing the outer all to the nearly rectangular temperature ing (yello curve in Fig. 7) account about 1.6 Kelvin in temperature difference at time of T o,i Boundary condition: H 5.7 W.m -2 K -1 ; K 1.43 W.m -1 K -1 x 40 mm; 6 nod T i K; T i 0; 0 ( 180, the 3 rd -order Runge-Kutta method) Time (econd) x 10 4 Fig. 7 Wall temperature ditribution ith convective boundary condition h 5.7 W.m -2.K -1 on both ide expoed to nearly rectangular ump of temperature (270 K K in time interval of 6 hour); the ame initial and boundary condition in the exterior all cheme in Fig. 5. Becaue the ratio of the ability to tore the heat energy to the ability to conduct the heat energy could be expreed a rep. ith thermal diffuivity: ρc k 2 a T Input tep at 0 : () (18) () (19) there could be etimated a time through the concrete lab egment in hich the tranient proce of heat conductance ould complete. With 86.5 % of the final value reached after to time contant, it reache 95 % after three time contant. Becaue the time contant of the homogenou -egment i from eq.(19) () T e K < 0 T e K 0

8 2 πa the time adequate to three time contant yield approximately () (20) () (21) a The diffuivity a hould reflect vapor penetration on each lab in horizontal direction hile fluctuance ithin the adacent layer of air in the vertical direction could be neglected. Intead the vertical temperature profile (T f(b y )), i neceary to acquire ith meaurement data in order to obtain convective heat tranfer coefficient h cv. The complementary calculation of tate equation for air (p/ρt cont.) yielded aement of the denity fluctuation, thu the ater content and ma-tranfer coefficient, h D. The meaured value ere ued in a computational model of air tate equation in order to fill the value of vertical temperature profile and ater vapor value. The partial preure of ater vapor (air denity change) along the vertical lab urface i een in Fig. 8 on the right ide: Fig.8 The air denity fluctuation map in cro-cut of ingle-zone pace ith non-uniformly cooled all (in 2D-meridian-cut, cale in kg.m -3 ) height H 1900 mm; ACH 0.5. The imulation tet, including periodical harmonic (ine function) diturbance (Fig.9) caued by expoing the uppermot all urface to the external heat load, allo to etimate the all egment thermal damping and thermal condition of revering the temperature and permeance gradient - in hich may occur de point on contruction fabric [7], [8]. Thermal damping over the all a characterized by a dumping coefficient T, e T ν T T, e (-) (22) - temperature amplitude difference on out-faced all urface, - temperature amplitude difference on interior-faced all urface and etimated at 0.19, the beginning of reveral permeance and temperature gradient indicate point P in Fig. 9:

9 295 T ref T o, x Boundary condition: h 5.7 W.m -2 K -1 ; k 1.43 W.m -1 K -1 x 40 mm; 6 nod T i K; T i 0; 0 T e ± 6.75 K P ( C) (Runge-Kutta 3rd-order) p Time (econd) x 10 4 Fig.9 Wall temperature ditribution ith convective boundary condition h 5.7 W.m -2 K nod, x 400 mm; time tep 180, p , T i K; T e ± 6.75 K ( C), harmonic (ine) temperature change. Runge-Kutta 3 rd -order, Bi Solution and concluion The all thermodynamic parameter ere etimated in the tranient period beteen to thermal teady and quai-teady (real experimental data) tate ith everal temperature and moiture varying diturbance. The concrete lab tranient time in hich ould complete 95 % of the final value a etimated (eq.(21)) ith time contant of the value ~ , Fig.6. The all partition perpendicular to the one-dimenional heat flux determined the model order ith boundary condition precribed through all function of convective heat tranfer coefficient h cv, temperature T and ma-tranfer coefficient, h D for each all egment energy balance equation. Moiture diffuion in free air caue minimal vapor preure differential beteen connected place a uch i minor driving force in increaing the convective heat tranfer coefficient until the condenation may uccur. The heat and ma flux revere caue a cut of all thermal capacity, available for the interior overall heat torage capacity. The virtual plane P-point in Fig. 9 over the all egment - moving through the all determine a ne location of the plane equation dt( ) 0. The Biot dx number for the P-plane hift from urface inide characterize the heat torage ma capability a the time lag ill become effectively horter ( h P ( - x p ) < k ). T x x P Nomenclature: a i - thermal diffuivity of i-layer (m ) A - urface area of -all (m 2 ) b - dimenionle parameter (-) c - pecific heat (J.kg -1.K -1 ) b y, x h - patial dimenionle height, idth (-) h c - convective heat tranfer coefficient (W.m -2.K -1 ) k - thermal conductivity (W.m -1.K -1 ) kinetic turbulent energy (J.kg -1 ), (Chapter 4) n - ytem order p - air preure (Pa) q - heat flux per unit urface area (W.m -2 ) R - thermal reitance (W -1 m 2.K) - all thickne (m)

10 t, T - temperature ( C), thermodynamic temperature T i - (indoor) air temperature T, - urface temperature of inner ide of -all T e - urface temperature of exterior all T(x,) - temperature at time in ditance x u - velocity component in x direction (m. -1 ) v - velocity component in y direction (m. -1 ) V - volume (m 3 ), x, y - Carteian (patial) coordinate (m) Greek letter ν (overall) dumping coefficient kinematic vicoity (m ) ρ denity (kg.m -3 ) ρ i.c i volumetric heat capacity of i-part (J.m -3.K -1 ) time () Bi, Gr, Biot, Grahof number Nu, Ra Nuelt, Rayleigh number Re Reynold number Reference [1] L. Hach and Y. Katoh. Application of Quai-Steady-State Thermodynamic Model of Conventional Heating Sytem ith Takahahi Control Method, IBPSA 2001, Brazil, Proceeding [2] C.O.R. Negrão. Integration of computational fluid dynamic ith building thermal and ma flo imulation, Energy and Building, vol.27, 1998, pp [3] D.E. Fiher and C.O. Pederen. Convective Heat Tranfer in Building Energy and Thermal Load Calculation, ASHRAE Tranaction (1997), 103 (2) [4] A. Tindale. Third-order lumped-parameter imulation method. Building Service Engineering Reearch and Technology, Vol. 14 No.3, 1933, pp [5] F. Bauman, A. Gadgil, R. Kammerud, E. Altmayer, and M. Nanteel (1983). Convective Heat Tranfer in Building: Recent Reearch Reult, ASHRAE Tranaction, 89 (1A) [6] E.R. Gilliland, Diffuion Coefficient in Gaeou Sytem. Ind. Eng. Chem., Vol.26, p.681,1934. [7] L. Hach, Y. Katoh. Simulation of Thermal Balance of Non-Air Conditioned Room under Winter Condition Uing Quai-Steady-State Simulation Model. International Journal JSME, Vol.46, No.1, [8] H. Recknagel, E. Sprenger, and E.-R. Schramek, Tachenbuch für Heizung und Klimatechnik, R. Oldenbourg Verlag, München, Germany, Contact information Dr.Eng. Lubo Hach, Ph.D., Intitute of Applied Phyic and Mathematic, Faculty of Chemical Engineering, Pardubice Univerity, Pardubice, Czech Rep., lubo.hach(a)upce.cz.