PAijpam.eu RELATIVE HEAT LOSS REDUCTION FORMULA FOR WINDOWS WITH MULTIPLE PANES Cassandra Reed 1, Jean Michelet Jean-Michel 2

Transcription

1 International Journal of Pure an Applie Mathematics Volume 97 No ISSN: (printe version); ISSN: (on-line version) url: oi: PAijpam.eu REATIVE HEAT OSS REDUCTION FORMUA FOR WINDOWS WITH MUTIPE PANES Cassanra Ree 1 Jean Michelet Jean-Michel 2 12 Department of Mathematics an Computer Science South Carolina State University Orangeburg SC USA Abstract: We erive an expression for the heat flux through a winow constructe with multiple panes of glass separate pairwise by air layers of a given thickness. We compute the relative heat loss reuction achieve in comparison to a winow with no air gap an the same amount of glass. We examine how the relative heat loss reuction function behaves when we scale the number of panes up to a large value. AMS Subject Classification: 65K10 78M50 80A20 Key Wors: heat flux reuction heat transfer heat flux equilibrium 1. Introuction In this moel warm air escapes from a heate builing through a winow. The winow comprises n glass panes of thickness separate pairwise by a layer of air of with. We assume that the insie temperature T 1 is greater than the outsie temperature T 2 reflecting the fact that heat flows from the insie to the outsie. In the case n = 2 we follow Mesterton-Gibbons [1]. If we inicate by x = 0 the interface of the insie air an the insie winow pane then the outsie pane an outsie air meet at x = 2+. et T (x) be the temperature at x. Thus we have T (0) = T 1 an T (2+) = T 2. We also write T () = T A an T (+) = T B for the other two air-glass interfaces in between. Receive: August Corresponence author c 2014 Acaemic Publications t. url:

2 544 C. Ree J.M. Jean-Michel The heat flows along x accoring to Fick s law F (x) = k(x) T x where F (x) stans for the heat flux per unit area across the winow at point x an the function k(x) represents the thermal conuctivity of the meium at x. Fick s law simply states that heat flux at a point x is proportional to the temperature graient at x; The steeper the graient the higher the heat flow. In the two-pane moel for example k is a piecewise constant function that takes the value K A for x between an + the points x that lie in the air gap. For x in the remaining intervals 0 < x < + < x < 2 + in the glass meium we assign to k the value k G. In this paper we are intereste in etermining the heat flux across the glass an air for an n-glaze winow once the flow has reache a constant equilibrium. In Section 2 some straight-forwar computations as foun in [1] yiel the equilibrium value the flux F settles to in the case n = 2. We use these computations in Section 3 to treat the case n = 3. Formulas for the flux an relative heat flux reuction for a winow with n panes follow in Section 4. We conclue our investigation with Section Double-Glaze Winow As in [1] integrating with respect to x the expression for Fick s law over the interval 0 < x < we obtain T 1 T A F (x). Repeating the process over the remaining intervals an equating the expressions for the respective heat fluxes so obtaine we have the chain of inequalities K G T 1 T A = K A T A T B T B T 2 Solving for T A an T B in terms of T 1 an T 2 gives = F. T A = T B = ( KG +1 )T 1 +T 2 T 1 + ( +2 KG )T

3 REATIVE HEAT OSS REDUCTION FORMUA Therefore the flux across across the ouble-pane winow eventuallys settles own to an equilibrium value F = F 2 which in the = 0 limit becomes F 2s T 1 T 2 2+ K G T 1 T 2 2 where F 2s is the rate of heat loss for a single-pane winow with the same amount of glass. Thus by setting the glass panes a istance of units apart we achieve a relative reuction of heat loss 2 = F 2S F 2 F 2S = 2+ K G The thermal conuctivity of glass at room temperature varies between an J/cm.sec. C [2] while the thermal conuctivity of ry air is approximately J/cm.sec. C [3]. Therefore the ratio K G K A varies between 16 an 32 which means that a winow with a conuctivity ratio of 16 an an air gap of four pane-withs alreay achieves a heat loss reuction of 97%. The heat reuction function 2 is a strictly increasing function of the gap aspect ratio / an approaches rapily the limiting value 1. Manufacturers know this which is why we o not often see winows with 4 or 5 glass panes on the market. Nevertheless how the heat loss reuction function scale for a winow with a large number n of glass panes remains an interesting theoretical question. How oes epen on n an the gap aspect ratio /? Specifically we aress the following situation. Instea of two we take n glass panes of thickness each an put an air gap of thickness between each successive pair to form an n glaze winow. Compare to a single-pane winow of thickness n how much heat loss reuction is achieve? What form oes the function n take? We take an inuctive approach to this question. Determining 3 will allow us to infer the form of n for a general n..

4 546 C. Ree J.M. Jean-Michel 3. Triple-Glaze Winow With a triple-glaze winow we have air-glass interfaces at x = that are kept at temperatures T 1 T A T B T C T D T 3 respectively. Integrating Fick s law as we i in the last section leas us this time to the string of equations K G T 1 T A K A T C T D = K A T A T B T D T 3 T B T c = F 3 where F 3 enotes the heat flux through the three-pane winow. Part of Equations (1) can be written as K G T 3 T D = K A T D T C T C T B = (1) = F 3 which we can as we i in the previous section solve for T D an T C to fin T D = (ρ+1)t 3 +T B ρ+2 T C = T 3 +(ρ+1)t B ρ+2 if we set ρ. On the other han in (1) we also have K G T 1 T A = K A T A T B T B T c = F 3 from which we euce T A = (ρ+1)t 1 +T C ρ+2 T B = T 1 +(ρ+1)t C. ρ+2 Next we use the expressions for T B an T C to obtain (ρ+2)t B = T 1 +(ρ+1)t C = T 1 + ρ+1 ρ+2 T 3 + (ρ+1)2 ρ+2 T B

5 REATIVE HEAT OSS REDUCTION FORMUA which yiels T B = ρ+2 2ρ+3 T 1 + ρ+1 2ρ+3 T 3. This allows us to fin F 3 T D = T 1 +2(ρ+1)T 3. 2ρ+3 Finally inserting T D into the last of the equations in (1) we arrive at the formula for the heat flux through a three-pane winow: ( ) T 1 +2T 3 1+ K G 3+2 K G T 3. It is as one woul expect a function of the the insie an outsie air temperatures. This time again the temperatures at the inner air-glass interfaces o not influence the flow at equilibrium. F n 4. Winow with n Panes From the expression obtaine in the last section for the heat flux for a winow with 3 panes we infer the flux for an n glaze winow to be of the form ( ) T 1 +(n 1)T n 1+ K G n+(n 1) K G T n where T 1 an T n are the insie an outsie air temperatures respectively. In the = 0 limit F n reuces to F ns T 1 T n n with F ns enoting the flux across a single-pane winow with the same amount of glass. The relative heat loss reuction is given by n = F ns F n F ns = KG (n 1) n+(n 1) K. G

6 548 C. Ree J.M. Jean-Michel As a sequence of functions of the gap aspect ratio n converges to the limit as n gets large. As for the behavior of that sequence that limit is 1+ reache in a monotonic fashion since we have n n = [ n+(n 1) K G which remains strictly positive for all positive values of the gap aspect ratio. Therefore the absolute ceiling on the amount of relative heat loss reuction achievable is 1+ ] 2 no matter the number of panes use. The monotonic convergence of the sequence n conforms to one s intuition that the more panes an the wier the gaps the more energy is save. 5. Conclusion In an inuctive approach using the expression for the heat flux through a ouble-glaze then the heat flux through a triple-glaze winow we inferre the expression for the heat flux through a winow with n panes of thickness set a istance of units apart. We obtaine a formula for the relative heat loss reuction n as the relative ifference between the heat flux through an n-glaze winow an that of a single-pane winow with the same amount of glass. As a function of the gap aspect ratio we foun that the amount of relative heat loss reuction n achieve starts at 2+ increases monotonically with n an approaches the limiting value when n equals References [1] M. Mesterton-Gibbons A Concrete Approach to Mathematical Moelling Wiley New York (1995). [2].D. anau A.I. Akhiezer E.M. ifshitz General Physics: Mechanics an Molecular Physics Pergamon Press New York (1967).

7 REATIVE HEAT OSS REDUCTION FORMUA [3] G.K. Batchelor An Introuction to Flui Dynamics Cambrige University Press Cambrige (1967).

8 550