NUMERICAL SIMULATION OF COLD AIR JET ATTACHMENT TO NON ADIABATIC WALLS

Transcription

1 Eleventh International IBPSA Conference Glasgow, Scotland July 27-30, 2009 NUMERICAL SIMULATION OF COLD AIR JET ATTACHMENT TO NON ADIABATIC WALLS Nikola Mirkov 1, Žana Stevanović 1, and Žarko Stevanović 1 Institute of Nuclear Sciences Vinča, Laboratory for Theral Engineering and Energy, Belgrade, Republic of Serbia ABSTRACT A jet of cold air is denser then abient air but it adheres to the ceiling of the roo over the given distance when it is blown horizontally close to it. Such behaviour of fluid jets is well-known as Coanda effect and it is widely used in practice like in the case of ventilation and air-conditioning of roos. This phenoenon is not sufficiently known both in ters of echanis and quantitative effects. The coplexity is arising in the case of non-adiabatic ceiling that is coon practice in real building structures, since the cold air jet, attachent distance is strongly influenced by heat flux thought the ceiling. The ai of this paper is to propose an algorith of nuerical siulation of fluid flow and heat transfer in the roo and ceiling siultaneously, known as conjugate heat transfer proble. Using proposed odel, uncertainty of heat transfer coefficient deterination is avoided integrating the sae general transport equation for fluid flow and heat transfer over the whole coputational doain including both air oveents in the roo and ceiling solids. INTRODUCTION When a cold air jet is issued into quiescent surroundings bellow a ceiling parallel to the axis of the jet discharge, the so-called Coanda effect forces the jet to deflect towards the wall boundary and attach to the ceiling wall. After the jet ipinges on the wall, the flow redevelops in the wall jet region. This type of flow is often called wall attaching offset jet and it occurs in any engineering applications other then air-conditioning such as environental discharges, heat exchangers, fluid injection systes, cooling of cobustion chaber wall in a gas turbine and others. Fro a coputational perspective, airflows in the roos are very coplex. Due to this characteristics, they present a great challenge for the available nuerical odels. Fig. 1 shows the schee of roo odel where the regions of interest are depicted including diffuser exhaust and ceiling wall. Due to the entrainent of fluid between the jet and the upper wall, there is a reduction of pressure in this region forcing the jet to deflect towards the boundary and eventually attach with it. Along the attachent length fluid jet and solid wall therally interact. Figure 1 Schee of Roo Model Legend: R1-R2-R3-R4: First Cross-Section of Roo (15 x 3 x 2.25) R2-S1-S2-R3: First Cross-Section of Wall (Ceiling: (15 x 3 x 0.25)) F1-F2-F3-F4: Exhaust of Rectangular Diffuser (0.5 x 0.25) M1-M2: Central-Vertical Line on x/h=5 M2-M3: Central-Vertical Line on x/h=10 M2-M3: Central-Vertical Line on x/h=20 The present study is aied at investigating the tridiensional turbulent fluid flow of wall attaching jet and conjugate heat transfer characteristics of fluid flow and solid wall. A conjugate heat transfer proble occurs when the fluid regie is coupled with the conducting solid wall of finite thickness. There have been nuerous studies on wall jets, one study of cold air jet attachent to walls is presented in Marchal (1999). Kanna and Das (2005a) have studied the conjugate heat transfer of lainar plane wall jet and reported a closed-for solution for the interface teperature, local Nusselt nuber and average Nusselt nuber. The conjugate heat transfer of a lainar incopressible offset jet is reported in other paper by Kanna and Das (2005b). The ideas of conjugate heat transfer as they were presented here appear in Patankar (1980)

2 MATHEMATICAL MODEL Syste of partial differential equations describing conservation of ass, oentu and energy, together with two transport equations for turbulence kinetic energy and turbulence kinetic energy dissipation rate constituting Chen-Ki k ε turbulence odel, has following for: ρ + i = t xi ( ρu ) 0 Ui P ( ρui ) U j ( ρ Ui ) µ + ef = t x j x j x xi µ T j t x j x j Pr x ef ( ρt ) + U ( ρt ) = 0 µ t k ( ρk ) + U j ( ρ k ) µ + = t x j xk σ k xk = ρ ( G ε ) µ t ε ( ρε ) + U j ( ρε ) µ + = t x j xk σε xk ε = ρ ( Cε 2 ) 1 G Cε 2 ε + Cε 3 G / k k Which is uniquely defined with additional relations: σ k (1) Ui Uk Ui G = νt + xk xi xk 2 k t = C µ (2) µ ρ ε µ = µ + µ ef P RT ρ = ; σ ε ; σ T ;C µ ; C ε 1; C ε 2 ; C ε 3 = 1.0; 1.314; 0.9; 1.44; 1.92; 0.25 Carefully looking at equations at this syste we can observe that they have the sae for of general equation of conservation of field, which was expected. Difference appears in additional ters on right hand side. If we generally nae the source ters and denote with S Φ, then we can alternatively write one represenative transport equation instead of writting syste of equations (1). This equation will have the for: Φ ( Φ ) + U j ( Φ) Γ = S t x j x j x ρ ρ Φ Φ t (3) First ter on left hand side represents local change of variable Φ, then follows a convective ter, and third ter defines a diffusion of variable Φ. The for of general transport coefficient Γ Φ depends on physical eaning of general variable Φ. This general for has significant adventage in defining nuerical odel because we need to define identical nuerical algoriths for solving all partial differential equations of conservation. Integrating syste of partial differential equations (1) coupled with additional realtions (2), with defined initial and boundary conditions for each case separatly, we attain required solution field of physical paraeters of fluid in defined space and tie doain. On the other hand this algorith didn t solve the proble of heat transfer between fluid and solid wall, and proble of heat conduction in solid wall. Later proble is governed by well known equation of heat conduction in solids: 2 T T c λ = 0 t x j x j (4) Where T stands for teperature of aterial, c and λ represent specific heat capacity and heat conduction coefficient of solid body respectively. Classical approach in solving proble of heat transfer between fluid and solid wall is aplication of Newton s law of heat transfer. ( w f ) q = α T T (5) Where T w is the solid wall teperature which is in contact with the fluid of teperature T f, and α represents heat transfer coefficient. According to above relations it is clear that we need to know the value of the heat transfer coefficient and teperature distributions in order to deterine the aount of heat exchanged between the solid wall and the fluid. Equation (5) has siple for but is very coplex in the essence. Heat transfer coefficient is a function of several variables. In the first place it depends of therophysical characteristics of fluid, velocity of fluid flow, flow character (lainar or turbulent) and heat exchange surface geoetry. As a consequence of these characterictics coefficient of heat transer has attributes of local variable. Previous theoretical and engineering practice of solving this proble have ostly been based on detering average coefficient of heat transfer - a process which leeds to reduction of solution accuracy. Most popular for of detering average coefficient of heat transfer is derived fro criterial analysis trough Nusselt nuber:

3 α L Nu = (6) λ Where L represents characteristic scale of flow field. On the other side Nusselt nuber is a function of Prandtl and Reynolds nuber, in general for this function has following for: b c Nu = a Pr Re (7) Where a, b, and c are epiricaly deterined. Shown classical way of solving this proble brings us to conclusion that even if the fields of physical paraeters in flow field and solid wall are given in differential for, epiris in detering coefficient of heat transfer faces us with uncertanity of solution accuracy. Quality of solution of partial differential equations of turbulent heat transfer and oentu in fluid (1) and differential equations of heat conduction in solid wall (4) cannot be neglected, especially because equation (4) itself has the for of general transport equation (3). This is the ain reason for searching alternative idea for the solution of given proble by discarding epiris connected with Newton s law. Conjugate differential odel of heat transfer in fluid and solid The idea of conjugate differential odel of heat transfer in fluid and solid lays in the possibility to treat the fluid and solid wall as a unique flow space. In order to present this odel in detail certain atheatical anipulations are executed on equation (4) so it is being recasted in the for of general transport equation and then added to the syste of partial differential equations (1). Intergrating such syste, epirical deterination of heat transfer coefficient is avoided and the solution is obtained both for the fluid and solid wall doain. Equation (4) can be rewritten in following for: ρλ T ( ρt ) + 0 ( ρt ) = 0 (8) t x j x j c x j Where it can be written that Γ = ρ λ / c. Second, convective ter is written intentionaly although it doesn t exist bacause we are dealing with solid doain. Therefore, if we soehow anage to define that part of flow space as belonging to solid, set velocity field equal to zero, and if we translate transport coefficient Γ and aterial density ρ to coresponding equivalent which holds for fluid, obatined differential equation will be valid for fluid space, and we have eans to add it to syste of partial differential equations (1). Equation (8) will undergo atheatical transforations. First it will be ultiplied by ρ / ρ getting: ρλ T t x j c x ( ρt ) = 0 And then adjusting transport coefficient we get: ρλ c λ T t x j c cp λ x ( ρt P ) = 0 (9) (10) Knowing that relation for the transport coefficient in fluid is expressed by Γ = ρλ / c, equation (10) becoes: T λ c T T t x j λc x P ( ρt ) Γ = 0 P (11) Final step in obtaining differential equation of interest is introducing porosity of integational cell coefficient. β λ cp V = (12) λc Finally we can conclude the procedure of transforing equation (4) which now has the for of equation valid for fluid flow doain. where Γ = β Γ. T T T t x j x ( ρt ) Γ = 0 V T (13) Forally and essentially equation (13) represents energy equation of syste (1), so the nuber of equations in syste (1) reained the sae. Integrating energy equation is siultanious for fluid and solid doain, while it is necesary to define following conditions in order to have succesful integration. 1. Set velocity field to zero in doain of flow space which belongs to solid wall. 2. Introduce new dependant variable the porosity coefficient β and set it to one in fluid part of doain, and having value of space which belongs to solid wall. V λ c / c λ flow NUMERICAL SETUP AND BOUNDARY CONDITIONS In this study atheatical odel has been solved nuerically using PHOENICS software. For that purpose physical doain is divided by 75 x 40 x 50 cells where 75 x 40 x 10 cells cover the solid wall. The flow is resolved down to viscous sub-layer and logarithic wall functions are used for the first row of the cells. Nuerical grid at central X-Z plane is shown in Figure 2. p

4 Three cases have been considered with difference in flow speed at diffuser exhaust. Velocities are equal to 1, 1.5, and 2 /s respectively. Teperature of issuing jet is 15 o C in all three cases. It is iportant to stress that wall surface teperature is not a boundary condition, it is calculated fro the inner region. For the other side of the wall it is assued that there are no heat sources or sinks. Brick is specified as a wall aterial. Figure 2 Nuerical grid at central X-Z plane (75 x 40 x 50 = total nuber of cells, where 75 x 40 x 10 = cells cover the solid wall) PRESENTATION OF RESULTS Two sets or results are presented, first concerning flow field, where attaching the air jet to the ceiling due to Coanda effect becoes apparent, and second concerning teperature distribution in central vertical plane and in horizontal cross section of the wall placed five centietres above the fluid-solid interface. Horizontal coponent of velocity at central x-z plane for all three respective cases is shown in Fig. 3. Teperature distribution in central vertical plane and in horizontal cross section of the wall placed five centietres above the fluid-solid interface are shown in Figures 4, 5 and 6. For the convenience of presenting the results in fluid flow, in order to coprise three-diensional effect of flow, we chose to visualise the fictive surfaces with equal teperature (teperature iso-surface). Fig. 7, 8 and 9 show contours of the cold air jet by teperature iso-surfaces with value of 16 o C, just one degree above the teperature of air jet at diffuser exhaust. Finally teperature distributions along central lines going trough both fluid and solid doain are shown on Fig. 10, Fig. 11. and Fig. 12 for each separate case respectively. CONCLUSION Our basic objective has been description of ethodology for siultaneous coputation of teperature distribution in roo and surrounding walls. Principal idea and benefit of suggested procedure is eliination of uncertainty in detering coefficient of heat transfer and epirical relations connected to Re and Nu nubers. Practically, anipulating differential equation of theral energy conservation and prescribing special boundary conditions in solid, odel allows siultaneous integration of coplete differential odel in fluid and solid. This paper doesn t have intention to verify quantitative results yet to present the ethodology. As an adequate exaple, one that has certain coplexion related with it, ceiling attaching wall jet exhibiting Coanda effect is siulated for different velocities of air flow. Results shown in previous section are illustrative and clearly display good properties of suggested procedure. One of the ain disadvantages of the procedure is ability to be carried out only in case of differential odel and corresponding nuerical integration with available software of this class. In this paper we used FLAIR - special HVAC odule of general code PHOENICS ( ACKNOWLEDGEMENT This paper is concerned by the Project of Technical Developent (No.TR-18004) funded by Serbian Governent. REFERENCES Kanna P. R., Das M. K (a). Conjugate forced convection heat transfer fro a flat plate by lainar plane wall jet flow, International Journal of Heat and Mass Transfer 48 pp Kanna P. R., Das M. K (b). Conjugate heat transfer study of two-diensional lainar incopressible offset jet flow. Nuerical heart transfer: Part A. 48 (7) pp Marchal D L'adhérence des jets d'air froid au plafond des locaux cliatises, International journal of Theral Sciences, pp Patankar S. V Nuerical Heat Transfer and Fluid Flow. Heisphere Publishing Corporation, Francis & Taylor Group, New York NOMENCLATURE G Voluetric production rate of k k Turbulence kinetic energy ε Turbulence dissipation rate Re Reynolds nuber Nu Nusselt nuber Γ Transport coefficient λ c T f T w α β V Heat conduction coefficient Specific heat capacity Fluid teperature Wall teperature Teat transfer coefficient Integrational cell porosity

5 Figure 3 Air Flow Field: U-velocity at central x-z plane (Top: Case 1, inlet velocity equal to 1 /s, Middle: Case 2, inlet velocity equal to 1.5 /s, Botto: Case 3, inlet velocity equal to 2 /s) Figure 4 Teperature Field for Case 1 (Top: Central x-z plane; Botto: solid x-y plane at z=2.3)

6 Figure 5 Teperature Field for Case 2 (Top: Central x-z plane; Botto: solid x-y plane at z=2.3) Figure 6 Teperature Field for Case 3 (Top: Central x-z plane; Botto: solid x-y plane at z=2.3)

7 Figure 7 Teperature Iso-Surface of 16 o C for Case 1. Figure 8 Teperature Iso-Surface of 16 o C for Case 2. Figure 9 Teperature Iso-Surface of 16 o C for Case

8 Wall - Solid Roo - Air Roo Height [ ] x / H = 5 U = 1.0 /s U = 1.5 /s U = 2.0 /s Teperature [ o C ] Figure 10 Teperature distribution at central line M1-M Wall - Solid Roo - Air Roo Height [ ] x / H = 10 U = 1.0 /s U = 1.5 /s U = 2.0 /s Teperature [ o C ] Figure 11 Teperature distribution at central line M3-M Wall - Solid Roo - Air Roo Height [ ] x / H = 20 U = 1.0 /s U = 1.5 /s U = 2.0 /s Teperature [ o C ] Figure 12 Teperature distribution at central line M5-M