Numerical analysis of heat and mass transfer in a compact finned.pdf

ORIGINAL Numerical analysis of heat and mass transfer in a compact fi nned tubes air heat exchanger under dehumidifi cation conditions Riad Benelmir Salim Mokraoui Received 15 June 2009 Accepted 28 September 2011 Published online 13 October 2011 Springer Verlag 2011 Abstract A simulation model of a fi n and tube heat exchanger is presented The effect of the relative humidity air speed fi n base temperature and inlet air temperature on the estimation of the overall heat transfer coeffi cient and fi n effi ciency under wet conditions is also investigated This model considers a non uniform airfl ow velocity as well as a variable sensible heat transfer coeffi cient List of symbols cp a Moist air specifi c heat at constant pressure J kg 1K 1 CCondensation factor K 1 dThermal diffusivity of vapor in air m2s 1 D Specifi c diffusion coeffi cient of vapor in air m2s 1 DtExternal tube diameter m DhHydraulic diameter m E Energy fl ow rate W gAcceleration of gravity m s 2 h Half fi n height m i Specifi c enthalpy J kg 1 jColburn factor for the heat transfer l Half fi n length m LeLewis number LvLatent heat of vaporization J kg 1 m00Flux mass density kg m 2s 1 m Mass fl ow rate kg s 1 NuNusselt number Pf Half fi n spacing m PlLongitudinal tube pitch m PtTransverse tube pitch m PrPrandtl number Qt id Qt r Heat fl ux for ideal and real fi n W q00 t Total heat fl ux density W m 2 q00 sen Sensitive heat fl ux density W m 2 q00 l Latent heat fl ux density W m 2 rExternal tube radius m ReReynolds number RHRelative humidity of the moist air TaMoist air temperature K Ta iInlet moist air temperature K Tdew aAir dew point temperature K Ta oOutlet moist air temperature K TfFin temperature K Tf bFin base temperature K Tc Condensate fi lm temperature K uiMoist air inlet velocity m s 1 VaElementary air volume m3 Vf Elementary fi n volume m3 WaMoist air humidity ratio Wa iInlet moist air humidity ratio Wa oOutlet moist air humidity ratio WS fSaturatedairhumidityratioatthe fi n temperature WS f b Saturated air humidity ratio at the fi n base temperature WS cSaturated air humidity ratio at the condensate fi lm temperature xLongitudinal distance from the tube center m ka kcMoistairand condensate fi lmthermal conductivities W m 2K 1 qaMoist air density kg m 3 R Benelmir dry ia i ia o m 00 cic 1 where m00 c is the condensate mass fl ux As indicated by the ASHRAE HANDBOOK 12 the error induced by neglecting the sensible heat of the con densate m00 cic is in the order of magnitude of 1 3 On the other hand as stated above the total heat fl ux results from both sensitive and latent heat components thus the fol lowing expression of total heat fl ux density yield q00 t m00 a dry ia i ia o q 00 sen q 00 l asen hum Ta Tc m00 cLv 2 According to the mass transfer law the mass fl ux of the condensate is expressed as m00 c am Wa WS c 3 As reported by Lin et al 8 most of the investigators applied the Chilton Colburn analogy to set a relationship between the mass transfer coeffi cient and the sensitive heat transfer coeffi cient hence thefollowingrelationis reported and used in our study am asen hum Le2 3 cp a 4 Combining Eqs 2 3 and 4 the following equation is obtained q00 t asen hum Ta Tc Lv Le2 3 cp a Wa WS c 5 Moreover the total heat fl ux density is related to the overall heat transfer coeffi cient by the following relation q00 t aO hum Ta Tc 6 Thus we obtain the expression bellow for the overall heat transfer coeffi cient aO hum asen hum1 Lv Le2 3 cp a Wa WS c Ta Tc 7 2 2 The physical problem The schematic diagram of the problem is shown in Fig 2 The rectangular fi ns are arranged around tubes lined up or ranked in staggered rows The refrigerant fl ows inside tubes and moist air streams outside The fi n temperature is considered lower than the dew point temperature of air which draws away a condensation on the surface of the tube and the fi n This study will consider the heat and mass transfer for a representative tube and fi n elementary unit The investigation of heat and mass transfer performance during the cooling of moist air through an extended sur face associated with dehumidifi cation should take into account the convective heat transfer process between the air fl ow and the condensate fi lm the conduction inside the fi n and the condensate fi lm and the mass transfer process between the air fl ow and the condensate fi lm Fig 3 As the fi ns spacing is very weak regarding the fi ns height and length heat and mass transfer along with the fi n plane s normal direction z direction is neglected The heat fl ux exchanged by the fi n is assumed to be identical on both faces Hence only one face is considered in the study of heat and mass transfer by the fi n as a result of symmetry condition applied on the fi n median plane Figure 4 shows the physical domain of the current study The elementary volumes are defi ned as follow Vf dfdxdy Vc dcdxdy Va pf dc dxdy Vt pf df dxdy 2 3 Governing equations In this study we consider the two dimensional problem of heat transfer through the fi n surface and condensate fi lm and the dynamic air stream according to the x y plane Ta Wa Moist air Inlet Ta i Wa i m a dry Outlet Ta o Wa o m a dry Condensate film WS c Tc Tc Cold wall Tw T Wa i m a dry Fig 1 Air dehumidifi cation by a cold wall fin tube Pt 2h 2l Pl 2h 2l Pl Pt air fin tube Pt 2h 2l Pl 2h 2l Pl Pt Fig 2 Schematic of fi ns arranged around tubes lined up or ranked in staggered rows Heat Mass Transfer 2012 48 663 682665 123 Fig 4 The problem of the vapor mass transfer from air to the fi n tube wall occurs according to the direction z The mathematical formulation is accomplished with respect to some basic assumptions The air fl ow is con sidered as incompressible and evolving in a laminar steady state The thermo physical properties of air condensate fi lm and fi n tube are temperature dependant On the other hand we consider that heat transfer through the conden sate fi lm is purely conductive and that the radiation transfer mode is always neglected Furthermore the condensate fi lm is assumed to be thin and uniform and due to the weak thickness of the fi n the adiabatic condition is assumed at the fi n end For convenience of heat and mass transfer analysis the following dimensionless parameters are introduced as T a Ta Tf b Ta i Tf b T f Tf Tf b Ta i Tf b 8 W a Wa WS f b Wa i WS f b W c WS c WS f b Wa i WS f b 9 x x r y y r h h r l l r r r r 1 10 P Pf r d c dc r u x ux ui u y uy ui 11 2 3 1 Continuity and momentum equations for air fl ow The two dimensional continuity and momentum equations for air fl ow are oux ox ouy oy 0 12 ux oux ox uy oux oy ma o2ux ox2 o2ux oy2 13 ux ouy ox uy ouy oy ma o2uy ox2 o2uy oy2 14 Introducing the dimensionless variables defi ned above into Eqs 12 14 leads to the following dimensionless equations ou x ox ou y oy 0 15 u x ou x ox u y ou x oy 2 ReD o2u x ox 2 o2u x oy 2 16 u x ou y ox u y ou y oy 2 ReD o2u y ox 2 o2u y oy 2 17 Air flow 2h 2 Condensate film fin 2l Heat flux Condensate mass flux 2 r tube Fig 3 Heat and mass transfer phenomenon around a fi n tube Control volume Condensate film Tc WS c 2pfpf c2 f c c f f c air Ta Wa fin Tf WS f fin Tf WS f fin Tf WS f z x y 2r Control volume Condensate film Tc WS c 2pfpf c2 f c c f f c air Ta Wa fin Tf WS f fin Tf WS f fin Tf WS f z x y 2r Fig 4 Physical domain of the present study 666Heat Mass Transfer 2012 48 663 682 123 where ReDindicates the Reynolds number based on the internal fi n diameter ReD qauiD la 18 The air speed at the inlet and outlet is settled uniform and parallel to the x axes which yields to the following boundary conditions x l 8y u x 1 u y 0 19 The upper and lower edges of the fi n are subject to the following boundary conditions y h 8x u y 0 20 Finally at the fi n base the non sliding condition is used ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi x 2 y 2 p 1u x u y 0 21 2 3 2 Mass balance equation for vapor in air fl ow The vapor fl ow mass balance equation in the elementary air volume represented by Fig 5 is expressed as follow D mv m00 cdxdy 0 22 In this equation the fi rst term represents the vapor fl ow rate variation in the air volume and the second term corre sponds to the mass fl ow rate between the moist air and the condensate fi lm According to Fig 5 the variation of the vapor mass fl ow is written as D mv ux oWa ox uy oWa oy qa dryVa 23 Using Eqs 3 22 and 23 yields to the following equation ux oWa ox uy oWa oy asen hum pf dc Le2 3 cp a qa dry Wa WS c 24 The dimensionless form becomes u x oW a ox u y oW a oy asen hum p d c Le2 3 c p a ui qa dry W a W c 25 And the corresponding boundary conditions are x l 8y W a 1 y h 8x oW a oy 0 26 2 3 3 Energy balance equation for air fl ow Referring to Fig 5 the energy balance equation held the same form as Eq 22 D Ea q00 sendxdy 0 27 This equation denotes that the most air sensible heat vari ation of the elementary volume is equilibrated by the sensible heat fl ow exchanged between air and the con densate fi lm The variation of the sensible heat of the elementary volume can be expressed as D Ea cp aux oTa ox uy oTa oy qa dryVa 28 Using Eqs 2 and 27 Eq 28 leads the following form ux oTa ox uy oTa oy asen hum pf dc Le2 3 cp a qa Ta Tc 29 After introducing the dimensionless variables we get u x oT a ox u y oT a oy asen hum p d c Le2 3 c p a ui qa T a T c 30 The related boundary conditions are x l 8y T a 1 31 y h 8x oT a oy 0 32 2 3 4 Energy balance equation for the condensate fi lm As mentioned above heat transfer through the condensate fi lm is assumed to be purely conductive Using the fact that the temperature of the condensate fi lm internal surface is the same as that of the fi n surface the heat fl ux transferred from the condensate fi lm to the fi n is q00 t kc Tc Tf dc 33 This fl ux is equal to the convective heat fl ow transferred by air to the condensate fi lm consequently Eq 33 can be written y z pf x dy dx Moist air y z Elementary air volume pf c x dy dx Moist air Fig 5 Vapor fl ow rate variation in an elementary air volume Heat Mass Transfer 2012 48 663 682667 123 aO hum Ta Tc kc Tc Tf dc 34 From this equation the condensate fi lm temperature is deduced Tc Ta Ta Tf 1 aO hum dc kc 35 2 3 5 Mass balance equation for condensate fi lm The fi lm wise condensation of a stationary saturated vapor on a plane vertical surface has been analyzed by Nusselt 13 by means of some assumptions The expression of the condensate fi lm thickness given by Nusselt 13 is dc 4lckc h y Tc Tf gLvqc qc qv 1 4 36 where the subscripts c and v refer to condensate fi lm and water vapor respectively Substituting Tcby its expression Eq 35 into Eq 36 leads to the following relationship dc 4lckc h y Ta Tf gLvqc qc qv kc kc aO humdc 1 4 37 2 3 6 Energy balance equation for the fi n surface The energy balance equation for the fi n is obtained from the heat conduction equation within the fi n surface thus the subsequent equation is obtained kf o2Tf ox2 o2Tf oy2 D Ef 0 38 where D Ef is the thermal energy fl ow rate received by the fi n elementary volume expressed as D Ef q00 tdxdy Vf aO humkc df kc aO humdC Ta Tf 39 Combining these equations yields o2Tf ox2 o2Tf oy2 aO humkc kfdf kc aO humdC Ta Tf 0 40 The dimensionless form of Eq 40 is o2T f ox 2 o2T f oy 2 r2aO humkc kfdf kc aO humdC T a T f 0 41 Using the adiabatic condition in the inlet and the outlet as well as the symmetry condition in both upper and lower of the fi n wall the following boundary conditions holds x l 8y oT f ox 0 42 y h 8x oT f oy 0 43 At the fi n base surface the temperature is considered equal to that of the tube ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi x 2 y 2 p 1 T f 0 44 2 4 Solving equations The two dimensional model developed above is based on the following equations the continuity and momentum equation Eqs 15 17 the mass balance equation for water vapor Eq 25 the energy balance equation for air stream Eq 30 the heat transfer equation in the fi n surface Eq 41 and the heat and mass transfer equations for the condensate fi lm Eqs 35 and 37 In our model the simultaneous infl uence of the local speed and heat transfer coeffi cient is considered for solving heat and mass transfer within the air fl ow Eqs 25 and 30 Moreover Eq 30 uses in its expression the mass fl ow of moist air qaui while in Eq 25 the dry air mass fl ow is used This allows the consideration of the effect of con densation on heat and mass transfer only once 2 4 1 Solving continuity and momentum equations The problem described by Eqs 15 17 is a classical fl uid fl ow problem as the fl ow around a cylinder However in our case the fl uid fl ows inside a rectangular channel In order to analyze the heat and mass transfer fi n perfor mance it is necessary to know the airfl ow pattern partic ularly the distribution of the airfl ow velocities The investigation of air velocity fi eld has been carried out either by using the analytical approaches given by Johnson 14 or by a numerical analysis using the fi nite volume method In the completion of this study as the Reynolds number based on fi n length is less than 2000 laminar case and as the air thermo physical properties are weakly temperature dependent except the kinetic viscosity the following expressions of the dimensionless velocities found by Johnson 14 are approved u x 1 1 x 2 y 2 2y 2 x 2 y 2 2 45 u y 2x y x 2 y 2 2 46 The distributions of these velocities over the physical domain where the half fi n length and high are settled to 2 5 are shown in Fig 6a b 668Heat Mass Transfer 2012 48 663 682 123 As shown in Fig 6a b the horizontal and vertical velocities fi elds present an apparent symmetry regarding x and y axes The horizontal dimensionless velocity at the inlet and outlet tends towards unity is maximal at the upper and lower fi n edges and is minimal close to the tube wall as a result of the channel reduction Likewise the vertical dimensionless velocity is close to zero when going up the inlet and outlet or the upper and lower fi n edges and is also minimal near the tube surface 2 4 2 Solving heat and mass transfer equations The heat and mass transfer problem has been solved using an appropriate meshing of the calculation domain and a fi nite volume discretization method Figure 7 illustrates the fi n meshing confi guration used In this study up to 11 785 nodes are used in order to take into account the effect of the mesh fi nesse on the process convergence and results reliability The deviations on the calculation results of the fi n effi ciency with the different meshing prove to be less than 0 3 The numerical simu lation is achieved using MATLAB simulation software 15 A global calculation algorithm for heat and mass transfer models is developed and presented in Fig 8 2 5 Heat performance characterization In order to evaluate the fi n thermal characteristics we need to defi ne the heat transfer coeffi cients the Colburn factor j and the fi n effi ciency gf 2 5 1 Colburn factor The sensible Colburn factor is expressed as jsen Nusen ReDh Pr1 3 47 The Reynolds number based on the hydraulic diameter is defi ned as follows ReDh qaumax aDh la 48 where the maximal moist air velocity umax ais obtained at the contraction section of the fl ow umax a 2h 2h 2 ui 49 By defi nition the hydraulic diameter is expressed as Dh 8h l p 2pp 4h l p pp 50 air 2 5 2 1 5 1 0 500 511 522 5 2 5 2 1 5 1 0 5 0 0 5 1 1 5 2 2 5 a b 0 2 0 2 0 2 0 2 0 4 0 4 0 4 0 4 0 6 0 6 0 6 0 6 0 6 0 8 0 8 0 8 0 8 0 8 0 8 0 8 1 1 1 1 1 1 1 1 1 2 1 2 1 2 1 2 1 2 1 2 1 4 1 4 1 4 1 4 1 6 1 6 x y 2 5 2 1 5 1 0 500 511 522 5 2 5 2 1 5 1 0 5 0 0 5 1 1 5 2 2 5 0 6 0 6 0 4 0 4 0 4 0 4 0 2 0 2 0 2 0 2 0 2 0 2 0 0 0 0 0 0 0 2 0 2 0 2 0 2 0 2 0 2 0 4 0 4 0 4 0 4 0 6 0 6 y x Fig 6 a Horizontal velocity u x distribution b Vertical velocity u y distribution 2 5 2 1 5 1 0 5 0 0 5 1 1 5 2 2 5 2 5 2 1 5 1 0 5 0 0 5 1 1 5 2 2 5 Fig 7 Fin meshing with 627 nodes h 2 5 l 2 5 Heat Mass Transfer 2012 48 663 682669 123 The Nusselt and Prandtl numbers are given by Nusen asen hum Dh ka 51 Pr la cp a ka 52 The Colburn factor takes into account the effect of the air speed and the fi n geometry in the heat exchanger Knowing the heat transfer coeffi cient the determination of Colburn factor becomes usual 2 5 2 Heat transfer coeffi cients Regarding the physical confi guration of the fi n and tube heat exchanger the condensate distribution over the fi n and tube is complex In this study the condensate fi lm is assumed uniformly distributed over the fi n surface and the effect of the presence of the tube on the fi lm distribution is neglected The average condensate fi lm thickness is cal culated as follow dc RAf At At dcds Af 53 where Af denotes the net fi n area Af 4lh pr2 54 and Atrepresents the total tube cross section At pr2 55 The condensate thickness dcis calculated using Eq 37 and can be estimated iteratively Assuming the temperature profi le of the condensate fi lm to be linear the heat transfer coeffi cient of condensation is obtained as follow ac kc dc 56 The theory of hydrodynamic fl ow over a rectangular plate associated with heat and mass transfer allows us to evaluate the sensible heat transfer coeffi cient In this case a hydro thermal boundary layer is formed and results from a non uniform distribution of temperatures air velocity and water concentrations across the boundary layer Fig 9 According to Blasius theory the hydraulic boundary layer thickness can be defi ned as follow dH 5 x Re1 2 L 57 Identify the fin temperature eq 41 Calculate air local velocity eqs 45 and 46 Calculate local sensible heat transfer coefficient eq 60 Calculate Taand Wa eqs 30 and 25 Calculate the condensate film thickness 53 no yes Condensate flow rate 3 heat flow rate 5 fin efficiency 67 Calculate the boundary layer thickness eq 59 Input parameters ui RHi Ta i Tf b pf l h Le Initialization of variables Ta RH c Calculate proprerties Lv cp Calculate local overall heat transfer coefficient 7 10 6 1 N ji aN ji a TT 10 6 1 N ji f N ji f TT 10 6 1