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1、Proceedings of the ASME 2013 International Mechanical Engineering Congress and ExpositionIMECE2013November 15-21, 2013, San Diego, California, USAIMECE2013-64514STUDY OF COMBINED HEAT AND MASS TRANSFER FROM FINS USING NON- DIMENSIONAL FINITE ELEMENT FORMULATION9Copyright © 2013 by ASMESun Pasha

2、hMechanical Engineering Department King Fahd University of Petroleum & MineralsDhahran, Saudi ArabiaSyed M. ZubairMechanical Engineering Department King Fahd University of Petroleum & MineralsDhahran, Saudi ArabiaAbul Fazal M. ArifMechanical Engineering Department King Fahd University of Pet

3、roleum & MineralsDhahran, Saudi ArabiaABSTRACTThe use of dimensional analysis and dimensionless parameters is very common in the field of heat transfer. The paper presents a non-dimensional finite element capable of m ing combined heat and mass transfer from fins. The aim of the formulation is t

4、o get solution of the fin problems that do not have a d form solution. The performance of a fin is described through its efficiency and numerous d form solutions for fin efficiency under combined heat and mass transfer are available in the literature. Deriving a d form solution for geometric or mate

5、rial complexities is somewhat a difficult task. An example is variable profile composite fin. A composite fin is composed of base material or substrate with apartially wet and fully wet operating conditions.A composite fin is the fin that has a coating layer over the fin base material (or substrate)

6、. The analytical d form solution for a variable profile composite fin is somewhat difficult task due to two materials and variable geometry. The objective of the present study is to develop a non-dimensional thermal finite element and to study the combined heat and mass transfer from variable profil

7、e composite pin fins or spines.NOMENCLATURESymbolDesignationUnitswa2aParameter defined in Eq.kg kg -1(5)coating layer. Finite element approach can handle such complexity with relatively ease, Therefore the main objective isb Parameter defined in Eq.2(6)kgw kga-1K-1to developed formulation for mass t

8、ransfer problems. The formulation is derived in dimensionless form to extend thec Specific heat of incoJ kgpmoist air stream-1 K-1applicability of finite element results to a class of problems with same governing dimensionless parameters. The derived formulation is then applied to study the combined

9、 heat and mass transfer for variable profile composite fins under fully wet condition.INTRODUCTIONFins or extended surfaces are commonly used for enhng heat transfer between a prime surface and its surrounding environment. However for the applications which involve cooling and dehumidification proce

10、ss, combined heat and mass transfer occurs between the fin surface and surrounding fluid. There are several d form solutions available in the literature for combined heat and mass transfer from fins underhHeat transfer coefficientW m-2 K-1hDMass transfer coefficientkg m-2 s-1ifgLatentheatofJ kg-1 ev

11、aporation for waterkThermal conductivityW m-1 K-1LLengthmm0Dry fin parameterm-1qHeat fluxW m-2RSpine radiusmTTemperatureoCtThicknessmx,y,zCartesian coordinatesNon-Dimensional ParametersBiBiot NumberLeLewis numberq *Specified heat flux4. The heat transfer and mass transfer coefficients are related by

12、 Chilton-Colburn analogy 2:x , y , zCartesian coordinatesqTemperaturezAspect ratio h = c hDpLe2 3 (3)NON-DIMENSIONAL FINITE ELEMENT FOR THERMAL APPLICATION:The objective of present work is to develop a generalized non-dimensional finite element formulation for fin problems with combined heat and mas

13、s transfer. The element should be capable of ming conduction phenomena with convection and condensation boundary conditions for orthotropic material.The governing equation and boundary conditions:5. The Lewis number Le, relating heat and mass transfer coefficients is equal to 1.6. The fin is operati

14、ng under completely wet condition.To solve Eq. (1) with boundary conditions (2), an additional equation for w is required. A relationship between w and T is given by Kloppers and Kroger 3, however that is a non-linear relationship. There are some linear relationships available in the literature and

15、a brief discussion is presented by Sharqawy and Zubair 2. We are using the linear relationship developed by Sarqawy and Zubair2:The governing differential equation for steady state heat conduction with no internal heat generation in an orthotropic material in three dimensions is 1.w = a2 + b2Twith(4

16、)¶2T¶2T¶2Ta = w- wd p - wb Tkx ¶x2 + ky ¶y2 + kz ¶z2 = 0(1)2bT - Tb(5)with following boundary conditions 1 2:d pbb = wd p - wb 2T - T(6)on surface S1 :T = Tbd pb22The parameters a and b can be calculated from the ambienton surface S :k ¶T l + kx ¶ x2¶T m

17、+ ky ¶ y¶T n + q* = 0z ¶ z(2)air conditions and fin base temperature. Therefore by using the Eqs. (3) to (6), the last boundary condition in (2), can be written as:on surface S3 :k ¶T l + k¶T m + k¶T n + h¢(T ¢ - T ) = 0k ¶T l + k ¶T m + k¶T n +

18、 h (T- T )x ¶ xy ¶ yz ¶ z¥(7)x ¶ x+h i (wy ¶ y- w ) = 0z ¶ z¥whereh¢(T¥¢ - T )is the combined heat transfer due toD fg¥where, kx , ky and kz are the thermal conductivities in x , y andz -directions respectively,. Tb and T¥ are specifie

19、d and ambient temperatures respetively, q* is the specified heat flux, h is theconvection and condensation. Therefo¢ and T ¢ are the corresponding equivalent heat transfer coefficient and equivalent ambient temperature given by:h¢ = h (1+ Bb )T ¢ = T¥ + B (wa - a2 )coefficie

20、nt of convective heat transfer, hD is the mass transfer1+ Bb2(8)2¥coefficient, ifg is the water latent heat of evaporation, w¥ iswith,the humidity ratio of ambient air and and l, msurface normals.Assumptions:and n are theB = ifgpc Le23 (9)Following assumption areto simplify the derivation

21、of the formulation:1. There is no internal heat generation2. The heat transfer coefficient and latent heat of condensation of water vapor are constant.3. The thermal associated with the presence of thin water film due to condensation is small and may be neglected 2.The parameter B can be considered

22、as constant having an average value of 2415 oC, because over the practical range of air temperature and relative humidity its variation is within± 1.6% of the average value 2.The Variational principle:éTTùVariational principle is used here for deriving the finite elementêò&#

23、242;ò éëBùûéëLùû éëDùûéëBùû dV + Còò éëNùûéëNùû dS ú =formulation. The variational principle specifies a scalar qu ty (functional P ), defined by an integral fo

24、rm for aêë VS3òò éëNùû T q*dS + ¡òò éëN ùû T dSûú(14)continuum problem. The solution of the continuum problem isS2S3a function that makes P stationary with respect to arbitrary changes in it 4.The above equation1 is

25、 of the form:k= f (15)The functional for heat transfer problem is given by 5:P = U +Pq + Ph(10)Therefore, the element stiffness matrix and load vectors may be deduced as follows: = + withThe functional has three terms which are associated withkkqkhqinternalenergy (U ),heatconduction(Pq )andheatk =&#

26、242;òòBT L D BdVand(16)convection(Ph ) . For the governing equation (1) withVSassociated boundary conditions (2); the terms may be written,as follows 6:31é æ ¶T ö2æ ¶T ö2æ ¶T ö2 ùSimilarly:f = f + f , with:U = 2 òòòê

27、;kx ç ¶x ÷+ k y ç ¶y ÷ + kz ç ¶z ÷ ú dVqhV êë èøèøèø úûf =NT q *dSand f = ¡ éNù T dS(17)Pq = -òò q*TdS(11)qòòS2hòò ë ûS3S2P =h æ 1 T -ö

28、It is worth mentioning that the non-dimensional form ofgoverning equations has resulted in an additional factor (i.e.hòò ç 2T¥ ÷TdSL) in stiffness matrix which does not have any counterpart in3SèøS andS are separate surface areas over which heat flux q *conventiona

29、l finite element formulation, whereas all other23matrices (i.e. B, D and N) are essential constituents of aand convection loss h (T - T¥ ) are specified because they cannot occur simultaneously on the same surface. Therefore for the case of equivalent heat transfer (cf. Eq. (7) the functional r

30、educes to:h¢(T¥¢ - T )finite element formulation. Similarly the factors C and ¡ are introduced that are related to heat transfer due to condensation.Non-dimensional finite element formulation: Solution to a class of problems:P = òòò êkx ç÷1é 

31、30; ¶T ö22¶xæ ¶T ö2+ ky ç ¶y ÷æ ¶T ö2 ù+ kz ç ¶z ÷ ú dVThe ability of non-dimensional finite element method to provide the solution to a class of problems can be investigatedV êë èøèø&#

32、232;ø úû¥-òò q*T dS + 1 òò h¢(T ¢ - T )2 dS(12)by identifying the governing non-dimensional parameters from non-dimensional finite element formulation (cf. Eq. (16) andSS22 3(17). The matrix Limplies to the fact that the non-Definition of non-dimensi

33、onal parameters:Following non dimensional parameters are defined:q = T , x = x , y = y and z = z(13)T¥LxLyLzwhere q is the non-dimensional temperature, x , y and z aredimensional finite element results will be valid for a considered aspect ratio of the studied domain. The second governingmatrix

34、 is D , which implies that the results will be valid forthe problems that would have the same Biot numbers. q * is the non-dimensional specified heat flux whereas the parameters C and ¡ are the parameters related to mass transfer bycondensation (see Annex). Thus, a single non-dimensionalnon-dim

35、ensional spatial variables,Lx , Lyand Lzare thefinite element simulation would provide the results to a class ofum dimensions of the domain to be discretized along x ,y and z -directions respectively.Inserting the non-dimensional parameters in (12) and minimization with respect to q yields:problems

36、for a particular set of L, D q * , C and ¡ values.Implementation of the non-dimensional finite elementThe non-dimensional finite formulation is implemented by develo a FE code in. In fact, if the non- dimensional formulation did not contain the geometric1 All matrices are given in Annexdimensio

37、ns ratio matrix L then the implementation would bevery straight forward by using any commercial finite element code (e.g. ANSYS) because the parameters for the matrix Dcan ily be deduced from the normalization defined inIt is obvious from Eq. (18) that the fin efficiency is a function of fin paramet

38、er mL only, however the non-dimensional finite element results have been obtained with the following operating conditions:equation (6) whereas the expressions for B and N are the same as for the dimensional form of the finite element formulation, with spatial variables x , y and z replaced with thei

39、r non-dimensional counterparts x , y and z .APPLICATION OF NON-DIMENSIONAL FINITE ELEMENT FORMULATION:The objective of this section is to apply the derived formulation to study the combined heat and mass transfer from compositePatm= 101.3 kPa, RH = 1, T0.6z= 27ooC, T = 7 Cbpin fins of vari

40、able profiles. A composite fin is composed of substrate (i.e. fin base material) with a coating layer over it a coating each having different materials.Validation of non-dimensional FE m:The two-dimensional axisymmetric form of the developed non- dimensional finite element formulation will be used t

41、o study the thermal performance of the composite pin fins. Therefore,0.20.100r0.81the accuracy of the non-dimensional FE m s should first be validated through a test case of study wit own d-form solution.The d-form solution for combined heat and mass transfer from a composite pin f

42、ins could not be found in the literature, however the efficiency of a one dimensional rectangular profile pin fin with an insulated tip under complete wet condition is given by 2:tanh(mL)Figure 1:Non-dimensional mesh for a rectangular profile spine having aspect ratio L / R = 10ND-FEAAnalyticalzLrR1

43、0.80.6h =wheremL1+ b2 Bm = m0(18)iswetspineparameterand0.42h kRbm0 =is the dry spine parameter.Therefore, we consider a rectangular profile spine with aspect0.2ratio L / R = 10 as the case of study for the validation purpose.The dimensionless mesh of the fin is shown in Figure 1. Fine mesh is used n

44、ear the fin base. The mesh size has been selected after assessing the convergence of the non-dimensional finite element results. The mesh independence was verified by obtaining a series of solutions with successive fine mesh sizes. The mesh size was then selected after which the grid independence wa

45、s observed. Note thatum dimensionless thickness and length are equal to u due to normalization (cf. Eq. (13).The fin efficiency based on non-dimensional finite element results is compared, with the d form solution (cf. Eq. (18), over a range of mL values in Figure 2. The graph shows an excellent agr

46、eement between two results which implies that the non-dimensional finite element m provides good accuracy results.0012mL345Figure 2: Comparison of non-dimensional finite element results with d form solution for a rectangular profile spine under complete wet condition.A relative humidity (RH) of 100%

47、 is considered to achieve the fully wet condition over the considered range of the fin parameter mL. The profile of efficiency curve does not change at lower RH values however the partially wet condition is reached at relatively lower mL values. To identify the limiting mL values corresponding to th

48、e end of fully wet condition, a series of simulations are run with RH ranging from 40 to 100%. The results are shown in Figure 3. It can be observed that all curves are overlapped however the range of completely wetcondition is a function of RH. For example the end of completely wet condition of RH

49、= 0.4 corresponds to mL = 1.zLrRRH= 1= 0.9= 0.8= 0.7= 0.6= 0.5= 0.4Patm = 101.3 kPa,T = 27 oC, Tb = 7 oC10.8the operating condition (i.e. temperature, atmospheric pressure and RH) are the same as for the validation study.Fin ProfileThe most commonly used fin profiles are schematically depicted in Fi

50、gure 5(a) to (d), whereas the configuration of a rectangular profile composite fin is shown in Figure 5(e). A0012mL345constant layer thickness substrate for all profiles.z LRztRtLrRbRbz RLtc will be considered over the finFigure 3: Efficiency of a completely wet rectangular profile spine (i

51、nsulated tip) at different RH valuesFor the completely wet condign the fin tip temperature must be less than the corresponding dew point temperature. Ther(a)(b)(c)rzR trRbL(d)RtcL(e)temperature difference between fin tip and air dew point is presented in Figure 4 which provides useful information ab

52、out the completely wet condition over a range of RH values.Figure 5: Schematic of profiles for spines. (a) Rectangular (b)Taper (c) Convex parabolic and (d) Concave parabolic(e) Schematic of rectangular profile composite finzLrRRH= 1= 0.9= 0.8= 0.7= 0.6= 0.5= 0.4Patm = 101.3 kPa,T = 27 oC, Tb = 7 oC

53、A fin without any coating will be considered as the base case of study. The results of the base case would serve as a reference for the assessment of coating effects. The geometry of the fin will be governed by two parameters which are fin aspect ratioz = L / Rband ratio of fin tip to fin base thick

54、ness i.e.12rtb = Rt / Rbsuch thatrtb < 1 .Following values will be used forTdp-Ttipfin substrate geometric parameters:108Fin substrate aspect ratio: z = 10Fin substrate tip to base radius ratio: rtb = Rt6Rb = 0.25420mL345The efficiency of different profile spines is shown in Figure 6, which shows

55、 that the parabolic concave profile has the highest fin efficiency.Figure 4: Temperature difference between air dew point and fin tip temperature as an indication of fully wet condition.After validating the formulation it will now be used to study the performance of variable profile composite fins.

56、The validation studies were for the insulated tip boundary condition d form analytical solution. The formulation has however no such limitation; therefore all the following studies will be done for convective tip boundary condition, whereasRectangular TaperConvex Parabolic Concave ParabolicPatm = 101.3 kPa, RH = 1T = 27 oC, Tb = 7 oCz = 10, rtb = 0.251transfer coefficient equal to hc = kc tc .The validity ofBicmust however be examined to0.8verify whether or not it completely governs the coating effects. This can be done by comparing the results of d