Modeling Of Wire On Tube Heat Exchangers Using Finite Element Method

Finite Elements in Analysis and Design 38 (2002) 417–434 www.elsevier.com/locate/ÿnel Modeling of wire-on-tube heat exchangers using ÿnite element method G.A Quadir ∗ , G.M Krishnan, K.N Seetharamu School of Mechanical Engineering, USM(KCP), 31750 Tronoh, Perak, Malaysia Abstract Wire-on-tube heat exchangers are analysed under normal operating conditions (free convection) using ÿnite element method Galerkin’s weighted residual method is used to minimise the errors The e ects of ambient temperatures and mass ow rates of the refrigerant are determined This is used to ÿnd out the length of tube required for phase change for its initiation and completion The methodology adopted is validated against the existing data for both sensible heat and latent heat transfer The analysis also leads to the information about the adequacy of the number of tubes for complete condensation of the refrigerant vapour under given operating conditions di erent from that of design conditions This methodology can be used as a design tool for the design of wire-on-tube heat exchangers The derating of heat exchangers under abnormal ambient conditions can also be predicted ? 2002 Elsevier Science B.V All rights reserved Keywords: Wire on-tube heat exchangers; Free convection; Finite element method; Phase change Introduction Wire-on-tube heat exchangers are more often used by refrigerator manufacturers as condensers mainly due to their simple construction, ruggedness, and low cost This type of heat exchanger consists of a single steel tube, bent into serpentine parallel passes carrying the uid, mainly refrigerant and solid steel wires are attached to the tube to increase the surface area The solid wires are spot welded on to diametrically opposite sides of the tubes as shown in Fig The refrigerant enters the tube in a vapour state and leaves the condenser in a liquid state undergoing a phase change The heat transfer takes place from the outer surfaces of the wires and tubes ∗ Corresponding author E-mail address: gaquadir@rocketmail.com (G.A Quadir) 0168-874X/02/$ - see front matter ? 2002 Elsevier Science B.V All rights reserved PII: S - X ( ) 0 - 418 G.A Quadir et al / Finite Elements in Analysis and Design 38 (2002) 417–434 Nomenclature A Cp D H h hf hfg L m N P S T; t U X x area, m2 speciÿc heat capacity at constant pressure, W=kg K diameter, m speciÿc enthalpy of saturated vapour, kJ=kg speciÿc enthalpy of a liquid, kJ=kg speciÿc enthalpy of saturated liquid, kJ=kg latent heat of vaporization, kJ=kg length, m mass ow rate, kg=s number, shape function perimeter, m pitch, m temperature, K overall heat transfer coe cient, W=m2 K dryness fraction(= (H − hf )=hfg ) distance along the length of an element, m Subscripts r sat sup t ti to w 1; ∞ refrigerant saturated condition superheated condition tube inside tube outside tube wire inlet and outlet conditions ambient temperature Fig Sketch of the wire-on-tube heat exchanger G.A Quadir et al / Finite Elements in Analysis and Design 38 (2002) 417–434 419 to the external environment by free or forced convection The analysis of heat transfer through wire-on-tube heat exchangers is very complex in view of the number of parameters involved such as the wire diameter Dw , the tube diameter Dt , the wire spacing Sw , the tube spacing St , the overall wire length Lw , the length of the tube Lt , the multilayer stacking of the coil, ow of refrigerant, etc Tanda and Tagliaÿco [1] gave a Nusselt number correlation as a function of the geometric and operating parameters to predict free convection heat transfer from a vertical wire-on-tube heat exchanger to ambient air based on their experimental results Hoke et al [2] carried out experiments to investigate the air-side convective heat transfer for wire-on-tube heat exchangers used in most refrigerators They were able to give a correlation valid for all the experimental data for seven wire-on-tube heat exchangers studied under forced convection regime The importance of angle of attack for locating the wire-on-tube exchangers that are cooled by forced convection is also highlighted [2] The air-side resistance to heat ow for such type of heat exchanger is around 95 percent of the total resistance between the refrigerant and the air when the refrigerant is in two phase region [3] New and e ective heat exchangers with tubes ÿnned with wires and spirals were conducted by Martynov [4] to investigate heat exchangers for cryogenic plants Hiroshi et al [5] performed experiments to study the ow and heat transfer characteristics during condensation of R-11 and R-113 in the annuli of horizontal double-tube test condensers The condensers were made up of a 19:1 mm O.D corrugated inner tube with ÿns soldered on the outer surface and three outer smooth tubes of 24.8, 27.2, and 29:9 mm I.D An empirical equation for the local heat transfer coe cient was developed, in which the dimensionless parameters based on the surface tension controlled ow and the vapour shear controlled ow models were introduced for the low and high vapour velocity regimes The wire-on-tube heat exchangers are also commonly used in immersion cooling of electronic equipments [6] The miniaturisation of electronic equipments has resulted in high heat uxes being produced The simplest type of immersion cooling system involves an external reservoir which supplies liquid continually to the electronic enclosure The vapour generated inside is simply allowed to escape to the atmosphere Due to several impracticalities associated with this system, a more sophisticated immersion cooling system is used In this system, the vapour is condensed and returned to the electronic enclosure by placing the condenser either externally or internally to the electronic enclosure In the external arrangement, the vapour leaving the enclosure is cooled by a cooling uid such as air or water outside the enclosure The condensate is returned to the enclosure for reuse With the internal arrangement, the condenser is placed in the electronic enclosure The cooling uid in this case circulates through the condenser tube, removing heat from the vapour surrounding the wire-on-tube heat exchanger The vapour that condenses drips on top of the liquid in the enclosure and continues to recirculate Finite element modeling for the analysis of general types of heat exchangers with and without phase change has been carried out by various researchers [7–14] However, to the authors’ knowledge, no attempt has so far been made to analyse numerically the performance of a wire-on-tube heat exchanger either under free convection or forced convection conditions The present paper is an attempt in that direction Free convection condition on the outside of the tubes normally exists and hence this case is considered for the analysis of wire-on-tube heat 420 G.A Quadir et al / Finite Elements in Analysis and Design 38 (2002) 417–434 Fig (a) One tube length of the wire-on-tube heat exchanger; (b) Bare portion of tube; (c) Tube with wire attached exchanger using ÿnite element method Galerkin’s weighted residual approach is adopted to minimise the residual errors Analysis The numerical analysis developed is based on the ÿnite element method FEM utilises discretisation of the domain over which the solution is sought and obtains the characteristic of one element in the form of an element matrix depending upon the weighting function assumed for minimising the residual error If need arises, the assembly of the element matrix is performed and the solution obtained for imposed boundary conditions Otherwise step-wise solution for each element is carried out over the entire domain Fig 2(a) shows one tube length of a heat exchanger consisting of bare portions of the tube and the wired portion A typical one-element discretised model of the wire-on-tube heat exchanger is shown in Figs 2(b) and (c) depending upon whether the element refers to the bare portion of the tube or wired portion of the tube, respectively Every element has two nodes, one at entry and the other at exit point of tube side uid Refrigerant 12 is considered as the uid entering the tube in a superheated vapour state The uid ÿrst follows desuperheating process bringing the state of the uid to dry saturated condition Then the uid starts condensing in the tube Depending upon the length of the tube available, the uid may be subcooled at the exit of the wire-on-tube heat exchanger Governing equations and ÿnite element formulation Since the refrigerant in a wire-on-tube heat exchanger may follow three distinct regions as described above, it is important to describe the governing equations and their FEM formulations separately for each region G.A Quadir et al / Finite Elements in Analysis and Design 38 (2002) 417–434 421 3.1 Desuperheating regime The di erential equation governing heat transfer from superheated vapour to ambient air (i.e free convection condition) over an elemental area dA in terms of enthalpy H is dH U H − T∞ = 0; (1) + dA mr Cpr where, mr is the mass ow rate of the refrigerant, Cpr is the speciÿc heat of the refrigerant, U is the overall heat transfer coe cient for the element, and T∞ is the temperature of the ambient air The enthalpy for the refrigerant in the element (which is fairly small; 0:0017 m) is assumed to vary linearly as H = N1 H1 + N2 H2 ; (2) where the shape functions are given by A A N1 = − and N2 = (3) A A and A is the area at any given location which varies from zero to A where A is the area of an element Applying this approximation to governing equation (1), two equations for the two unknowns H1 and H2 are obtained as follows: U dH H N1 − T∞ dA = 0; (4) + dA mr Cpr dH H U − T∞ dA = 0: (5) + dA mr Cpr Minimising the residual error by Galerkin’s method, the following element matrix results: 2C − 0:5 C + 0:5 H1 U AT∞ =(2mr ) = ; (6) C − 0:5 2C + 0:5 U AT∞ =(2mr ) H2 N2 where C= U A : (6mr Cpr) The above formulation is valid till the enthalpy of the refrigerant equals to that corresponding to dry saturated condition 3.2 Phase change regime As the temperature of the refrigerant remains constant and equal to the saturation temperature in this region for given pressure, the di erential equation governing heat transfer is written as dH U (7) + (Tsat − T∞ ) = 0: dA mr Assuming linear variation in enthalpy of the refrigerant in the element as before: H = N1 H1 + N2 H2 (8) 422 G.A Quadir et al / Finite Elements in Analysis and Design 38 (2002) 417–434 and following the same procedure as that in earlier region, the following element matrix is found: +1 −1 H1 U A(Tsat − T∞ )=mr = : (9) +1 −1 H2 U A(Tsat − T∞ )=mr The above formulation will be valid till the quality of the wet vapour is zero or its enthalpy equals to that of saturated liquid enthalpy at the operating condenser pressure 3.3 Sub-cooling region The di erential equation governing heat transfer from liquid refrigerant to ambient air in this region in terms of temperature is written as: dT mr Cpr (10) + UP(T − T∞ ) = 0; dx where P is the equivalent perimeter of the tube element Assuming linear variation for the temperature of the refrigerant in the tube: T = N1 T1 + N2 T2 (11) and adopting the same procedure as for the earlier regions, the following element matrix is found: 2C − 0:5 C + 0:5 T1 3CT∞ = ; (12) C − 0:5 2C + 0:5 T2 3CT∞ where, C = UP l=6mr Cpr = U A=6mr Cpr Here Cpr should refer to the liquid condition It may be noted that the above element matrix is the same as that derived for the desuperheating region The only di erence is in terms of the variable selected A computer programme is developed to take into consideration all the three regimes discussed earlier The programme is able to locate the position where the change of region=regime takes place Validation Since the methodology used for the analysis of wire-on-tube heat exchangers consist of sensible heat transfer and latent heat transfer, it is essential to validate the above method against two similar heat transfer cases for which results are available One such case is the analysis of shell and tube heat exchanger where sensible heat transfer only takes place The other case considered is an evaporator where latent heat is transferred For both the cases, the initial task is to develop the element sti ness matrix and to assemble the global sti ness matrix which are explained below 4.1 Case I In order to ensure the accuracy of the computer programme and the methodology adopted in the present analysis, a double tube heat exchanger for which experimental data is available as G.A Quadir et al / Finite Elements in Analysis and Design 38 (2002) 417–434 423 Fig (a) Schematic diagram of an evaporator; (b) A discretised element of evaporator shown in Fig 3(a) is considered [15] Refrigerant 12 evaporates in the central tube by the hot water owing in the annulus This example will serve the purpose of validation in the phase change region As usual, the evaporator is divided into number of elements for the purpose of discretisation Here every element, as shown in Fig 3(b), has four nodes, two of the nodes at entry and exit points of the tube side uid and other two nodes at entry and exit points of the annulus side uid Assuming H to represent the enthalpy of the refrigerant and hw for enthalpy of water, the di erential equations for the heat transfer between the two uids may be written as dH UP hw − − Tsat = (13) dx mr Cpw and dhw UP hw − Tsat = 0: + dx mw Cpw (14) With the introduction of linear ÿeld variables as H = N1 H1 + N2 H2 and hw = N1 hw1 + N2 hw2 and using Galerkin’s approach for minimising error, the following element matrix is established:      −0:5 0:5 −2E − −E H B          −0:5 0:5  H     −B   − E − 2E   = ; (15)     0 2G − 0:5 G + 0:5   D  hw1              D 0 G − 0:5 2G + 0:5 hw2 where E= UP l ; 6mr Cpw B= UP lTsat ; 2mr G= UP l 6mw Cpw and D= UP lTsat : 2mw 424 G.A Quadir et al / Finite Elements in Analysis and Design 38 (2002) 417–434 Fig Variation of dryness fraction along length for an evaporator The element sti ness matrices are assembled together to form the global sti ness matrix Boundary conditions are incorporated for the evaporating uid inlet and the heating water outlet given in the following example [15]: Tube parameters: Inner tube bore, m Inner tube o.d., m Outer tube bore, m Refrigerant 12: Mass ow rate of refrigerant, kg=s Inlet pressure of refrigerant, bar Outlet pressure of refrigerant, bar di = 0:011 887 = 0:012 700 D = 0:019 050 mr = 0:024 570 p1 = 4:102 21 p2 = 3:943 625 Water: Mass ow rate of water, kg=s mw = 0:653 94 Outlet temperature of water, K T2 = 288:67 The enthalpy of the evaporating refrigerant is obtained along the length of the evaporator by the proposed ÿnite element method A constant value of U = 4000 W=m2 K was taken, which is the average value of U in the region considered (X ¡ 0:9) and is given in Fig of [15] Then the dryness fraction is calculated and plotted against the length of the evaporator as shown in Fig The results are in excellent agreement with those described in [15] The present method, therefore, demonstrates the validity and accuracy of the FEM formulation in the phase change region G.A Quadir et al / Finite Elements in Analysis and Design 38 (2002) 417–434 425 Fig (a) Shell and tube heat exchanger with discretised elements; (b) The ÿrst discretised element with nodal points 4.2 Case II For the purpose of validating the numerical procedure adopted in the subcooling region where only sensible heat is exchanged, a shell and tube heat exchanger is analysed by FEM using Galerkin’s approach The discretised model of a shell and tube heat exchanger (1 shell pass-2 tube passes) is shown in Fig 5(a) The ÿrst element shown in Fig 5(b) is considered for the element matrix formulation The di erential equation governing the heat transfer for the hot uid in the element is written as dT mh Cph h + U (Th − tc ) = (16) dA and dtc mc Cpc − U (Th − tc ) = (17) dA for the cold uid Here tc refers to the temperature of the cold uid Assuming linear isoparametric elements for the ÿeld variables and area, Th = N1 T35 + N2 T36 ; tc = N1 T1 + N2 T2 426 G.A Quadir et al / Finite Elements in Analysis and Design 38 (2002) 417–434 and A = N1 Ai + N2 Ao ; (18) the ÿnal set of equations for element matrix formulation using Galerkin’s method for error minimisation are: (2S2 − Y )T1 + (Y + S2 )T2 − 2S2 T35 − S2 T36 = 0; (19) (S2 − Y )T1 + (Y + 2S2 )T2 − S2 T35 − 2S2 T36 = 0; (20) −2S2 T1 − S2 T1 + (2S2 − S1 )T35 + (S1 + S2 )T36 = 0; (21) −S2 T1 − 2S2 T2 + (S2 − S1 )T35 + (S1 + 2S2 )T36 = 0; (22) where mh Cph mc Cpc U A and Y = : ; S2 = 2 The global sti ness matrix is formed depending upon the number of elements considered Boundary conditions are imposed and thereafter the temperature distribution is obtained from the solution of the assembled global matrix To show the prediction accuracy, the following ow and uid properties for the analysis of shell and tube heat exchanger is considered [8] S1 = Heat transfer coe cient, U = 600 W=m2 K Heat capacity rate of cold uid = 200; 000 W Heat capacity rate of hot uid = 17; 000 W Total area of heat transfer = 64:0 m2 ◦ Inlet temperature of hot uid = 190 C ◦ Inlet temperature of cold uid = 35:0 C Number of elements = 32 The calculated temperature distribution at di erent node points using the present model is shown in Table The table also shows three other sets of solutions corresponding to a FEM model using subdomain collocation method [8], the model given by Gaddis and Schlunder [16] and the analytical solution obtained from expressions given in [17,18] It is observed that the temperature distribution obtained by the present model based on Galerkin’s method is very close to that given by the analytical solution Thus the accuracy of the present method for the sensible heat transfer region is demonstrated and validated Since ÿnite element formulation for desuperheating region is the same as for subcooling region, there is no need to demonstrate the validity for the desuperheating region separately Results and discussion The FEM formulations given earlier for both sensible and latent heat transfer are coded to study the performance of the wire-on-tube heat exchanger Refrigerant 12 enters in the G.A Quadir et al / Finite Elements in Analysis and Design 38 (2002) 417–434 427 Table Temperature distribution in shell and tube heat exchanger (Shell side) Node points Analytical result [17,18] ◦ T ( C) Present model ◦ T ( C) Ref [8] model ◦ T ( C) Ref [16] model ◦ T ( C) 33 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 46.355 170.342 153.276 138.45 125.597 114.431 104.736 96.319 89.013 82.669 77.162 72.379 68.228 64.623 61.494 58.776 56.417 46.364 170.31 153.277 138.438 125.599 114.416 104.738 96.31 89.014 82.663 77.162 72.376 68.228 64.621 61.493 58.775 56.415 46.357 170.308 153.258 138.419 125.57 114.388 104.704 96.278 88.979 82.629 77.127 72.343 68.195 64.589 61.463 58.746 56.387 46.246 170.98 154.424 139.942 127.334 116.306 106.704 98.306 90.994 84.599 79.028 74.159 69.916 66.209 62.976 60.152 57.689 ◦ superheated state (10 bar, 70:69 C) The required properties are taken from [19] Table shows the number of runs carried out for various ambient temperatures and number of coils The coil dimensions are tabulated in Table which are taken from Ref [2] The overall heat transfer coe cient was taken to be 10 W=m2 K under the free convection condition For any tube, the outside surface area in the wired portion of the tube is di erent from that in the bare portion of the tube Therefore each tube is divided into three distinct sections The ÿrst one is related to the bare portion of the tube whereas the second one refers to the portion of the tube where wires are welded The third section is similar to the ÿrst section It is also to be noted that due to the gap between the wires in the second section, the outside surface area of elements are to be taken properly For this section, two methods were attempted during the calculations The ÿrst took into account the exact area for each element and the second method considered an average area for an element in that section Calculations were performed with 70 elements in the ÿrst, 348 elements in the second and 70 elements in ◦ the third section for coil with mass ow rate of 0:0008 kg=s and ambient temperature of 10 C The results are shown in Table The two calculations, however, show very little di erences (less than 2%) between the two results as regards the onset of phase change and sub-cooling as well Therefore the average element area method in the second section was considered for all calculations that are reported in this paper Further, in order to have the results independent of the element size, three di erent number of elements (size) 70-300-70, 70-600-70 and 140-600-140 are used to calculate the enthalpies, dryness fractions and temperatures for the coils as mentioned above for the same conditions of mass ow rate and ambient temperature for the purpose of comparison The notation 70-300-70 means that 70 elements are taken for the bare portions of the tube, whereas 300 refers to the wired portion of the tube The results are shown in Table It is noticed that with the increase 428 G.A Quadir et al / Finite Elements in Analysis and Design 38 (2002) 417–434 Table Parametric studies undertaken by varying mass ow rates of refrigerant, ambient temperatures, tube diameters and other dimensionsa No 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 a mr 0.0002 0.0004 0.0006 0.0008 0.0010 T∞ (K) 283.00 288.00 293.00 298.00 303.00 308.00 313.00 283.00 288.00 293.00 298.00 303.00 308.00 313.00 283.00 288.00 293.00 298.00 303.00 308.00 313.00 283.00 288.00 293.00 298.00 303.00 308.00 313.00 283.00 288.00 293.00 298.00 303.00 308.00 313.00 Coil # 1 1 No 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 mr 0.0012 0.0008 0.0008 0.0008 0.0008 T∞ (K) 283.00 288.00 293.00 298.00 303.00 308.00 313.00 283.00 288.00 293.00 298.00 303.00 308.00 313.00 283.00 288.00 293.00 298.00 303.00 308.00 313.00 283.00 288.00 293.00 298.00 303.00 308.00 313.00 283.00 288.00 293.00 298.00 303.00 308.00 313.00 Coil # 1∗ 1∗∗ T∞ —Ambient temperature in Kelvin, mr —mass ow rate of refrigerant in kg=s in number of elements, the maximum di erence between the results of the coarse and ÿne elements is less than 1.9 percent in relation to the prediction of the beginning of phase change as well as the sub-cooling As the computation with a ÿne element is not cost e ective, the discretisation 70-300-70 is used for the present calculation Having decided about the element size and also the average area method to be considered for the wired portion of the tube, it is thought of tabulating the calculated results for a typical set of runs numbering 22–28 of Table The results in Table show the values of enthalpies, dryness fractions and temperatures depending upon whether the ow condition is in the desuperheating region, phase change region G.A Quadir et al / Finite Elements in Analysis and Design 38 (2002) 417–434 429 Table Wire-on-tube heat exchanger dimensionsa Variable Coil #1 Ref [10] Coil #1∗ (Modiÿed coil #1) Coil #1∗∗ (Modiÿed coil #1) Coil #4 Ref [10] Coil #5 Ref [10] Dw (mm) Sw (mm) Nw Dto (mm) Dti (mm) St (mm) Lt (mm) Nt 1.22 5.34 176 4.80 3.18 25.4 660 16 1.22 5.34 176 4.92 3.57 25.4 660 16 1.22 5.34 176 6.21 5.16 25.4 660 16 1.62 4.57 168 4.92 3.57 31.8 578 18 1.21 4.56 166 6.21 5.16 31.8 575 12 a Values in bold indicate the di erence in dimensions of the coil with respect to coil #1 Table Exact and average area methoda Mass ow rate of refrigerant, mr = 0:0008 kg=s Ambient temperature, T∞ = 283 K Method Location of starting of phase change in tube # 1.40 1.45 Exact area Average area a Location of starting of sub-cooling in tube # 12.63 12.61 1.4 indicates 40% location in tube #2 in the direction of ow of the refrigerant Table Variation in element size Mass ow rate of refrigerant, mr = 0:0008 kg=s Ambient temperature, T∞ = 283 K No Element members size 70-300-70 (1:672=1:55=1:672) mm 70-600-70 (1:672=0:776=1:672) mm 140-600-140 (0:836=0:776=0:836) mm Location of starting of phase change in tube # Location of starting of sub-cooling in tube # 1.45 12.62 1.44 12.64 1.45 12.62 or sub-cooling region These values are shown at the end of the tubes for a refrigerant mass ◦ ◦ ow rate of 0:0008 kg=s The variation of ambient temperature from 10 C to 40 C is considered in order to cater to di erent climatic conditions The location of the start of phase change or ◦ sub-cooling is also shown in this table for di erent ambient temperatures For example at 10 C ambient temperature, the desuperheating of refrigerant vapour continues till 1.45 tube length (tube #2) from the start The phase change then follows and is completed at 12.62 tube length (tube #13) Then the sub-cooling starts and the subcooled temperature of the refrigerant at the ◦ ◦ end of the heat exchanger is 293:08 K (i.e 20:08 C) which is 10 C higher than the ambient 430 G.A Quadir et al / Finite Elements in Analysis and Design 38 (2002) 417–434 Table Calculated values of enthalpies (H ), dryness fraction (X ) and temperatures (T ) at the end of tubes for coil #1 ◦ Refrigerant initial conditions: P = 10 Bar, Tsup = 70:69 C; Mass ow rate of refrigerant, mr = 0:0008 kg=sa Ambient temperatures, T∞ (K) 283.00 288.00 293.00 298.00 303.00 308.00 313.00 Tube number H=X=T H=X=T H=X=T H=X=T H=X=T H=X=T H=X=T 10 11 12 13 14 15 16 209.2723 0.9495 0.8604 0.7713 0.6823 0.5932 0.5041 0.4151 0.3260 0.2369 0.1478 0.0588 312.0056 303.4877 297.397 293.0831 210.6779 0.9717 0.8962 0.8208 0.7453 0.6699 0.5945 0.5190 0.4436 0.3681 0.2927 0.2173 0.1418 0.066 314.4752 306.7452 212.0934 0.9926 0.9308 0.8690 0.8072 0.7454 0.6835 0.6217 0.5599 0.4981 0.4363 0.3745 0.3127 0.2508 0.1890 0.1272 213.5126 205.5525 0.9634 0.9135 0.8671 0.8190 0.7705 0.7227 0.6745 0.6264 0.5782 0.5301 0.4819 0.4338 0.3856 0.3375 214.9137 207.7447 0.9927 0.9582 0.9237 0.8892 0.8547 0.8201 0.7856 0.7511 0.7166 0.6821 0.6475 0.6130 0.5785 0.5440 216.2972 209.9989 206.1812 0.9957 0.9748 0.9539 0.9330 0.9121 0.8913 0.8704 0.8495 0.8286 0.8077 0.7868 0.76590 0.7450 217.672 212.3564 208.8455 206.8611 205.6815 204.8169 0.9991 0.9919 0.9846 0.9773 0.9700 0.9627 0.9555 0.9482 0.9409 0.9336 Location of starting of phase change at tube # 1.450 1.60 1.80 2.30 2.730 3.734 6.807 Location of starting of sub-cooling at tube # 12.620 14.791 a NSC NSC NSC NSC NSC NSC—no sub-cooling takes place temperature As the ambient temperature is increased, it is observed that the beginning of the ◦ phase change and sub-cooling is delayed At 20 C, there is no sub-cooling taking place and the refrigerant remains in two phase condition This state of a air is repeated for other higher ambient temperatures This suggests that additional tube length is required in order to complete condensation of refrigerant vapour for such cases Further results for other mass ow rates and di erent ambient temperatures for coil #1 (corresponding to runs 1– 42) are shown in Fig In this ÿgure PC10, PC15, etc represent curves ◦ ◦ showing the location of phase change taking place in tubes at 10 C; 15 C, etc., respectively The other notations SC10, SC15, etc refer to the starting of the sub-cooling similar to the phase change notations Fig does not show the variation of locations for sub-cooling corresponding ◦ ◦ ◦ to 35 C and 40 C because at 35 C, sub-cooling takes place only for 0:0002 kg=s of refrigerant, ◦ whereas there is no sub-cooling for 40 C for all mass ow rates of refrigerant It is observed as expected, that phase change or sub-cooling are delayed with either the increase in mass ow G.A Quadir et al / Finite Elements in Analysis and Design 38 (2002) 417–434 431 Fig Location of starting of phase change and sub-cooling with variation in mass ow rates and ambient temperatures for coil #1 rates or increase in ambient temperatures The variations in this ÿgure, however, follow almost straight line behaviour Next, the calculations are performed with mass ow rate of refrigerant = 0:0008 kg=s for coils #1∗ and #1∗∗ which have the same dimensions except for the tube diameters The results are shown in Table With a small increase (4.80 –4:92 mm) in the outer tube diameter (coil #1∗ ), there is very slight change (less than 1%) in the prediction of location for phase change or sub-cooling However, with larger increase (4:80–6:21 mm) in the outer tube diameter (coil #1∗∗ ), there is a change (10%) in the results, indicating much earlier occurrence of phase change and sub-cooling Finally, calculations are carried out with mass ow rate of refrigerant, mr = 0:0008 kg=s for coils #4 and #5 which are di erent as compared to coil #1 and the results are tabulated in Table It may be pointed out that coil #4 has more number of tubes compared to coil #1 or coil #1∗ ; its tube diameter is the same as that of coil #1∗ and the length of the tube is shorter For coil #5, the number of tubes is less (12 only); its tube diameter is the same as coil #1∗∗ and the length of the tube is shorter It is observed that for both the coils, phase change is predicted ◦ earlier as compared to coil #1 and 1∗ at all temperatures considered (10 –40 C) However, in coil #5, the phase change is delayed as compared to coil #4 and this delay increases as the ambient temperature becomes higher This may be due to the smaller diameter wires and less number of tubes Both these factors result in smaller outside surface area, thus delaying phase ◦ change For coil #4, the sub-cooling takes place in tube number 14 at 20 C while the same could not take place in coil #1 for the same temperature This may be due to the more number ◦ of tubes in coil #4 The sub-cooling does not take place for coil #5 at 15 C whereas it had taken place at that temperature for coil #1 This may be attributed to less number of tubes in 432 G.A Quadir et al / Finite Elements in Analysis and Design 38 (2002) 417–434 Table Variation of location of phase change and sub-cooling with coil #1∗ and coil #1∗∗ Refrigerant initial conditions: ◦ P = 10 Bar, Tsup = 70:69 C Mass ow rate of refrigerant, mr = 0:0008 kg=sa Coil #1∗ No Ambient temperature, T∞ (K) 283.00 288.00 293.00 298.00 303.00 308.00 313.00 a Coil #1∗∗ Location of starting of phase change in tube # Location of starting of sub-cooling in tube # Location of starting of phase change in tube # Location of starting of sub-cooling in tube # 1.44 (1.45) 1.59 (1.60) 1.79 (1.80) 2.28 (2.30) 2.71 (2.73) 3.70 (3.73) 6.75 (6.81) 12.53 (12.62) 14.68 (14.79) NSC (NSC) NSC (NSC) NSC (NSC) NSC (NSC) NSC (NSC) 1.33 (1.45) 1.47 (1.60) 1.66 (1.80) 2.03 (2.30) 2.51 (2.73) 3.43 (3.73) 6.26 (6.81) 11.40 (12.62) 13.40 (14.79) NSC (NSC) NSC (NSC) NSC (NSC) NSC (NSC) NSC (NSC) NSC—no sub-cooling takes place; Values given in parentheses are results for coil #1 Table Variation of location of phase change and sub-cooling with coil #4 and coil #5 Refrigerant initial conditions: ◦ P = 10 Bar, Tsup = 70:69 C Mass ow rate of refrigerant, mr = 0:0008 kg=sa Coil #4 No Ambient temperature, T∞ (K) 283 288 293 298 303 308 313 a Coil #5 Location of starting of phase change in tube # Location of starting of sub-cooling in tube # Location of starting of phase change in tube # Location of starting of sub-cooling in tube # 1.18 (1.45) 1.29 (1.60) 1.43 (1.80) 1.64 (2.30) 2.19 (2.73) 2.77 (3.73) 5.25 (6.81) 9.50 (12.62) 11.24 (14.79) 13.53 (NSC) 17.31 (NSC) NSC (NSC) NSC (NSC) NSC (NSC) 1.29 (1.45) 1.42 (1.60) 1.60 (1.80) 1.84 (2.30) 2.48 (2.73) 3.31 (3.73) 5.83 (6.81) 10.79 (12.62) NSC (14.79) NSC (NSC) NSC (NSC) NSC (NSC) NSC (NSC) NSC (NSC) NSC—no sub-cooling takes place; Values given in parentheses are the results for coil #1 G.A Quadir et al / Finite Elements in Analysis and Design 38 (2002) 417–434 433 Fig Location of starting of phase change and sub-cooling with variation in ambient temperatures for di erent coils coil #5 At all other higher temperatures, sub-cooling does not take place for coil #5 which was observed in coil #1 as well It may also be noted from the above results that at lower temperatures, the occurrence of phase change in the tubes does not vary considerably for all the coils considered However, this observation is not true for sub-cooling The above results on the location of phase change and sub-cooling with variation in ambient temperatures for di erent coils are plotted in Fig The digits 1, 1∗ , 1∗∗ , and used in the notations for this ÿgure refer to the coil number whereas PC and SC stand for phase change and sub-cooling as mentioned earlier Conclusions Galerkin’s ÿnite element method is formulated to analyse the performance of a wire-on-tube heat exchanger under free convection cooling condition The present method is validated successfully against the available data where latent heat transfer or the sensible heat transfer takes place The proposed method is used to study the e ect of parameters like mass ow rates of the refrigerant, ambient temperature, etc on the beginning and end of the phase change or sub-cooling taking place inside the tubes of the heat exchanger These predictions are carried 434 G.A Quadir et al / Finite Elements in Analysis and Design 38 (2002) 417–434 out successfully using a locally developed computer programme The present method can be used to check the adequacy of the number of tubes provided in an existing wire-on-tube heat exchanger when the operating conditions are di erent from design conditions This methodology can also be used as a design tool to design new wire-on-tube heat exchangers and also predict deratings under abnormal ambient conditions References [1] G Tanda, L Tagliaÿco, Free convection heat transfer from wire-and tube heat exchangers, Trans ASME J Heat Transfer 119 (1997) [2] J.L Hoke, A.M Clausing, T.D Swo ord, An experimental investigation of convective heat transfer from wire-on-tube heat exchangers, Trans ASME J Heat Transfer 119 (1997) 348–356 [3] D.M Admiraal, C.W Bullard, Heat transfer in refrigerator condenser and evaporators ACRC TR-48, University of Illinois at Urbana-Champaign, Il, 1993, pp 34 –35 [4] V.A Martynov, New and e ective heat exchangers with tubes ÿnned with wires and spirals, Chem Petroleum Eng J 25 (1989) 3– 4, 124 –128 [5] H Hiroshi, N Shigeru, M Yoichi, A Tohru, N Haruo, Condensation of refrigerants R-11 and R-113 in the annuli of horizontal double-tube condensers with an enhanced inner tube, Exper Thermal Fluid Sci (2) (1989) 173–182 [6] Y.A Cengel, Introduction to Thermodynamics and Heat Transfer, McGraw-Hill, 1997, pp 809 –811 [7] C.M Dakshina Moorthy, S.G Ravikumar, K.N Seetharamu, FEM applications in the phase change exchangers, Warme und sto ubertragung 26 (3) (1991) 37–140 [8] K.N Seetharamu, FEM Analysis of Heat Exchangers, in: J.N Reddy, C.S Krishnamoorthy, K.N Seetharamu (Eds.), Finite Element Analysis for Engineering Design, Lecture Notes in Engineering, Vol 37, Springer, Berlin, 1988 [9] K.N Seetharamu, Invited talk on modelling of heat exchangers using ÿnite element method, in: Proceedings of the International Conference On Computational Mechanics, Tokyo, 1986, pp 89 –95 [10] S.G Ravikumar, K.N Seetharamu, P.A Aswatha Narayana, Performance evaluation of cross ow compact heat exchangers using ÿnite elements, Int J Heat and Mass Transfer 32 (1989) 889–894 [11] S.G Ravikumar, K.N Seetharamu, P.A Aswatha Narayana, Finite element analysis of shell and tube heat exchangers, Int Commun H M T 15 (1988) 151–163 [12] S.G Ravikumar, K.N Seetharamu, P.A Aswatha Narayana, Application of ÿnite elements in heat exchangers, Appl Numer Methods (1986) 229–234 [13] S.G Ravikumar, K.N Seetharamu, P.A Aswatha Narayana, Analysis of compact heat exchangers using ÿnite element method, in: Proceedings of the 8th International Heat Transfer Conference, Vol 2, San Francisco, 1986, pp 379 –384 [14] S.G Ravikumar, K.N Seetharamu, P.A Aswatha Narayana, Analysis of network of heat exchangers using ÿnite element, in: Proceedings of the International Conference On Computational Mechanics, Bombay, India, 1985, pp 861–871 [15] E.M Smith, Thermal Design of Heat Exchangers: A Numerical Approach—Direct Sizing & Stepwise Rating, Wiley, New York, 1997, pp 298–301 [16] E.S Gaddis, E.U Schlunder, Temperature distribution and heat exchange in multipass shells and tube exchangers with ba es, Heat Transfer Eng (1979) 43–52 [17] D.Q Kern, Process Heat Transfer, McGraw-Hill Book Co., New York, 1961 [18] D.Q Kern, A.D Kraus, Extended Surface Heat Transfer, McGraw-Hill Book Co., New York, 1972 [19] C Borgnakke, R.E Sonntag, Thermodynamics and Transport Properties, Wiley, New York, 1997 [...]... portion of the tube is di erent from that in the bare portion of the tube Therefore each tube is divided into three distinct sections The ÿrst one is related to the bare portion of the tube whereas the second one refers to the portion of the tube where wires are welded The third section is similar to the ÿrst section It is also to be noted that due to the gap between the wires in the second section,... when the operating conditions are di erent from design conditions This methodology can also be used as a design tool to design new wire- on- tube heat exchangers and also predict deratings under abnormal ambient conditions References [1] G Tanda, L Tagliaÿco, Free convection heat transfer from wire- and tube heat exchangers, Trans ASME J Heat Transfer 119 (1997) [2] J.L Hoke, A.M Clausing, T.D Swo ord,... P.A Aswatha Narayana, Analysis of compact heat exchangers using ÿnite element method, in: Proceedings of the 8th International Heat Transfer Conference, Vol 2, San Francisco, 1986, pp 379 –384 [14] S.G Ravikumar, K.N Seetharamu, P.A Aswatha Narayana, Analysis of network of heat exchangers using ÿnite element, in: Proceedings of the International Conference On Computational Mechanics, Bombay, India,... mentioned earlier 6 Conclusions Galerkin’s ÿnite element method is formulated to analyse the performance of a wire- on- tube heat exchanger under free convection cooling condition The present method is validated successfully against the available data where latent heat transfer or the sensible heat transfer takes place The proposed method is used to study the e ect of parameters like mass ow rates of the... dimensions of the coil with respect to coil #1 Table 4 Exact and average area methoda Mass ow rate of refrigerant, mr = 0:0008 kg=s Ambient temperature, T∞ = 283 K Method Location of starting of phase change in tube # 1.40 1.45 Exact area Average area a Location of starting of sub-cooling in tube # 12.63 12.61 1.4 indicates 40% location in tube #2 in the direction of ow of the refrigerant Table 5 Variation... Performance evaluation of cross ow compact heat exchangers using ÿnite elements, Int J Heat and Mass Transfer 32 (1989) 889–894 [11] S.G Ravikumar, K.N Seetharamu, P.A Aswatha Narayana, Finite element analysis of shell and tube heat exchangers, Int Commun H M T 15 (1988) 151–163 [12] S.G Ravikumar, K.N Seetharamu, P.A Aswatha Narayana, Application of ÿnite elements in heat exchangers, Appl Numer Methods 2 (1986)... the average area method to be considered for the wired portion of the tube, it is thought of tabulating the calculated results for a typical set of runs numbering 22–28 of Table 2 The results in Table 6 show the values of enthalpies, dryness fractions and temperatures depending upon whether the ow condition is in the desuperheating region, phase change region G.A Quadir et al / Finite Elements in Analysis... number of elements (size) 70-300-70, 70-600-70 and 140-600-140 are used to calculate the enthalpies, dryness fractions and temperatures for the coils as mentioned above for the same conditions of mass ow rate and ambient temperature for the purpose of comparison The notation 70-300-70 means that 70 elements are taken for the bare portions of the tube, whereas 300 refers to the wired portion of the tube. .. 8 Variation of location of phase change and sub-cooling with coil #4 and coil #5 Refrigerant initial conditions: ◦ P = 10 Bar, Tsup = 70:69 C Mass ow rate of refrigerant, mr = 0:0008 kg=sa Coil #4 No Ambient temperature, T∞ (K) 1 283 2 288 3 293 4 298 5 303 6 308 7 313 a Coil #5 Location of starting of phase change in tube # Location of starting of sub-cooling in tube # Location of starting of phase... area of elements are to be taken properly For this section, two methods were attempted during the calculations The ÿrst took into account the exact area for each element and the second method considered an average area for an element in that section Calculations were performed with 70 elements in the ÿrst, 348 elements in the second and 70 elements in ◦ the third section for coil 1 with mass ow rate of ... saturated condition superheated condition tube inside tube outside tube wire inlet and outlet conditions ambient temperature Fig Sketch of the wire- on- tube heat exchanger G.A Quadir et al / Finite Elements... performance of a wire- on- tube heat exchanger either under free convection or forced convection conditions The present paper is an attempt in that direction Free convection condition on the outside of. .. #5 Location of starting of phase change in tube # Location of starting of sub-cooling in tube # Location of starting of phase change in tube # Location of starting of sub-cooling in tube # 1.18