ANALYSIS OF ENTROPY GENERATION IN A CIRCULAR TUBE WITH SHORT LENGTH TWISTED TAPE INSERTS

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1 Proceedings of the th National and 11 th International ISHMT-ASME Heat and Mass Transfer Conference December 8-31, 013, IIT Kharagur, India HMTC13006 ANALYSIS OF ENTROPY GENERATION IN A CIRCULAR TUBE WITH SHORT LENGTH TWISTED TAPE INSERTS Vishal Anand Infotech Enterrises Hyderabad, Andhra Pradesh, India vish.anand.iit@gmail.com Krishna Nelanti Infotech Enterrises Hyderabad, Andhra Pradesh, India Krishna.Nelanti@infotechenterrises.com ABSTRACT The current aer resents the analysis of entroy eration in a circular tube with short length twisted tae inserts under uniform heat flux boundary condition. To investigate the effect of fluid roerties, two different fluids, water and Freon have been chosen for study. Effect of the length ratio(lr) and heat flux on the dimensionless entroy eration, Bejan number and uming ower to heat transfer ratio have been illustrated though grahs. The reasons behind these trends have been discussed in detail. Finally, a comarison has been made between the entroy erated in water and in Freon. The total entroy erated is higher for Freon, but the entroy erated only due to fluid friction is greater for water. It is hoed that the results resented in this aer will aid in the designing of better heat transfer enhancement techniques. NOMENCLATURE A: Cross-sectional area (m ) Be: Bejan number C : Secific heat (J/kgK) D: Inner diameter of tube (m) f: Friction factor FF: Dimensionless entroy erated due to fluid friction h: Convective heat transfer coefficient (W/m K) k: Thermal conductivity (W/mK) L: Length of tube (m) lf: Length of full length twisted tae (m) ls: Length of short-length twisted tae insert (m) LR: Length ratio =ls/lf m& : Mass flow rate (kg/s) Nu: Nusselt number P: Pressure (Pa) PPR: Puming ower to heat transfer ratio Pr: Prandtl number Q & : Heat transfer rate (W) '' : Heat transfer rate flux (W/m ) Re: Reynolds number s: Secific entroy (J/kg K) S & : Rate of entroy (W/K) T: Temerature (K) TT: Twisted tae U: Average velocity (m/s) y: Pitch ratio Greek Symbols ψ : Dimensionless entroy eration µ : Viscosity (Ns/m ) ρ : Density (kg/m 3 ) P : Pressure difference (Pa) Subscrits: avg: Average : Generated i: Inlet out: Outlet ref: Reference w: Water wall: Wall P : Pressure difference T : Temerature difference INTRODUCTION There are many techniques described in literature and used in industry to enhance the rate of heat transfer in tubes. For examle, the rate of heat transfer can be imroved by making internal fins in a tube, using a twisted tae (TT) insert, changing the shae of tube into a helix and various other methods. Among these methods, the use of TT s warrants considerable interest because it enhances heat transfer by two mechanisms: firstly TT s

2 act as turbulators, enhancing the heat transfer by turbulence; secondly they also thin the boundary layers near the wall of the tube which further enhances the heat transfer rate. But the enalty of this method is that the friction exerienced by the flow also increases, which results in more uming ower to be required for the same mass flow rate. In the ast few years, a lot of research has gone into redicting the heat transfer coefficients and ressure dro in a tube with TT and its various other modifications. A brief review of literature is resented here due to lack of sace. Manglik and Bergles [1-] gave the correlation for friction factor f and Nusselt number Nu in a circular tube with TT for both laminar and turbulent flow. Eiamsa-ard et al [3] comared the heat transfer enhancement in a circular tube with single TT, full length dual TT and regularly saced dual TT. Eiamsa-ard et al [4] investigated the heat transfer enhancement in a tube with loose fit TT inserts. Effect of clearance ratio on Nu and f were studied in detail. Thianong et al [5] studied emirically heat transfer enhancement in a dimled circular tube fitted with a twisted tae. It was found that the dimled tube with TT is better at transferring heat than the lain tube alone and the dimled tube without TT. Promvonge and Eiamsa-ard [6] investigated the heat transfer and ressure dro characteristics of a circular tube fitted with a conical ring and a twisted tae. Eiamsaard et al. [7] found the correlations for Nusselt number and friction factor for flow through a circular ie fitted with a twisted tae with delta-winglets. In eral, they found that the tubes with oblique delta winglet are more efficient at heat transfer enhancement than those with straight delta winglet. Eiamsa-ard et al [8] studied the convective heat transfer in a circular tube with shortlength twisted tae insert. It was reorted that the circular tube with short-length twisted tae insert had worse heat transfer enhancement characteristics than that with full length TT. It is to be noted that almost all the research in TT s till now has been aimed at measuring the heat transfer and ressure dro characteristics. Introduction of TT s leads to a decrease in entroy eration due to heat transfer and an increase in entroy eration due to friction. This suggests that the total entroy erated, due to both heat transfer and fluid friction, is a good indicator to judge the effectiveness of a heat transfer enhancement technique. This aer deals with entroy eration in a circular tube with short-length twisted tae of different length ratios (LR) under uniform heat flux boundary condition. To study the effect of fluid roerties on entroy eration, two different fluids, water and Freon have been used. The effect of length ratio and heat flux on the dimensionless entroy eration ( ψ ), Bejan number (Be) and uming ower to heat transfer ratio (PPR) have been investigated analytically. The reasons behind these trends have been exlained in detail. Geometry: The geometry of the circular tube with the twisted tae insert has been shown in Fig. 1. The length of short length twisted tae is denoted by ls, while the length of the full length tae is denoted by lf. The tubes with four different length ratios, (LR =ls/lf) are used in the analysis, namely LR= 0.9, 0.43, 0.57, 1.0. All the tubes have the same twist ratio (y =4.0) same diameter (D= 0.06m) and length (L=1.0m). Figure 1: GEOMETRY OF SHORT LENGTH TT s WITH THREE DIFFERENT LENGTH RATIOS. ANALYSIS The flow of a fluid inside a circular duct fitted with short-length twisted tae insert and subjected to uniform heat flux is considered. The fluid enters the tube at a uniform temerature T i. Under these conditions, neglecting changes in kinetic and otential energy and neglecting axial conduction, the energy balance for an infinitesimally small control volume of length dx, is given by: mc & dt = '' πddx (1) Solving this differential equation for T(x), the following is obtained: '' π Dx T ( x) = + Ti mc &. () In terms of Reynolds number, the T(x) is given by: 4 '' x T ( x) = + Ti µ ReC. (3) The temerature at the outlet of tube is given by: 4 '' L Tout = + Ti µ ReC. (4) The infinitesimal entroy erated inside this control volume is given by: q '' Ddx d π = mds & & Twall. (5) For an incomressible fluid, C dt dp mds & = m& m& T ρt. (6) Substituting Eq. (6) in Eq. (5) the following is obtained: dt mdp q" Ddx d & & π = mc & T ρ T T wall (7) Also,

3 '' = h( T T ) wall '' Twall = + T h And, dp= fρ D U dx Substituting the value of T wall and dp from Eq. (8) and Eq.(9) resectively into Eq.(7), the following is obtained: d mc & dt mfu & dx '' πddx = + T DT ( "/ h+ T ) (8) (9). (10) Substituting the value of T(x) from Eq. (3) into Eq. (10), and integrating from x=0 to x=l, the following equation for total entroy eration rate in the tube is obtained in final form: & 4 '' L + Ti µ Re C ''/ h+ Ti = mc & ln( ) + T 4 '' L i ''/ h+ Ti + µ ReC PPR= 4 '' L + Ti µ µ ln( ) 8 D '' T mc f U Re ReC i (11) FF = Dimensionless entroy erated due to fluid friction only limψ '' 0 = = Re µ Lf D ρ C T 3 i (14) So, ψ FF Be= ψ (15) Puming ower to heat transfer ratio: Another arameter which is a good indicator of the erformance of a heat exchanger tube is the ratio of uming ower to heat transfer (PPR), which can be exressed as: A PU PPR= Q& (16) P along the length of the duct Here the ressure dro can be obtained from the infinitesimal ressure dro dp, given by Eq. (9), Substituting we get: 3 fu ρ 8 ''. (17) In terms of Re, the following exression for PPR is obtained: PPR= µ f Re ρ D '' (18) In literature, dimensionless entroy eration has been used to quantify the entroy erated, which is given by: ψ = (1) mc & Bejan number: The exression for dimensionless entroy eration (Eq. (1)) does not convey, out of the two entroy eration mechanisms: heat transfer and fluid friction, which one dominates. To resolve this, Paoletti et al. [9] defined a new dimensionless number known as Bejan number. Bejan number is given as: BejanNumber = EntroyGeneratedDueToHeatTransfer TotalEntroyGenerated (13) At Be = 1, all the entroy erated is due to heat transfer. At Be = 0, all the entroy erated is due to fluid limψ '' 0 friction. In the current analysis, if we take, it gives the dimensionless entroy erated due to fluid friction only, i.e. Nusselt number and Friction Factor: The correlations for Nusselt number and friction factor for flow in a circular tube with short-length TT insert is taken from literature [8] and reroduced below: Nu f LR = Re Pr (19) =.8Re LR (0) Fluid roerties: To study the effect of fluid roerties, two different fluids: water and Freon are considered. Water: The thermo-hysical roerties of water are calculated using the following equations, given in [10]: 3 ( T )( T ) ρw( kg / m ) = 1000(1 ) ( T ) (1)

4 k ( W / mk) = ( T+ 73) w ( T+ 73) () 73 µ Ns m w = + ( T+ 73) + ( / ) (ex[.10 ( 4.45)( ) 73 ( T+ 73) 6.55( ) ])( ) C ( J / kgk ) ( w 1 = ( T 73) ( T + 73) ( T + 73) 6 3 (3) ( T 73) ) R / (4) In the above equations, T is the temerature of water in deg C and R is the universal gas constant. Freon: The density of Freon is taken as constant, since it varies roughly by only 1% in the temerature range considered. For determining the viscosity of Freon, the following emirical correlation given by Sherman [11] is used: T n 1 1 µ ( T ) = µ ( Tref )( ) ex[ B( )] Tref T Tref (5) In Eq.(5) above, T ref is the reference temerature whose value is taken as 73 K. B and n are fluid deendent arameters, whose numerical values (for Freon) are found out using regression. To estimate thermal conductivity and secific heat, fourth degree olynomial equations are used. The coefficients of these olynomial equations are also determined using regression. The data for regression is taken from Incroera and DeWitt [1]. The regression is carried out using the commercially available software - Microsoft Excel. The fluid roerties are evaluated at T avg = (T out +T i )/ using an iterative rocedure. A value of T out is guessed. Using this guessed value, the fluid roerties are found out at T avg. These fluid roerties are then used to find out the new T out using Eq. (4).This rocess is reeated till the difference in the successive values of T out is less than.1 K. The T out for the last iteration is taken as the final value. The T avg, which is calculated using the final value of T out, is used to estimate the fluid roerties. RESULTS AND DISCUSSION For a given set of arameters (shown in Table 1), the effect of LR, '' and Re on dimensionless entroy eration (ψ ), Bejan number (Be) and uming ower. to heat transfer ratio (PPR) was lotted in Fig. to Fig. 7 for both water and Freon. The results are described below. Effect on ψ : Fig. shows the effect of a.) LR and '' on dimensionless entroy eration ψ vs Re for water. As can be seen from the figure, ψ decreases when Re increases. This is because as Re increases, T(x) T i, i.e. lim T ( x) = Ti Re (6) This means that the temerature gradients inside the fluid decrease as Re increases, due to which the ψ decreases. Even though, as Re increases, entroy eration due to ressure also increases, but the contribution of ressure to total entroy erated is miniscule ( Be is close to 1, see next sub-section). So the net result is that the total entroy erated decreases with increase in Re. It is also seen from the same figure that as LR increases, the ψ decreases. This is because an increase in LR causes an increase in Nu, which means the wall-bulk temerature difference decreases, so the entroy erated due to temerature difference also decreases. Also from Fig. (b), it is seen that ψ increases with increase in ''. This is because as '' increases, the temerature gradients inside the fluid also increase, which leads to an increase in entroy eration. '' on Fig. 3 shows the effect of a.) LR and dimensionless entroy eration vs Re for Freon. Trends similar to the case of water can be seen here. Effect on Be: '' on Be vs Re Fig. 4 shows the effect of a.) LR and for water. As can be seen from the figure, Be is close to 1. It means that almost all the entroy erated is due to temerature difference/heat transfer only. This is corroborated by Jarungthammachote [10] for lain hexagonal ies. It is also seen from the figure that Be decreases with increase in LR. This is because as LR increases, the Nu increases so the wall-bulk temerature difference decreases, which leads to decrease in entroy erated due to temerature difference. In the same way, Be increases with increase in '', because an increase in '' causes the temerature gradients to increase. '' Fig. 5 shows the effect of a.) LR and on Be vs Re for Freon. Trends similar to the case of water can be seen here. Table 1: VALUES OF CONSTANT PARAMETERS USED IN THE ANALYSIS. Parameters T i D L Numerical Values 30 K 0.06 m 1 m

5 a.) a.) Figure : EFFECT OF a.) LR AND q ON DIMENSIONLESS ENTROPY GENERATION ( ψ ) FOR WATER Figure 3: EFFECT Of a.) LR AND q ON DIMENSIONLESS ENTROPY GENERATION ψ OF FREON.

6 a.) a.) Figure 4: EFFECT OF a.) LR AND q ON Be FOR WATER. Figure 5: EFFECT OF a.) LR AND q ON Be FOR FREON

7 a.) a.) Figure 6: EFFECT OF a.) LR AND q ON PPR FOR WATER. Figure 7: EFFECT OF a.) LR AND q ON PPR FOR FREON.

8 Effect on PPR: Fig. 6 shows the effect of a.)lr and '' on PPR vs Re for water. It can be seen from the figure that PPR increases with increase in Re, because to um a fluid at higher Re, more uming ower is required. Also, as can be seen from the figure, PPR increases with increase in LR. This is because as LR increases, the friction exerienced by the fluid also increases, which means more uming ower is required. Also it is seen that as '' increases, PPR decreases. This is because '' aears in the denominator of the exression for PPR. Similar trend is seen in the Fig.7 which lots the effect of a.) LR and '' on PPR vs Re for Freon. Comarison between two fluids: Table shows the comarison of total entroy erated, entroy erated due to heat transfer and entroy erated due to ressure difference between water and Freon for the same mass flow rate. The table shows that the total entroy erated and entroy erated due to temerature difference is higher for Freon while the entroy erated due to ressure dro is higher for water. This can exlained as follows. The C of Freon is lower than that of water, due to which the temerature at the exit is higher for Freon. This means that the temerature gradients inside Freon are larger. So the entroy erated due to temerature difference is more for Freon. The entroy erated due to ressure difference is greater for water because viscosity of water is greater than that of Freon. Fluid Table : COMPARISON OF ENTROPY GENERATION BETWEEN WATER AND FREON. 10 3, T 10 3, P ( W / K) ( W / K) ( W / K) Water Freon CONCLUSION: 10 3 An analysis of entroy eration in a circular tube with short length twisted taes has been carried out in this aer. Water and Freon have been taken as the working fluids. The following conclusions can be drawn from the study: 1. Both ψ and Be decrease with increase in Re, while PPR increases.. While ψ and Be decrease with increase in LR, PPR increases. 3. ψ and Be increase with increase in '', while PPR decreases. 4. Between Freon and water, Freon has higher rate of total entroy eration and higher rate of entroy eration due to temerature difference, while entroy erated due to ressure difference is higher for water. The reasons behind these trends have been discussed in detail. It is hoed that the results resented in this aer will go a long way in the design of heat exchanger ies which are efficient in transfer of heat but do not destroy the available work. REFERENCES [1] Manglik RK, Bergles AE, 1993, Heat transfer and ressure dro correlations for twisted tae inserts in isothermal tubes: art I-laminar flows. ASME J. Heat Transfer. 115(4), [] Manglik RK, Bergles AE, 1993, Heat transfer and ressure dro correlations for twisted tae inserts in isothermal tubes: art II- transition and turbulent flows. ASME J. Heat Transfer. 115(4), [3] Eiamsa-ard, S., Thianong C., Eiamsa-ard, P., Promvonge, P., 010, Thermal characteristics in a heat exchanger tube fitted with dual twisted tae elements in tandem. International Communications in Heat and Mass Transfer. 37(1), [4] Eiamsa-ard, S., Wongcharee K., Sriattanaiat S., 3-D Numerical simulation of swirling flow and convective heat transfer in a circular tube induced by means of loose-fit twisted tae. International Communications in Heat and Mass Transfer. 36(9), [5] Thianong, C., Eiamsa-ard, P., Wongcharee K., Eiamsa-ard, S., 009, Comound heat transfer enhancement of a dimled tube with a twisted tae swirl erator. International Communications in Heat and Mass Transfer. 36(7) [6] Promvonge, P., Eiamsa-ard, S., 007, Heat transfer behaviors in a tube with combined conical ring and twisted tae insert. International Communications in Heat and Mass Transfer. 34(7), [7] Eiamsa-ard, S., Wongcharee, K., Eiamsa-ard, P., Thianong, C., 010, Heat transfer enhancement in a tube using delta winglet twisted tae insert. Alied Thermal Engineering. 30(4), [8] Eiamsa-ard, S., Thianong, C., Eiamsa-ard, P., Promvonge, P., 009, Convective heat transfer in a circular tube with short length twisted tae insert, International Communications in Heat and Mass Transfer. 36(4), [9] Paoletti S., Risoli F., Sciubba E., 1989, Calculation of exergetic losses in comact heat exchanger assages. ASME-AES. 10, [10] Jarungthammachote S Entroy eration analysis for fully develoed laminar convection in hexagonal duct subject to constant heat flux. Energy. 35(1), [11] Sherman FS Viscous Flow. McGraw Hill Co. New York, [1] Incroera FP, DeWitt DP Fundamentals of heat and mass transfer. John Wiley and Sons (Asia), Singaore.