Two-Layer Model for the. Heat Transfer to Supercritical CO 2. E. Laurien, S. Pandey, and D. M. McEligot*

Transcription

1 5 th International Supercritical CO Poer Cycles Symposium March 8-31, 016 San Antonio, USA To-Layer Model for the Heat Transfer to Supercritical CO E. Laurien, S. Pandey, and D. M. McEligot* University of Stuttgart / Institute of Nuclear Technology and Energy Systems Pfaffenaldring 31, D Stuttgart, Germany , Laurien@ike.uni-stuttgart.de * Nuclear Engineering Division, Univ. Idaho, 995 University Blvd., Idaho Falls, Idaho USA Topic: sco Fluid Mechani & Heat Transfer Astract: Heat transfer to CO at super-critical pressure (sco ) ithin smalldiameter (-5 mm) pipes or channels of a compact heat exchanger must e predicted accurately in order to ensure its reliaility under various operation conditions. Complex nonlinear ehaviour of the heat transfer is oserved due to the large variations of the sco properties ith the local fluid temperature. Deterioration of the heat transfer may occur under conditions hen a layer of lo-conductivity laminar fluid near the channel alls increases in thickness due to flo acceleration and/or uoyancy. The classical to-layer model, taking account of the turulent core layer and a laminar su-layer, is extended to capture these effects. Ne parameters of the extended model, in particular a modified thickness of the laminar su-layer, are calirated using results of Direct Numerical Simulations of heated sco -pipe flos. Our ne model is applied to small channels under heating or cooling conditions at 8 and 0 MPa in the temperature range eteen 3 C and 165 C, as relevant for a sco poer cycle. It is shon that predictions for the heat transfer and the all shear stress in vertical channels of the lo-temperature recuperator and the reject-heat exchanger deviate significantly from standard correlations. 1. Introduction The modified recuperative Brayton cycle ith sco estalishes an attractive thermal energy-generation process ith high net efficiency of 47 % at a maximum temperature of only 650 C [1]. The cycle has a high-pressure section at 00 ars (0 MPa) and a lo-pressure section at 77 ars (7.7 MPa). It requires the design of compact heat exchangers ith cooled channels on the lo-pressure side having ulk temperatures eteen 165 C and 3 C and ith heated channels on the high-pressure side having temperatures eteen 61.1 C and C. These compact heat exchangers consist of a metal lock ith small channels or pipes (hydraulic diameter aout 1

2 mm) in hich a turulent flo of sco exists. Some of them operate near the critical point of CO (3.9 C and 7.48 MPa). The flo in theses channels can e vertically upard, donard or horizontal. Heat transfer and friction should e predicted accurately for the design method. Typically, experiments ith sco in small vertical pipes have een performed ith a constant all-heat flux induced y electrical heating of the pipe alls [, 3]. As a consequence of the strong property variation of sco ith temperature such flos may, under the influence of uoyancy and acceleration, exhiit complex and sometimes unexpected local ehaviour such as heat transfer enhancement or deterioration [4]. Similar phenomena may also exist in turulent cooled pipe flo, ut only fe experiments under cooling conditions are availale. To predict the local all temperature empirical correlations of experimental data are availale [5]. Hoever, uncertainties of these methods exist. Measurements of the local all shear stress as functions of the longitudinal coordinate are not feasile. Criteria for deterioration are discussed in [6]. An important parameter to characterize fluid turulence locally at an axial position in the pipe is the local ulk-reynolds numer Re umd/, here and are the local fluid density and the viscosity in the turulent core, and u m is the mean velocity. The Reynolds numer is in the order of 6000 to A ay to investigate turulent flos in this Reynolds-numer range is y Direct Numerical Simulation (DNS), see [7, 8]. DNS is ased on an accurate numerical integration of the unsteady conservation equations of mass, momentum and energy on a very fine numerical grid. Since all scales of turulence are numerically resolved, a turulence model is not necessary. Hoever, DNS requires extensive computer resources and can today only e performed to study a small numer of characteristic cases. A practical ay to model flo and heat transfer of circular pipe flos is the classical to-layer model, see e.g. [9]. Here, the assumption is made that the shear stress and the heat flux are constant in the direction perpendicular to the pipe alls. The flo and heat transfer in the so called turulent core layer and the laminar sulayer can e modelled analytically for constant fluid properties. In comination ith a mixing-length turulence model, this method has provided valuale insight into the heat transfer of constant-property flos [10]. Results of this theory are used today for a ide range of different fluids [11]. Accurate results are otained only hen a distinction is made eteen the thicknesses of the laminar viscous su-layer for the velocity and the laminar conducting su-layer for the temperature [10, 11]. The to-layer model is also attractive for variale-property flos. The idea to extend it to super-critical ater has first een presented in [1] for ater flos ith rough alls. The capaility of the method to capture heat transfer deterioration has een demonstrated for supercritical ater [1]. In the present paper the to-layer model is applied to sco flos ith some modifications, hich are introduced to improve its accuracy. The coupling of the momentum and energy equations in the to-layer model requires information oth aout the all shear stress and the all temperature, simultaneously. A detailed determination of important parameters such as the thickness of the viscous and the conducting su-layers had so far failed, ut ne

3 DNS data have no ecome availale [8], hich can serve as ases for model improvement. In the present ork e use the ne DNS data [8] for the determination of parameters in a modified version of the to-layer model. Our ne method is applied to various cases of flo conditions relevant for the advanced design of the supercritical caron-dioxide poer cycle [1] in order to investigate the characteristi of these flos. A comparison of the to-layer model to sco experiments [3] is also presented.. To-Layer Model for Circular Pipe Flos of sco.1 Model Equations At each axial position, corresponding to its average or ulk enthalpy h, the to- layer model integrates the momentum and energy equations locally in direction y perpendicular to the all under simplifying assumptions. The flo conditions for each calculation are defined y five additional parameters: the pressure p, the mass flux per unit area G, the all heat flux per unit area q, the pipe diameter D R, and the inlet temperature T. For small relative roughness, the all of narro channels in can e regarded as hydraulically smooth. The flo is sudivided into a turulent core layer ( logarithmic-layer ) and a laminar su-layer adjacent to the all. In [10, 11] a laminar su-layer for the velocity, denoted as the viscous su-layer, and a laminar su-layer for the temperature, denoted as the conducting su-layer, each ith its on thickness, are defined. Non-dimensional quantities are used and referred to as all units. For their definitions in the context of the core layer e use the ulk properties [1], i.e. the fluid properties at ulk temperature. For their definition in the context of the viscous/conducting su-layers e use the all properties [1], i.e. the fluid properties at all temperature. The all temperature and the all-shear stress necessary to define these units, are a priori unknon. Therefore, iteration must e employed. Our Excel procedure, here explained for a heated case, egins ith the initialization of the iteration procedure for each ulk enthalpy h. First, the ulk temperature T, the ulk dynamic viscosity, the ulk density, the ulk heat conductivity k, the ulk heat capacity for constant pressure c, the ulk thermal expansion coefficient, the mean velocity u m, the local ulk-reynolds numer Re umd/, and the ulk Prandtl numer Pr c p / k are determined using the program REFPROP [13]. Estimates for all shear stress u / 8 from the empirical Blasius equation [10] for the pipe friction coefficient 0.31/ 4 Re and for all temperature T T qd /( Nu k ) from the empirical Dittus-Boelter correlation [10] for the Nusselt numer Nu.03Re Pr are provided. p m 3

4 The next step is the determination of the changes of the velocity and the temperature across the turulent core layer. Since the ulk properties are used, therefore oth quantities are independent of the all temperature. The velocity at the edge of the viscous su-layer and the temperature at the edge of the conducting su-layer are computed. Let us define the all units for the turulent core layer as u u u ; y y u u u (1) ; m 8 According to the constant-property logarithmic la of the all, the velocity difference across the turulent core layer is u tur 1 ln R ln y ith the von-karman constant and the thickness of the viscous su-layer y 11.8 [10]. For a poer-la velocity profile u 8.6 ( y ), hich is a good approximation of the logarithmic la of the all at high Reynolds numers [10], the relation eteen the mean and the centre-line velocity u m 8.6 R 0 u cl is 1 7 R r 1 7 r d ( y ) u R The mean velocity can e calculated y assuming contact mass flux. The velocity at the edge of the viscous su-layer can then e determined from the velocity u cl at the centre-line of the pipe y u u u (4) cl tur Next, the temperature T at the edge of the conducting su-layer is determined. The non-dimensional temperature in all units is defined as T ( T T ) u q c p According to the to-layer model, the temperature difference across the turulent core layer is T tur PrT ln R ln y cl () (3) (5), (6) here y is the non-dimensional thickness of the conducting su-layer and PrT 0.85 is the turulence Prandtl numer. With the assumption su-layer T T, e get the temperature at the edge of the conducting cl 4

5 T T T (7) tur is modelled consistent ith the con- The non-dimensional su-layer thickness stant-property theory [7, 8] as 1 3 y y y (8) Pr The all-prandtl numer Pr is determined from the estimation of the all temperature aove. Eqs. (1) (8) estalish the initialization part of our method. In the folloing the iteration part of our method is descried. For improved accuracy the viscosity at the all is replaced y its average across the viscous su-layer. The all shear stress follos from the laminar friction la ithin the viscous su-layer du u y y (9) The non-dimensional thicknesses of the viscous su-layer y y and of the conduction su-layer are ased on all-properties, i.e., the index in eq. (1) is replaced y the index. The temperature difference across the conducting su-layer is most accurately estimated y using the thermal-resistance analogy [14]. A ne state variale is defined T (10) ( T ) k( T) dt T ref The relation eteen and T is provided y a lookup tale ith an aritrary reference temperature T ref. In this ork e use T ref = -0 C. The all temperature follos indirectly from q y The quantities T ) ( T ) (11) ( y and y are modified elo in sect.3 in order to take account of important physical effects due to property variation such as acceleration and uoyancy. Eqs. (8) (10) must e repeated until converged, i.e., the all temperature and the all shear stress remain constant during final iterations. We use a constant numer of nine iterations, hich appeared to e sufficient at a first glance. To avoid a possile instaility of the iteration, under-relaxation ith a factor of 0.5 is used, ut no detailed investigation or optimization of this iteration procedure as performed. In some computations the all shear stress ecame negative, in particular under cooling conditions near the critical pressure, e.g. at 7.7 MPa. These results are not presented in this paper, ecause the limitations of the theory are exceeded. 5

6 . Comparison to Experimental Data The aove discussed theory is used in [15] to model the experiments of Kim et al. [3] at 7.75 MPa, see Figure 1. The model is ale to predict the measured sharp rise of the all temperature only qualitatively. This deterioration occurs, hen the all temperature reaches the pseudo-critical value of = 33. C, here the changes of the sco -properties ith temperature and the heat capacity are at their maximum. T pc Figure 1: Comparison of the all temperature (full line) otained ith the to-layer model to experiments [3] (symols) [3], D = 4.4 mm, G = 400 kg/m and q = 50 kw/m. The dashed lines indicates the ulk temperature and the pseudocritical temperature Hoever, the maximum temperature of 10 C predicted y the model is higher than the measured maximum temperature of 95 C and there is a shift of 0 kj/kg in the predicted onset of deterioration toards a larger enthalpy. The reason for this may e the inaility of the model, descried so far, to capture important physical effects such as acceleration and uoyancy. For example, acceleration may lead to relaminarization of the flo and associated to an increase of the thickness of the conducting su-layer. Furthermore, it ecame ovious [15], that for loer all-heat flux the onset of deterioration cannot e predicted accurately. Therefore, the accuracy and reliaility of the model must e improved. Since the present case is not relevant for the sco poer cycle due to its large pipe diameter, it is not presented further. T pc.3 Model Improvement In order to improve our model generally, the Direct Numerical Simulation (DNS) results of [8] serve as a data ase. Compared to experiments DNS results have the advantage that the local all shear stress is availale in addition to the all temperature. 6

7 Figure : Comparison of our predictions using the to-layer model (full lines) ith results of the DNS [8] (dashed lines) at p 8 MPa, D mm, G 166 kg/ m s, q 10.8 kw / m and T in 8 C. The flo is ith uoyancy upard, donad or ith no uoyancy. The non-dimensional thicknesses of the laminar su-layer su-layer y y and the conducting are modified from their original values of sect.1. Each modification aims to model a particular physical effect, hich is oserved in the DNS: (i) relaminarization (due to flo acceleration), (ii) the internal (structural) and (iii) the external (on the mean velocity profile) effect of uoyancy. Figure shos the all shear stress and the all temperature for to cases, hich ere simulated ith DNS [8]. An additional case considers acceleration, ut no uoyancy, i.e., the gravity acceleration g is artificially set to zero in the simulation. This is done for reference in order to separate the acceleration effects from the uoyancy effects. The folloing modifications of the model of sect.1 ere applied: To model (i) re-laminarization due to flo acceleration the thickness of the viscous su-layer is increased proportional to the non-dimensional acceleration parameter K v [6] to fit DNS data empirically, ased on ulk properties: y 11.8 c K ; K v v 4q u m v ; (1) GD h h cp 7 ith c v for oth upard and donard flo. This leads to a decrease of the all shear stress compared to a non-accelerated case (not shon). The result is presented in Figure marked ith no uoyancy. The conducting su-layer is not modified. The all temperature remains the same as in the original model. Next (ii), the internal (structural) effect of uoyancy is modelled y a modification of the thickness of the conducting su-layer y 11.8 Pr 1 3 c uoyin, Ri Gr Ri (13) ; Re 7

8 ith c 140 for donard c 30 for upard flo. This leads to an increase uoyin, uoyin, (upard flo) or a decrease (donard flo) of the all temperature. The sign of the parameter c, is explained as follos: for donard flo the flo is thermally uoyin unstale resulting in a production of thermal turulence and therefore in a decrease of the thickness of the conducting su-layer. For upard flo the flo is thermally stale and turulence is damped, hich is modelled y an increase of the conducting su-layer thickness. To model these effects the ulk Richardson numer is taken as the relevant model parameter. Due to the change in temperature the shear stress is also affected, ut the changes ere small (not shon). In the upard case the DNS data sho a ave (or oscillation) ith a ave-length of aout 10 D. The origin of this ave is not fully understood yet and therefore it is not modelled here. In the donard case a local maximum of the all temperature near z/d = is also not modelled, ecause it is considered an inlet effect, hich is eyond the capailities of the to-layer model for developed flos. The external effect of uoyancy on the mean flo profile (iii) is modelled y modification of the all-shear stress. We consider a modified all shear stress as the consequence of a shear-induced contriution as calculated in eqs. (9) and an additional uoyancy-induced contriution ith 7 z 5D c 10 Gr e, Gr, mod uoyex, 1 c, 0. Pa for upard cuoy, 0. 05Pa uoyex ex g D 3 T T, (14) for donard flo. The uoyancy-induced all shear stress may e interpreted as a result of a circulating natural convection flo superposed to the longitudinal forced convection, leading to M-shaped velocity profiles in upard flo. The convection flo is modelled as a function of the all-grashof numer. In upard-flo the DNS data are ell approximated. In the donard case, hoever, some details are not captured accurately. Here, further improvement of our model is necessary. The term ith the exponential function in eq. (14) models a short distance, hich is assumed to e necessary for the convection flo to ecome estalished. A comparison of the upard and donard flo cases in the region far donstream in Figure suggests, that the upard flo leads to a further increase of the all temperature, denoted as deterioration hereas the donard-flo case does not. This is an often-oserved phenomenon in flos at super-critical pressure. The numerical values of c uoy, in and c uoy, out fit the DNS in the est possile ay. We are aare, that using only to DNS cases may not e sufficient to estalish an accurate model for a ide range of flo conditions. Hoever, the computation of additional cases to provide a larger DNS-data ase is underay [16]. 3. Application to a sco-compact Heat Exchangers In this chapter our method is applied to cases relevant for the supercritical CO poer cycle [1]. We consider vertical upard or donard flo ithin channels ith 8

9 a hydraulic diameter of mm. The all is either upard or donard and heated or cooled, leading to the staility ehaviour summarized in Tale 1. Tale 1: Thermal staility tale of vertical pipe flo heated cooled upard flo stale unstale donard flo unstale stale The stale cases tend toard development of an M-shaped velocity profile. For these, the additional all shear stress is positive, otherise negative. The heated flo is accelerated and therefore re-laminarization occurs. In order to promote the understanding of the flo, e further assume constant all heat flux, although this is not the case in a heat exchanger. Our method is not applicale for horizontal flo, ecause strong density stratification occurs, as the DNS [16] indicates. 3.1 Lo-Temperature Recuperator at the High-Pressure Side The flo is heated at 00 ars [1], here sco ehaves practically like a gas flo. The flo may e upard or donard, depending on the design of the recuperator. The all temperature y proposed to layer model is shon in Figure 3 (for upard flo) and Figure 4 (for donard flo) Figure 3: a) Wall shear stress and ) all temperature for upard flo, to all heat fluxes, p = 0 MPa, D = mm, G = 166 kg/m s, T in = 61.1 C, a) all shear stress (full line) and comparison to the Blasius correlation (dashed line), ) all temperature (full line) and comparison ith the Dittus-Boelter correlation (dashed line) The all temperature is almost the same as predicted y the Dittus-Boelter correlation, ut the all shear stress shos significant deviation from the constantproperty prediction due to uoyancy. 9

10 Figure 4: same as figure 3 for donard flo 3. Lo-Temperature Recuperator at the Lo-Pressure Side The flo of the lo-pressure side is cooled, i.e., in Figure 5 the flo is from right to left. Compared to the previous cases e have loered the all heat flux, ecause for higher heat flux negative values of the all shear stress occur, for hich our theory does not apply. We also changed the pressure from 77 to 80 ars for the same reason. The all temperature predicted y our method deviates significantly from the prediction of the Dittus-Boelter correlation. For the all shear stress a large deviation occurs as ell. These changes are due to the comined effects of uoyancy and variale properties. Re-laminarization due to acceleration does not occur, ecause a cooled flo is not accelerated. The all temperature is the same for upard or donard flo. The reason for same temperature is that effects of uoyancy are not significant. We have calculated Gr /Re as criterion for uoyancy, if its value is more than 10 -, then uoyancy effects are significant [17]. But in oth the cases its magnitude is in the range of Reject-Heat Exchanger at the Lo-Pressure Side The flo on the sco -side is cooled. We also changed the pressure from 77 to 80 ars. Figure 6 shos a sharp rise of the all shear stress eteen 400 and 450 kj/kg for donard flo. In this case, uoyance parameter Gr /Re is ranging from 10 - to 10-4, hich explains difference in temperature at the outlet eteen 375 and 306 kj/kg. 10

11 Figure 5: a) Wall shear stress and ) all temperature ith a cooled all at p = 8 MPa, D = mm, G = 166 kg/m s, q = 30.8 kw/m, T in =165.8 C. Comparison th the Blasius and Dittus-Boelter Correlations ased on ulk properties. Figure 6: a) Wall shear stress and ) all temperature ith a cooled all at p = 8 MPa, D = mm, G = 166 kg/m s, q = 15.4 kw/m, T in =69.8 C. Comparison th the Blasius and Dittus-Boelter Correlations ased on ulk properties. 4. Conclusion The extended to-layer model has the potential to predict the flo in compact heat exchangers for heated or cooled alls, oth for upard and donard flo. Due to using only a small numer of cases for the caliration of the model, additional validation ork remains necessary to improve its reliaility and accuracy. In order to promote the understanding of the flo in the lo-temperature recuperator and the reject-heat exchanger of the sco poer cycle [1], e have made sample calculations ith constant all heat flux. At a pressure of 80 ars the predictions of the all temperature and the all shear stress deviate significantly from constantproperty correlations. 11

12 The decision hether the flo in a heat exchanger should designed to e upard or donard can e supported y our data. Horizontal flo is not ithin the capaility of our theory, hoever see [15]. For the high-pressure side, donard flo is favourale ecause of the loer all shear stress compared to upard flo. For the lo pressure side donard flo may lead to very small (almost zero) all shear stress at 80 ars. Systematic computations of donard flo at 77 ars could not e performed due to negative all shear stress in the case of donard flo ith cooling. This does not occur in the case of upard flo. 5. References [1] V. Dostal, M.J. Driscoll, and P. Hejzlar: A supercritical caron dioxide cycle for next generation nuclear reactors, MIT-ANP-TR-100, (004) [] J.D. Jackson, and W.B. Hall: Forced convection heat transfer to fluids at supercritical pressure. Turulent forced convection in channels and undles, Hemisphere, , (1979) [3] H. Kim, Y.Y. Bae, H.E. Kim, J.H. Soong, and B.H. Cho: Experimental Investigation on the Heat transfer characteristi on a vertical upard flo of supercritical CO, Proc. ICAPP 06, Reno, NV, June 4-8, 006 [4] J. Y. Yoo: The turulent flos of supercritical fluids ith heat transfer, Ann. Rev. Fluid Mech., , (013) [5] I.L. Pioro, H.F. Khartail and R.B. Duffey: Heat transfer to supercritical fluids floing in channels - empirical correlations (Survey), Nucl. Eng. and Design 30, 69-91, (006) [6] D. M. McEligot, and J. D. Jackson: Deterioration criteria for convective heat transfer in gas flo through non-circular ducts, Nucl. Eng. and Design 3, , (004) [7] J. H. Bae, J.Y. Yoo, and H. Choi, Direct numerical simulation of turulent supercritical flos ith heat transfer, Physi of Fluids 17, , (005) [8] Xu Chu, and E. Laurien: Direct numerical simulation of heated turulent pipe flo at supercritical pressure, Journal of Nuclear Engineering and Radiation Science, to appear 016 [9] L. Prandtl, Eine Beziehung zischen Wärmeaustausch und Strömungsiderstand der Flüssigkeit, Physik. Z. 11, , (1910) [10] W. Kays, M. Craford, and B. Weigand: Convective heat and mass transfer, McGra-Hill, Ne York, International Edition 005 [11] B.A. Kader: Temperature and concentration profiles in fully turulent oundary layers, Int. J. Heat and Mass Transfer 4, (1981) [1] E. Laurien: Implicit model equation for hydraulic resistance and heat transfer including all roughness, Journal of Nuclear Engineering and Radiation Science, to appear 016 [13] E. W. Lemmon, M. L. Huer, and M. O. McLinden: Thermophysical Properties Division, National Institute of Standards and Technology, Boulder, CO (010) [14] D.M. McEligot, and E. Laurien, Insight from simple heat transfer models, The 7th International Symposium on Supercritical Water-Cooled Reactors (ISSCWR-7), Helsinki, Finland, March, 015 1

13 [15] S. Pandey, and E. Laurien, Heat transfer analysis at supercritical pressure using to layer theory, The Journal of Supercritical Fluids, Volume 109, Pages 80-86, ISSN , March 016. [16] X. Chu, and E. Laurien: Inflo Effect to Heat Transfer of Pipe Flo ith CO at supercritical Pressure, this conference. [17] P.X. Jiang, Y Zhang, C.R. Zhao, R.F. Shi, Convection heat transfer of CO at supercritical pressures in a vertical mini tue at relatively lo reynolds numers, Experimental Thermal and Fluid Science 3, , (008) NOMENCLATURE Non-dimensional parameters Gr - Grashof numer, ρ D 3 gβ T -T /µ K v - acceleration parameter 4qum Kv ; GD h h c Nu - Nusselt numer, q D/ T -T /k Pr - Prandtl numer, c p / k / Re - Reynolds numer, D Ri - Richardson numer, Ri = Gr/Re Latin symols c p kj/kg K specific heat capacity at constant pressure D m pipe diameter, D = R g m/s gravity acceleration G kg/m s mass flux per unit area h kj/kg specific enthalpy k W/m K thermal conductivity p MPa pressure q W/m heat flux per unit area r m radial coordinate R m pipe radius T C, K temperature u m/s streamise velocity y m distance from the pipe all Greek symols β 1/K thermal expansion coefficient κ - von-karman constant, 0.41 λ - darcy pipe friction coefficient kg/m 3 density τ Pa shear stress u m p 13

14 µ Pa s dynamic viscosity Indices ulk + all unit ased on ulk properties cl centreline laminar, conducting su-layer in pipe inlet m mean, cross-sectional average mod modified to take account of uoyancy pc pseudo-critical, here c p has its maximum t turulence tur turulent all layer laminar, viscous su-layer all + all unit ased on all properties τ all friction SHORT BIOGRAPHY INTRODUCTION OF AUTHOR(S) Dr. Eckart Laurien is full time professor at Institute of Nuclear Technology and Energy System (IKE), University of Stuttgart, Germany. Mr. Sandeep Pandey is currently orking as a doctoral research scholar at IKE, University of Stuttgart, Germany. He holds Master of Technology in Energy Studies from Indian Institute of Technology Delhi, India. Dr. D.M. McEligot is Distinguished Visiting Professor, Nuclear Engineering, University of Idaho. 14