5.1 Heat removal by coolant flow

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1 5. Convective Heat Transfer 5.1 Heat removal by coolant flow Fel pellet Bond layer Cladding tbe Heat is transferred from the srfaces of the fel rods to the coolant. T Temperatre at center of fc fel pellet Temperatre at otside srface of fel pellet T f Coolant flow T b Temperatre at inside srface of fel coolant T c Temperatre at otside srface of cladding tbe Coolant temperatre T m

2 5. Properties of coolant Reqirement of nclear properties 1. Low netron absorption cross section. Low radiation-indced radioactivity 3. Low damage by irradiation 4. High moderating power for thermal netron reactor, and low moderating power for fast netron reactor

3 Reqirement of thermal-hydralic properties 1. High thermal condctivity and specific heat, and as a reslt high heat transfer coefficient. Low melting point in case of coolant which is solid at room temperatre 3. Low viscosity (low friction pressre drop and strong trblence) 4. No thermal decomposition 5. Chemically inactive with air and water

4 Characteristics of gas coolant -Low specific heat and thermal condctivity -Reqired heat transfer is obtained by pressrization Characteristics of water: light water (H O) and heavy water (D O) -High pressre operation needed becase of high satration vapor pressre -Good thermal properties -High moderating power

5 Characteristics of liqid metal: alkali metals, sodim and lithim -Low satration pressre (Low pressre operation is possible at high temeratre) -Low Prandtl nmber, high thermal condctivity, and low density -Low corrosion rate nder good control of imprity (Steel corrosion in Li is higher than in Na.) -Chemically active with water and oxygen in case of alkali metals (Li reacts with nitrogen mildly)

6 Characteristics of molten salts: Flibe (LiF-BeF ), HTS (NaNO 3 -KNO 3 -NaNO ), Flinak (LiF-NaF-KF), Li CO 3 -Na CO 3 -K CO 3, etc. -High specific heat (Good for heat transportation) -High viscosity (Laminarization) -High Prandtl nmber (Not good for heat transfer media) -Nickel-base alloy (Hastelloy) is compatible with molten salt

7 Table 5.1 Properties of varios coolants CO He H O D O Na Li Flibe Melting pint ( ) Boiling point ( ) at 0.53Pa Density (kg/m 3 ) Thermal condctivity (W/mK) Specific heat (kj/kgk) Viscosity (10-6 Pa s) Kinematic viscosity (10-6 m /s) Thermal diffsivity (10-6 m /s) Pr Note: Temperatre and pressre dependency

8 5.3 Approach to convective heat transfer (Single-phase flow) Solid Heat flx q Velocity v Temperatre T How to calclate the heat transfer coefficient, or the temperatre profile close to the solid srface, i. e. the fel rod srfaces in coolant? Coolant flow

9 Calclation of velocity and temperatre field Navier-Stokes eqations where Continity eqation Momentm eqation Energy eqation Dρ ρ( v) Dt Dv p v Dt DT ρc v k T Dt ρ μ ρg μ : Dynamic viscosity of flid k : Thermal condctivity of flid Dφ φ φ φ φ D vx vy vz Dt t x y z Dt v vy v x y φ φ φ v φ x y z x y z v t Bondary conditions v 0, given q at solid srface

10 ' ',, ' w w w v v v y y v M T ρε μ ρ τ ' ' Eddy diffsivity of momentm y Mixing length, κ ρ μ m m T l y l j i i j j i j i i v v x v x v x x P Dt Dv ' ' 1 ν ρ Reynolds eqation (Time average) Average velocity and flctation Prandtl s mixing length model Reynolds stress (additional terms) ε M Approach to trblent flow Time t Trblence model 0.41 κ in pipe flow (by Nikladse), R y R y R l m

11 Calclation of velocity profile in trblent flow The 1/7th law of velocity profile max r 1 R 1/ ( y ) / Δp Δx τ ρ 14 Re / ( y ) Derived τ τ, w ρ n, y τ y, ν 1 D r/r

12 von Karman s niversal velocity profile (Log law) Viscos sblayer 5 < 0 y < y y < 30, < 5, Transition region ln( y Trblence region 30 < where y, / 5) 5.5.5ln( y ) ln(y /5) y Non - dimensional velocity Friction velocity τ 5.5.5ln(y ) τ w / ρ, / τ y Derived from Prandtl s mixing length model, y ( τ Wall shear stress / v) y, τ w

13 Varios approach Solid (Fel rod) Heat condction Flid (Coolant) Empirical heat transfer correlations Theoretical heat transfer eqations Navier-Stokes eqations Trblence models Experiment Approximation of Bondary layer Mixing length model Two eqation model Reynolds stress model LES DNS LES:Large eddy simlation DNS: Direct nmerical simlation

14 5.4 Theoretical and empirical heat transfer coefficient Heat from fel rods is transferred to coolant flowing along the fel rods. Heat transfer coefficient W/m K q'' h( T T ) w q : Heat flx, W/m T w : Srface temperatre T b : Blk temperatre of coolant (Mixed average temperatre) b

15 5.4.1 Measrement of heat transfer coefficient Smooth channel with niform heat flx or niform wall temperatre Uniform heat flx q' ' δ Flid L D T in Tot Hydro-dynamically and thermally flly developed flow Heat balance ρ c p πd 4 ( T ot T in ) q' ' πdl

16 Determination of heat transfer coefficient Heat flx q determined from inpt power, radial temperatre gradient in channel wall, and enthalpy increase from inlet to otlet: temperatre difference and flow rate Inner srface temperatre T w determined from extrapolation of radial temperatre gradient in channel wall Flid temperatre (T in T ot )/ h T w ( T in q' ' T ot ) /

17 5.4. Non-dimensional correlations Reynolds nmber:ratio of inertia term to diffsion term in momentm eqation ρd Re μ c pμ Pr k N hd k Hydralic diameter: Characteristic length of flow channel A De 4 F D ν Prandtl nmber:ratio of velocity bondary layer thickness to temperatre one Nsselt nmber:non-dimensional heat transfer coefficient ν α

18 Empirical heat transfer correlations Laminar flow -Heated plate -Constant heat flx N Re Pr > 0.5 1/ Pr 1/ 3 Trblent flow (Ditts-Boelter s eqation) -Uniform wall srface temperatre 5 Re > < Pr < 100 L / D > 60 N Re Pr n Heating n 0.4 Cooling n 0.3

19 Liqid metal flow (Sbbotin s eqation) Pe Re Pr > 5 10 N Pe Pr < In case of ndeveloped thermal bondary layer Length of entrance region L 1/ / D 0.63Re [ ] /( L / ) N m / N 1.11 Re D 4

20 Table 5. Heat transfer correlation for forced convective laminar flow Correlation Commemts Pipe Plate Constant q Constant T w Constant T w Pr>0.6 N4.36 N3.65 N0.664Re 1/ Pr 1/3 Avarage N hl / k Flly develop ed Constant q Pr>0.5 N0.916Re 1/ Pr 1/3

21 Table 5.3 Heat transfer correlation for forced convective trblent flow Conditions Correlation Comments Plate N0.088Re x 0.8 Pr 1/3 Developed, Local N x h x x/k Pipe Constant T w 0.7<Pr<100 Re>10 5 L/D>60 N0.03Re x 0.8 Pr n Developed, Ditts- Boelter s eqation n0.4 for heating n0.3 for cooling Constant q 0.1>Pr PeRePr>10 5 N50.05Pe 0.8 Developed, Sbbotin s eqatin N m /N [Re 0. /(L /D)] 0.75 Undeveloped, Entrance length L/D0.63Re 1/4

22 Nsselt nmber N N0.03Re 0.8 Pr 0.4 (Trblent flow, Ditts-Boelter) N4.36 (Laminar, q'':const.) N3.65 (Laminar flow, T:Const) Reynolds nmber N50.05Re 0.8 Pr 0.8 (Trblent flow, Sbbotin, Liqid metal) Re

23 5.4.3 Check points for se of heat transfer correlations - Experimental conditions in derivation of the correlation - Applicability range in Re and Pr - Laminar flow or trblent flow? - Flly developed or ndeveloped? - Constant wall temperatre or constant heat flx? - Definition of characteristic length sch as hydralic diameter

24 5.5 Calclation of temperatre increase in flow direction Heat balance condition in steady state condition Increase of cross-sectional average temperatre coolant along fel rod c p mdt & ( z) q'( z) dz πrq''( z) dz πr q'''( z) dz dz z T q 1 z T c m& T ( z) 1 H / q' ( z) p dz Fel rod T 1

25 5.6 Calclation of Pressre drop in fel bndle Basic theory for friction shear stress Prandtl s bondary layer theory for laminar flow Momentm eqation in x-direction x v y dp dy 0 gc dp ρ dx Bondary layer approximation ν y y x τ s Continity eqation v 0 x y V

26 Friction pressre loss coefficient in laminar flow c f ρ V τ s / C f 1 L c f dx L Re L Relation between c f τ w c f ρ and λ τ w D πd ΔP 4 x D ΔP ρ τ w Δ λ πd 4 Δx 8 λ c f 4 τ w Friction pressre loss coefficient in trblent flow (Blasis s eqation) τ w d ρ, Re ( / 4 Re ν < Re < 10 )

27 Friction pressre loss in fel bndle Δp 1 1 λ ρ Δz D e : Flid density p : Cross sectional average pressre : Cross sectional average velocity De: Hydralic diameter λ : Frictional pressre loss coefficient ρ Laminar flow Re < 300 λ x : Distance in flow direction 64 Re Δz Trblent flow(blasis s eqation) Re > 300 λ Re Fel rod z p

28 Friction loss coefficient λ λ64/re (Laminar, Theory) Rogh wall λ0.3164/re 0.5 (Trblent flow, Blasis) λconstant Reynolds nmber Re Frictional pressre loss coefficient

29 Pressre loss in varios channel geometries Δp Δz 1 ξ ρ For grid spacer, orifice, ventri, elbow, etc.

30 Problem 3: A single-phase liqid flow in an annlar channel A coolant flows throgh an annlar channel between a straight circlar heater rod and a straight circlar tbe at a mass flow rate of W, where the heater rod is installed axis-symmetrically at the center of the straight tbe with two spacers. The length of the channel is L, the oter diameter of the heater rod is d, and the inner diameter of the tbe is D. 1)Write expressions for the cross sectional area A, the hydralic eqivalent diameter D e and the thermal eqivalent diameter D h of the annlar channel. ) Determine an expression for the mean velocity, and then an expression for the pressre loss in the channel for a flly developed flow sing the mean velocity, the flid density ρ, the pressre loss coefficient for one spacer ζ, and the friction factor λ.

31 Problem 4 : Heat removal in core APWR core consists of fel rods with the arrangement given in Table 5.4. Heat is generated niformly in the fel pellets with the heat generation rate of 400MW/m 3, and the otside srfaces of the cladding tbes are cooled by the coolant. The properties of the coolant are given in Table 5.5. Calclate the following variables to three significant digits: P 13 mm 1) Hyralic diameter, Reynolds nmber and Prandtl nmber. )Nsselt nmber and heat transfer coefficient at the cladding srface sing the Ditts-Boelter s eqation. )Temperatre at the srfaces of fel rods.

32 Table 5.4 Cooling conditions Average coolant temperatre Total coolant mass flow rate in core Diameter of fel pellet Diameter of cladding tbe, D Pitch in the arrangement of fel rods, P Arrangement of rod bndle Nmber of fel bndles in core kg/s 9 mm 10 mm 13 mm 17x Table 5.5 Physical properties of coolant at 300 Density, ρ 714 kg/m 3 Specific heat c p 5.73 J/kg K Dynamic viscosity, μ Thermal condctivity, k 9.6x10-5 Pa s 0.54 W/m K

UNIT V BOUNDARY LAYER INTRODUCTION

UNIT V BOUNDARY LAYER INTRODUCTION The variation of velocity from zero to free-stream velocity in the direction normal to the bondary takes place in a narrow region in the vicinity of solid bondary. This