5. Convective Heat Transfer
5.1 Heat removal by coolant flow
Fuel pellet
Bond layer Heat is transferred from the surfaces of the fuel rods to the coolant. Cladding tube
T Temperature at center offc fuel pellet
Temperature at outside Tsurfacef of fuel pellet
Temperature at insideTb surface of fuel coolant Coolant flow Temperature at outsideTc surface of cladding tube
Cool ant temperatureTm 5.2 Properties of coolant
Requirement of nuclear properties
1. Low neutron absorption cross section 2. Low radiation-induced radioactivity 3. Low damage by irradiation 4. High moderating power for thermal neutron reactor, and low moderating power for fast neutron reactor Requirement of thermal-hydraulic properties
1. High thermal conductivity and specific heat, and as a result high heat transfer coefficient 2. Low melting point in case of coolant which is solid at room temperature 3. Low viscosity (low friction pressure drop and strong turbulence) 4. No thermal decomposition 5. Chemically inactive with air and water Characteristics of gas coolant -Low specific heat and thermal conductivity -Required heat transfer is obtained by pressurization
Characteristics of water: light water (H2O) and heavy water (D2O) -High pressure operation needed because of high saturation vapor pressure -Good thermal properties -High moderating power Characteristics of liquid metal: alkali metals, sodium and lithium
-Low saturation pressure (Low pressure operation is possible at high temerature) -Low Prandtl number, high thermal conductivity, and low density -Low corrosion rate under good control of impurity (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) Characteristics of molten salts:
Flibe (LiF-BeF2), HTS (NaNO3-KNO3-NaNO2), Flinak
(LiF-NaF-KF), Li2CO3-Na2CO3-K2CO3, etc.
-High specific heat (Good for heat transportation) -High viscosity (Laminarization) -High Prandtl number (Not good for heat transfer media) -Nickel-base alloy (Hastelloy) is compatible with molten salt Table 5.1 Properties of various coolants
CO2 He H2O D2O Na Li Flibe Melting pint (℃) 0 3.9 97.9 180.7 458 Boiling point (℃) －56.6 at －269 100 101 881 1342 1435 0.53Pa Density (kg/m3) 28 2.4 712 770 832 483 2040 Thermal conductivity (W/mK) 0.059 0.28 0.536 0.536 67.4 52.7 0.929 Specific heat (kJ/kgK) 1.17 5.27 5.69 6.19 1.26 4.23 2.3 Viscosity (10-6Pa･s) 34 39 89 98 250 360 14900 Kinematic viscosity (10-6m2/s) 1.2 16 0.13 0.13 0.30 0.74 7.3 Thermal diffusivity (10-6m2/s) 1.8 22 0.13 0.11 64 26 0.19 Pr 0.67 0.73 0.98 1.2 0.0046 0.029 38
Note: Temperature and pressure dependency 5.3 Approach to convective heat transfer (Single-phase flow) Velocity v How to calculate the heat transfer coefficient, or the temperature Solid profile close to the solid surface, i. e. the fuel rod surfaces in coolant? Heat flux q” Temperature T
Coolant flow Calculation of velocity and temperature field Navier-Stokes equations Dρ =−ρ() ∇⋅v Continuity equation Dt Dv Momentum equation ρμρ=−∇p + ∇2 vg + Dt DT Energy equation ρC = k∇2T v Dt where μ : Dynamic viscosity of fluid k : Thermal conductivity of fluid D ∂ Dφ ∂φ ∂φ ∂φ ∂φ = + v ⋅∇ ≡ + vx + vy φ + vz φ Dt ∂t Dt ∂t ∂x ∂y ∂z φ 2 2 2 ∂v ∂vy ∂vy ∂ ∂ ∂ φ ∇⋅ v = x + + ∇2 ≡ + + ∂x ∂y ∂z ∂x2 ∂y 2 ∂z 2 Boundary conditions v = 0, given q” at solid surface Approach to turbulent flow u’ Average velocity and fluctuation u u = u + u', v = v + v', w = w + w' Reynolds equation (Time average) Time t ρ ⎧ ⎛ ⎞ ⎫ Dvi 1 ∂P ∂ ⎪ ⎜ ∂vi ∂v j ⎟ ⎪ = − + ⎨ν + − v'i v' j ⎬ Dt ∂x ∂x ⎜ ∂x ∂x ⎟ i j ⎩⎪ ⎝ j i ⎠ ⎭⎪ Reynoldsτ stress (additional terms) Turbulence model ρ ∂u ∂u = − u'v' = μT ∂ = ρε M y ∂y
ε M Eddy diffusivity of momentum μ Prandtl’s mixingρ ∂ length model
u 2 4 2∂ l ⎛ y ⎞ ⎛ y ⎞ T = lm , m = 0.14 − 0.08⎜1− ⎟ − 0.06⎜1− ⎟ , in pipe flow (by Nikuladse) y R ⎝ R ⎠ ⎝ R ⎠
Mixing length lm = κy κ = 0.41 Calculation of velocity profile in turbulent flow
The 1/7th law of velocity profile 1/ 7 u ⎛ r ⎞ = ⎜1− ⎟ uy+ = 857.(+ )17/ umax ⎝ R ⎠
1 Derived 0.8
Δp 0. 3164 1 1 = ρu 2 0.6 Δx Re14/ 2 D 0.4 n u + = (y + ) , u u u + ≡ , y+ ≡ y, 0.2 u τ τ τ ν w 0 uτ ≡ 0 0.2 0.4 0.6 0.8 1 ρ r/R von Karman’s universal velocity profile (Log law) Viscous sublayer 25 + 0 < y < 5, + + 20 u =5.5+2.5ln(y ) u + = y+ u+=5.0+5.0ln(y+/5)
+ 15 Transition region u 5 < y+ < 30, 10 + + + + u = 5.0 + 5.0ln(y / 5) 5 u =y Turbulence region 0 0 1 2 3 30 < y+ , 10 53010 10 10 y+ + + u = 5.5 + 2.5ln(y ) τ Derived from ρPrandtl’s mixing length model Non - dimensional velocity u + ≡ u / u , y+ ≡ (u / v)y, where Friction velocity u ≡ / , Wall shear stress τ τ w τ w
τ Various approach Solid (Fuel rod) Heat conduction
Fluid (Coolant)
Empirical Theoretical Navier-Stokes equations heat transfer heat transfer correlations equations Turbulence models
・Two equation Approximation Mixing length model ・LES Experiment of Boundary model ・Reynolds stress ・DNS layer model
LES：Large eddy simulation DNS: Direct numerical simulation 5.4 Theoretical and empirical heat transfer coefficient
Heat from fuel rods is transferred to coolant flowing along the fuel rods. Heat transfer coefficient W/m2K
qh''= (Twb− T ) q”: Heat flux, W/m2
Tw: Surface temperature
Tb: Bulk temperature of coolant (Mixed average temperature） 5.4.1 Measurement of heat transfer coefficient
Smooth channel with uniform heat flux or uniform
wall temperature Uniform heat flux q' ' δ
Fluid u D L T Tin out Hydro-dynamically and thermally fully developed flow ρ π Heat balance D2 c u (T − T ) = q''πDL p 4 out in Determination of heat transfer coefficient Heat flux q’’ determined from input power, radial temperature gradient in channel wall, and enthalpy increase from inlet to outlet: temperature difference and flow rate
Inner surface temperature Tw determined from extrapolation of radial temperature gradient in channel wall
Fluid temperature (Tin+Tout)/2 q'' h = Tw − (Tin + Tout ) / 2 5.4.2 Non-dimensional correlations Reynolds number：Ratio of inertia term to diffusion term in momentum equation ρuD uD Re ≡= μν Prandtl number：Ratio of velocity boundary layer thickness to temperature one
c pμ ν Pr ≡= k α Nusselt number：Non-dimensional heat transfer coefficient hD Nu ≡ k Hydraulic diameter： Characteristic length of flow channel 4 A D ≡ e F Empirical heat transfer correlations
Laminar flow Nu = 0.916 Re1/ 2 Pr1/ 3 -Heated plate -Constant heat flux Pr > 0.5
Turbulent flow (Dittus-Boelter’s equation) -Uniform wall surface temperature 5 Re >10 0.7 < Pr <100 L / D > 60 Nu = 0.023Re0.8 Pr n Heating n = 0.4 Cooling n = 0.3 Liquid metal flow (Subbotin’s equation) 5 Pe ≡ Re Pr >10 Pr < 0.1 Nu = 5 + 0.025Pe0.8
In case of undeveloped thermal boundary layer Length of entrance region L / D = 0.623Re1/ 4
0.2 0.8 0.275 Num / Nu∞ = 1.11[Re /(L / D) ] Table 5.2 Heat transfer correlation for forced convective laminar flow
Correlation Commemts Pipe Constant q” Nu=4.36 Fully Constant Tw Nu=3.65 develop Plate Constant T Pr>0.6 Nu=0.664Re1/2Pr1/3 Avarage w ed Nu ≡ hL / k Constant q” Pr>0.5 Nu=0.916Re1/2Pr1/3 Table 5.3 Heat transfer correlation for forced convective turbulent flow
Conditions Correlation Comments
0.8 1/3 Plate Nu=0.0288Rex Pr Developed, Local
Nux=hxx/k 0.8 n Constant Tw 0.7
Nu=4.36 （Laminar, q'':Const.） 101 Nusselt number Nu=5+0.025Re0.8Pr0.8 Nu=3.65 (Turbulent flow, （Laminar flow, T:Const） Subbotin, Liquid metal) 100 102 103 104 105 Reynolds number Re 5.4.3 Check points for use of heat transfer correlations
- Experimental conditions in derivation of the correlation - Applicability range in Re and Pr - Laminar flow or turbulent flow? - Fully developed or undeveloped? - Constant wall temperature or constant heat flux? - Definition of characteristic length such as hydraulic diameter 5.5 Calculation of temperature increase in flow direction Heat balance condition in steady state condition Increase of cross-sectional average temperature coolant along fuel rod z c p m& dT(z) = q'(z)dz T
= π2 Rq''(z)dz dz q’ = πR 2q'''(z)dz
1 z T(z) = T + ∫ q'(z)dz 1 −H / 2 T c p m& Fuel rod 1 5.6 Calculation of Pressure drop in fuel bundle
Basic theory for friction shear stress
Prandtl’s boundary layer theory for V∞ laminar flow y ∂ u ∂ x τ Momentum∂ equation in x-direction s ∂ u u gρ dp ∂ 2u u + v = − c +ν x y dx ∂y 2 Boundary layer approximation Continuity equation ∂u ∂v dp = 0 + = 0 dy ∂x ∂y Friction pressure loss coefficient in laminar flow c ≡ τ s 1 L 1.328 f 2 C f ≡ ∫0 c f dx = ρ∞V∞ / 2 L Re L
Relation between cf and 2 λ τ τ π ρu w = c τ w f 2 u D 2 D πΔP τ w 2 4 Δx D ΔP λρu c f w = = = ∴λ = D 4 Δx 8 4 τ Friction pressure loss coefficient in turbulent flow (Blasius’s equation)ρ 0.03955 ud = u 2 , Re ≡ (3×103 < Re < 105 ) w Re1/ 4 ν Friction pressure loss in fuel bundle
Δp λ 1 1 z − = ρu 2 Δz 2 De
Laminar flow 64 Δz Re < 2300 λ = Re Turbulent flow（Blasius’s equation） 0.3164 Re > 2300 λ = p Re0.25 ρ : Fluid density Fuel rod p : Cross sectional average pressure u : Cross sectional average velocity De: Hydraulic diameter λ : Frictional pressure loss coefficient x : Distance in flow direction λ λ=64/Re
(Laminar, Theory)
-1 10 Rough wall λ=Constant
0.25 -2 λ=0.3164/Re
Friction loss coefficient loss Friction 10 (Turbulent flow, Blasius) 102 103 104 105 Reynolds number Re
Frictional pressure loss coefficient Pressure loss in various channel geometries
Δp ξ 1 2 = ρu Δz 2
For grid spacer, orifice, venturi, elbow, etc. Problem 3: A single-phase liquid flow in an annular channel A coolant flows through an annular channel between a straight circular heater rod and a straight circular tube at a mass flow rate of W, where the heater rod is installed axis-symmetrically at the center of the straight tube with two spacers. The length of the channel is L, the outer diameter of the heater rod is d, and the inner diameter of the tube is D.
1)Write expressions for the cross sectional area A, the hydraulic
equivalent diameter De and the thermal equivalent diameter Dh of the annular channel. 2) Determine an expression for the mean velocity u, and then an expression for the pressure loss in the channel for a fully developed flow using the mean velocity u, the fluid density ρ, the pressure loss coefficient for one spacer ζ, and the friction factor λ. Problem 4 : Heat removal in core APWR core consists of fuel rods with the arrangement given in Table 5.4. Heat is generated uniformly in the fuel pellets with the heat generation rate of 400MW/m3, and the outside surfaces of the cladding tubes are cooled by the P = 13 mm coolant. The properties of the coolant are given in Table 5.5. Calculate the following variables to three significant digits:
1) Hyraulic diameter, Reynolds number and Prandtl number. 2）Nusselt number and heat transfer coefficient at the cladding surface using the Dittus-Boelter’s equation. 2）Temperature at the surfaces of fuel rods. Table 5.4 Cooling conditions Average coolant temperature 300 ℃ Total coolant mass flow rate in core 1.9 kg/s Diameter of fuel pellet 9 mm Diameter of cladding tube, D 10 mm Pitch in the arrangement of fuel rods, P 13 mm Arrangement of rod bundle 17x17 Number of fuel bundles in core 193
Table 5.5 Physical properties of coolant at 300 ℃ Density, ρ 714 kg/m3
Specific heat cp 5.73 J/kg･K Dynamic viscosity, μ 9.6x10-5 Pa･s Thermal conductivity, k 0.54 W/m･K