Hydrodynamics and heat transfer in nuclear reactor

Hydrodynamics and heat transfer in nuclear reactor

Lecture 7. CONVECTIVE HEAT

TRANSFER IN NUCLEAR REACTOR

Lecturer: prof. Alexander Korotkikh

Contents

Introduction

Factors influencing the heat transfer

Modeling of process of convective heat transfer

Heat transfer in single-phase fluid with free flow (natural convection)

Heat transfer in single-phase fluid with forced flow (forced convection)

Heat transfer in forced cross flow pipes and bundles of pipes

The heat transfer to liquid metals

Heat transfer to water boiling

Conclusion

Introduction

Convection heat transfer is always accompanied by thermal conductivity, since the motion of the fluid or gas unavoidably contact the individual particles with different temperatures. In the practical calculations of heat transfer using Newton- Richman law:

( ) , [W]s lQ t t F

Specific conditions are taken into account by the coefficient of proportionality , which call the heat transfer coefficient.

Factors which influence on the heat transfer

1. The origin of motion (free or forced). Free movement or natural convection occurs under the influence of the density difference of cold and hot fluid particles (gas). Forced movement (forced convection) occurs under the influence of the pressure difference created by a pump, compressor. In some cases, along with forced, at the same time can develop free traffic. The relative influence of the latter the greater the greater the temperature difference in the individual volumes of liquid (gas) and less than the speed of the forced movement.


2. The flow regime (laminar, transition, turbulent). In laminar flow the particles of fluid (gas) are moving quietly, without mixing (Fig. a), the heat transfer from surface to fluid (gas) is thermal conductivity. The turbulent regime is characterized by continuous mixing of all liquid layers (Fig. b). In this case, the wall surface is formed of a laminar sublayer of liquid with thickness ls . The heat transfer from wall surfase to fluid and turbulent flow is performed by conduction and convection. Intermediate regime of fluid motion between laminar and turbulent is called transitional.

a) Laminar b) Turbulent


3. Hydrodynamic and thermal boundary layers. In any mode of motion of particles of fluid immediately adjacent to a solid surface, as would stick to it. As a result close to the streamlined surface under the action of viscous friction forces formed a flat liquid layer near retarded which the fluid velocity changes from zero (at the surface of the body) to the velocity of the undisturbed flow (away from body). This retarded layer of fluid is called hydrodynamic boundary layer. Similarly, the concept of hydrodynamic boundary layer there is the concept of thermal boundary layer. This liquid layer , adjacent to a solid surface, within which the temperature liquid varies from the wall temperature ts to the temperature of the stream away from the surface tl. The ratio of the thickness of thermal and hydrodynamic boundary layers is determined by the value of the Prandtl number Pr = v/a, where v – kinematic viscosity, a - is the thermal diffusivity of liquid.


4. Thermophysical properties of the fluid. The heat transfer coefficient depends on the heat conductivity , heat capacity cp, kinematic viscosity , density , the coefficient of volume expansion and other fluid properties. In particularly, in the presence of phase transitions (boiling, condensation), from heat of evaporation r, surface tension etc. 5. Geometric size, shape, orientation of heat transfer surface. Is established that the heat transfer coefficient depends on the geometrical shape of the body surface (planar, cylindrical, spherical or other), sizes (length of the surface, the diameter of the tube or sphere, etc.), orientation of the heat transfer surface (vertical, horizontal with heat transfer upwards, horizontal with heat transfer down, sloped, etc.).


the Nusselt number it characterizes the intensity of convective heat exchange, where – thermal conductivity of liquid environment; l is the geometric dimension (length);

Nul

the Reynolds number it characterizes the ratio of forces inertia and forces of viscosity, where w is the velocity of the flow, – kinematic viscosity;

Rewl

the Prandtl number it characterizes the thermophysical properties of the fluid, where a is the thermal diffusivity;

Pra


the Peclet number here the numerator characterizes the heat carried by convection and the denominator is the heat transported by conduction;

Pe Re Prwl

a

the Grashof number it characterizes the attitude lift force occurs due to the difference of the densities of the liquid due to temperature changes t , to viscous forces, where is the coefficient of volume expansion;

3

2Gr

g t l

The Nusselt number Nu is determined by the number of tasks convective heat transfer, because it contains the sought value of heat transfer coefficient . The other similarity number (Re, Pr, Gr, Pe ...) are crucial and include quantities which depend on the heat transfer coefficient.

Nu = f(Re, Pr, Gr, Pe …).


When modeling the heat transfer, the experimental results are processed in the similarity and relationship between them are in the form of the similarity equations

Nu Re Pr ,n m

ld ld lC

Nu ,ld

d

Re ,ld

w d

ν

where C, m, n – constant coefficients determined experimentally.

PrNu (Gr Pr ) ,

Pr

m

n llх lх l

s

C

3

2Gr .lх

g t x

where

or

Heat transfer at free convection of fluid

Heat transfer coefficient

with the free motion of fluid

3 910 Gr Pr 10lx l – laminar regime

– transition regime

– turbulent regime

9 1010 Gr Pr 6 10lx l

10Gr Pr 6 10lx l

3

2Gr ,lх

g t x

Grashof number

s lt t t – the temperature difference

= f(t) are given in manuals for liquids. For gases are calculated 1

.lT


0.25

0.25 PrNu 0.60(Gr Pr )

Pr

llх lх l

s

Laminar regime

local value of

0.25

0.25 PrNu 0.75(Gr Pr )

Pr

llh lh l

s

average value of lam

Turbulent regime

0.25

1/3 PrNu 0.15(Gr Pr )

Pr

llх lх l

s

local and average value of turb

Transition regime

.2

turb lamtrans

Vertical surface


For horizontal flat surfaces (plates) the above formula is applicable for calculation, but in this case decrease by 30 %, if the heat transfer surface facing downt. The calculated heat transfer coefficient should be increased by 30 %, if the heat-transfer surface facing upward.

For horizontal pipes with laminar regime of fluid flow to calculate the average heat transfer coefficient

0.25

0.25 PrNu 0.5(Gr Pr )

Pr

lld ld l

s

Horizontal surface

Heat transfer at forced convection of fluid

The boundary layer and modifying the heat

transfer coefficient along the surface

Horizontal surface

The forced fluid flow (forced convection) occurs under the action of pressure difference, which in conjunction with the thermal properties determines the fluid velocity w.

4Re 10lх – laminar regime 4 610 Re 4 10lх – transition regime

6Re 4 10lх – turbulent regime 0.25

0.5 0.33 PrNu 0.66 Re Pr

Pr

lll ll l

s

0.25

0.8 0.43 PrNu 0.037 Re Pr

Pr

lll ll l

s

– laminar, Re<5·105

– turbulent, Re>5·105

Rell

w l


Reld

w d

Re 2300ld

42300 Re 10ld

4Re 10ld

Channel

– laminar regime

– transition regime

– turbulent regime

At laminar flow regime of fluid in pipe distinguish viscous and viscous-gravitational modes.

0.140.33

Nu 1.55 Pe sld ld l

l

d

l

– viscous regime

Pe Re Prw d

a

– the coefficient of dynamic viscosity, d, l – inner diameter and length of the pipe,

5(Gr Pr ) 8 10 ,ld l Re 2300ld


5(Gr Pr ) 8 10 ,ld l Re 2300ld Viscous-gravitational regime – 0.25

0.33 0.43 0.1 PrNu 0.15 Re Pr Gr

Pr

lld ld l ld l

s

l/d 1 2 5 10 15 20 30 40

l 1.9 1.7 1.44 1.28 1.18 1.13 1.05 1.02

0.25

0.8 0.43 PrNu 0.021 Re Pr

Pr

lld ld l l

s

Turbulent regime – 4Re 10ld

1 2

2

l ll

t tt

– average temperature

of liquid or gas

Reld 2300 3000 5000 6000 8000 10000

0.40 0.57 0.72 0.81 0.96 1.00 tr

42300 Re 10ld Transition regime –

tr turb tr

turb

4equ

Fd

P


а)

b)

с)

Forced cross flow pipes

Rewd

Re < 40 40 < Re < 1000 Re > 105

= (80 90) = (120 140)

is the separation angle

The character of changes of heat transfer coefficient on the regime of fluid flow


0.25

0.4 0.37 PrNu 0.76Re Pr

Pr

lld ld l

s

Re < 40

40 < Re < 1000 0.25

0.5 0.38 PrNu 0.5Re Pr

Pr

lld ld l

s

103 < Re < 2105 0.25

0.6 0.38 PrNu 0.25Re Pr

Pr

lld ld l

s

2105 < Re < 107

0.25

0.8 0.37 PrNu 0.023Re Pr

Pr

lld ld l

s

If the angle of attack < 90 , it was found according to the formulas the heat transfer coefficient should be multiplied by a correction factor

21 0.54cos

To calculate the average heat transfer coefficient around the perimeter of the pipe

average value of


0.25

0.5 0.36 PrNu 0.52Re Pr

Pr

lld ld l

s

40 < Re < 103

103 < Re < 105 0.25

0.65 0.33 PrNu 0.26Re Pr

Pr

lld ld l s

s

0.15

2s

s

d

is the correction coefficient taking into account the density the location of the pipes in the beam

Re > 105 0.25

0.84 0.36 PrNu 0.021Re Pr

Pr

lld ld l

s

average value of 3

The first row corridor beam 1 30.6

The second row corridor beam 2 30.9


0.25

0.5 0.36 PrNu 0.71Re Pr

Pr

lld ld l

s

40 < Re < 103

103 < Re < 105 0.25

0.60 0.33 PrNu 0.41Re Pr

Pr

lld ld l s

s

is the correction coefficient taking into account the density the location of the pipes in the beam

Re > 105 0.25

0.84 0.36 PrNu 0.021Re Pr

Pr

lld ld l

s

average value of 3

The first row corridor beam 1 30.6

The second row corridor beam 2 30.7

1/6

1

2

s

s

s

sin If the angle of attack < 90


, [W]bund l sQ t t F

Convective heat flux

where

1

1

n

i i

ibund n

i

i

F

F

1 2 3( 2)bund

n

n

or

For axial flow bundles of tubes which are cooled by gases or fluids, the average Nusselt number is calculated

0.1 0.4Nu Nu Re Pr 1 2expeq

eqld tur ld l B

0.25

0.8 0.37 PrNu 0.023Re Pr

Pr

ltur ld l

s

where /eqB d d

2

1.1 1s

Bd

2

1.27 1s

Bd


Liquid metals are applied in those cases when it is necessary to ensure intensive heat removal from the heating surface or when low pressure is required to have a high temperature working fluid.

Liquid metals have a high boiling point that allows to raise their temperature without applying high pressure.

Pr 0.005 0.05 The Prandtl number

In laminar flow the heat is transferred by the thermal conductivity. In the turbulent is thermal conductivity and convection.


When ts=const the values of the Nusselt number from the Peclet number at the laminar regime the flow of liquid metal

1 10 100 1000 104

4.04 3.86 3.74 3.68 3.66

2Peld

Nuld

When q = const the number Nu in the field of stable laminar flow is independent of the number Pe and equal to a constant value:

Nu 4.36ld

0.8Nu 5 0.025Peld ld

When ts=const and the turbulent regime the flow of liquid metal

When q = const 0.8Nu 7 0.025Peld ld

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Hydrodynamics and heat transfer in nuclear reactor