HEAT PROCESSES

Heat transfer

Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2010

HEAT PROCESSESHP4

Mechanisms of heat transfer. Conduction, convection (heat transfer coefficients), radiation (example: cooling cabinet). Fourier’s law of conduction, thermal resistance (composed wall, cylinder). Unsteady heat transfer, penetration depth (derivation, small experiment with gas lighter and copper wire). Biot number (example: boiling potatoes). Convective heat transfer, heat transfer coefficient and thickness of thermal boundary layer. Heat transfer in a circular pipe at laminar flow (derivation Leveque). Criteria: Nu, Re, Pr, Pe, Gz. Heat transfer in turbulent flow, Moody’s diagram. Effects of variable properties (Sieder Tate correction for temperature dependent viscosity, mixed and natural convection).

Mechanisms of heat transferHP4

There exist 3 basic mechanisms of heat transfer between different bodies (or inside a continuous body)

Conduction in solids or stagnant fluids

Convection inside moving fluids, but first of all we shall discuss heat transfer from flowing fluid to a solid wall

Radiation (electromagnetric waves) the only mechanism of energy transfer in an empty space

Aim of analysis is to find out relationships between heat flows (heat fluxes) and driving forces (temperature differences)

Heat flux and conductionHP4

Benton

( )

reaction heat, heat fluxonly for gas phase change enthalpy accumulation

conduction(compressive and electric heatof enthalpyheating).

( ) Rp

T pc T Q

t t

General form of transport Fourier equation for temperature field T(t,x,y,z) in a solid or in a stagnant fluid taking into account internal heat sources and an adiabatic temperature increase during compression of gas.

Heat flux and conductionEnergy balance of a closed system dq = du + dw (heat delivered to system equals internal energy increase plus mechanical work done by system) tells nothing about intensity of heat transfer at the surface of system, neither about relationships between heat fluxes and driving forces (temperature gradients). This problem is a subject of irreversible thermodynamics.

Intenzity of heat transfer through element dA of boundary is characterized by vector of heat flux [W/m2]

n

dA q q

Direction and magnitude of heat flux is determined by gradient of temperature and thermal conductivity of media

Tq

Heat flow through boundary is projection of heat flux to the outer normal qn

Fourier’s law of heat conduction

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Thermal conductivity

Material [W/(m.K)] a [m2/s]

Aluminium Al 200 80E-6

Carbon steel 50 14E-6

Stainless steel 15 4E-6

Glas 0.8 0.35E-6

Water 0.6 0.14E-6

Polyethylen 0.4 0.16E-6

Air 0.025 20E-6

Thermal and electrical conductivities are similar: they are large for metals (electron conductivity) and small for organic materials. Temperature diffusivity a is closely related with the thermal conductivity

Memorize some typical values: pca

Thermal conductivity of nonmetals and gases increases with temperature (by about 10% at heating by 100K), at liquids and metals usually decreases.

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Distribution of temperatures and heat fluxes in a solid can be expressed in differential form, based upon enthalpy balancing of infinitesimal volume dv

Integrating this differential equation in a finite volume V the integral enthalpy balance can be expressed in the following form using Gauss theorem

Conduction – Fourier equation

hq

t

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V V S

hdv qdv n qds

t

n

dA q

yx zqq q

qx y z

Accumulation of enthalpy in unit control volume Divergence of heat fluxes (positive

if heat flows out from the control volume at the point x,y,z)

Accumulation of enthalpy in volume V

Heat transferred through the

whole surface S

Heat flux q as well as the enthalpy h can be expressed in terms of temperatures, giving partial differential equation – Fourier equation

Conduction – Fourier equation

( )

( ) ( ) ( )

g

p g

p g

hq Q

tT

c T QtT T T T

c Qt x x y y z z

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Thermal conductivity need not be a constant. It usually depends on temperature, and for anisotropic materials (e.g. wood) it depends also on directions x,y,z – in this case ij should be considered as

the second order tensor.

Internal heat source (e.g. enthalpy change of a chemical reaction or a volumetric heat produced by passing electric

current or absorbed microwaves)

Let us consider special case: Solid homogeneous body (constant thermal conductivity and without internal heat sources). Fourier equation for steady state reduces to the Laplace equation for T(x,y,z)

Boundary conditions: at each point of surface must be prescribed either temperature T or the heat flux (for example q=0 at an insulated surface).

Solution of T(x,y,z) can be found for simple geometries in an analytical form (see next slide) or numerically (using finite difference method, finite element,…) for more complicated geometry.

Conduction - stationary

2

2

2

2

2

220

z

T

y

T

x

TT

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2

2

22

10 ( )

10 ( )

T Tr

x r r rT

rr r r

The same equation written in cylindrical and spherical

coordinate system (assuming axial symmetry)

Calculate radial temperature profile in a cylinder and sphere (fixed temperatures T1 T2 at inner and outer surface)

Example temperature profile in a cylinder

2 1 1 2 2 11 1 2 1 2

2 1 2 1

2 1 1 2 2 2 1 11 2 1 2 1 2

2 1 2 1

ln ln, T=c ln c

ln / ln /

c, T=- c ( )

T T T R T RTr c r c c

r R R R R

R R T R T RTr c c T T c

r r R R R R

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Sphere (bubble)

cylinder

R1

R2

Knowing temperature field and thermal conductivity it is possible to calculate heat fluxes and total thermal power Q transferred between two surfaces with different (but constant) temperatures T1 a T2

Conduction – thermal resistance

TR

TTQ 21

L

RRRT 2

/ln 12)(1

2

2

1

1

hh

SRT

RT [K/W] thermal resistance

h1 h2

S

T1 T2

h

S1

S2

T1 T2

L

R1

R2

T1 T2

L

R1

T1

T2

h

Q

2211 SS

hRT

L

RhRT 2

/2ln

Serial Parallel Tube wall Pipe burried under surface

In this way it is possible to express thermal resistance of windows, walls,

heat transfer surfaces …

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Time development of temperature field T(t,x,y,z) in a homogeneous solid body without internal heat sources is described by Fourier equation

with the boundary conditions of the same kind as in the steady state case and with initial conditions (temperature distribution at time t=0).

This solution T(t,x,y,z) can be expressed for simple geometries in an analytical form (heating brick, plate, cylinder, sphere) or numerically.

Conduction - nonstacionary

)(2

2

2

2

2

2

z

T

y

T

x

Ta

t

T

The coefficient of temperature diffusivity a=/cp is the ratio of

temperature conductivity and thermal inertia

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Theory of penetration depth

Result is ODE for thickness as a function of time

4at

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Development of temperature profile in a half-space. Use the acceptable approximation by linear temperature profile

Tw

T0

x

δ

t+tt

+Δ

0

0

0

|

( )2

x

w

w w

T Tdx a

t x

TTdx a

t

T Ta

t

Integrate Fourier equations (up to this step it

is accurate)

Approximate temperature profile by line

Using the exact temperature profile predicted by erf-function, the penetration

depth slightly differs =(at)

Theory of penetration depth=at penetration depth. Extremely simple and important result, it gives us prediction how far the temperature change penetrates at the time t. This estimate enables prediction of thermal and momentum boundary layers thickness etc. The same formula can be used for calculation of penetration depth in diffusion, replacing temperature diffusivity a by diffusion coefficient DA .

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Wire Cu

=0.11 m

=398 W/m/K

=8930 kg/m3

Cp=386 J/kg/K

ConvectionHP4

Benton

( )

convection viscousconductionheatingthis term is zero for

incompressible liquid

( ) : Rp

T Dpc u T q Q

t Dt

General form of transport Fourier Kirchhoff equation

ConvectionCalculation of heat flux q from flowing fluid to a solid surface requires calculation of temperature profile in the vicinity of surface (for example temperature gradients in attached bubbles during boiling, all details of thermal boundary layer,…).

Engineering approach simplifies the problem by introducing the idea of stagnant homogeneous layer of fluid, having an equivalent thermal resistance (characterized by the heat transfer coefficient [W/(m2K)])

Tf is temperature of fluid far from surface (behind the boundary of thermal boundary layer), Tw is wall temperature. Thickness of stagnant boundary layer δ, f thermal conductivity of fluid.

Tq

)( wf TTq ( ) ( )ff w f f w

Tq T T T T

y

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n

dA

q

Boiling (bubbles)

Outer flow-thermal boundary layer

,yTf Tf

f

Example heating sphereHP4

Temperature distribution inside a solid sphere

22

( )s s sp

T Tc r

t r r r

It is correct only as soon as the heat flux q or the temperature is uniform on the sphere surface

Boundary condition (convection)

( )ss f s

TT T

r

Heat flux calculated from Fourier law inside the

sphere equals the flux in fluid

Fourier equation can be integrated at the volume of body (sphere in this case)

2sp s s

V V V

Tc dv T dv qdv

t

The integrals can be evaluated by the mean value and by Gauss theorem, assuming uniform flux at the surface

( )sp f s

S

TMc n qdS n qS T T S

t

Example heating sphereHP4

For the case that the temperature inside the sphere is uniform (as soon as the thermal conductivity s is very high) the mean temperature is identical with the surface temperature

/0

( )

( )

sf s

p

ts f f

p

dT ST T

dt c M

T T T T e

c M

S

This exponential solution works only for small values of Biot number

Thermal resistance of fluid >> thermal resistance of solid1.0s

DBi

Convection – Nu,Re,PrHeat transfer coefficient depends upon the flow velocity (u), thermodynamic parameters of fluid () and geometry (for example diameter of sphere or pipe D). Value is calculated from engineering correlation using dimensionless criteria

Nusselt number (dimensionless , reciprocal thickness of boundary layer)

Reynolds number (dimensionless velocity, ratio of intertial and viscous forces)

Prandl number (property of fluid, ratio of viscosity and temperature diffusivity)

uD

Re

D

Nu

a

Pr

Rem: is dynamic viscosity [Pa.s], kinematic viscosity [m2/s], =/

And others

Pe=Re.Pr Péclet numberGz=Pe.D/L Graetz number (D-diameter, L-length of pipe)

RayleighDe=Re√D/Dc Dean number (coiled tube, Dc diameter of curvature)

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Convection in a pipe

Liquid flows in a pipe with the constant wall temperature Tw that is different than the inlet temperature T0. Temperature profile depends upon distance from inlet and upon radius r (only thin temperature boundary layer of fluid is heated). Heat flux varies along the pipe even if the heat transfer coefficient is constant, because driving potential – temperature difference between wall and the bulk temperature Tm depends upon the distance x. Tm is the so called mean calorific temperature

A

m uTdAm

xT 1

)(

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Basic problem for heat transfer at internal flows: pipe (developed velocity profile) and a constant wall temperature

( ) ( )( ( ))w mq x x T T x

Heat flux from wall to bulk ( is related to the calorific

temperature as a characteristic fluid

temperature at internal flows)

Convection in a pipe

0 ( ( ) ( )) ( ( ))p m m w mmc T x T x dx T T x Ddx

0 ( ( ) ( )) ( ( ))w mm h x h x dx T T x Ddx

dxDTT

dTcm

mw

mp

x dx

DT0

Tw

Tm

Q

Axial temperature profile Tm(x) follows from the enthalpy balance of system, consisting of a short element of pipe dx :

0

( )ln w m

pw

T T xmc D x

T T

Solution Tm(x) by integration

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Convection in a pipePrevious integration is correct only if and the wall temperature are constant.

This doesn’t hold in laminar flow characterized by gradual development of thermal boundary layer (at entry this layer is thin and therefore =/ is high, decreases with increasing distance). Typical correlations for laminar flow

is almost constant at turbulent flows characterized by fast development of thermal boundary layer. Typical correlation (Dittus Boelter)

More complicated are cases with mixed convection (effect of temperature dependent density and gravity), variable viscosity and first of all influence of phase changes (boiling/condensation).

0 0

outletT Lm

pw mT

dTmc D dx

T T

31.6 Re PrD D

NuL

0.8 1/30.023Re PrD

Nu

L

DT0

Tw

Toutlet

Q

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general formula for variable wall temperature and variable heat

transfer coefficient

ln2/3

0.06683.66

1 0.04

D GzNu

Gz

Leveque Haussen

Convection Laminar LevequeHP4

x

DT0

Tw

yumax

Leveque method is very important technique how to estimate thickness of thermal boundary layer and the heat transfer coefficient in many internal flows (not only in circular pipes). This theory is applicable only for “short” channels, in the region of developing temperature profile.

3ln 1.6D

Nu Gz

Graetz number Gz=Re.Pr.D/L

D

uux

8)(

velocity at bouindary layer

( ) 8x

x xDt

u u

time of penetration

at 2 3

8

axD

u

Mixed convection, Sieder TateHP4

0,141/31/3 1/3 4/9

3

2

1,618 1 0,0111 ,

,

W

Nu Gz Gz Gr

g TDGr

Temperature dependendent properties of fluid are respected by correction coefficients applied to a basic formula (Leveque, Hausen, …similar corrections are applied in correlations for turbulent regime)

Temperature dependent viscosity results in changes of velocity profiles. In case of heating the wall temperature is greater than the bulk temperature, and viscosity of liquid at wall lowers. Velocity gradient at wall increases thus increasing heat transfer (look at the derivation of Leveque formula modified for nonnewtonian velocity profiles). Reversaly, in case of cooling (greater

viscosity at wall) heat transfer coefficient is reduced. This effect is usually modeled by Sieder Tate correction (ratio of viscosities at bulk and wall temperature).

Temperature dependent density combined with acceleration (gravity) generate buoyancy driven secondary flows. Resulting effect depends upon orientation (vertical or horizontal pipes should be distinguished). Intensity of natural convection (buoyancy) is characterized by Grashoff number Gr)

Leveque

Mixed convection (Grashoff)

Sieder Tate correction

Convection Turbulent flowHP4

Boccioni

Convection Turbulent flowHP4

Turbulent flow is characterised by the energy transport by turbulent eddies which is more intensive than the molecular transport in laminar flows. Heat transfer coefficient and the Nusselt number is greater in turbulent flows. Basic differences between laminar and turbulent flows are:

Nu is proportional to in laminar flow, and in turbulent flow.

Nu doesn’t depend upon the length of pipe in turbulent flows significantly (unlike the case of laminar flows characterized by rapid decrease of Nu with the length L)

Nu doesn’t depend upon the shape of cross section in the turbulent flow regime (it is possible to use the same correlations for eliptical, rectangular…cross sections using the concept of equivalent diameter – this cannot be done in laminar flows)

3 u 0.8u

The simplest correlation for hydraulically smooth pipe designed by Dittus Boelter is frequently used (and should be memorized)

0.80.023Re PrmNu m=0.4 for heating

m=0.3 for cooling

Similar result follows from the Colburn analogy 3/18.0 PrRe023.0Nu

Pressure drop, friction factorHP4

21

2 f

Lp u

D

Friction factor f depends upon Re

and relative roughness

Pressure drop is calculated from Darcy Weissbach equation

Turbulent boundary layerHP4

Velocity profile

Laminar sublayer

e-roughness

Buffer layer

Thickness of laminar sublayer is at value y+=5

Rougness of wall has an effect upon the pressure drop and heat transfer only if the height of irregularities e (roughness) enters into the so called buffer layer of turbulent flow. Smaller roughness hidden inside the laminar (viscous) sublayer has no effect and the pipe can be considered as a perfectly smooth.

**, ,

4w

w

u y dp Dy u

dx

Friction velocity

Dimensionless distance from wall

y

Example smooth pipeHP4

Calculate maximum roughness at which the pipe D=0.1 m can be considered as smooth at flow velocity of water u=1 m/s.

Re 10000uD

4

0.3160.0316

Ref

211600

2 f

pu

L D

5 10 25L

e mD p

Blasius correlation for friction factor (smooth

pipes)

Thickness of laminar sublayer (y+=5)

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EXAMHP4

Heat transfer (Fourier Kirchhoff transport equation

explained in more details in the course Momentum Heat and Mass transfer)

2

( )

convection viscousconduction reaction heatheatingthis term is zero for phase change enthalpy( )

incompressible liquid electric heat

( ) : Rp

T

T Dpc u T q Q

t Dt

What is important (at least for exam)HP4

Dimensionless criteria

Nusselt

Biot

Fourier

Reynolds

Prandtl

Peclet

Graetz

Rayleigh

fluid thermal

D DNu

solid

DBi

2

atFo

D

ReuD

Pr pc

a a

Re.PrPe

Re.Pr . /Gz D L

a

THgRa

3

reciprocal thermal boundary layer

thermal resistance in solid / thermal resistance in fluid

dimensionless time related to the penetration time through distance D

ratio of inertial and viscous forces

ratio of momentum and temperature diffusivities

What is important (at least for exam)HP4

)(2

2

2

2

2

2

z

T

y

T

x

Ta

t

T

Conduction - temperature field in solids

4at Penetration depth (distance travelled by temperature disturbance in time t)

Steady heat transfer2 2 2

2 2 20

T T T

x y z

Unsteady heat transfer (wave of thermal disturbance)

TR

TTQ 21

L

R1

R2

T1 T2

L

RRRT 2

/ln 12

Thermal resistance RT

22

cylindrical c.s. spherical c.s.

1 10 ( ) 0 ( )

T Tr r

r r r r r r

What is important (at least for exam)HP4

Convection – heat transfer from fluid to solid (-heat transf.coef.)

( ) ( )ff w f wq T T T T

31.6 Re PrD D

NuL

Forced heat transfer in a pipe

Laminar flow (Leveque)

Turbulent flow (Dittus Boelter)0.8 1/30.023Re Pr

DNu

21

2 f

Lp u

D

Pressure drop in pipes, effect of roughness and Moody diagram