[PDF] Heat transfer

127917_3E_learningsupport_EN.pdf

Heat transfer

1 Lectures and seminars in EN: prof. Fatima Hassouna, room B139

Lectures in CZ: prof. Pavel Hasal, room B III

Seminars in CZ: prof. Pavel Hasaland prof. VladislavNevoral, room B139

Lectures and seminars

2

Fatima Hassouna

Email: Fatima.Hassouna@vscht.cz

tel. +420220443251/ +420220443104

Office: B 033

3Problems will be solved in Maple and COMSOL

Recommended books

Scalar, vector and tensor quantities

Scalar product, vector product, vector differential operators, material derivatives, ǀolumetric and surface integrals, mass balance in general ǀolume͙ Transformation of equations into dimensionless shape, scaling of quantities

Characteristic heat conduction time

Lecture 1

4

Scalar, vector and tensor

Scalar:

Anelementofafield,usuallyarealnumber

-Itisnotspatiallyoriented -Intheselectedspace(time)itcanbeexpressedbyonevalue -Typicalscalars:temperature,pressure,concentration a

Vector: A vector hasmagnitude(size) anddirection

-has a direction -in 3D space it can be characterized by three values -typical vectors: velocity, gradient of pressure, gradient concentration Tensor of 2nd order: geometric objects that describelinear relationsbetweengeometric vectors, scalars and other tensors. -has a direction -In 3D space it can be characterized by nine values -Typical tensor: velocity deformation tensor in the liquid 5

Edžample͗ cube of material subjected to an arbitrary load ї measure the stress on it in ǀarious

directions ў measurements form a second rank tensor; the stress tensor 6

Tensors

Tensorsexpresstensionsintheliquidorsolidmedia.

The tensor can describe what changes of characteristic property (the change invelocity in liquids or the change of shape in solid materials) in the direction perpendicular to some surface which are caused (initiated) by applyingtangential or normal forces to that area. 7

Scalar product

The scalar product is the product of the vector size bprojected into vector aand vector size aand / or vice versa.

¾Properties of the scalar product of vectors

-Commutative -Distributive

¾The product of two vectors = scalar

ș 8

Scalar product

¾Product of vector and tensor = vector

¾The property of the scalar product of the vector and the tensor

Associative

Proof will be done by students separately!

9

Vector differential operators

¾Gradient operator

¾Gradient of scalar field = vector field:

whose magnitude is the rate of change and which points in the direction of the greatest rate of increase of the scalar field.Ifthe vector is

resolved, its components represent the rate of change of the scalar field with respect to each directional component.

Thegradientis a multi-variable generalization of the derivative.

While a derivativecan be defined as a function of a single variable, for functions of several variables, the gradienttakes its place. The gradient is

a vector-valued function, as opposed to aderivative, which isscalar-valued. The given vector must be differential to apply the gradient phenomenon.

¾Gradient of vector field = tensor

10

Divergence of a Vector Field

Consider air as it is heated.

The velocity of the air at each point defines a vector field F. When the air is heated in a region, it expands in all directions, and thus the velocity field F points outward from that region. This expansion of fluid flowing with velocity fieldFis captured by the divergence ofF The divergence of the velocity field in that region should have a positive value.

Example:

The divergence represents the volume density of the outward flux of a vector field from an infinitesimal

volume around a given point. 11

The divergence of a vector field F = is defined as the partial derivative of P with respect to x plus the partial derivative of

Q with respect to y plus the partial derivative of R with respect to z. ׏

In Cartesian

׏

In Cylindrical

׏

In Spherical

׏

Divergence of a Vector Field

12

Vector differential operators

¾Vector field divergence = scalar

¾Laplace operator = operator divergence

¾Divergence of tensor field = vector

13

Vector differential operators

Exercise -prove it is true

14

Material derivative

Usingadifferentialfunction(scalarorvector)wecanmonitortheapproximateincrementofthisfunctionaroundtheselected

point.Forcalculatingthedifferentialfunctionitisnecessarytoknowthederivative(tangents)ofthisfunctionattheselected

pointforallindependentvariables(spatialcoordinatesandtime).

Thematerialderivativeindicatestherateofchangeofaspatialvariableasitisperceivedbyanobservermovingalongwiththe

fluid.

By dividing the differential by time increment,

we obtain a material derivative of the function where v is the flow velocity vector

Local partConvection part-

Non-zero even in steady state

Thematerialderivativecomputesthetimerateofchangeofanyquantitysuchasheat(temperature)ormomentum(velocity,

(whichgivesaccelerationforaportionofamaterialmovingwithavelocityv).Ifthematerialisafluid,thenthemovementis

simplytheflowfield. da: differential change or total differential of, a of function of several variables (t, x, y, z) 15

Integral transformation

ExpressconservationlawsfortransportablequantitiesinthecontrolvolumeV,whichissurroundedby thecontrolsurfaceS ¾Thecontrolvolumestillcontainsthesameamountoffluidandatthefluidflowisdeformed

¾nisanormalvectorperpendiculartothedSarea,౵౵n౵౵=1,nx2+ny2=1(nx=(1,0,0)andny=(0,1,0)in2D)

¾Theamountofphysicalquantity(property)thatincreases/decreases(accumulates)inthecontrol volumeVisequaltotheamount(measured)ofthequantity(property)thatenter/exitthroughtheS- areaofthecontrolvolume

For scalar type variables, vector (tensor):

ĺĺĺ

16 ¾Theflowofmassoverthecontrolvolumeisequaltomass accumulationinthecontrolvolume.

¾Massflowacrossthesystemboundarycanbewrittenas

productofdensity,velocityandareaboundarysystem. ¾Thesurfaceintegralcanbeconvertedtoavolumeintegral usingintegraltransformation.

¾Equationsofcontinuityindifferentialform

¾Forconstantdensityfluids,theequationcanbefurther simplified Continuity equation in physics is an equation that describes the transport of some quantity

Continuity equation -mass conservation law

17 Transformation of equations into dimensionless form oThe method serves to reduce the number of parameters oThe following equations can be used for any system of physical units (SI or other)

Procedure:

1-identify all dependent, independent variables: -Dependent (temperature, velocity, pressure)

-Independent (time, space coordinates)

2-For each variable we choose a characteristic magnitude that is the same dimension as this variable

3-We introduce dimensionless variables by dividing the dimensional by characteristic variables.

X X = X0 ๝ dimensional variable characteristic variable

4-We substitute the dimensionless variables in the equation and we divide the equation by the constant before a selected

term of equation

5-We obtain equations in dimensionless shapes. Coefficients before the members are also dimensionless -Dimensionless

criteria 18 Example: Transform the following equation into dimensionless number: Independent variables: t, x, y (time, coordinates x, y) Dependent variables: vx, vy, p (x, y, component of vector velocity, pressure ) Parameters (constants): Ș, ȡ(dynamic viscosity, density)

Introducing dimensionless variables

We derive, how derivatives of dimensional variables depend on dimensionless derivatives 19

And wesubstitutethemin the original equation:

So far we usedgeneralnotationforscaling factors. At thispoint, let͛s define some of them͗ v0с U ͙ aǀerage flow ǀelocity

dž0 с y0 с d ͙ tube (pipe) diameter

t0 с dͬU ͙ conǀectiǀe time (lengthͬǀelocity)

We divide the equation by a factor

Dimensionless Reynolds criterion Redimensionless form of equation 20 Furthermore, we define characteristic pressure as Then:

1 parameter instead of two

Pa.s.m.s-1.m-1= Pa

Viscous term

Pressure drop

Inertial term

Navier-Stokes

equation

Fluideflow

Reфф1 ї ǀelocity ǀery low and ǀiscosity high ї inertial term close to 0

Reхх1 ї ǀiscosity ǀery low ї inertial forces high ї ǀiscous term can be neglected

Re is the number that tells us whether a flow is turbulent (inertial forces dominate) or not

We can use scaling argument to get the ratio

Re = ʌd U / ɻ

ratio of inertial forces to viscous forces 21

Scaling

A special case -dimensionless-characteristic properties, in other words scaling factors, are defined in a

way that dimensionless variables (both dependent and independent) and their amount of changes are equal to values in the order of 1. If we obtain In many cases, it is rather difficult to find scaling factors.

If the equation is well scaled, the values of dimensionless criteria determine the weight of particular terms of the

equation. Some terms can thus be neglected.

Scaling is an important aid (tool) in the derivation of the theoretical criterion equation for calculating Nusseltor

Sherwood numbers

Derivatives also acquire values in order of 1

22
Example: Perform scaling of the Fourier equation for heat conduction T͙ temperature in K t͙ time in s x͙ spatial coordinate in m a ͙ thermal diffusiǀityFirst, we transform the equation to dimensionless one:

Now we have to define the scaling factors for t0and x0. Usually the size of the system is known, for example the wall thickness,

tube diameter, ͙ For edžample, for pipeline the characteristic dimension is the diameter d. If we set x0= d, then it is assured

that (and that͛s what we want) As weassumethat-We did not have to define scaling of T0but it is usually the difference betweenthe maximum and minimum tempereture of the system. 23

Furthemore,we know that

and Whatremainsatthispoint isto definethescalingfortimeto hold true: and

Weidentifytimescaling

24

Diffusion/conductivetime

¾Timeduringwhichtheheatistransferred

atadistancedbyheatconductivity.

¾Timeisproportinaltothesquareofthis

distance.

¾Itisratheraroughestimateoftheorder

thanitsexactvalue. 25
26

Lecture 2

Heat control, Fourier's law

Fourier equation, derivation for general control volume boundary conditions

Biot's number

Steady heat conduction in the thickness variable plate (seminar)

Steady rods in a circular cross section (seminar)

Thermal resistance

Mechanism of heat conduction

27

Particles/Moleculescan:

¾vibrate:wigglefromafixedposition

¾translate:movefromonelocationtoanother

¾rotate:revolveonanimaginaryaxis

Thesemotionsgivetotheparticles/moleculeskineticenergy. Solids can not move through space. They only vibrate. Liquids and gases are free to move around in space. They can have all three modes of motion. Three modes of molecular motion:Heat transfer: Basic concepts https://chem.libretexts.org 28
Temperatureisameasureoftheaverageamountofkineticenergypossessedbytheparticlesinasampleofmatter. Themoretheparticlesvibrate,translateandrotate,thegreaterthetemperatureoftheobject.

Heattransferisatransferofkineticenergyofmolecules

Itisthetemperaturedifference(temperaturegradient)betweenthetwoneighboringobjectsthatcausesthisheat transfer. Heatflowsindirectionofdecreasingtemperaturessincehighertemperaturesareassociatedwithhighermolecular energy. Theheattransfercontinuesuntilthetwoobjectshavereachedthermalequilibriumandareatthesametemperature Heattransfercanbegroupedintothreebroadcategories:conduction,convection,andradiation.

Heat transfer: Basic concepts

Heat transfer modes

ConductionConvectionRadiation

29

Conduction

Conductiontransfersheatviadirectmolecularcollisionwithoutanymotionofthematerialasawhole. Anareaofgreaterkineticenergywilltransferthermalenergytoanareawithlowerkineticenergy. Heattransferbyconductionappliesinsolid,liquidandgaseousmaterials,insystemsatrestaswellin motion Conductionisthemostcommonformofheattransferandoccursviaphysicalcontact. Exampleswouldbetoplaceyourhandagainstawindoworplacemetalintoanopenflame.

Example:

In commercial heat exchange equipment,heat is conducted through a solid wall (often a tube wall) that separates two fluids

having different temperatures. https://phys.org

Heat transfer: Basic concepts

Convection

Convective heat transfer occurs when a gas or liquid flows past a solid surface whose temperature is

different from that of the fluid.

Example:

Whenafluid(e.g.airoraliquid)isheatedandthentravelsawayfromthesourceofheat,itcarriesthethermalenergyalong.

Thefluidaboveahotsurfaceexpands,becomeslessdense,andrises.

Astheimmediatehotfluidrises,itpushesdenser,colderfluiddowncausingconvectioncurrentswhichtransportenergy

Forced convectionNatural convection

Fluid motion is caused by an external

agent such as a pump or blower.

Fluid motion is the result of buoyancy forces

created by temperature differences within the fluid. 30

ConductiveHeat Transfer

The time rate of heat flow(or heat transfer)is proportional to the temperature gradient. The constant of proportionality is

a coefficientofthermalconductivity.

qx... time rate of heat flow [W m2]-heat transferred per unit of time through the cross-sectional area

͙ coefficient of thermal conductivity [W m-1 K-1]: depends on the thermodynamic state of the material

T ͙ temperature ΀K΁

dž ͙ spatial

Fourier's Law of Heat Conduction

Temperature gradient: Driving force for heat conduction (negative) Conductive heat transfer can be expressed with "Fourier's Law" Fourier͛s law is ǀalid in this form only if thermal conductiǀity can be assumed constant. 31
Heat flow Qx[W] in the perpendicular direction to the area A In a general case (form) -heat conduction can occur in all directions: -coefficient of thermal conductivity -an important property of materials -W m-1 K-1

Metals -good thermal conductors -101-102W m-1 K-1

Thermal insulators (cork, foam plastic, cotton) -10-2Wm-1 K-1 bricks1 W m-1 K-1 water 0.6 W m-1 K-1

Air0.025 W m-1 K-1

T ͙ temperature gradient -difference -over the material (oC,oF) ѐ

A -area

qx... time rate of heat flow [W/m2]

A ͙ heat transfer area m2

Partial derivatives used:

temperature varies in all three directions. 32

Heat flow

From the first law of thermodynamics, it follows that for isobaric system (constant pressure) performing only volume work the

change of enthalpy is equal to the heat exchanged between the system and its surrounding We will express the change of enthalpy of the system using specific enthalpy: h ͙ specific enthalpy ΀J kg-1]

V ͙ system ǀolume ΀m3]

͙ density ΀kg m-3]

In the case when the system volume as well as density (or mass) are constant, then Cp ͙ specific heat capacity -an important property of materials [J kg-1 K-1] (the amount of heat required to increase the temperature of 1 kg of a material by 1 K (1°C)). 33
Non-stationary heat conduction in one dimensional (1D) system

q* -volume source of heat [W m-3΁ (Joule heat (electric current through conductor), reaction heat (e.g. nuclear reaction)͙)

Heat energy balance in the element V

INPUT + SOURCE = OUTPUT + ACCUMULATION

Fourier equation

thermal diffusivity [m2s-1] Area

Control volume ȴV

(source) Area Example -Plug flow reactor model (PFR) (piston flow reactor)

Controlarea S

ControlvolumeV

34
Derivation of the Fourier equation for a general volume element

Balance of thermal energy in infinitesimalelement

INPUT + SOURCE = OUTPUT + ACCUMULATION

INPUT -OUTPUT+ SOURCE = ACCUMULATION

Totalheat transfer across the

boundaries of the system Totalheat transfer across the boundaries of the system qn͙the normal component of the heat flow intensity across the boundaries of the system

Unit vector

35

Accumulationin thevolumeelement

Weassumeand Cpare constant

Balance:

Gauss transform:

q* -volume source of energy [W m-3] The balance must hold also for elementary volume dV, then: 36

Thermal diffusivity:

Fourier equation for heat conduction

General form for all coordinate

systems 37

Example to solve

Seminar II to solve

Edge

Zero flux

38

Typical boundary conditions

‰First-type (Dirichlet) boundary condition

The value of temperature on the edge is defined as:

At Z=0 їTedge= T0

‰Second-type (Neumann) boundary condition

Z=0Z=L

X=0 X=W

L1) Zero heat flow over the edge

2) Use of Neumann's boundary conditions at the symmetry

axis and in the semi-infinite domains

Usage -axis of symmetry

Semi-infinite domains

materialImaterialII 39
Condition of continuity of heat flow on the phase interface

Convective boundary condition, condition of continuity of heat flow on the phase-interface (one of the phases conveys

heat by conduction and convection = heat transfer), Robin boundary condition (third type boundary condition) phase interphasesolid phaseflowing fluid -coefficient of heat transfer [Wm-2K-1]depends on -geometry -flow type -material properties of the fluid

From courses of CHI

No accumulation of the heat at the interface.

Flux continuity. Used for all system coordinates

40

Steady heat conduction

Modelling of the heat transfer through the walls as: -steady state, -one dimensional For one-dimensional heat conduction (temperature depending on one variable only), we can devidea basic description of the process. Measure the temperatures of an exposed surface of a plane wall For 1 D heat transfer through a plane wall of thickness L, specified temperature

Boundary conditions are expressed as:T(x=0) = Tw1

T(x=L) = Tw2

T1, ɲ1

Tw1 Tw2 ʄ

T2, ɲ2

X=0X=LL

Heat transfer through a wall is

one-dimensional when the temperature of the wall varies is one direction only 41

Biot͛s number

Biotnumbershowshowconvectionandconductionheattransferphenomenaarerelated. Smallvaluesofthisnumbershowsthattheconductionisthemainheattransfermethod,whilehighvaluesofthisnumber indicatesthattheconvectionisthemainheattransfermechanism.

Bi = (L/Ȝ)/(1/Į)

= Į.L/Ȝ Biotnumber = Internal conductive resistance within the body External convective resistance at the surface of the body

Bi > > 1External resistance is very small

Bi < <1: Internal resistance is very small: high conductive

0.1

ɲ: Heat transfer coefficient: Intensity of sharing the heat by transport by convection from the surface to the surrounding

ʄ: Thermal conductivity: Intensity of heat transport inside the solid by conduction to the surface

42

Thermal resistance

The concept of thermal resistance is based on the observation that many diverse physical phenomena can be described by a

general rate equation that may be stated as follows:

Flow rate =

Driving force

resistance

The quantity that flows is heat (thermal energy) and the driving force is the temperature difference. The resistance to heat

transfer is termed the thermal resistance, and is denoted byRth. Thus, the general rate equation may be written as:

A: cross-sectional area, across which the heat flows

T1-T2: temperature difference

B: thickness of the material.

In rectangular system:

ʄ

Thethermalresistanceconceptpermitssomerelativelycomplexheat-transferproblemstobesolvedinaverysimplemanner.

Thereasonisthatthermalresistancescanbecombinedinthesamewayaselectricalresistances.Thus,forresistancesin

series,thetotalresistanceisthesumoftheindividualresistances:

Resistance in series:

Resistance in parallel:

43

Mechanisms of Heat Conduction

Processes responsible for conduction take place at the molecular or atomic level.

Heat conduction: random molecular motion

Thermal energy is the energy associated with translational, vibrational, and rotational motions of the molecules comprising

a substance.

high-energy molecule moves from a high-temperature region of a fluid toward a region of lower temperature (and, hence, lower

thermal energy), it carries its thermal energy along with it.

When a high-energy molecule collides with one of lower energy, there is a partial transfer of energy to the lower-energy

molecule Molecular motions and interactions is a net transfer of thermal energy from regions of higher temperature to regions of lower temperature. Insolids:resultofvibrationsofthesolidlatticeandofthemotionoffreeelectronsinthematerial. Inmetals:freeelectronsareplentiful,thermalenergytransportbyelectronspredominates. Innon-metallicsolids:thermalenergytransportoccursprimarilybylatticevibrations. Moreregularthelatticestructureofamaterialis,thehigheritsthermalconductivity(e.g.Quartz). Materialsthatarepoorelectricalconductorsmayneverthelessbegoodheatconductors(diamond). Insulatingmaterials,bothnaturalandman-made,owetheireffectivenesstoairorothergasestrappedinsmall

compartmentsїrelativelylowthermalconductivityofair(andothergases),therebyimpartsaloweffectivethermal

conductivitytothematerialasawhole. 44
45

Heat transfer in ribbed surface

Thin film approximation

Heat exchange efficiency over

Ribbed surface

Lecture 3

46

Heat conduction overa ribbed surface

Significance: AEheaters, heatingelements

AEengine coolers

AECPU coolers (Central Processing Unit coolers)

Schematic figureAssumptions:

H>>LAEchanges in the direction of yare negligible

Rib material is an excellent heat conductor AE

temperature changes in the base of the body in the z direction are negligible Ribs are sufficiently distant AEno mutual influence of heat transfer between individual ribs The system is in steady state and inside ribs there is no source of heat Rib

Body base

Source of heat

Heating fluid

CPU body

Symmetry area

47

Fourier equation

Laplace equation

4 boundary conditions:

It is possible to analyze heat transfer for each rib separately symmetry

Transferconduction/convection

Temperatureofsurroundingfar awayfromribs

fixedtemperature transfer/conduction 48

Thin layer approximation of the ribbed surface

¾Consider steady heat transfer from an edžtended surface or ͞fin" to the surrounding (e.g. air).

¾L and W are such L/W>>1.

¾It is assumed that Bi<<1.

¾It is assumed that y direction is large enough to make the problem 2D.

¾Thus we assume that T=T(x,z).

¾Due to the symmetry of the ribї Consider half of the object.

¾Difference between a fin and fully submerged object is that the temperature at one end of the fin is fixed.

¾Though the small Bi does not make the fin isotherm, it allows us to eliminate one of the independent variables: Importance of

resulting ͞fin approdžimation" is that it is prototype for reducing 2D model into a 1D one. ¾T is not function of x (No change of T in x axis at constant Z).

¾Given that the temperature field is approximately 1D, the local value can be replaced by the cross-sectional average (at

constant Z).

Z=0Z=L

X=0 X=W Z

Bi<< 1 (conduction)

Z X T=T0 Tь

T(x, z)

Problem 4

49

Ribefficiency

= heatflowfromthesurfaceoftheribto thesurroundingsatthemaximum drivingforce, i.e. the surface temperature of the rib is everywhere T0and the drivingforce is (T0-T) heat flow from the surface of the rib

We will express it using dimensionless quantities

Ⱥ= (T-Tь)/(T0-T)

ɻ= L/W

Z=Z/W

Problem 4

50
Only for systems where there is no conduction (stationary)

Fourier equation

(y direction) 51
52
53

LECTURE 4

Steady heat conduction in multiple

dimensions (rectangular cross-section body) and uninterrupted heat conduction (heat transfer over a membrane) Steady heat conduction in multiple spatial dimensions

Fourier equation

In the simplest case, a long rectangular cross-section (beam, wire, ...) can be considered, which is placed in an environment

with constant properties

In the body, heat can be released due to the passage of electric current (resistance wire) or due to chemical reaction (type of

plug/piston reactor).

The nature of the solution [T (x, y)] depends on the choice of boundary conditions. At all edges, for example, the constant

temperature T0can be considered. 54

Fourier equation + boundary conditions:

Before the solution, it is appropriate to modify the equation to make the boundary condition homogeneous.

So we define:

Boundary condition

Equation

Edges

T= T-T0^

In the edges

55
Analyticalsolutioncanbeobtainedby finiteFourier transform(FFT) methodorothermethod The FFT method will not be discussed in the basic course and will not part oftheexamination. 56
Transientheatconductionin a spatially distributed system

Large flat plate of the thickness

For the solution of the problem we need two boundary conditions and one initial condition

Transient heat conduction

The temperature of a body, in general, varies with time as well as position Let us consider spatially 1D system without heat source

We consider the variation of temperature with time and position in one-dimensional problems such as those associated with

a large plane wall, a long cylinder, and a sphere, e.g. membrane, planar heat transfer surface, brick wall͙

57
Let͛s consider that eǀerywhere inside the plate, initial temperature is T0

At time t> 0, we will increase the temperature on left edge to a value of T1. On the right edge we will keep the

temperature T0.

By solution of the Fourier equation we will obtain the temperature as a function of time and coordinate x.

The student should be able at this point to answer:

1)What temperature profile will be established in the plate?

2)Order estimate of the time required to establish the temperature profile.

58
We will transform model equations to a dimensionless form

Initial condition

boundary conditions The exact solution can be found for example by FFT method: a: Thermal diffusivity (m2s-1) 59
( 60
61

Combined heat by conduction and convection

Deriving the Fourier-Kirchhoff equation for general control volume

Péclet'snumber

Lecture 5

62

Heat transfered by macroscopic

movement of matter (fluids)

Example:

Whenafluid(e.g.airoraliquid)isheatedandthentravelsaway fromthesourceofheat,itcarriesthethermalenergyalong. Thefluidaboveahotsurfaceexpands,becomeslessdense,and rises. Astheimmediatehotfluidrises,itpushesdenser,colderfluid downcausingconvectioncurrentswhichtransportenergy

Forced convectionNatural convection

Fluid motion is caused by an external

agent such as a pump or blower.

Fluid motion is the result of buoyancy forces

created by temperature differences within the fluid.

HeattransferbyConvection

Convective heat transfer occurs when a gas or liquid flows past a solid surface whose temperature is different from that of the fluid.

Convective heat transfer

Let us assume that the volume flow of the substance is perpendicular to the plane

Each mass carries a certain thermal contentQ.

In a usualcase: heatflowisequalto enthalpyflow(ሶܳൌሶܪ

We will further assume thatCpis constant

(i.e. Cpis not a function of temperature in a given temperature range Trefto T) 63
Fromthefirstlawofthermodynamics,itfollowsthatforisobaric system(constantpressure)performingonlyvolumeworkthe changeofenthalpyisequaltotheheatexchangedbetweenthe systemanditssurrounding h-specific enthalpy[kg/s][J/kg]

Average specific heat capacity [J kg-1K-1]

[W]

݀ܳൌ݀ܪ

݀ܳ

݀ݐൌ݀ܪ

݀ݐ

. A V. The intensity of the heat flow through the convection of the surface A is thus: v-Velocity of convective flow[ms-1] The intensity of the heat flow is oriented in space: 64
The transfer by conduction and convection typically takes place simultaneously. The overall intensity of heat flow qis a sum of that by conduction qvand convection qk

Generally:

For coordinate x:

65
Transient heat transfer by conduction and convection in 1D system We balance the heat energy in the control volume V

INPUT + SOURCE = OUTPUT + ACCUMULATION

Volumetric source of heat

66
constants Thecontinuityequationapplies.If we consider an incompressible flow(liquid), then

Itfollowsthatvxisconstantinthespatial1Dsystem.Ifthecross-sectionofthesystemalongtheaxischanged,vxwouldalso

change.However,itwouldbea2Dsystem!

Fourier-Kirchhoffequationin 1D-system

67

Incompressibleflowimpliesthatthedensityremainsconstantwithinaparceloffluidthatmoveswiththeflowvelocity

їdivergenceofflowvelocityiszero

Slide 15

3D-system

1D-system

Derivation of the Fourier-Kirchhoff equation in general form

V-controlvolume

S-area enclosing the control volume

The heat flux Qpasses through the boundary of the system. The flowleaving(entering) thesystem flow is the flow

perpendicular to the surface.

Thus, the heat fluxthrough the area dSis: qn.dS

qn-the normal component of the vectorq

INPUT -OUTPUT + SOURCE = ACCUMULATION

Sum offlowsacross

allboundariesSum ofsourcesovertheentirevolume

Accumulationofthermal

energyin thewholesystem68 massof element dVsum ofaccumulationsofheatin theentirevolume

Wewillwritethebalance:

If the volume element does not depend on time, we can change the order of integration andderivation in the accumulation term:

Thebalance holdsalsoforthevolumedV:

69
- the scalar product is distributive

Bothquantitiesdependon

spatialcoordinates

It holds true that:

-Proofas a homework f -scalarfunction a-vectorfunction -Continuity equation for an incompressible fluid70

ї operator of the material derivative

Fourier-Kirchhoffequation

In Cartesian coordinates,the equation can be written as: 71
Material derivative describes the time rate of change of some physical quantity (like heat or momentum) of a material element that is subjected to a space-and- time-dependent macroscopic velocity field variations of that physical quantity TransformationofFourier-Kirchhoff (FK) equation into dimensionless form

Let-convectivetime

Pe-Pécletnumber

Letq* -dimensionlessvolumesource ofheat

72

Fourier-Kirchhoffequation

Physicalsignificance of Pécletnumber (Pe)

velocity of heat sharingby convection (velocity of convection) Velocity of heat sharingby conduction(velocity of conduction)

Conductivetime:

Conductivevelocity:

73

V0 = u

X0 = L

When a/x0tends to0: Petends to infinite: Convection mechanism dominates If V0tends to 0, Petends to 0. Almost no flow of the fluid: Conduction mechanism dominates 74
75

Lecture 6

Transient Heat transfer

Combined heat transfer by conduction and convection

Newton's law of cooling

Nusseltcriterion

Nusseltcriterionin body wrapping

Qualitative Behavior of Nusselt's Criterion in a Limited Area in Laminar Flow

Graetzproblem

76

Transient Heat transfer

WallFluid

Thermal sublayer

Heat transfer by

convectionand conduction

Fluid core

Heat transfer by

convection Combinedheat transfer by conduction and convection -very common

Typical examples:

heat transfer between the heat exchange surface andfluid in recurrent exchangers flowaroundparticles(particle drying)

free convection (natural convection: by density differences in the fluid occurring due to temperature gradients.) (gas

heating over heating elements) Typical temperature distributionforfluid flowalongat the surface

Constant flux

77
Theimportanceofconductionincreasesin a directiontowardsthewallas the flowvelocity approaches

0 at the wall.

At thelayerclosestto thewallthefluid velocityiszero, hence the heat transfer at the phase interface takes place only by conduction.

At steady state, the heat flow in the most immediate layermust be equal to the heat flow over the entire

thermal sublayer, therefore: wall

Newton's law of cooling

thermalconductivityoffluid heattransfer coefficient 78
Equations can be transformed to dimensionless forms Characteristic dimension of the system such as tubing diameter wall

Afterre-arrangement:

Nusseltcriterion

Nu -how many times the heat transfer (conduction+ convection) is more intense than in the case of stationary fluid (where heat transfer isonlydueto conduction).

For stationary fluid, it holds that:

In heat transfer at a boundary (surface)

within a fluid, Nu: Ratio of convective to conductive heat transfer across (normal to) the boundary.wall wall

Dimensionless form of heat transfer coefficient

79

For stationary fluid, it holds that:

TheNusseltnumberdependson thenatureofthefluid flow, thefluid propertiesand thegeometricarrangement. position geometricsimplexes

The nature of fluid flow:

vkinematicviscosity[m2 s-1]

Properties of the fluid:

Prandtl number

Reynolds number

(ratio of momentum diffusivity to thermal diffusivity) (ratio of inertial forces to viscous forces) wall 80
In some cases, the dependency of the Nusselt criterion can be written: To calculatetheheattransfer coefficient, itisnecessaryto finda suitabledependence forcalculatingthe

Nusseltcriterion

Dependenciescanbeobtained:

By the solution of the Fourier-Kirchhoff equation and possibly other transport equations empirically

Position

81
ThevalueoftheNusseltcriterionispositiondependent.Forexample,incaseofbodywrapping,the Nussletcriterionisdifferentateachsurfacelocation-thevalueofthenormaltemperaturederivative changestothesurface.

Body at high temperature

Flow direction

Thickness of thermal sublayer

82
Itisthereforeadvantageousto definetheaveragevalueoftheNusseltnumberon theentiresurfaceoftheobject. Thescalarproductexpressesthetemperaturederivativevaluein thedirectionofthenormalvector, a vectorperpendicularto thebody surface. TherelationshipsforcalculatingNu canbefoundin theform: 83
Qualitative Behavior of Nusselt's Criterion in a Limited Area in Laminar Flow

Piping systems

Flow between flat plates

Flowing liquid films

Let͛sconsiderpipewithradialcoordinatesrandaxialz.Wealsoconsiderthatthefluidinthepipelineflowsinlaminarregime

accordingtotheaxisz,thefluidhasatemperatureofT0attheinletandthewallshaveconstanttemperatureTW.

Flow direction

Entrance region Thermal developedregion

Thecoreofthefluid (thehatchedarea) remainsata distance fromunheatedinput -not yetaffectedby theheatflowfrom

thewallsconduit. Thisistheso-calledentryarea. Oncetheheatfromthewallsarrivesto the middlepart, theentire

volumeoffluidis affectedby heat(warmed/ cooled). Thenwe talk aboutso-calledthermaldevelopedarea. 84

Qualitativecharacterofthetemperaturefield

along the zcoordinate decreases The value TW-T0remains constant throughout the entranceregion The value Nu in the entranceregiondecreases along the axis z

Entrance area

85

EntranceregionThermally developedregion

decreasing constant decreasing decreasing decreasing constant ¾Nu is very large at z=0 and declines with increasing z

¾Typically, Nu initially varies as some inverse power of z, so that a log-log plot of Nu(z) is linear at small z.

¾For long enough tubes or films, Nu approaches a constant, even though the temperature may continue to depend

on z.

¾The position at which Nu becomes essentially constant separates the thermal entrance region from the thermally

fully developed region. 86

Graetz'sproblem

Steady heat transfer, steady laminar flow without heat inside(no heat source)

Fourier-Kirchhoff's(FK) equation

solution of the Navier-Stokes equation for the laminar flow of incompressible fluid in a circular cross section called "Poisevilleflow"

Uistheaverageflowrate.

87

vr= 0 AEin laminar flow the fluid moves only in the direction of z sincePe>> 1 in zdirection(assumption)

3 boundaryconditions:

Temperature at pipe inlet

Symmetry, the heat does flow through the center in thedirectionr

Pipe wall temperature

Transformation to dimensionless form:

Uis themean velocity (velocity in the middle of the pipeline)

R: tube radius

Differential operations in

cylindrical coordinates

Convection mechanism dominates

Neglecting axial conduction

Called Graetzproblem

88
ThenoteworthyaspectoftheindependentvariablesisthatPehasbeen embeddedintheaxialcoordinate. Thischoiceofismotivatedbythefactthatwhenalltermsinequation aremadedimensionless,zandPeappearonlyasaratio. Peisbasedonlyonmeanvelocityandthediameter,whichisusualconvention forcirculartubes. 89

See definition of Pe

Transformationofboundaryconditions

to dimensiolessform: r 90
Because the solution to Graetz's problem is quite complicated, it will be solved numerically during the seminar. The main conclusion of Graetz analytical solution is that the Nu value in the thermally developedregion is constant. Specifically, in a circularcross-sectiontube for Here, an asymptotic (approximate) solution allowing finding the value of the Nusselt criterion in the entranceregion will be presented:

Not affectedcoreofthefluid

91
A substitutionwillbeintroducedso thatthecoordinateaxis has a beginningon thepipe walland isequalto oneatthecenter ofthepipe

Boundaryconditions:

=-1 92

In place offromthetube entry, theheatwastransferredto a distance offromthewall. Derivativeswillbereplaced

by differencesand anestimateofthesizeoftheindividualmembersoftheequationwillbemade.

This is the order estimate,

therefore the coefficient 2 will be neglected 93
Nowallthreeoftheresultingtermswillbeputintotheoriginalequation.

Becauseisvery small

We will calculate the Nusselt criterion:

x small ߜ ߜ

ο෤ݕൎͳ

ߜ ͳ

اߜ

ߜ 94

Example: Calculate the Nusselt criterion

ExactrelationshipforNu forgivenboundaryconditionsand geometry.

Estimated length of the inlet/entranceregion:

, where LT-The distance from the pipe inlet, when Nu reaches a constant value. 95
96

Lecture 7

Laminar flow alonga solid object

97

Laminar flow alonga solid object -Heat transfer

Flow alonga solid object ї heat transfer coefficient ɲ Heat transfer fluid-solid wall depends on flow-object orientation

Velocityprofile 2D or 3D

Pe » 1 ї temperatureedge sublayer: temperature change from surface (then constanttemperaturein bulk)

Solid object = sphere їvelocityprofile, heat transfer

Shape of velocityprofile,

spherical coordinates

Decomposition of average velocity U

Velocity profile dependent on r, ɲcoordinates

Velocity profile symmetry: based on rotational axis ੮

Velocity U composed of two parts

Based on Navier-Stokes:

Radius

Average velocityfar from sphere

98
FK equation in a steady state without heat source: FK equation in spherical coordinates (transformation formula (1) and (4) in page 85):

Temperature boundary conditions:

Scaling factors:

So far, angle ɲis not scaled

·(R/a)

Pe = (U.R)/aR

R2 R 99

Heat transfer for Stokes flow(creeping flow) (inertial forces are small compared with viscous forces)(Re ї 0)͗

Pe ا

-For Pe ї 0 heat transfer by conǀection is negligible -Heat transfer occurs primarily by conduction -Simplified form of FK equation:

-Since convection is negligible, fluid flow does not deform the temperature profile around sphere ї symetrical according to ɲ

cooridanate -Another simplification:Solution: << 1 100
Nu criterion for sphere with symetrical temperature profile: -If: -Or:

Nu = 1

Nu = 2

Heat transfer for Stokes flow (Re ї 0)͗ Peب a: thermal diffusivity [m2s-1]

ʆ: Kinematic viscosity [m2s-1]

High viscosity fluidsї ǀelocity profile is deǀeloped Ƌuite far from sphere

Low thermal diffusivity ї temperature changes only realy close to the surface = Temperature sublayer

Temperature and heat conductivity are linearly dependent

Thus:

101

Velocity profile

Temperature sublayer

Figure: Shape of velocity and temperature profiles. -Even though Pe ب -Heat conduction is the only mechanism of heat transport from sphere surface to fluidsurroundings -Velocity U = 0 on the surface -(Mechanism of radiation will be discussed later)

Two parts of solution: INNER and OUTER

1) OUTER solution

Description of temperature outside the sublayer

No meaning for calculation of Nu criterion

Without conduction parts + BC

Outer solution:

Temperature is not

a function of ɲangle. 102

1) INNER solution

Description of temperature inside the sublayer

Necessaryfor calculation of Nu criterion

Both effects -conduction and convection

Scaling ї all ǀariables ǀalue of appros. 1

FK equation:

-Temperature ɽ, velocities vR, vɲу 1 103

A)Scaling of ɲ(0-ʋ)

x /divided by Pe criterion

Left side of F.K. equation

Right side of F.K. equation

F.K. equation

(0, 180) Now scaling of radial coordinates should be performed ʇfrom <1, 1+x> to the interval <0,1> x: sublayer thickness

New radial coordinate <0,1>

When ʇ= 1їY = 0With increasing Pe, sublayer width is decreasing: conduction velocity very low compared to convection velocity. Therefore Y depends on Pe bis unknown constant. It must be positive in order to make new radialbetween <0,1> 105

B) Scaling of ʇ(1;1+sublayer width)

-New Y coordinate -ʇ= 1, Y = 0

-With increasing Pe, sublayer width is decreasing: conduction velocity very low compared to convection velocity

-b constant is positive

Value comparison of individual parts of FK

1. part

уPe-b

size

Size ~ 1

1stterm is in the order of1+x-1x=Y/Peb

106

Value comparison of individual parts of FK

2. part

уPe-b

3. part

уPe-b

уPeb-1

In thermal sublayer ʇis close to1, therefore x is close to 0 ї dž2 << x 107

Value comparison of individual parts of FK

4. part

уPe2b-1

5. part

уPe-1

All parts:

Peب

Pe-b = Pe2b-1-b = 2b-1b = 1/3

108
We can express Nu criterion as a function of ɲ:

Kis function of object shape and position:

Average Nu criterion of all surface:

For sphere (Peب

Constant size ~ 1

109
Heat transferin a laminar sublayer:Reب1, Peب

For Re ب1, Pe ب

Width may be various

-Prا -ʆ ї 0, a їь -Prب -ʆ ї ь, a ї0

Prу 1

ʆ у aɷT = ɷV

wall wallTemperature profile

Velocity profile

Velocity profile

Temperature profile

110

Nu criterion for laminar flow

arbitrary Stokes flow around solid object (small particles, aerosol͙)

Laminar flow around solid object

Laminar flow along solid object

One liquid along another (emulsions, bubbles͙)

The dependence is always in the form

and must hold Pe>> 1Nu= k ReaPrb 112

Lecture 8

Heat transfer by Natural (Free) convection

113

Heat transfer by Natural (Free) convection

LiƋuid temperature change ї density change (temperature dilatation) ї influences ǀelocity and pressure profile

FK eq. + NS ew. (2D or 3D) + continuity eq. solved together

Balance of momentum is given by NS:

ʌ с ʌ (T), ɻ= ɻ(T)

3 parts of NS equation influenced by temperature change!

Definition of dynamic pressure (including gravity) for reference temperature T0 and density ʌ0:

Dynamic pressure combined with NS eq.:

Driving force of natural convection

Inertial term

Navier-Stokes

equation

Fluideflow

Viscous term

114
-Assumption: Liquid density is influenced only by temperature change. It is not dependent on concentration of compounds. -Then, we are allowed to use Boussinesq approximation (buoyancy) (2 parts):

1)Density(T) changes lineary around T0 temperature. We can use Taylor series:

Definition of temperature dilatation coefficient (coefficient of expansion of the fluid) ɴ: Combination of Taylor series with formula of temperaturedilatationcoefficient:

Then, combined with NS and divided by density:

kinematic viscosity

ȡȡ0= ȕȡ(T-T0)

Boussinesqapproximation:

Thedensityisassumedtobeconstantinalltheconservationequationsexceptinthebody forceterminY-momentumequation,wherethetemperaturedependentdensitythat drivestheflowinnaturalconvectioniscapturedbytakingBoussinesqapproximationinto account:

Boussinesq approximation (buoyancy)

This approximation is accurate as long as changes in actual density are small

ȡȡ0= ȕȡ(T-T0)

116
-Then, we are allowed to use Boussinesq approximation (2 parts):

2) Density change is negligible around reference temperature: ʌ(T) уʌ(T0).

Density change is negligible if:

Then:

Temperature dilatation coefficient for ideal gas:

oror For gazB can be close to 0In liquids delta rho can be very small

Characteristic properties are constants:

A, rho 0, nu 0

117
Heat transfer by Natural convection near a vertical infinite wall

Constant

wall temperature

Heating of

liquid

Liquid

movement upwards -Heat transport influences density

ї change of ǀelocity and pressure profile

-FK eq. + NS eq. + continuity eq.:

ї for ǀelocity, pressure, temperature

Steady state without heat source:

Continuity eq.

118
Heat transfer by Natural convection in avertical infinite wall -Characteristic variables in value interval <0,1>: -Natural convection:

Intertial+pressure+free convection

parts of NS eq. approx. the same value(magnitude).

The viscosity part of NS

is negligible. T0 -reference temperature, ȴT -maximum temperature difference Ub -reference velocity of natural convection (now unknown)

L -characteristic length (often unknown)

This has to be set.Far from the wall

Nondimensionalization of NS eq.:

-For natural convection, all above mentioned parts have approx. the same value -Then: -Viscosity part of NS eq. ismultiplied by -Reynolds criterion:

Grashof criterion

NS eq. with Grashof criterion:

By analogy, dimensionless form of NS eq. in case of vyis:

Continuity eq.:

FK eq.:

-For description of natural convection, Rayleighnumber is used: -The goal of calculations is Nusselt criterion:

Péclet criterion for natural convection

Geometry simplex

Geometry simplex, Position

Destabilizing forces (natural convection)

Stabilizing forces (viscosity forces)

-For ї ǀelocity sublayer occurs (similar to laminar flow around solid objects)

Heat transfer by Natural convection near a vertical infinite wall: Gr1/2ب wall bulk edge layer

1)INNER solution: in edge layer.

-intertial+viscosity+free convection parts of NS eq. approx. the same value

2) OUTER solution: bulk.

-intertial+free convection parts of NS eq. approx. the same value -viscosity part of NS is negligible INNER solution is necessaryor Nu criterion calculation: љ Simplificationї lower number of used eƋuations Heat transfer by Natural convection near a vertical infinite wall: Gr1/2ب wall bulk very thin velocity sublayer

1)Second deriǀatiǀes in dž direction ї 0 near infinite wall.

2)In sublayer: pressure is not changing in y direction.

1)+2) Similar approximation for flow between two infinite walls.

-We know pressure P=P(x) = inserted pressure ȴp ї we do not haǀe to calculate it ї we need 3 eq. for : NS, FK, continuity eq. * It is difficult to prove simplifications 1),2) mentioned previously. In general, it is allowed to reduce one equation: P=P(x). Heat transfer by Natural convection near a vertical infinite wall: Gr1/2ب -Final equations: љ

For ї 1 it is necessary to rescale y coordinate and velocity vy (Gr cannot reach infinity)

y Heat transfer by Natural convection near a vertical infinite wall: Gr1/2ب љ It is necessary to rescale y coordinate and velocity vy (Gr cannot reach infinity): ї all parts haǀe approdž. the same value(approx. same magnitude)

To eliminate Gr

Heat transfer by Natural convection near a vertical infinite wall: Gr1/2ب Sublayer properties are influnced by Pr criterion too: љ

What are the ǀalues of indiǀidual parts of all used eƋuations͍ ї For limit ǀalues of Pr criterion͗ Pr ї 0

Pr ї ь

-Thickness of the thermalsublayer is influenced by Pr, but it should stay in interval from 0 to 1 a)DimensionlessY coordinate: b)Rescaled velocity (to keep velocity between values 0 and 1 in tempertature sublayer):

Thin velocity sublayer (low viscosity)

Wide temperature sublayer (high thermalconductivity)

Wide velocity sublayer (high viscosity)

Thin temperature sublayer (low thermalconductivity)

Unknown index

Heat transfer by Natural convection near a vertical infinite wall: Gr1/2ب

Rescaled equations:

Velocity sublayer

(important equation parts):

1)Viscosity

2)Intertial

3)Natural

convection

Outside of velocity

sublayer: Viscosity neglected

1)Intertial

2)Natural

convection Thermaland velocity sublayers near toinfinity wall.

Therefore:

N-S equation

In the thermal sub-layer

Heat transfer by Natural convection near a vertical infinite wall: Gr1/2ب

In momentum balance, viscosity

and natural convection prevail. From From

From continuity equation

oror From

From continuity equation

not usable From Heat transfer by Natural convection near a vertical infinite wall: Gr1/2ب

Now, we can calculate Nu criterion near to

wall surface: Local ǀalue near to the surface is approdž. у 1 Nu criterion for vertical wallNu criterion for vertical wall 131

Lecture 9

Heat transfer in turbulent and restricted environment 132
Heat transfer in turbulent and restricted environment

Character of flow is described by Rynolds number:

1)Re

2)Re >Rec:transient flow= laminar flow is changing with turbulent flow

3)Re ب

Characterization of turbulent flow:

Fluctuation of velocity, temperature and other parameters

Time dependent3D (always) flow

Swirling of fluid increases mass and heat transfer

Pressure drop is increasing

Fanning friction coefficient: ratio between the local shear stress and the local flow kinetic energy density

Describes dimensionless pressure drop

NS equation:

ܴ݁ൌܨ

ܨ

Inertial partPressure partViscosity part

Dynamic pressure

Velocity and pressure at a point

fluctuate with time in a random manner http://www.fponthenet.net/article/135747/Heat-exchanger-considerations.aspx

Inertial forces بviscosity forces (Re ب

ї left and right sides of NS eƋuation eƋual ї inertial part у pressure part

Rescaling/Dimensionless formula:

Fanning friction coefficient:

Dimensionless pressure drop

Darcy friction coefficient (Chemical Engineering I, II): (friction coefficient) 133

Inertial partPressure part

Average flow velocity

Tube radius

Tube length

Pressure drop over tube length L

where:Negligiblein thecentre,

ј nearto thewall

ߩȉݒ௬ȉ߲ ߲ݕൎെ߲ ߲ ෦ݒ௫ൌݒ௫ ܷ ܷ ܦ ܮ

ο݌

ߩȉܷȉ෦ݒ௬ȉܷ

ܦȉ߲

߲

݌ȉ߲

߲ ߲ ߲ ߲෦ݒ௫߲ ିଵ ൌο݌ȉܦ

ܮȉߩȉܷ

p L

TheFanningfrictionfactor,namedafter

JohnThomasFanning,isadimensionless

number,thatisone-fourthoftheDarcy frictionfactor.

Attentionmustbepaidtonotewhichoneof

theseisusedasthefrictionfactor.Thisthe onlydifferencebetweenthesetwofactors.

Inallotheraspectstheyareidentical,andby

applyingtheconversionfactorof4the frictionfactorsmaybeusedinterchangeably.

Fanning friction factor

emprici 135

Pressure drop in tube

Empirical formula

Circular cross section (laminar flow):

Turbulent flow:

Karman-Nikuradse:

Blassius equation:

Laminarflow

(slope-1)

Increasedpressure

Lossdueto turbulence

slope

Blassius equation

Dependence of friction factor f on Re criterion.

or

Karman-Nikuradse

136
Time fluctuationoftemperaturein a particularpositionx,y,z: -Averagetemperatureisanintegraltemperature -Integrationin aninterval from0 to ta: tf+ ɽ -Averagevelocityhas thesimilarexpression: ԦݒൌԦݒ൅ܷ

Reynoldsaveraging

Vectorofvelocityfluctuationin timetAverage velocity in theinterval ta

Temperature fluctuations attwo different scales

Time scale of changes in the

whole system tS

Time scale of one fluctuation tF

t-time

Time fluctuation of temperature

137

1) The mean value of fluctuations in ta interval is equal to zero:<ɽ>= 0

2) RMS (root mean square) fluctuation = quality indicator of fluctuation size:

3) Nextaveragingofaveragedquantitydoesnot changethevalue:

4) Orderofaveragingand derivationmaybeinterchanged:

Basedon definition:

Forsimplificationonlyx and y spatialcoordinates

Intensity of fluctuation

Averaging

Properties of averaged quantities (T = + ɽ)

138

A)Averagingofcontinutiyequation:

B)AveragingofFK equationwithoutsource:

Averagedcontinuityequation

Otherproperties:

Divergence ofvectoroffluctuation= 0

Nextslide

Everypart individuallyaveraged:

Onlyin 2D to simplifytheproplem:

139

B) AveragingofFK equationwithoutsource:

Meanvalueof

Fluctuations=0

Then, FK:

Fourier lawAveragedintensity

ofheatflow by conduction 140

B) AveragingofFK equationwithoutsource:

Now, wewillanalyse

multiplication:

Definition of heat flow caused by turbulent flow:

FinalformofaveragedFK eq.:

Accelerated heat conduction by turbulent flow

Describes fluctuation

Estimated from averaged values

142
Constitutive equation for heat flow caused by turbulence: wall coordinateHeat istransferedby turbulentvortex

Turbulent vortex near to a wall + heat transfer.

Turbulentvortextransfersheatfromplace with

highertemperatureto place withlower temperature

Turbulentheattransfer = acceleratedheat

conductionї heatflowisproportionalto negative gradient ofaveragedtemperature

Both heat and momentum are carried by the same

vortex ɸH is not constant -it is dependent on the distance to thewall

Nearto wall: ɸH ї 0 (no vortex)

Usually: ɸM у 0.85ΎɸH

Consitutive equation:

ɸH -diffusivity of heat vortex [m2s-1]

ɸH уɸM (diffusivityof heat vortex is comparable to momentumdiffusivity )

Diffusivity model

Diffusivity model

143
Constitutiveequationforheatflowcausedby turbulence: wall coordinateHeat istransferedby turbulentvortex

Turbulent vortex near to a wall + heat transfer

Consitutiveequation:

ɸH -difusivityofheatvortex[m2s-1]

ɸH уɸM (diffusivityofheatvortexiscomparableto diffusivity of momentum)

How to findthe turbulentdifusivity of vortex?

їPrandtlmethod:

-ɸM/ɸH = 0.85 -Firstly, wecalculateɸM ї thenɸH -l -distance ofmixedvolume -l с ʃΎy (y isperpendiculardistance fromtube) -Usuallyʃ у 0.4

Normal(perpendicular) velocityderivative

relatedto tube surface

Diffusivityof

momentum:

Diffusivityof

heatvortex:

ݍ்ൌ݂ሺܶ

ݍ்ൎݍכ

ݍ஼ൌെɉȉοܶ

Conduction

Turbulence

144
Correlation from table data (propertiesof flow, type of liquid, geometry) ref. Blassiuseq.

Eq. isvalidfor:

Colburneq.:

Bhattiand Shaheq.:

Eq. isvalidfor:Accuracy:

&

Valid for&

&

Nu criterion for turbulent flow in tube

146

Lecture 10

Heat transfer in boiling liquid

147

Heat transfer in boiling liquid

Duringboilingї phasetransitionliquid-vapour

Significantchangeofdensity-approx. 3 ordersofmagnitude (1 000 kg/m3 ї 1 kgͬm3) Densitychangecausesflowofbothliquidphaseand vapourphase(similarto natural convection) Coefficientofheattransfer ismuch higherdueto much higherdifferencein densities(comparedto natural convection)

Boilingcurve

Describesintensity ofheatflowbetweenliquidphaseand heatingplate on theirtemperaturedifference

ȴTe = Ts-Tb

Temperature

difference (overheating)Temperature ofheatingplate

Temperature

ofboilingpoint (forspecificpressure) 148

Wecandescribe4 phasesofboiling:

Heat flow

Boiling phase is dependent on the temperature

difference between heating plate and bulk liquid.

Hu H, Xu C, Zhao Y, Ziegler KJ, Chung JN 2017. Boiling and quenching heat transfer advancement by nanoscale surface modification. Scientific Reports 7(1):6117.

S. Nukiyama, Maximum and minimum values of heat q transmitted from metal to boiling water under atmospheric pressure. J. Soc. Mech. Eng.Jpn. 37 (1934) 53-54, 367-374

Boiling phases

149

1)Natural convection (ȴTe ш 0)

-liquidiswarmingup nearto heatingplate -liquidiscirculatingdueto changesin densityї newcoldliquidisgettingnearto heatingplate -theresi no occurringvapourphase -temperatureofliquidisincreasing -Nu criterionї correlationfornatural convection: ɲу103 Wm-2K-1

2)Nucleateboiling(ȴTe >0)

-ifthetemperaturedifferenceissufficient, smallbubblesstartsto occuron theheatingplate -bubblesoccuron thenucleationspotsї growї riseupwards -propertiesofbubblefloware dependenton thevalueoftemperaturedifference

-more intensiveflow/vortexingcausesbetterheattransfer ї coefficientɲdramaticallyincreases(withincreasingȴTe )

useful in heat exchangers

Individual boiling phases (ȴTe >0)

150

2)Nucleation boiling (ȴTe >0)

3)Transientboilingphase(ȴTe ب

-hightemperaturedifferenceї heatingplate isfullycoveredwithoccuringbubblesї colderliquidcannotbein contact

withheatingplate -heatconductivityofvapourissignificantlylowerthanconductivityofwater -"boilingcrisis͞ ї itmaydamageheatingplate/exchanger ȴTe у 102 K ї ɲgoesfrom104to 102 Wm-2K-1

Heatingplate

Individual boiling phases (ȴTe >0)

151

4) Film boiling (ȴTe >burningtemperature)

-thewholesurfaceofheatingplate iscoveredwithvapour -vapourfromthewholesurfacestartsgettingawayin thesametimeї coldwatergetsnearto heatingplate again -heattransfer coefficientisminimalin theLeidenfrostpoint (ɲу 102 Wm-2K-1) -with further temperature increaseї ɲisincreasingtoo -film boilingisnot desirabledin exchangers Bubble creation near to a heating plate(Nucleate boiling)

Individual boiling phases (ȴTe >0)

152
Dependence ofheattransfer coefficienton theoverheating(ȴTe): -Fornucleateboilingї Rohensowcorrelation:

Temperaturedifference= overheating

Heat flow

G. Nellis, S. Klein,

Heat transfer, Cambridge University Press (2009), 782

Calculatedforboiling/condensation

temperature.

Dynamicviscosityofliquid

Densityofliquid

Densityofvapour

Surfacetensionliquid-vapour

Specificheatcapacityofliquid

Prandtlnumberofliquid

Exponent -usually1-1.7

Gravitationalacceleration

Enthalpydifferenceequil. liquid-vapour

Constantdependenton theheatplate

materialand liquidtype (TABLES) 153

Calculationofburningtemperature(film boiling):

-Lienhardand Dhiraequation: A: surfaceofheatingplateKRIT у CRIT у critical Constantdependenton geometry ofdescribedsystem(TABLES). http://fchart.com/ees/heat_transfer_library/boiling/hs2010.htm

Regimes of heat transfer and two-phase flow

in a heated channel

Figure 2

Figure 1

Example

http://www.thermopedia.com/content/605/ 157

Lecture 11

Heat transfer in condensation

Condensationoccurswheneveravapourcomesintocontactwithasurfaceatatemperaturelowerthanthesaturation temperaturecorrespondingtoitsvapourpressure.

1)Drop condensation -wanted, difficult to achieve, non-wetting surface, high heat transfer coefficient

2)Film condensation -usual, wetting suface

Thenatureofcondensationdependsuponwhethertheliquidthusformedwetsordoesnotwetthesolidsurface. Iftheliquidwetsthesurface,thecondensateflowsonthesurfaceintheformofalmandtheprocessiscalledfilm condensation.

Ifontheotherhand,theliquiddoesnotwetthesolidsurface,thecondensatecollectsintheformofdroplets,whicheither

growinsizeorcoalescewithneighbouringdropletsandeventuallyrollofthesurfaceundertheinfluenceofgravity.This

processiscalleddropcondensation.

Heat transfer in condensation

Film wise condensationDropwise condensation

Jumping-droplet super-

hydrophobic condensation

Flooding condensation

https://www.researchgate.net/figure/Condensation-heat-transfer-modes-and-performance-Images-of-a- filmwise-condensation_fig1_259433420 The most common and best understood case of condensation heat transfer is that of filmcondensationof a pure quiescent vapor on a solid surface. Application: electric power generation, process industries, refrigeration and air-conditioning͙

Film condensation

Liquid film starts forming at the top of the plate and flows downward under the influence of gravity.

ɷincreases in the flow directions x

Heat in the amount hlgis released during condensation and is transferred through the film to the plate surface Tsmust be below the saturation temperature for condensation. The temperature of the condensate is Tsatat the interface and decreases gradually to Tsat the wall. 160

Liquid film on the surface

Film thicknessɷ

Heat transfer overthisfilm

Film condensation

Liquidthermalconductivity (film)

Temperatureofcondensation

Temperatureofheatexchange

surfaceTW Liquidfilm

161

Heat flow͗ Ƌ с ɲ(TK-TW)

Assumption: Ƚൎ஛

ఋ(ɲisheattransfer coefficient) ɷ(thicknessoffilm) shouldbesmall

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