6.1 Heat exchange - Treccani, il portale del · PDF file · 2017-10-30In the case...

303VOLUME V / INSTRUMENTS

6.1.1 Introduction

The study of heat exchange phenomena can be reduced totwo variables: temperature and heat flux. Temperature isan indication of the average molecular energy of a system;the heat flux indicates the thermal energy exchange fromone body to another. On the microscopic scale, thermalenergy is related to molecular kinetic energy; as thetemperature of a material increases, the thermal agitationof its molecules also increases. As an effect of the secondlaw of thermodynamics, bodies that have a bigger averagemolecular kinetic energy spontaneously transfer thisenergy to the bodies that have a smaller average molecularkinetic energy. Heat fluxes between two regions depend onseveral different material properties, among which thermalconductivity, specific heat, density and surface emissivity.In the case of heat transfer in a fluid, its flux velocity andviscosity are very important. Heat transfer mechanismscan be of three different types: by conduction, convectionand radiation.

Transfer by conduction takes place in solids and fluids atrest or in laminar motion. It is defined as a process ofmolecular character in which part of the energy of moleculesin the hotter portion of a material medium (therefore with abigger average molecular energy) gets transferred to themolecules in the colder portion, without any mass transfertaking place. Such a process takes place as an effect of directmolecular collisions, as an effect of the action of vibrationsand collisions. In metals a significant portion of thermalenergy gets transferred thanks to the movement ofconduction electrons.

Convection is defined as the macroscopic process thattakes place only in fluids where the hotter fluid moves (or isforced to move) in the regions of the fluid at a lowertemperature, where it gets remixed and transfers part of itsthermal energy. If the motion of the fluid is solely generatedby density differences caused by temperature differences,convection is called natural; if the fluid motion is producedmechanically, convection is called forced.

Radiation is the mechanism of heat transfer from onebody to another that takes place by radiating energyemission and absorption. All bodies emit radiating energy inall directions, in quantities depending on their temperature.

In this case, energy is transported by photons ofelectromagnetic radiation, mainly in the infrared and visibleregion of electromagnetic spectrum. When temperatures areuniform, radiative flux between objects is in equilibriumconditions and therefore no net transfer of thermal energytakes place. When temperatures are not uniform, radiativeflux generates a thermal energy transfer from hotter tocolder surfaces. In many cases, thermal exchange processescan take place thanks to a combination of all the three abovementioned processes.

6.1.2 Conduction

The fundamental law regulating heat transfer by conductionwas stated in 1822 by Jean-Baptiste-Joseph Fourier:

[1]

where q represents the heat transferred for unit time, A is thearea of the section of the body normal to the axis alongwhich heat flows, �dT�dx is the temperature gradient in theheat flux direction, and k is a physical parameter, calledthermal conductivity, that depends on the materialcharacteristics of the body and that usually varies withtemperature.

Table 1 shows conductivity values of some solids and thevalue of temperature at which such values have beenmeasured. In many cases, it is assumed that k varies withtemperature according to a linear law such as:

[2]

It is often convenient to refer to an average value of k in thetemperature interval of interest.

Coefficient k has much smaller values for fluids thanthose measured for solids, as can be observed from Table 2.In the cases where experimental values are missing, thermalconductivity can be estimated with good approximationthrough empirical relationships correlating k with specificheat, molecular weight and fluid viscosity.

Equation [1] can be used to derive non-stationarytridimensional equation for heat transmission in solids andfluids at rest:

k k T= +( )0 1 α

q kAdTdx

=−

6.1

Heat exchange

[3]

where cp is the specific heat at constant pressure of the body,r is its density, x, y and z represent rectangular coordinatesand q� indicates the heat generation rate – by unit volume ofthe body – due to chemical or nuclear reaction, or caused bythe passage of an electric current. It is necessary to specifysuitable initial and boundary conditions to obtain thesolution of [3], which gives the temperature of the body as afunction of time and position. Equation [3] can betransformed into spherical or cylindrical coordinates tobetter adapt to the geometry of the problem in question.Notice that if k is considered temperature dependent, [3]becomes non-linear and therefore it cannot be solvedanalytically anymore, with the exception of a few particularcases.

Steady conductionWhen the heat flux is stationary, q in the Fourier

equation is a constant. In the same way, �T��t in [3] is zero.By assuming k constant, [3] becomes:

[4]

With no heat generation (q��0), it is possible to directlyintegrate [1], thereby obtaining:

[5]

By suitably varying coordinates, it is possible to derivethe solution of heat conduction in different situations ofpractical importance.

In order to study the case of radial conduction in ahollow cylinder, the length of the cylinder is indicated byL, its internal and external radius are indicated by Ri andRe, whereas Ti and Te are the temperatures of the internaland external face. Heat flux through an elementary layer ata distance R from the cylinder axis that has a thickness dR(Fig. 1) is given by

[6]

By separating variables and integrating, it is possible toobtain:

[7]

It is possible to express the radial flux as if it took placethrough an equivalent plane layer that has a width equal to(Re�Ri) and an area equal to Aeq:

[8]

being

[9] A LReq lm= 2p

qkA T TR Req i e

e i

=−( )−

qk L T T

RR

i e

e

i

=−( )2p

ln

q k RL dTdR

= − 2p

q kAT Tx x

= −−

1 2

1 2

∇ =−2T qk�

c Tt x

k Tx y

k Ty z

kpr∂∂= ∂∂

∂∂

+ ∂∂

∂∂

+ ∂∂

∂TTz

q∂

+ �

PROCESS ENGINEERING ASPECTS

304 ENCYCLOPAEDIA OF HYDROCARBONS

Ti

Ri

R

dR

Te

Re

L

Fig. 1. Heat transmission by conduction in a hollow cylinder.

Table 1. Thermal conductivity of some solids

Material Temperature (°C)Conductivity

(kJ/h�m�°C)

Gold18 1,053.785

100 1,060.07

Pure iron18 243.02

100 228.355

Steel (1% C)18 163.41

100 161.315

Magnesium 0-100 574.03

Cork – 0.155

Table 2. Thermal conductivity of some fluids

FluidTemperature

(°C)Conductivity

(kJ/h�m�°C)

Liquid ammonia �15/�30 1.80

Benzene30 0.59

60 0.54

Glycerine 20 1.02

Ethyl alcohol 20 0.65

Methyl alcohol 20 0.78

Water

0 2.14

38 2.26

93 2.45

Hydrogen�200 0.18

0 0.60

Carbon dioxide 0 0.05

where Rlm is average logarithmic radius defined as

[10]

In order to solve the problem of radial conductionthrough a spherical layer, it is common to indicate theinternal and external radius of the layer with Ri and Re, andthe temperatures of the corresponding surfaces with Ti andTe. The heat transmitted through an elementary layer at adistance R from the centre of the sphere that has a thicknessdR is given by:

[11]

By separating the variables and integrating, one gets:

[12]

It is also possible in this case to express the heattransmitted through an equivalent plane layer that has awidth equal to (Re�Ri). The equivalent surface is given by:

[13]

that is, the geometric average of the internal and externalsurface of the spherical layer.

Another case of great practical interest is that oftransmission through a series of conductors that have thesame surface; refer for instance to three consecutive planewalls that have areas A1, A2, A3, widths x1, x2, x3,conductivities k1, k2, k3, and indicate with DT1, DT2, DT3 thetemperature increase between the surfaces delimiting thewalls. Since the heat flux through each of the three wallsmust be the same, it is

[14]

By defining a thermal resistance for each wall:

[15]

one gets

[16]

and summing together the single DTi, it is possible to derive:

[17]

or in other terms:

[18]

where rT represents the overall resistance given by the sumof the single resistances:

[19]

Equation [17] is similar to Ohm’s law for a system ofresistances in series.

An example of heat transmission through a seriesof conductors that have a variable section is given bythe system of two concentric cylinders A and B ofequal length L that have, respectively, conductivities k1and k2, both considered to be constant. The temperatureof the most internal surface is Ti, whereas thetemperature of the surface at which the two cylinders

are in contact is Tn and that of the most externalsurface is Te.

For cylinder A it is:

[20]

and then

[21]

and for cylinder B:

[22]

and then:

[23]

It is therefore possible to derive:

[24]

However, if a system of two hollow spheres is taken intoconsideration, it is possible to derive:

[25]

If the temperature of a material is a function of twospatial coordinates, bidimensional conduction occurs and therelative equation, having assumed k constant, is:

[26]

If q� is equal to zero, [26] becomes the Laplace equation,the analytical solution of which is only possible for specialboundary conditions and geometries (Carslaw and Jaeger,1959). When it is not possible to obtain an analyticalsolution, graphic or numerical solutions are employed byusing finite differences methods, for instance.

Non-stationary conduction When the temperatures of the material are functions of

space and time, equations more complex than thosepreviously introduced are necessary. The most generalequation is [3], tridimensional non-stationary. In most cases,the solutions to practical problems involving non-stationaryconduction phenomena need to use numerical solutions to bedeveloped on the computer. It is possible to find many casesfor problems corresponding to several different geometriesand boundary conditions in the literature.

One-dimensional problems are expressed by thefollowing equations:

[27]

valid for rectangular coordinates;

∂∂= ∂∂+T

tTx

qcp

α2

2

�

r

∂∂+∂∂=−

2

2

2

2

Tx

Ty

qk�

qT T

R Rk R R

R Rk R R

i e

n i

i n

e n

n e

=−( )

−+

−4

1 2

p

qL T T

kRR k

RR

i e

n

i

e

n

=−( )+

2

1 1

1 2

p

ln ln

T Tq

RR

k Li e

e

n− =ln

22p

qk L T T

RR

n e

e

n

=−( )2

2p

ln

T Tq

RR

k Li n

n

i− =ln

12p

qk L T T

RR

i n

n

i

=−( )1

2p

ln

r r r rT n= + + +1 2 ...

qT

rT Tr

i

T

i e

T

= = −∑∆

q r r r T T T T1 2 3 1 2 3+ +( )= + + =∆ ∆ ∆ ∆

∆ ∆ ∆T qr T qr T qr1 1 2 2 3 3= = =

r xkA=

q k A Tx

k A Tx

k A Tx

= = =1 1 1

1

2 2 2

2

3 3 3

3

∆ ∆ ∆

A R R A Aeq i e i e= =4p

qkR R T TR Ri e i e

e i

=−( )

−4p

q k R dTdR

= − 4 2p

RR RR Rlme i

e i

=−

( )ln

HEAT EXCHANGE


[28]

for cylindrical coordinates;

[29]

for spherical coordinates. In the last three equations, parameter a, also called

thermal diffusivity, was introduced, defined as:

[30]

These equations have been solved in the case of a planeslab, a cylinder, and a sphere immersed in a fluid. Solutionsare obtained as infinite series, and results are usually shownin a graph as curves characterized by four adimensionalratios that, when q��0, are (Gurney and Lurie, 1923):

[31]

[32]

where T� indicates environmental temperature; T0 is the initialuniform temperature of the body; T is the temperature of thebody in a given point at time t, measured from the beginningof the cooling or heating process; k is the thermal conductivityof the body assumed uniform and constant with varyingtemperature; r is the density of the body assumed constant; cpis the specific heat of the body; hT is the global heat transfercoefficient between the environment and the surface of thebody, expressed as heat transferred per unit time, unit surfacearea and unit temperature difference between the surface andthe environment; R is the distance in the direction alongwhich heat is transmitted between the point or the plane in themiddle of the body and the point in question; Rm is the radiusof the sphere or the cylinder, half of the width of the slab

heated from both sides, the width of the slab heated on oneside and perfectly isolated on the other; x is the distance in thedirection along which heat is transmitted between the surfaceof a semi-infinite body to the point in question.

Usually, when integrations are performed, cp, hT, k, R,Rm, T�, x and r are assumed constant.

In Fig. 2 the results in the case of an indefinite slab areshown; these results are expressed by plotting families ofcurves, corresponding to different values of m and n, whereY is plotted in logarithmic scale on the y-axis, and X in linearscale on the x-axis; similar results are obtained in the case ofa cylinder and a sphere.

The problem of conduction through a slab immersed in afluid is greatly simplified if the slab is so thin and thematerial has an electric conductivity so high that temperatureon the entire width of the slab can be considered constant.Suppose that the slab has a volume V, total surface A, initialtemperature T0, and is in contact with cooler air at a uniformtemperature T�; at every instant, the quantity of heat dQtransferred over time dt is proportional to surface A, to thedifference between the temperature T of the slab and that ofthe air through the coefficient h:

[33]

Under the imposed conditions, the numerical value of his relatively small and the corresponding heat transfervelocity per unit area is therefore also small. As an effect ofthis, if the value of conductivity k is high and the slab is thin,the temperature of the slab can be considered uniform, andby developing a thermal balance, it is possible to obtain:

[34]

By assuming hA/V rcp to be constant, [34] can be easilyintegrated giving:

[35]T TT T

ehAV c

tp−

−=

−�

�0

r

dQ hA T T dt V c dTp= −( ) = −� r

dQdt

hA T T= −( )�m k

h Rn R

Rn x

RT m m m

= = =��or��

Y T TT T

X ktc Rp m

= −−

=�

� 02��

r

α = kcpr

∂∂=

∂∂

∂∂

+

Tt R R

R TR

qcp

α2

2 �

r

∂∂=

∂∂

∂∂

+

Tt R R

R TR

qcp

α �

r



0.002

0.003

0.005

0.007

0.01

0.02

0.03

0.05

0.07

0.1

0.2

0.3

0.5

0.7

1.0

0 5.04.03.02.01.0

m�0

n�0.8n�0.6n�0.4n�0.2

n�0

n�0.8n�0.6n�0.4n�0.2

n�0

n�1

n�0.8n�0.6

n�0.4n�0.2

n�0

n�1

n�0.8n�0.6

n�0.4n�0.2

n�0

n�1

m�6

m��

m�0.5

m�

1m

�2

(T'�

T)/

(T'�

T0)

a�t/R2m

Fig. 2. Solution of the problem of non-stationary conduction for an undefinedslab that has a thicknessequal to 2Rm.

6.1.3 Convection

Heat transfer in fluids is helped by the possibility ofremixing in the fluid itself. In the case of natural convection,such remixing movements derive from density differences inthe fluid mass due to the different temperatures present inthe various points. Movements, however, can also beproduced by stirring, or by the movement of a fluid in a pipe:in these two cases, heat is said to be transmitted by forcedconvection. The latter is the most common case in processindustries where often a hot and cold fluid separated by awall are pumped through the heat exchange equipment. Inthe study of mechanisms controlling heat exchange in thiskind of equipment, it is necessary to account for the fact thatin the motion of a fluid around a solid, even when the bulk ofthe fluid is in turbulent motion, a film in which the fluid is inlaminar motion is created near the surface. In the laminarfilm zone, heat gets transferred mainly by molecularconduction. Resistance of laminar layer to heat flux varieswith its width, and it can be responsible for 95% of totalresistance for some fluids, or only 1% for others. Theturbulent fluid bulk and the transition layer included betweenthe turbulent zone and the laminar film offer a resistance toheat transfer that is a function of turbulence and of thermalproperties of the fluid. It is therefore possible to concludethat in this case the overall heat flux is really the effect ofseveral different mechanisms, where conduction acts inconjunction with convective effects.

The study of natural convection problems wouldrequire the simultaneous solution of coupled equations ofmomentum and energy balance, but this rigorous approachis only possible for the geometrically very simple problemof a vertical plate. Therefore it is common to usedimensional analysis methods, widely applied whenstudying heat exchange problems. They allow theexpressions for the local transfer coefficients to bedetermined by using adimensional parameters: geometricparameters of the problem and fluid properties are groupedin adimensional groups that represent the significantvariables of the problem, so that the effect of each factordoes not have to be independently determined. This methodonly operates on dimensions, and therefore does notprovide numerical values, but it allows the relativeinfluence of the various parameters to be estimated. Bydoing so, the investigation on the experimental correlationsamong data is enormously simplified.

Dimensional analysis allows us to derive that thecharacteristic parameter for the natural convection problemsis given by the Grashof number, NGr, an adimensional ratiogiven by:

[36]

where l represents a characteristic dimension of the system,g is the acceleration of gravity, b is the thermal expansioncoefficient of the fluid, m its dynamic viscosity, and DT thedifference between the temperature at the fluid-wall interfaceand that in the fluid bulk. The following correlation, calledNusselt equation, can be derived:

[37]

(where a is an adimensional parameter) in which two otheradimensional groups appear, the Nusselt number

[38]

and the Prandtl number

[39]

In [38] and [39] k represents the fluid thermalconductivity and cp its specific heat at constant pressure.

The properties of the fluid get evaluated at a temperatureTf that is the arithmetic average between the temperature in thefluid bulk T� and the temperature of the wall Ts. From equation[37] it is also possible to derive a dimensional equation, wherethe fluid properties are grouped in a single factor b:

[40]

Also in the case of forced convection, rigorousapproaches are limited to simple geometries and to laminarflow situations, whereas the analyses in turbulent fluxsituations are usually based on deterministic models, but arenot able to produce correlations that can be applied forprocess design. To study complicated geometries, empiricalcorrelations are used most of the time which have limitedvalidity, however. In forced convection, transfer coefficientsare strongly influenced by the flow characteristics, which inturn depend on several factors, such as turbulence intensity,conditions at the inlet, and conditions at the wall. Also in thiscase, therefore, dimensional analysis is used widely.

In order to study the problem of heat exchange towardsor from fluids flowing within cylindrical tubes, three regimesof motion can be distinguished, depending on the value takenby the Reynolds number:

[41]

where v is the linear velocity within the tube and D is thediameter of the tube itself. Fluid properties are evaluated inthe bulk of the fluid phase. If NRe�2,100, the regime ofmotion is laminar: the velocity of fluid particles keepsparallel to the tube walls and takes on a parabolic profile,zero at the walls and maximum at the tube axis. WhenNRe10,000, the regime of motion is turbulent andcorresponds to a situation where the transversal componentof velocity is also present. In laminar regime, the fluidmotions keeps regular and its particles move withoutremixing along stream lines that are easy to identify, whereasin turbulent regime it is characterized by chaotic motions(vortexes) that cause remixing. The presence of fluctuationsgenerated by motion irregularity makes momentum andenergy transfer in the direction normal to the tube wallsstronger, causing an increase of heat transfer coefficients.For 2,100�NRe�10,000 the so-called transition regimeoccurs where the two motions, laminar and turbulent, coexistsimultaneously in an unstable way and the situation is notclearly determined.

In the literature several theoretical solutions have beendeveloped for various geometries and boundary conditionsin relation to the problem of heat transfer in a fluid flowingin a tube in laminar regime conditions, supposing that it ismostly due to conduction. However, these solutions mostlyneglect natural convection phenomena, which in practice areoften quite important, and it is therefore advisable to useexperimentally derived empirical relationships. Dimensional

N vDRe =

r

µ

h b T lm m= ( ) −∆ 3 1

Nckp

Pr =µ

N hlkNu =

N a N NNu Gr

m= ( )Pr

N l g TGr =

3 2

2

r βµ∆

HEAT EXCHANGE


analysis suggests correlating data in terms of the Nusseltnumber, NNu, or in terms of the Graetz number:

[42]

whereas the effects of natural convection are collected in theGrashof number, NGr.

For horizontal circular tubes with NGr�100, it isrecommendable to use the Hausen (1943) correlation:

[43]

that approximately accounts for the variations of fluidproperties through the adimensional factor (mbmw), where mbis the viscosity at the average fluid temperature (at whichalso cp, r and k are evaluated), whereas mw is the viscosity atthe wall temperature.

If NGr100, for small DT and diameters, it is possible touse the Sieder and Tate (1936) equation:

[44]

A more general expression, valid for all diameters andDT, is obtained by adding a factor 0.87[1�0.015(NGr)

1�3] atthe second member of [44]. For vertical tubes, however, it isadvisable to use the graphs derived by Pigford (1955).

The flux around a body immersed in a fluid is known aslaminar if the boundary layer is laminar on the entire body,even if the main flow is turbulent. In this case, the generalcorrelation stands:

[45]

where the values of Cr and m depend on the body geometryand on its orientation with respect to the main flux.

In order to predict heat exchange for a fluid flowing withturbulent motion in a tube, several correlations have beenproposed. The use of dimensional analysis allows thefollowing relationship to be derived:

[46]

where e and f are empirical coefficients and must beexperimentally derived.

The Sieder and Tate equation for turbulent motion is:

[47]

where, similarly to what was done for laminar motion, theadimensional coefficient (mbmw) was introduced.

If the Stanton number, NSt, defined as

[48]

is introduced, the two Sieder and Tate equations can also bewritten as

[49]

in the case of laminar regime, and

[50]

in the case of turbulent regime.

It is therefore possible to draw a logarithmic scalediagram (Fig. 3) from which it is possible to derive thetransfer coefficients for all motion regions, by plotting(h�cpG)b(cpm�k)b

2�3(mw �mb)0.14 on the y-axis and (NRe)b on

the x-axis; G is the mass rate, equal to rv. Equation [49]is represented here by a series of curves correspondingto different values of the (L/D) ratio and terminating atNRe�2,100. The portions of curves corresponding to thetransition regime are drawn by hand, so that they canstart at these terminal points and then tangentially linkwith the curve corresponding to the turbulent regime.This diagram is widely applied, even if it has thefollowing limitations: • In laminar regime, it underestimates the transfer

coefficient when DT is quite high and natural convectionstreams disturb the velocity profile.

• In laminar regime, it underestimates the transfercoefficient when DT is so high that the film temperaturegoes slightly above the fluid boiling point, causingvapour bubble formation on the tube surface. If thetemperature of the fluid mass is below boiling point,these bubbles condense but their presence causes aturbulence that increases the value of the transfercoefficient.

• In turbulent regime, when 10�L/D�400, inleteffects become relevant and a more adequatecorrelation is:

[51]

• If the fluid is a liquid metal, NPr takes on low valuescompared to those of other fluids; resistance to heattransfer due to the film on the wall becomes low, due tothe high thermal conductivity and therefore the exponent1/3 for NPr does not give a correct estimate of the heattransfer coefficient. On the basis of theoreticalcalculations and experimental data (Rohsenow andHartnett, 1973), it is advisable to use the followingcorrelation:

[52]

All the equations so far refer to cylindrical pipes but theycan easily be extended to pipes that have a non-circularsection, by replacing diameter D with the equivalentdiameter Deq, defined by the ratio:

N N NNu = + ( )4 82 0 01850 827

. ..

Re Pr

N N N LDNu =

−

0 036 0 8 1 3

0 054

. ( ) ( ).

.

Re Pr

N N NStb

wPr Re( )

= ( )−2 3

0 14

0 2

0 023µµ

.

.

.

N N DL

NStb

wPr Re( )

=

(2 3

0 14 1 3

1 86µµ

.

. ))−2 3

NN

N Nh

c vStNu

p

= =Pr Re r

N N NNub

w

= ( ) ( )

0 023

0 8 0 33

0 14

.. .

.

Re Pr

µµ

N A N NNu

e f= ( ) ( )Re Pr

N C N NNu r

m= ( ) ( )Re Pr

1 3

N NNu Gzb

w

=

1 86 1 3

0 14

. ( )

.

µµ

NNNNuGz

Gz w

b= ++

3 66

0 085

1 0 047 2 3

0

..

. ( ) /

µ

µ

..14

N N N DLGz = Re Pr



(h/c

pG) b

�(c p

m/k

) b2/3 �

(mw

/mb)

0.14

0.0005

0.001

0.01

0.05L/D�50

L/D�100L/D�200

L/D�400

102 103 104 105 106 107

(NRe)b

Fig. 3. Evaluation of the transfer coefficient for the forced convection problem for a fluid flowing in a tube.

[53]

where A indicates the area of the pipe section and P the wetperimeter, i.e. the perimeter touched by the liquid inside thepipe.

Sometimes fluids are heated by making them passorthogonally to a tube layer and the calculation of h can beperformed by using McAdams equation (1954):

[54]

In this case, fluid properties are estimated at the surfacefilm temperature, calculated as the arithmetic average of Tband Tw; De is the tube diameter and umax is the fluid ratethrough the minimum free area between rows of tubes;finally, the D value depends on the number of layers oftubes, as shown in Table 3.There are also graphs that can be used to obtain initialestimates of transfer coefficients for liquids and gases usedin single phase heating and cooling operations inside tubes,or externally to tube layers. The values given by thesegraphs, however, must then be verified with more detailedcalculations, properly accounting for fluid velocity,temperature difference and equipment geometry.

6.1.4 Condensation

Exchange phenomena taking place in a fluid undergoing aphase change require a specific approach. Phase change ofa saturated vapour in its liquid state is calledcondensation. Heat transfer associated with condensationis usually classified as convective but, often, conductionalso provides significant contributions. Heat exchangefrom a condensing vapour is a phenomenon that can beencountered very often in practical applications in thechemical industry. Vapour heaters and head condensers ofdistillation columns are some of the most knownexamples. It is possible to distinguish two types ofcondensations: film and drop condensation. In the firstcase, condensing vapour forms a liquid film on the solidsurface that flows downwards, wetting it completely. Whenthe condensing fluid scarcely adheres to the surface, itforms drops that do not wet it completely, thereby leaving asubstantial portion in direct contact with the vapour. Dropcondensation gives much higher transfer rates (up to 6 oreven 18 times) than those obtained with film condensation.Liquid film, in fact, gives higher resistances to heattransfer. However, since drop condensation is veryunstable, it is seldom used, even though it can be helped inprinciple by adding particular substances to the vapour; inequipment design calculations, it is often assumed thatcondensation takes place with a film mechanism.

When developing equations to calculate the heat transfercoefficient for the condensation on a vertical wall, it isassumed that the deposited liquid film flows downwards witha laminar motion. This liquid film thus represents the biggestresistance to the flow of the heat freed by the condensingvapour, and the film width is a function of the flow length, tobe calculated according to appropriate hydrodynamic laws.By integrating on the entire flow length, Nusselt (1916)derived the following expression to calculate h:

[55]

The 4G�m ratio represents the Reynolds number that canbe applied to this kind of flux. For vertical walls, G isdefined as the flow of condensed fluid per unit of wetperimeter (for tubes) or wet width (for plates).

For tubes:

[56]

where Nt represents the number of tubes and WF is the flow inthe film, evaluated at the bottom of the tube, where it takes itsmaximum value. However, it was experimentally demonstratedthat [55] underestimates h, and therefore it is preferable to usea coefficient equal to 1.85 rather than 1.47 (McAdams, 1954).This discrepancy is probably due to the fact that somediscontinuities at the exchange surface, or some stressescaused by the vapour at the vapour-film interface, or any otherkind of disturbance can create turbulences in the film.

Reported correlations, however, neglect the influence ofthe Prandtl number, which can actually be relevant. This isthe reason why Dukler (1959) proposed an improved versionof Nusselt’s classical theory, obtaining as a result the graphshown in Fig. 4, which correlates the effects of the Prandtl

Γ =W N DF tπ

hk gµ

µ

2

3 2

1 3 1 3

1 474

r

=

.Γ

−hDk

Du ck

e p

=

0 33

0 6 0

.

.

∆ maxr

µµ

..33

D APeq =

4

HEAT EXCHANGE


0.01

0.1

1.010

52

1

0.5

0.1DuklerNusselt

NRe, 4G/m0 100,00010,0001,000

h[m

2/k

3r

2g

]1/3

NPr�cpm/k

Fig. 4. Evaluation of the transfer coefficient for the film condensation of a pure vapour on a vertical wall, calculated by the Nusselt theory and Dukler theory.

Table 3. Values of D to be used in the equation [54] as a function of the number of tube rows

Numberof tubes

1 2 3 4 5 6 7 8 9 10 andabove

D 0.7 0.72 0.83 0.87 0.92 0.94 0.96 0.97 0.99 1

and Reynolds numbers on the transfer coefficient. It isinteresting to observe that this graph does not clearly show awell determined transition Reynolds number. Notice alsothat deviations from the results of Nusselt’s theory decreaseas the terminal Reynolds number decreases. Furthermore,when NPr�0.4, smaller values of h are derived compared tothose calculated with Nusselt’s theory.

However, for condensation on a single horizontal tube,Nusselt’s theory allows the following correlation to bederived:

[57]

In this case, it is:

[58]

When the problem involves a set of tubes where the fluiddrips from the upper tubes to the lower rows thus causing anincrease of the width while moving downwards, even thoughlaminar flow is maintained, it is necessary to redefine G. Inthis case, the total condensed flux should be divided by thenumber of vertical rows of tubes Nvtr on which thecondensed liquid drips:

[59]

If tubes are circular, the value of Nvtr can be estimated as:

[60]

The value of the Fp1 factor depends on the geometricalarrangement of tubes (Table 4). In real conditions,condensate dripping from row to row causes the insurgenceof a certain turbulence, and [57] therefore underestimates h.For this reason, at the second member a factor of 1.85 or theDukler correlation is often used (see again Fig. 4), eventhough this was originally developed for vertical walls.When Reynolds numbers of films fall in the turbulent field,if no specific correlation is available, equation [57] or Fig. 4is used.

6.1.5 Boiling

Phase change of a saturated liquid to its vapour state isusually called boiling or evaporation. Boiling is often used totransfer heat in chemical processes. For instance, the heatnecessary to perform a distillation is generally providedusing a boiler. Heat exchange taking place during boiling isclassified as convective, since bubbles cause significantturbulence but, generally, several mechanisms play a role.

For instance, vaporization can take place as an effect of heatabsorption by radiation and convection at the surface of aliquid mass, or as an effect of heat absorption by naturalconvection from a heated wall placed below the free surfaceof the liquid; in the latter case, vaporization takes placewhen the supersaturated liquid reaches the free surface.Vaporization can also take place from falling films (and inthis case, the process is the opposite of condensationpreviously described), or by expansion of superheatedliquids via forced convection under pressure.

When a liquid mass is heated, not only the heat flux rate,but also the heat transfer coefficient h vary as the differencebetween the fluid-wall interface temperature Tw and that inthe liquid bulk Tb varies. For water under atmosphericpressure, the coefficient varies slowly as the temperaturedifference increases, if Tw�Tb�DT�5°C. Much faster is theincrease for DT5°C, as long as a narrow maximum occurswhen the temperature difference is equal to a critical value.A further temperature difference increase would cause thecoefficient to decrease.

This is the explanation: when DT�5°C, the phasevariation occurs by evaporation at the vapour-liquid interfacebut when DT5°C, several bubbles form at the heatingsurface, from which they subsequently detach. This last heattransfer mechanism is called nucleated boiling, and produceshigh transfer coefficients. However, when DT becomesgreater than a critical value DTc, vapour bubbles tend toform a continuous layer around the surface, insulating it andtherefore worsening thermal exchange conditions. Thismechanism is called film boiling. The effect of DT on thetransfer coefficient for water at atmospheric pressure isshown in Fig. 5. In general, the value of DTc depends on thenature of the solid surface and the liquid, but it usually fallsbetween 20 and 30°C.

At the critical temperature difference, the maximum heatflux is obtained. In real applications, usually values of DTlower than DTc are used in order to avoid falling in thecondition of film boiling, where the heat exchange surface

N F Nvtr p t= 1( ) .0 5

Γ =W N Lvtr

Γ =W LF

hk gµ

µ

2

3 2

1 3 1 3

1 51 4r

=

−

. Γ



tran

sfer

coe

ffic

ient

(kW

/m2 �

K)

2

4

6

8

10

20

40

60

DT (K)1 2 4 6 8 10 20 40 60

Fig. 5. Transfer coefficient for boiling water, outside a single horizontal tube submerged in a mass of water.

Table 4. Values of factor Fpl according to the variationin geometric disposition of the tubes

Tube arrangement Fpl

Triangular 2.10

Turned traingular 1.21

Square 1.05

Turned square 1.49

works with a low efficiency. Moreover, since the film ofvapour acts as an insulating layer, it is possible to superheatthe metal tube on the hot side. In order to design a boiler, itis very important to have an estimate of the maximum heatflux, and this largely depends on the density of generatedvapours. Low pressure and vacuum operations have a biggerpotential compared to the possibility of operating inconditions of nucleated boiling. The maximum heat flux fora mass of fluid heated by a single horizontal tube can becalculated on the basis of a correlation, expressed in terms ofthe operating pressure p and of the critical pressure of thefluid pc, which can be applied to most fluids, including waterand hydrocarbons (Mostinski, 1963):

[61]

where pressures are expressed in kPa. According to [61] themaximum heat flux increases when p increases starting from0, until it reaches a maximum when it is equal to 28% of pc,to then decrease back to zero as p approaches pc. Equation[61] is valid when heating is provided by a single horizontaltube, but when it is given by a group of horizontal tubes,vapour release from heating elements is less immediate dueto the proximity of adjacent tubes. In general, by increasingthe number of tubes and decreasing their distance, (q�A)maxdecreases. In some cases, (q�A)max goes down to the 10% ofthe value obtained with a single tube.

Heat transfer coefficient in nucleated boiling conditionscan be calculated by using, among others, the followingcorrelation (Mostinski, 1963):

[62]

This correlation does not account for the nature of theheating surface and it can be considered acceptable forcommercial tubes, whereas very smooth tubes offer fewersites for bubble nucleation, and therefore give lower valuesof the transfer coefficient. Turbulence created by boiling ingroups of horizontal tubes can give values of the coefficientthat are at least double those obtained for a single tube.

An expression was also proposed for the heat exchangein film boiling conditions:

[63]

6.1.6 Heat exchangers

In most industrial equipment for heat transmission, heat getstransferred from a hot to a cold fluid through a solid surface.The simplest equipment used for this purpose is the doubletube exchanger (or coaxial fluids exchanger), in which oneof the fluids flows through the external tube, while the otherflows through the internal tube. For double tube exchangers,two different exchange modes are possible, defined ascountercurrent when the two fluxes proceed in oppositedirections or cocurrent flow when the two fluids proceed inthe same direction.

Let’s suppose Tc and Th are the cold fluid temperatureand the hot fluid temperature, respectively, Tci and Tco theinlet and outlet temperature of the cold fluid, and Thi and Tho

the inlet and outlet temperature of the hot fluid. In cocurrentflow, the outlet cold fluid temperature can never surpass thehot fluid outlet temperature (ThoTco). As the heat exchangesurface gets higher, Tco gets closer to Tho.

In countercurrent it can happen that TcoTho, since thereis a bigger thermal recovery here. While the temperatureincrease in a countercurrent exchanger does not varysignificantly, whereas for cocurrent flow it goes from amaximum at the inlet section down to a minimum at the exitsection (Fig. 6).

Heat transfer between two fluids separated by a planewall is usually described using the equation:

[64]

according to which the heat q transferred in the unit timefrom the hot to the cold fluid is equal to the product betweena proportionality factor U, called global transfer coefficient,the area of the surface A through which heat flows and thedifference between the temperatures of the two fluids, Th�Tc.

Equation [64] applies only locally, or when exchangeconditions are kept constant on the whole equipment,otherwise it should be expressed in differential form:

[65]

The global transfer coefficient accounts for a series ofresistance to heat flow that appear in a system like this,including the fluids themselves (rh and rc), the deposits dueto the scaling on the hot and cold side of the wall thatseparates the fluids (rhs and rcs), and the solid wall itself (rw).The overall transfer coefficient can be expressed as the sumof the resistances in series:

[66]

The inverse of each one of these resistances is calledconductance and is indicated by h. The values ofconductance of the two fluids, he and hc, represent the heattransfer coefficients. The heat flux (q/A) through each of thesingle resistances is the same; temperature increase througheach resistance is inversely proportional to the value of h:

[67]

The difference between the temperature of the hot fluidand the cold fluid is given by the sum of DTs for the singleresistances.

∆T qAr q

A hh hh

= = 1

Ur r r r rh hs w cs c

=+ + + +

1

dq U T T dAh c= −( )

q UA T Th c= −( )

hk g

D Tv l v v

v o=

−( )

0 6203

.r r r

µ ∆

h p qA

ppcc

=

+0 00441 1 80 69

0 7 0 17

. ..

. .

44 10

1 2 10

pp

ppc c

+

.

q A pp

pp

pc c

c( ) =

−

max367 1

0 35 0 90. .

HEAT EXCHANGE


A B

ThiThi

ThoTho

TciTci

TcoTco

Tc

Tc

T

outletinletcocurrent

T

outletinletcountercurrent

Fig. 6. Temperature trends for the two fluids in a cocurrent flow exchanger (A) and in a countercurrent flow exchanger (B).

Usually the solid wall, which has a width equal to y, ismade up of conductor material, and the quantity of heat thatruns through it is given by:

[68]

Very often rw is negligible, since the thickness of thewall is very small, but also because the value k ofconductivity is very high.

Scale is a term defining any layer, or deposit, ofextraneous material on a surface that must exchange heat.These materials usually have low thermal conductivity, andtherefore they display a high resistance to thermal exchange.These layers can be created as an effect of the deposition offinely subdivided substances present in the process fluids, orby crystallization of some substance whose solubility at thetemperature of the wall is lower than the solubility at thetemperature in the fluid bulk. The formation of rather thickand hard layers of scales at the wall is caused bypolymerization reactions and sometimes by corrosionproducts of the walls themselves.

As noted before, when exchange conditions vary alongthe equipment, it is necessary to use equation [65]. Anexpression of heat flux as a function of inlet and outlettemperatures of the hot fluid (Thi and Tho) and of the coldfluid (Tci and Tco) can be derived with the followinghypotheses: a) the overall transfer coefficient U is constanton all the equipment; b) the exchanger operates in stationaryconditions; c) the specific heat of the two fluids is constantalong the whole length of the exchanger; d ) no phasevariations occur in the system; and e) heat dispersions arenegligible.

In particular, if the flows of the fluids take place incountercurrent, it is possible to derive:

[69]

The effective temperature difference, called logarithmicaverage temperature difference, is then given by:

[70]

If, however, the two fluids flow in parallel, it is possibleto derive:

[71]

In this case, the average logarithmic temperaturedifference is given by:

[72]

The overall transfer coefficient U can be directlyevaluated by [66] only if the exchange area is constant, ashappens in the case of transfer through plane surfaces.

However, most equipment has tubular exchange surfaces,where the external area Ao is larger than the internal area Ai.Equation [66] must therefore be modified in order to referall resistances to the same surface, especially the surfaceused to define A. Usually, in the case of tubular equipment,one refers to the external surface, and the overall transfercoefficient is therefore given by:

[73]

The resistance of the metallic wall of the tube ismultiplied by the ratio between the external surface and anaverage surface, calculated by the average logarithmic valueof the radius of the tube defined in [10].

Finally, it is necessary to take into consideration the casewhere U is not constant but varies with the temperature ofthe two fluids. In this case, most of the time it is assumedthat the dependency is linear, such as:

[74]

By integrating [65], the following expression isobtained:

[75]

where Uc and Uh are the values of the transfer coefficient atthe coldest point and the hottest point, respectively, of theexchanger.

In practice, cocurrent flow heat exchangers are seldomused, since they offer a limited exchange area. The mostwidely used exchangers are those of the shell-and-tube typethat are versatile and can be used also in the presence ofphase changes (for instance, condensation and boiling) withoperating pressures ranging from values close to vacuumconditions to values in the order of 40 MPa. In practice,shell-and-tube exchangers are composed of two plates towhich a certain number of tubes (tube bundle) are connected,inserted within a shell. With this set-up, it is possible tocompact a great exchange surface into a small space. Fig. 7shows two schematic drawings of exchangers. One of thetwo fluids runs through the tube bundle: in the simplest case,it enters at one head and exits at the other; in the moregeneral case, it runs through some of the tubes in onedirection, and then inverts the flux direction, and runsthrough other tubes in the inverse way, to then invert thedirection again, and so on, until n runs within the tubebundle are performed (n stages). The exchanger shown inFig.7A is of the 2-stage type, that shown in 7B is of the4-stage type. The other fluid runs through the cylindricalshell containing the tubes, hitting the tubes in a rathercomplex way. Usually, in order to improve heat exchange,some baffles are inserted within the shell that partially closethe shell section, allowing the tubes to cross them: thesebaffles are fastened alternatively on opposite sides of theshell. The fluid therefore zigzags, hitting the tubes in adirection that is approximately perpendicular in the spacebetween two baffles, whereas it moves more or less parallelto the tubes when it is close to the baffles’ openings. Sectionreduction, which causes flow rate increase, and the fact thatthe fluid partially hits the tube bundle frontally, cause a

q AU T T U T T

U T TU T

c hi co h ho ci

c hi co

h h

=−( ) − −( )

−( )ln

oo ciT−( )

U T= +α β∆

UAAr

AAr

AA

r r ro

ii

o

iis

o

lmw os o

=+ + + +

1

∆TT T T T

T TT T

lmhi ci ho co

hi ci

ho co

=−( ) − −( )

−−

ln

q UAT T T T

T TT T

hi ci ho co

hi ci

ho co

=−( ) − −( )

−−

ln

∆TT T T T

T TT T

lmhi co ho ci

hi co

ho ci

=−( ) − −( )

−−

ln

q UAT T T T

T TT T

hi co ho ci

hi co

ho ci

=−( ) − −( )

−−

ln

q ky A Tw= ∆



turbulence increase and therefore an improvement of thetransfer coefficient. Usually, the openings in the bafflesaccount for about 25% of the total section of the shell.

The calculation of the transfer coefficient for the tubeside is immediate, since it is possible to use the criteriaalready seen for the flux in tubular pipes (see again Fig. 3).Less simple is the calculation of the coefficient for the fluidflowing within the shell, since the regime of motion is rathercomplex in this case. However, a few correlations have beenproposed, such as the one by Kern (1950):

[76]

similar to [54]. Equivalent diameter Deq is calculated by [53]. Geq

indicates an equivalent mass rate. At a baffle opening, wherethe flux is parallel to the tubes, a mass rate Gw is estimatedby the ratio between the fluid flow rate and the free section,Sw, representing the total area of the opening from which it isnecessary to subtract the sum of the sections of the tubes thatcross it. Moreover, in the space included between twobaffles, where the flux is normal to the tubes, the free sectionarea, Ss, can be approximately calculated using theexpression yLxpDs�yT, where Ds is the shell diameter, yT is thedistance between the axes of two neighboring tubes locatedon the same horizontal row, yL is the distance between twoneighbouring tubes on the same horizontal row (yL�yT�De)and xp is the distance between subsequent baffles. Notice thatDs�yT is the approximated value of the number of tubes andtherefore also of the number of free spaces at the shelldiameter, whereas the product yLxp gives the area of the freespace between two neighboring tubes. Therefore in the spacebetween baffles, it is possible to estimate a mass velocity Gs,given by the ratio between the fluid mass flow and Ss. Theequivalent mass rate Geq, introduced in [76], can becalculated by the geometric average of Gw and Gs:

[77]

Equations such as [69], derived for cocurrent flowsystems, are also valid for shell-and-tube exchangers, as longas a corrective factor Y is introduced, since the two flows are

neither countercurrent nor cocurrent in this kind ofequipment. The value of Y is always smaller than 1 and itdepends on the type of exchanger, and on the inlet and outlettemperatures of the two fluids. The values of Y have beencalculated (Kern, 1950) for a series of fluxes of commoninterest, based on the assumptions mentioned above, thatwere introduced to derive equation [69]. In general, values ofY smaller than 0.8 are considered unacceptable and thecorresponding exchanger configurations is consideredinefficient.

In exchangers the tendency is to maintain the fluid rateshigh, so as to obtain transfer coefficients that are as high aspossible, without increasing pressure drops above reasonablelimits.

6.1.7 Radiation

A heated body emits electromagnetic radiations at anintensity and frequency that depends on its temperature. Forinstance, if the wire of an incandescent lamp is electricallyheated, both the energy quantity emitted in unit time and theportion of visible radiation increase as temperatureincreases. At a temperature below 400°C, the radiationemitted by the lamp is not perceived by the human eye, butgets felt by human skin as heat.

The quantitative description of this phenomenon is basedon quantum mechanics, but in qualitative terms it can beexplained by observing that, when transferring energy to asolid body, some of its atoms and molecules jump to excitedstates, but they tend to spontaneously go back to lowerenergy states. When this occurs, energy gets emitted aselectromagnetic radiations, which depend on the changesthat happen in the electronic, vibrational and rotational stateof atoms and molecules, and are thus distributed on differentwavelengths. The inverse process, called absorption, takesplace when some radiating energy is transferred to a body,and thus it moves to an upper energy level. Thisphenomenon occurs when radiating energy hits a bodythereby.

Radiating energy incident on the surface of a solidsurface is partially absorbed, partially reflected and partiallytransmitted. It is therefore possible to write:

[78]

where a indicates absorptance, i.e. the energy fractionabsorbed by the body, r is reflectivity, i.e. the reflectedfraction, and t is transmittivity, i.e. the transmitted fraction.

In most cases, materials of practical interest arerepresented by opaque substances with zero transmittivity.Bodies absorbing all incident energy are called black bodies.No real surface in practice behaves like a black body. Thebest example of a black body is an empty cavity with opaquewalls in communication with the environment through a holethat has a negligible area compared to the surface of thecavity. All the radiating energy entering the cavity remainstrapped in it, and the hole behaves like a black body.

Two bodies with areas A1 and A2 contained in aninsulated cavity externally emit radiating energies equal toA1�e1 and A2�e2, where e represents the total emissive power,i.e. the energy emitted per unit time and unit surface in thehemisphere located above each elementary surface. Thus, theenergy incident on the unit surface of the two bodies coming

a r+ + =τ 1

G G Geq w s=

hdk

D G ck

e eq eq p

f

=

0 36

0 55 1 3

..

µµ µ

µ

0 14.

HEAT EXCHANGE


A

B

Fig. 7. Shell-and-tube exchangers: A, 2-stage type; B, 4-stage type.

from the cavity surface is indicated with eB, and theabsorbing powers of the two bodies, which is the ratiobetween the absorbed and the incident radiation, is indicatedwith a1 and a2. In thermodynamic equilibrium conditions,the energy incident on and absorbed by each body must beequal to the energy emitted, and therefore it is:

[79]

or, in other terms:

[80]

This relationship allows to state that in equilibriumconditions, the ratio between absorbing and emitting poweris equal for all bodies (Kirchhoff ’s law). For a black body,a�1 and therefore e�eB. The emitting power of the blackbody depends exclusively on temperature.

The ratio between the emissive power of a genericsurface and that of a black body is called surface emissivityand is indicated by:

[81]

In thermal equilibrium conditions, the emissivity of abody is equal to its absorbing power. Besides the totalemissive power, it is also possible to define the specificemissive power el, that represents the energy emitted in thehemispace for unit surface, unit time, and in a wavelengthinterval included between l e l�dl. Total emissive powernot only depends on thermodynamic temperature T, but alsoon the wavelength l. For the black body, this dependencywas discovered by Planck in 1900 (Planck, 1923):

[82]

where c is the speed of light in vacuum, equal to 2.9979�108

m/s, kB is the Boltzmann constant, equal to 1.3807�10�23 J/K, hP is

Planck’s constant equal to 6.6261�10�34 J�s, whereas c1 andc2 are the so-called first and second constant of Planck’s lawthat are equal to 3.7403�10�16 J�m2/s and 1.4387�10�2 m�K,respectively.

Fig. 8 shows the dependency of el on the wavelength, atdifferent values of temperature. Each one of the curvesobtained at a given temperature shows a maximum for awavelength that varies as temperature varies, according toWien’s law:

[83]

with the constant expressed in cm�K.The total emissive power of a black body can be derived

from the specific power through an integration on the wholewavelength spectrum.

By using the expression of el given by Planck’s law andcalculating the integral, the Stefan-Boltzmann law isobtained: eB�c1T

4�c24�sT 4, where s is equal to 5.67�10�8

W/(m2�K4).

Calculation of radiating energy transferred between two surfaces

In order to calculate the radiating energy transferredbetween two surfaces, it would be necessary to consider the

exchange of monochromatic radiations and integrate over thewhole wavelength spectrum. For the sake of simplicity, it ismore convenient to calculate the energy that the two bodieswould exchange if they were black, by using Stefan’s law,and multiply the result thus obtained by the emissivity of thesurface involved. In general, the emissivities of bodiesdepend on temperature and on the nature of surfaces.Metallic surfaces at low temperature can reach emissivityvalues around 0.05, which can increase up to 0.70 underhigh temperatures. The emissivity of an oxidized metallicsurface can vary between 0.65 and 0.95, depending on thetemperature of the surface itself. Bricks, glass, marble andpaper usually have emissivities larger than 0.90.

The absorptance of a surface not only depends on thesurface nature and its temperature, but also on the spectraldistribution of the incident radiation. If the body receivingthe radiation absorbs all the wavelengths with the sameabsorptance, it is called grey and its absorptance isindependent from the energetic distribution of the incidentradiation. In this case, absorptance a is equal to theemissivity of the black body at its temperature.

At this point, it is possible to calculate the radiatingenergy transmission between two surfaces of solidsseparated by a non-absorbing medium. If they are two

λmax =0 2898.

T

eh c

ed

ce

Ph c k T c TP Bλ λ λ

π λλ

λ=

−=

−

− −2

1 1

2 5

1

5

2

ε= eei

B

e e eB1

1

2

2α α= =

e A A e e A A eB B1 1 1 1 2 2 2 2α α= =;



2,040°C

1,540°C

1,200°C

0

10

20

30

40

50

60

70

80

100

90

l (mm)2 30 1 4 5

el (a.u.)

Fig. 8. Dependency of el the specific emissive power on the wavelength for different values of temperature.

parallel surfaces with an infinite extension, grey, withemissivity e1 and e2 at temperature T1 and T2, body 1 emitsthe radiating energy e1�e1sT1

4 for unit time and surface, ofwhich the fraction e2 is absorbed by body 2 and the fraction(1�e2) is reflected again. This in turn will be partly absorbedand partly reflected, and so on.

It is possible to demonstrate that the energy absorbed bysurface 2, per unit of emitting surface, is given by:

[84]

and similarly:

[85]

Therefore the energy transferred in total from 1 to 2 is:

[86]

If the two surfaces are black, since e1�e2�1, [86] becomes:

[87]

If, however, the two surfaces are not parallel and have aninfinite extension, it is necessary to introduce a geometricfactor F12, called view factor, defined as the fraction ofenergy emitted by surface A1 in all directions intercepted bysurface A2.Therefore it is:

[88]

In the presence of different surfaces that together receiveall the radiating energy emitted by a single surface A1, thefollowing relation stands:

[89]

If A1 does not see any part of itself, then F11�0. Thevalues of the geometric factors involved in the solution ofthe most common problems have been calculated and shownin diagrams. The practical calculation of the view factor is,in any case, often quite complicated. The diagrams derivedby Hottel (1950, 1954) allow to derive the view factors in thecase of direct radiation between two adjacent rectangleslocated on perpendicular planes, and in the case of directradiation between identical geometrical forms that arelocated in opposite positions on parallel planes. When thetwo parallel radiating surfaces are connected by a reflectingsurface, the energy does not only transfer directly betweenthe two surfaces, but in part also by reflection through thereflecting surface. In this case, the quantity of exchangedheat can be then expressed by the equation:

[90]

where the values of the geometric factor F_

are always biggerthan the values of F since the presence of a reflecting surfacebetween two parallel surfaces increases the quantity of heatexchanged in the unit time.

If a closed cavity is taken into consideration, as theinside of a furnace, it can be subdivided into a certainnumber of radiating surfaces A1, A2, …, whereas the rest of

the cavity can be considered like a single reflecting surfaceAR at uniform temperature TR.

If there are two radiating surfaces A1 and A2, byindicating their geometrical factor with F12, the one betweenthe first surface and the reflecting one with F1R, the onebetween the reflecting surface and the second surface withFR2 and, finally, with FRR the reflecting surface with itself (ifthe surface is plane it is FRR�0), it is possible todemonstrate:

[91]

in fact, the fraction F12 of the energy emitted by A1 isdirectly absorbed by A2, and the latter also receives thefraction FR2�(1�FRR) of the radiation initially incident on AR.

The considerations developed above are valid providedthe radiating surfaces involved are black. An exactcalculation of the heat transferred between two non-blacksurfaces is, however, too complicated and therefore it issimplified by assuming that surfaces are grey. If oneconsiders two radiating surfaces A1 and A2 with emissivitiese1 and e2 in a cavity, it is possible to demonstrate that theexchanged heat, due to the combined mechanism of directradiation, reflecting surface and multiple reflection insidethe cavity, can be expressed by the equation

[92]

where factor F12 is given by:

[93]

Equation [92], which is the most general among thosepresented above to describe heat exchange, can be used toderive a heat transfer coefficient. In fact, it can be rewrittenin the following form:

[94]

from which the following expression for the transfercoefficient can be derived:

[95]

Gas radiation When studying solid surfaces radiations, it is possible to

assume with good approximation that bodies underexamination are grey, which means that emissivity orabsorptance are independent from the frequency of theradiation. In the study of gas radiation, it is necessary toaccount for this dependency. In fact, if the radiation of ablack body passes through a mass of gas, absorption takesplace only in some frequency intervals of the infraredspectrum. Similarly, if the gas gets heated, it irradiatesenergy in the same frequency interval. These infraredfrequencies are produced by molecular transitions betweenrotational and vibrational quantum levels.

In practice, among the gases contained in combustionproducts, carbon monoxide and dioxide, water vapour,sulphur dioxide, hydrogen chloride, and hydrocarbons havean appreciable value of emissivity. On the other hand, gaseswith symmetric diatomic molecules such as nitrogen, oxygen

hT TT T

=−−

σΦ12

1

4

2

4

1 2

q AT TT T

T T=−−

−( )σ1 12

1

4

2

4

1 2

1 2Φ

Φ12

12 1

1

2 2

11 1 1 1 1

=+ −

+ −

F

AAε ε

q A T T A T T= −( ) = −( )σ σ1 12 14

24

2 21 14

24Φ Φ

F FF FF

R R

RR12 12

1 2

1= +

−

q F A T T F A T T= −( ) = −( )σ σ12 1 14

24

21 2 14

24

F F F F11 12 13 14 1+ + + + =...

A F A F1 12 2 21=

qA T T

114

24= −( )σ

qA T T

114

24

1 2

11 1 1

= −( )+ −

σ

ε ε

qA

T2 1 2

4

1 2

1 11

→ =+ −

σ

ε ε

qA

T1 2 1

4

1 2

1 11

→ =+ −

σ

ε ε

HEAT EXCHANGE


and hydrogen do not show a significant emissivity.Emissivity and absorptance of a gaseous mixture depend onits radiating components, and they are proportional to theirconcentration and to the average width L of the gaseousmass.

Emissivity values of the most important gases depend ontemperature, on their partial pressure, and on the width L ofthe gaseous mass, which can be calculated by using thefollowing equation:

[96]

where V is the gas volume and S is the total surface. Total emissivity, or absorptance, of a gas mixture is

smaller than the sum of those of the pure components. If, forinstance, one considers a mixture of CO2 and H2O, you get:

[97]

where eg is the emissivity of the gas at a given temperature,eCO2

is CO2 emissivity at the same temperature, eH2O is wateremissivity, C is a correction term depending on both eCO2and eH2O.

Ovens and furnacesOne of the most important industrial applications of heat

transfer by radiation is represented by ovens, which arelargely used in oil refineries, for crude oil atmosphericpressure and under vacuum distillation, in thermal crackingand many service operations. Ovens are equipment in whichmost of the total heat gets transferred by direct radiationfrom a flame towards a surface capable of absorbing heat.Since radiation essentially moves in a straight line, thesesurfaces must be capable of seeing the sources that emits theradiation. Usually in industrial ovens, the bodies receivingthe heat are tubes containing the fluid that must be heated,and these tubes in their turn, are contained within wallsmade of a high reflectivity material; therefore it is not easyto theoretically determine the paths covered by radiation andthe thermal transfer rate.

Generally for ovens, heat flux can be expressed throughthe following equation:

[98] q �ℑsA�(T14�T2

4)

where T1 is the temperature of the source, T2 is thetemperature of the tubes, A� is an effective surface of thetubes, and ℑ is an adimensional transfer factor.

In most ovens, tubes are located on one or two rows, rightin front of the walls. Part of the radiation directly hits thetubes, which absorb it, whereas the remaining portion hits thewalls, which partly reflect it and partly disperse it in theatmosphere. It is possible to estimate an effective planesurface, Acp, to which a unit emissivity is assigned, in order toreplace the total tube surface in design calculations,multiplying the number of tubes by their length exposed toradiation and by the distance between their centres. Since theset of tubes does not entirely absorb the energy emitted by theflame, it is necessary to also calculate an absorptionefficiency factor a. A method to calculate a was developedby Hottel (1931). Exchange factor value ℑ mostly depends onthe emissivity of the gases produced in the combustion andon the radiation reflected by the walls. Important parameters,on the basis of what was discussed above, are therefore theconcentration of CO2 and H2O, gas temperature and the

width of the gaseous mass. Since most of the energy incidenton the walls gets reflected in the tubes direction, ovens thathave large parts of walls exposed to radiation are capable oftransferring more heat per unit surface by radiation than thosewhere walls are greatly shielded by tubes.

Bibliography

Fanaritis J.P. et al. (1980) Heat exchange technologies (heat transfer),in: Grayson M. (executive editor) Kirk-Othmer encyclopedia ofchemical technology, New York, John Wiley, 1978-1984, 26v.;v.XII, 129-170.

Graetz L. (1880) «Mathematik für Physiker», 25, 316.

Holman J.P. (2007) Heat transfer, New York, McGraw-Hill.

Incropera F.P., DeWitt D.P. (2001) Fundamentals of heat and masstransfer, New York, John Wiley.

References

Carslaw H.S., Jaeger J.C. (1959) Conduction of heat in solids,Oxford, Clarendon.

Dukler A.E. (1959) Dynamics of vertical falling film systems,«Chemical Engineering Progress», 55, 62-67.

Gurney H.P., Lurie J. (1923) Charts for estimating temperaturedistributions in heating or cooling solid shapes, «Journal ofIndustrial and Engineering Chemistry», 15, 1170-1172.

Hausen H. (1943) Presentation of heat transfer in tubes by means ofgeneralized exponential functions, «Zeitschrift des VereinesDeutscher Ingenieure», 4, 91-98.

Hottel H.C. (1931) Radiant heat transmission between surfacesseparated by non-absorbing media, «American Society ofMechanical Engineers. Transactions», 53, 265-273.

Hottel H.C. (1950) Radiant heat transmission, in: Chemical engineers’handbook, New York, McGraw-Hill, 493-498.

Hottel H.C. (1954) Radiant heat transmission, in: McAdams W.H.,Heat transmission, New York, McGraw-Hill, Chapter 4.

Kern D.Q. (1950) Process heat transfer, New York, McGraw-Hill.

McAdams W.H. (1954) Heat transmission, New York, McGraw-Hill.

Mostinski I.L. (1963) Application of the law of corresponding statesto the calculation of heat transfer and critical heat flux for boilingliquids, «Teploenergetika», 4, 66-71.

Nusselt W. (1916) Die Oberflachen-Kondensation des Wasserdampfes,«Zeitschrift des Vereines Deutscher Ingenieure», 60, 541.

Pigford R.L. (1955) Nonisothermal flow and heat transfer insidevertical tubes, «Chemical Engineering Progress», 51, 79-92.

Planck M. (1923) Vorlesungen über die Theorie der Wärmestrahlung,Lipsia, Barth.

Rohsenow W.M., Hartnett J.P. (1973) Handbook of heat transfer,New York, McGraw-Hill.

Sieder E.N., Tate G.E. (1936) Heat transfer and pressure drop ofliquids in tubes, «Journal of Industrial and Engineering Chemistry»,28, 1429-1435.

List of symbols

A area of the section of the body normal to the heatflow direction; section of a tube

a absorbivity of a body Ae external surface of a spheric layer Aeq equivalent surface for heat exchange problems

through a cave cylinder, or a spherical layer Ai internal surface of a spherical layer

ε ε εg C= + −CO H O2 2

L VS

=

0 85 4.



c speed of light in vacuumcp specific heat at constant pressure D tube diameter Deq equivalent tube diameter Ds diameter of the cylindrical shell of an exchanger e total emissive power of a body el specific emissive power of a bodyFi,j view factors Fp1 parameter in equation [60]g gravity accelerationG mass velocity of a fluid in an exchangerh heat transfer coefficient hT global heat transfer coefficient between the

environment and the surface of a body, in a non-stationary conduction problem

k thermal conductivity kB Boltzmann constant L length of a cylinder or a tube; width of a gaseous

mass l characteristic dimension of the system in natural

conduction problems NGr Grashof numberNGz Graetz numberNNu Nusselt numberNPr Prandtl numberNRe Reynolds number NSt Stanton numberNt number of tubesNvtr number of vertical rows of tubes on which the

condensed fluid drips p pressurepc critical pressure of a fluid q heat exchanged per unit time q� heat generation rate r thermal resistance of a wall R in a non-stationary conduction problem, distance, in

the direction along which heat is transmitted,between the point or the plane in the middle of thebody and the point under examination

r reflectivity of a bodyrc in a heat exchanger, the resistance offered to heat

exchange by the cold fluid rcs in a heat exchanger, the resistance offered to heat

exchange by the scale on the cold fluid side Re external radius of a hollow cylinder, or of a

spherical layer rh in a heat exchanger the resistance offered to heat

exchange by the hot fluid rhs in a heat exchanger the resistance offered to heat

exchange by the scale on the hot fluid sideRi internal radius of a hollow cylinder, or of a

spherical layerRlm logarithmic average radius Rm in problems of non-stationary conduction, radius of

the sphere or of the cylinder, or half of the width ofthe slab heated from both sides, or width of the slabheated on one side and perfectly insulated on theother

rw in a heat exchanger, the resistance offered to heatexchange by the wall

Ss in a heat exchanger, area of the free section in thespace between two baffles.

Sw area of the free section in the shell of an exchanger T thermodynamic temperaturet timeT� environmental temperature T0 in a non-stationary conduction problem, initial

temperature of a body Tb temperature in the fluid bulk Tc in an exchanger, temperature of the cold fluid Tci in an exchanger, inlet temperature of the cold fluid Tco in an exchanger, outlet temperature of the cold fluidTe temperature of the external surface of a hollow

cylinder, or of the external surface of a sphericallayer

Th in an exchanger, temperature of the hot fluidThi in an exchanger, inlet temperature of the hot fluidTho in an exchanger, outlet temperature of the hot fluidTi temperature of the internal surface of a hollow

cylinder, or of the internal surface of a sphericallayer

Tn temperature of the contact surface between twoconcentric cylinders, or spheres

Ts wall temperature U global transfer coefficient umax fluid flow rate through the minimum free area

between tube rows V volume of a body v linear velocity of a fluid within a tube WF flow in a condensing film xp distance between two subsequent baffles in an

exchanger Y correction factor to use when the fluxes in a heat

exchanger are neither parallel nor in countercurrent Y adimensional ratio (eq. [31]) yT distance between the axes of two neighbouring

tubes in an exchanger

Greek lettersr densitya thermal diffusivity b coefficient of thermal expansion of a fluid m dynamic viscosity of a fluid mb viscosity of a fluid at its average temperature mw viscosity of a fluid at the wall temperature D parameter of equation [54]G flux of condensed fluid per perimeter unit t transmittivity of a body e emissivity of a surface l wavelengths Stefan-Boltzmann constant

Stefano Carrà

MAPEI

Milano, Italy

HEAT EXCHANGE


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