I020 Heat Transfer

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I2. HEAT TRANSFER

Practically all the unit operations of chemical engineering involve heat effects. Thus, theprinciples governing the transfer of heat are important to chemical engineers in the design and

analysis of chemical processes.

I2.1. MECHANISMS OF HEAT TRANSFER

Heat is energy that is transferred from one body to another when there is a temperature

difference between them, as when a hot body is brought into physical contact with a coldbody. In accordance with the second law of thermodynamics, the transfer of heat is from the

hot body to the cold body. As required by the first law of thermodynamics, the heat given up

by the hot body equals the heat gained by the cold body. The transfer of heat stops when the

two bodies reach the same temperature. Even when the bodies are not in physical contact,

heat transfer is still possible through the exchange of thermal radiation. The threefundamental heat transfer mechanisms are (1) conduction, (2) convection, and (3) radiation.

Heat transfer by conductiontakes place when a temperature gradient exists within a solid orquiescent fluid medium. The internal energy is transferred from the more energetic particles

of the medium to adjacent, less energetic particles without appreciable displacement of the

particles. In solids, heat conduction occurs by two mechanisms: (1) transmission ofvibrational energy in the lattice structure and (2) transport by free electrons.

Heat transfer by convectiontakes place between a solid surface and an adjacent moving fluid

that are at different temperatures. The fluid motion brings hot and cold fluid parcels into

contact causing energy transport at a greater number of sites than in the absence of fluidmotion.

Heat transfer by radiationoccurs when a cold body absorbs the radiant energy emitted by a

hot body that is not in physical contact with it. Energy transfer by radiation does not require

the presence of an intervening material medium and is fastest at the speed of light in a

vacuum.

I2.2. STEADY-STATE CONDUCTIVE HEAT TRANSFER

I2.2.1. Fouriers Law of Heat Conduction (One-Dimensional)

Under steady-state condition, the conductive heat flux(the rate of heat transfer per unit timeper unit surface area) q/Ain the positivexdirection and across an isothermal surface of areaA

in a homogeneous conducting medium is proportional to the temperature gradient ( dxdT )at that surface:


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dx

dTk

A

q (I.21)

In the absence of convection and radiation, Fouriers law is also valid for liquids and gases.

I2.2.2. Thermal Conductivity

The proportionality constant kin the Fourier equation is a physical property called thermalconductivitythat indicates how well a medium conducts thermal energy.

In general, k varies with temperature. However, for engineering applications k may beconsidered constant when the temperature drop in the conducting medium is not more than

200oC (Hagen, 1999). For larger temperature ranges, kmay be evaluated either by using the

average of the individual values of kfor the two surface temperatures, or by calculating thearithmetic average of the surface temperatures and using the value of k at the average

temperature and treating it as a constant in the Fourier equation.

The temperature dependence of k may be taken into account using the linear relationship(McCabe, 2001):

bTak (I.22)

where aand bare empirical constants and Tis the temperature of the conducting medium.

I2.2.3. Conduction Through a Solid

During steady-state heat conduction, the temperature distribution within the conductingmedium does not change with time, there is neither accumulation nor depletion of heat within

the material, and the rate of heat transfer is constant.

A. Conduction Through a Plane Wall

Consider the steady-state conduction of heat from point 1to point 2within a large plane wall

of constant cross sectional area A. After separation of variables in the Fourier equation andintegration,

RT

xxTTkA

xTkAq

)()(

12

12 (I.23)

where )( 12 xxx , thickness of plane wall

)(kAxR , conductive thermal resistance of wall

)( 21 TTT , temperature drop across wall


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Although mathematically improper, it is common practice to use the expression T instead of)( T to represent the thermal driving force.

B. Conduction Through a Hollow Cylinder

Consider the radial conduction of heat in a long hollow cylinder from its inside surface at

temperature iwT , to its outside surface at temperature owT , . The area across which heat flows

is not constant but is proportional to the radius ( rLA 2 ). Thus, for a long hollow cylinderwith inside radius ri, outside radius roand lengthL(after separation of variables in the Fourier

equation and integration):

R

T

rr

TTAk

rr

TTLkq

io

oi

L

io

oi

)(

)(

)/ln(

))(2( (I.24)

where:

L

io

Ak

rrR

)( (I.25)

LrA LL )2( (logarithmic mean surface area) (I.26)

)r/rln(

)rr(r

io

io

L

(logarithmic mean radius) (I.27)

C. Conduction Through a H ollow Sphere

Consider the outward flow of heat through a hollow sphere with inside surface temperature Ti,inside radius ri, outside temperature To, and outside radius ro. The surface area perpendicular

to the heat flow path is proportional to the square of the radius (

2

r4A

). The heat transferrate through a hollow sphere is (after separation of variables in the Fourier equation and

integration):

R

T

)rr(

)TT(Ak

)rr(

)TT()rr4(kq

io

oiG

i0

oi

oi

(I.28)

where:

G

io

Ak

rrR

)( (I.29)

oioiG AArrA 4 (geometric mean surface area) (I.210)

I2.2.4. Conduction Through a Multilayer Wall

A. Conductive Resistances in Series

Consider the steady-state transfer of heat through a wall consisting of a series of layers ofdifferent conducting media A, B, and C that are in excellent thermal contact. This is


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analogous to current flow through several resistances in series where thermal resistance

corresponds to electrical resistance; temperature drop to voltage drop; and heat transfer rate to

electric current.

1. Total Temperature Drop:

The total temperature drop across the multilayer wall is equal to the sum of the temperaturedrops across each layer,

CBA TTTT (I.211)

2. Heat Flow Rate:

The rate of heat flow in each layer is the same,

qqqq CBA (I.212)

3. Total Wall Resistance:

The total resistance of the multilayer wall is equal to the sum of the individual resistances,

CBA RRRqTR / (I.213)

where: (a) for a multilayer plane wall:

)( AkxqTR AAAAA (I.214))( AkxqTR BBBBB (I.215)

)( AkxqTR CCCCC (I.216)

(b) for a multilayer cylindrical wall:

)/()( ,,, ALAAiAoAAA AkrrqTR (I.217)

)/()( ,,, BLBBiBoBBB AkrrqTR (I.218)

)/()( ,,, CLCCiCoCCC AkrrqTR (I.219)

where:LrALrALrA CLCLBLBLALAL )2(,)2(,)2( ,,,,,, (I.220)

)/ln(

)(,

)/ln(

)(,

)/ln(

)(

,,

,,

,

,,

,,

,

,,

,,

,

CiCo

CiCo

CL

BiBo

BiBo

BL

AiAo

AiAo

ALrr

rrr

rr

rrr

rr

rrr

(I.221)

CiBoBiAo rrrr ,,,, , (I.222)


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These results can be generalized to includeNlayers in the multilayer wall.

B. Conductive Resistances in Parall el

Consider the steady state heat conduction through a wall consisting of plane slabs of different

materialsA,B, and Carranged side by side in parallel. This situation is analogous to current

flow through several resistances in parallel.

1. Total Heat Flow Rate:

The total heat flow rate is equal to the sum of the heat flow rates through each slab,

CBA qqqq (I.223)

2. Overall Temperature Drop:

The temperature drop is assumed equal across each material,

TTTT CBA (I.224)

3. Overall Resistance:

The reciprocal of the overall wall resistance is equal to the sum of the reciprocals of the

resistance of each layer,

CBA RRRTqR /1/1/1//1 (I.225)

This result can be generalized to includeNnumber of materials in the composite wall.

I2.2.5. Thermal Contact Resistance

If two materials are fitted perfectly together (with no intervening air spaces) to form acomposite wall, there will be no thermal resistance at the interface and the adjacent surfaces

will be at the same temperature. However, when adjacent layers do not fit tightly together,

thermal contact resistance (or interface resistance) will arise due to: (1) the presence of

fewer contact points through which heat can be transferred by conduction, and (2) thepresence of an insulating stagnant fluid layer trapped between the two surfaces. The heat

transfer rate through an interface is given by

c

cR

TTAhq

(I.226)

where: T = temperature drop across a solid-solid interface

cR = thermal contact resistance across interface


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ch = thermal contact resistance coefficient at interface

A= interfacial area

I2.2.6. Thermal Insulators

Thermal insulators are materials that, because of their low thermal conductivities, retard the

transfer of heat in pipes, vessels, buildings, etc.

A. R-value

Theparameterused to classify the thermal performance of a thermal insulation isitsR-value,which is defined as follows:

1. For a flat insulation:k

x

Aq

TvalueR

/ (I.227)

2. For a cylindrical insulation:i

oo

o r

r

k

r

Aq

TvalueR ln

/

(I.228)

B. Criti cal Radius of I nsulation

At steady state, the heat transfer rate qacross a layer of insulation that surrounds the outer

surface of a cylinder equals the rate of convection from the outer surface of the insulation(characterized by the outside convective heat transfer coefficient ho). As more insulating

material is added, the outside surface area of the insulation increases while its outside surface

temperature decreases. The heat transfer rate through the insulation is a maximum when theouter radius of the insulation reaches a critical radius rcr.

1. For a cylindrical insulation:o

crh

kr (I.229)

2. For a spherical insulation:o

crh

kr

2 (I.230)

When the outer radius of the insulation is less than the critical radius, adding more insulation

will increase the heat transfer rate. However, when the outer radius of insulation is greaterthan the critical radius, adding more insulation will decrease the heat transfer rate.

I2.3. TRANSIENT HEAT TRANSFER ACROSS A SOLID BOUNDARY

Consider a solid material initially at a uniform temperature Tabeing suddenly immersed in a

heating or cooling fluid medium, which is assumed to be at a uniform temperature T .


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Before an equilibrium temperature can be reached in the solid, some time must elapse during

which the temperature within the solid changes both with position and with time an

unsteady-stateor transientperiod.

I2.3.1. Dimensionless Parameters

The definition of the following dimensionless quantities reduces the number of parameters

involved in the formulation and the solution of transient heat conduction problems.

A. Fourier Number

The Fourier numbergives the ratio of the rate at which heat is conducted through a body to

the rate at which heat is stored in the body. A large value of the Fourier number indicates fast

propagation of heat through a body (Cengel, 2003).

1. For a flat solid slab: 22p

Fos

t

sc

ktN

(I.231)

2. For a solid cylinder or a solid sphere:22

p

For

t

rc

ktN

(I.232)

where s= half-thickness of slab

r = radius of sphere or cylinder

t = time of heating or cooling

= thermal diffusivity of conducting medium

pck / (I.233)

B. Biot Number

The Biot number is the ratio of the conductive resistance within a solid to the convectiveresistance at its surface. It is a measure of the importance of the internal resistance relative to

the external resistance.

k

hL

h

kLN cc

Bi

/1

/ (I.234)

where Lc is the surface area to volume ratio (characteristic dimension) of the solid of

arbitrary shape:

sc AVL / (I.235)


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I2.3.2. Transient Heat Conduction in Slabs, Cylinders, and Spheres

The time-temperature history of a solid during unsteady-state heating or cooling is given

below for the two cases of very small and very large number Biot Number corresponding tovariable surface temperature and constant surface temperature of the solid, respectively.

A. Variable Sur face Temperature ( 0BiN )

A very smallNBiindicates that the conductive resistance within the solid is much smaller than

the external convective resistance at the solid-fluid boundary. Thus, the temperature gradient

within the solid is so small that its temperature is essentially uniform spatially but varies withtime as the solid exchanges heat by convection with the surrounding fluid medium. This

idealized system with spatially uniform but time-dependent temperature is known as a

lumped-heat-capacity system. The smaller the size of the system, the more realistic this

assumption will be. Lumped-system analysis is generally applicable for 1.0BiN (Cengel,

2003).

1. Large Flat Solid Slab (McCabe, 2001):

sc

Utln

TT

TT

p

1

a

b

(I.236)

k

s

hU 2

11 (I.237)

2. Long Circular Solid Cylinder (McCabe, 2001):

rc

Ut2ln

TT

TT

p

1

a

b

(I.238)

k

r

hU 3

11 (I.239)

3. Solid Sphere (McCabe, 2001):

rc

Ut3ln

TT

TT

p

1

a

b

(I.240)


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k

r

hU 5

11 (I.241)

where: bT = average temperature of solid at time t

T = temperature of heating or cooling fluid medium, which is assumed to be uniform

aT = initial temperature of solid

U= overall heat-transfer coefficient

B. Constant Sur face Temperature )1.0,( FoBi NN

A very largeNBimeans that the convection resistance at the solid-fluid interface is negligible

relative to the conduction resistance within the solid. Thus, the surface of the solid is quickly

brought to, and maintained at, the ambient fluid temperature T . Furthermore, since most of

the resistance is internal to the solid, the temperature within the solid changes both with time

and with position from its initial temperature Ta to the temperature of the surrounding fluid

medium T at the end of the transient period.

1. Large Flat Solid Slab (Middleman, 1998):

)N47.2(exp81.0TT

TTFo

a

b

(I.242)

This equation is also applicable to a slab heated from one side only, provided that no heat is

transferred at the other side and 0dxdT at that surface.

2. Long Solid Cylinder (Middleman, 1998):

)N78.5(exp692.0TT

TTFo

a

b

(I.243)

3. Solid Sphere (Middleman, 1998):

)N87.9(exp608.0TT

TTFo

a

b

(I.244)

I2.3.3. Transient Heat Conduction in Semi-Infinite Solids

Consider solid of infinite thickness that is at a uniform temperature Ta at time t = 0. The

surface temperature is brought quickly to, and held at, the temperature of the surrounding


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fluid medium so that at time t > 0, Ts= T . The temperature within the solid will change with

time but these temperature changes will be confined to the region near one surface only. The

thermalpenetration distance px at time t> 0 is that distance from the surface of the solid

beyond which not enough heat has penetrated to affect the temperature significantly. It is

arbitrarily defined as that distance from the surface at which the temperature change at time t

> 0 is 1 % of the initial change in surface temperature (McCabe, 2001):

txp 64.3 (I.245)

I2.3.4. Amount of Heat Transferred

The total amount of heat transferred QT through a unit surface area A during heating or

cooling of a solid in time tis:

A. For a flat solid slab: )( abpT

TTcsA

Q

(I.246)

B. For a solid cylinder:2

)( abpT TTcr

A

Q

(I.247)

C. For a solid sphere:3

)( abpT TTcr

A

Q

(I.248)

D. For a semi-infinite solid:

t)TT(k2

A

Qas

T (I.249)

I2.4. CONVECTIVE HEAT TRANSFER

When a fluid flows over a solid surface, most of the thermal resistance is present in a thin

layer of fluid adjacent to the solid surface. Heat is transferred mainly by conduction acrossthis film. Thermal convection is thermal conduction with the added complexity of thermal

energy transfer by moving fluid parcels.

I2.4.1. Newtons Law for Convective Heat Transfer

Consider the transfer of heat from a solid material to an adjacent flowing fluid. At a given

location in the solid-fluid interface, the convective heat flux is proportional to the difference

between the surface temperature of the solid sT and the temperature of the fluid T along the

heat flow path that is sufficiently far from the surface:


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)( TThA

qs (I.250)

where: h = convective heat transfer coefficient (film coefficient)

A = area of the solid surface adjacent to the fluid

The convective heat transfer coefficient, in general, varies along the heat flow direction. Theheat transfer rate over the entire heat transfer area is obtained by using an average convective

heat transfer coefficient in the convective heat transfer equation. This average coefficient is

determined by averaging the local convection heat transfer coefficients over the entire heattransfer surface.

Unlike thermal conductivity, the convective heat transfer coefficient is not a thermal property

of the heat transfer medium. It is dependent not only on the thermal properties of the fluidmedium but also on the system geometry, temperature difference, and fluid flow pattern.

I2.4.2. Heat Exchange Equipment

Typically, the function of a heat exchanger is to increase the temperature of a cold fluid and

decrease the temperature of a hot fluid.

A double-pipe heat exchanger consists of two concentric pipes with one fluid flowing

through the center pipe while the other fluid flowing through the annular space.

A shell-and-tube heat exchangerconsists of tube bundles enclosed in a cylindrical casing

(the shell) with one fluid flowing through the tubes and the other fluid through the space

between the tube bundles and the casing.

A cross-flow heat exchanger has rows of tubes enclosed within an unbaffled rectangular

shell. The number of tubes in each row is the same and flow in the shell is directly across the

tubes.

A plate heat exchanger consists of many corrugated stainless-steel sheets separated by

polymer gaskets and clamped in a steel frame. Inlet portals and slots in the gaskets direct thehot and cold fluids to alternate spaces between the plates.

A finnedheat exchanger is one in which the outside area of the tubes in contact with the

fluid stream having the lower heat transfer coefficient is made much larger than the insidearea to enhance the transfer of heat with the use of fins, pegs, disks, and other appendages.

A scraped-surface heat exchanger is adouble-pipe heat exchanger with a central tube, theinside surface of which is wiped by longitudinal blades mounted on a rotating shaft. This type

of heat exchanger is used for heat transfer to or from viscous liquids, especially food products

and other heat sensitive liquids.


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I2.4.3. Basic Heat Exchanger Design Equations

A. Average Temperature of a F lu id Stream

During heating or cooling of a fluid stream in a tube, a temperature gradient exists across anygiven stream cross section (from tube wall surface to the center of the fluid stream). The

energy-average (or bulk, also mixing cup) fluid stream temperature, T is the uniformtemperature that the fluid stream at a particular section in the tube would attain if all the fluid

elements across the stream cross section were mixed adiabatically and allowed to come to

equilibrium.

B. Energy Balance for a Flu id Stream

The general energy balance for a system in steady state flow is

SKP wqeehm )( (I.251)

where: m = mass flow rate of system

Ke = kinetic energy change per unit mass of system

h = specific enthalpy of system

Pe = potential energy change per unit mass of system

Sw = rate of shaft work

q = heat transfer rate into or out of the system

For a fluid in steady-state internal flow, 0

Sw and hqee KP

,, . The energy balancereduces to

qhm (I.252)

C. Overal l H eat Balance

1. Heat Exchanger

When the outer surface of the heat exchanger is well insulated, sensible heat transfer isconfined between the hot fluid stream and the cold fluid stream only. Thus,

0 hc qq (I.253)

)()(hbhahcacbc

hhmhhm (I.254)

When the fluids have constant specific heat capacities,


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)()( hbhaphhcacbpcc TTcmTTcm (I.255)

where: Tcb= bulk temperature of cold fluid stream leaving the heat exchanger

Tca= bulk temperature of cold fluid stream entering the heat exchanger

Tha = bulk temperature of hot fluid stream entering the heat exchanger

Thb= bulk temperature of hot fluid stream leaving the heat exchanger

2. Condenser

In a condenser, the cold fluid stream absorbs the latent heat released by the hot fluid streamas it condenses:

hcacbpcc mTTcm )( (I.256)

If the vapor at the inlet is superheated and/or if the condensate leaving the condenser is

subcooled, appropriate sensible heat terms must be added in the right-hand side of the overall

heat balance equation.

D. Overall Heat Transfer Coeff icient

In heat exchanger analysis, it is convenient to express heat transfer in terms of an overall heat-

transfer coefficient U, which is defined in an analogous manner to Newtons law for

convective heat transfer as

TUAq (I.257)

A comparison of UandRshows that

TUAR

Tq

(I.258)

UAR

1

(I.259)

For a tubular heat exchanger, the heat transfer areas on both sides of the tube wall, AiandAo,

are not equal, giving rise to two overall heat transfer coefficients, Ui and Uo:

TAUTAUq iioo (I.260)

A plate heat exchanger, on the other hand, has only one Ubecause the areas on both sides of

the plate are the same.

1. Plate Heat Exchanger


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Consider a plate being exposed to a hot fluid on one side and a cold fluid on the other. Heat

transfer from the hot fluid to the cold fluid through this plate is analogous to current flow

through several resistances in series with two convective thermal resistances and one

conductive thermal resistance. At a particular section of the exchanger where the temperatureof the hot fluid is Tiand the temperature of the cold fluid is To:

Local heat flow rate: qqqq owi (I.261)

Local overall temperature drop: owi TTTT (I.262)

where:

oi TTT (I.263)

i,wii TTT (I.264)

o,wi,ww TTT (I.265)

oo,wo TTT (I.266)

Local overall thermal resistance: RRRR owi (I.267)

where:

)Ah(1qTR iiii (I.268)

)( AkxqTR wwwww (I.269)

)Ah(1qTR oooo (I.270)

Local overall heat-transfer coefficient:ow

w

i h

1

k

x

h

1

U

1

(I.271)

2. Tubular Heat Exchanger

Consider the steady-state transfer of heat from a hot fluid stream inside a tube to a cold fluid

stream outside the tube. The process consists of three steps: (1) convection from the hot fluid

to a tube wall surface, (2) conduction through the tube wall, and (3) convection from thesurface on the other side of the tube wall to the cold fluid.

At a section in the heat exchanger where the temperature of the hot fluid is Ti and thetemperature of the cold fluid is To:

Local heat flow rate: qqqqowi

(I.272)

Local overall temperature difference: owi TTTT (I.273)

where:

iwii TTT , (I.274)

owiww TTT ,, (I.275)


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oowo TTT , (I.276)

oi TTT (I.277)

Local overall thermal resistance: owi RRRR (I.278)

where:)(1 iiiii AhqTR (I.279)

)( Lwwwww AkxqTR (I.280)

)(1 ooooo AhqTR (I.281)

Local overall heat-transfer coefficient:

ooLw

w

iiiioo AhAk

x

AhAUAUR

1111

(I.282)

where:oLw

ow

ii

o

o hDk

Dx

hD

D

U

11

(I.283)

oo

i

Lw

iw

ii hD

D

Dk

Dx

hU

11 (I.284)

Special Cases:(a) For large-diameter thin-walled tube, the inner and outer surfaces are almost

identical, oi DD and

owwi

oih/1k/xh/1

1UUU

(I.285)

(b)When the thin heat exchanger tube is made of a highly conductive material, its

thermal resistance is negligible and the overall heat transfer coefficient further

simplifies to

oi hhU

111 (I.286)

(c) When the fluid outside the tube has a much higher thermal resistance than the tube

wall and the fluid inside the tube, it will control the rate of heat transfer, i.e.,

outside resistance controlling, thus

oo hU (I.287)

Values of the overall heat transfer coefficient range from about 10 W/m2-oC for gas-to-gas

heat transfer to about 10,000 W/m2-

oC for heat transfer that involves phase changes (Cengel

2003).


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E. Tube Wall Sur face Temperatures

The thermal resistance concept can be used to determine iT or oT , which when added to,

or subtracted from, Tior Towill give the inside and outside tube wall surface temperatures

wiT and woT , respectively:

owi qqqq (I.288)

o

o

Loww

w

iio

i

o h/1

T

)D/D)(k/x(

T

hD/D

T

U/1

T

(I.289)

F. Fouling

In actual practice, heat transfer surfaces do not remain clean. The accumulation of solid

deposits on one or both sides of the heat exchanger tubes causes a reduction in the rate of heat

transfer and a deteriorating performance of the heat exchanger with time. It can be minimizedby avoiding large temperature differences and low fluid velocities and by adding chemical

inhibitors.

The fouling factor accounts for the additional resistance of scale deposits on the surface of a

heat exchanger tube. It is defined as the thermal resistance due to fouling for a unit area of a

heat exchange surface:

of

of

if

ifh

Randh

R,

,

,

,

11 (I.290)

where hf,iand hf,oare the fouling coefficients for the inside and outside surfaces of the tube,

respectively.

The total thermal resistance in the heat exchanger becomes:

oofooLw

w

iifiiiioo AhAhAk

x

AhAhAUAUR

,,

111111

(I.291)

The fouling factor can be determined from experimental overall heat-transfer coefficients forboth clean and dirty heat exchanger tubes:

tubecleantubedirty

fUA

1

UA

1R

(I.292)

I2.4.4. Heat Exchanger Analysis


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A. LMTD Method

1. Parallel-Flow and Counter-Flow Heat Exchangers

The rate of heat transfer in a differential section of a simple heat exchanger is

dATTUdq ch )( (I.293)

where dAis the element of surface area required to transfer heat at the rate dqat a point in the

exchanger where the local overall heat transfer coefficient is U and the local overall

temperature difference between the two fluid streams is (Th- Tc).However, fluids become heated or cooled as they pass through a heat exchanger. Thus, in

general, the local temperature difference between the two fluid streams will vary with position

in the heat exchanger and some mean temperature driving force must be used when

calculating for the heat transfer rate over the entire heat exchanger. When the overall heat-transfer coefficient, the mass flow rates and specific heat capacities of the fluids are constant,

the heat-transfer rate Tq over the entire heat exchanger is given by

LTT TUAq (I.294)

where:T

A = total heat-transfer area in the heat exchanger

21, TT = temperature difference between hot and cold fluids at two heat exchangerterminals

LT = logarithmic mean temperature difference (LMTD)

)T/Tln(

TTT

12

12L

(I.295)

2.Multipass and Cross-Flow Heat Exchangers

Many commercial heat exchangers are not simple double-pipe systems and involve a

combination of mixed fluid flows: counter flow, parallel flow, and cross flow. The correction

factorFG, defined with LT based on the equivalent counter flow, is used to account for this

as well as for the geometry of the exchanger as follows:

LG TUAFq (I.296)

where LT = logarithmic mean temperature difference for counter-flow double-pipe heat

exchanger with same inlet and exit temperatures as the heat exchanger being considered.

FG is correlated in terms of two dimensionless temperature ratios, Z and .ParameterZgives the ratio of the fall in temperature of the hot fluid to the rise in temperature

of the cold fluid (McCabe, 2001):


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ha hb

cb ca

T TZ

T T

(I.297)

Parameter is the heating effectiveness of the heat exchanger, which is the ratio of the

actual temperature rise of the cold fluid to the maximum possible temperature rise (obtainable

if the warm end approach were zero based on counter flow) (McCabe, 2001):

cb caH

ha ca

T T

T T

(I.298)

B. Effectiveness-NTU M ethod

1. Number of Heat Transfer Units

In a heat exchanger operating at steady-state, the rate at which heat is transferred across the

heat-conducting wall is equal to both the rate at which heat is absorbed by the cold fluid

stream and the rate at which heat is lost from the hot fluid stream:

hc qqq (I.299)

)TT(cm)TT(cmq hbhaphhcacbpcc (I.2100)

hbhahcacbcL TTCTTCTUA (I.2101)

L

cacb

c T

TT

C

UA

(I.2102)

L

hbha

h T

TT

C

UA

(I.2103)

where pccc cmC and phhh cmC are the heat capacity rates of the cold and hot fluid

streams, respectively.

The number of heat transfer units, HN ,gives the ratio of the temperature change of the

fluid stream with the smaller heat capacity rate, minC , (hence, with the greater temperature

change) to the average temperature driving force in the heat exchanger. Thus,

minC

UANH (I.2104)

2. Heat Transfer Effectiveness

The maximum possible heat transfer rate in a heat exchanger occurs when the fluid having the

smaller heat capacity rate undergoes the maximum possible temperature change in a heatexchanger. This maximum temperature change is equal to the difference between the inlet

temperature of the hot and cold fluids:


19/50

I2 - 19

maxminminmax )( TCTTCq caha (I.2105)

This occurs in either of the following two cases in a counter-flow heat exchanger with an

infinite heat transfer area:(a) the cold fluid, having the smaller heat capacity rate, is heated to the inlet

temperature of the hot fluid, or

(b)

the hot fluid, having the smaller heat capacity rate, is cooled to the inlet

temperature of the cold fluid.

The heat transfer effectiveness, , gives the ratio of the actual heat transfer rate in a given

heat exchanger to the maximum possible heat transfer rate:

maxq

q (I.2106)

where is the heat transfer effectiveness. When the cold fluid has the smaller heat capacity

rate, minCCc and hacb TT . Thus,

)(

)(

min caha

hbhah

TTC

TTC

(I.2107)

Correspondingly, when the hot fluid has the smaller heat capacity rate, minCCh and

cahb TT . Thus,

)(

)(

min caha

cacbc

TTC

TTC

(I.2108)

The heat transfer effectiveness is useful in determining the heat transfer rate in a heat

exchanger without knowing the outlet temperatures of the fluids as well as in predicting outlet

temperatures when inlet temperatures are known.

The heat transfer effectiveness can be expressed in terms of the number of heat transfer units

and heat capacity ratio as follows:

Counter-flow double pipe heat exchanger:

)]1(exp[1)]1(exp[1cNc

cNH

H

(I.2109)

Parallel-flow double pipe heat exchanger:

c

cNH

1

)]1(exp[1 (I.2110)


20/50

I2 - 20

where: maxmin/ CCc , heat capacity ratio

maxmin ,CC = smaller and larger heat capacity rates, respectively

I2.4.5. Finned Heat Exchangers

A heat transfer fluid whose heat transfer coefficient is much smaller than the other fluid

creates a bottleneck on the path of heat flow. Fins made of highly conductive material are

used to extend the heat transfer area out into this fluid to compensate for its high thermal

resistance, thus enhancing the heat transfer.

A. F in with Constant Cross Section

1. Fin with Insulated Tip

The heat transfer from an insulated fin tip will be negligible. The heat transfer rate from the

fin Fq is given by (Cengel, 2003):

aLtanh)TT(pkAhq fbcoF (I.2111)

where: ho= convective heat transfer coefficient at fin surface

Ac= uniform cross sectional area of fin

p= perimeter of the fin cross sectionk= thermal conductivity of fin

T = temperature of surrounding fluid medium

fbT = temperature of fin base

c

o

kA

pha (I.2112)

Since heat transfer from a fin is proportional to its surface area, this equation is applicable

when the surface area of the fin tip is a negligible fraction of the total fin area.

2. Infinitely Long Fin

When the fin is very long, the temperature of the fin tip will approach the temperature of thesurrounding fluid medium. Since 1tanh aL as L thus, for an infinitely long fin(Cengel, 2003):

)( , TTpkAhq owcoF (I.2113)


21/50

I2 - 21

3. Fin with Convection at Tip

In this case, in addition to the lateral fin area, the fin tip area is also subjected to convection.

Thus, the total surface area of fin subjected to convection is:

tipfinsideslateralF AAA (I.2114)

Dividing each term in the preceding equation by the perimeter p gives the equivalent fin

length, eL :

p

ALL ce (I.2115)

where:

4

DLLe for cylindrical fin of diameterD (I.2116)

)(2 wt

twLLe

for rectangular fin of thickness tand width w (I.2117)

2

tLLe for thin rectangular fin (I.2118)

Thus, this case can be treated as fin with insulated tip by replacing the actual fin length withthe equivalent length, which takes into account heat loss from the edges as well as from the

tip of the fin:

eowcoF aLTTpkAhq tanh)( , (I.2119)

B. Fi n Ef fi ciency

The temperature drops along the length of a fin and because of the decreasing temperature

driving force toward the fin tip, the heat transfer rate from the fin will also drop. The fin

efficiency, F , gives the ratio of the actual heat transfer rate to the maximum heat transfer rate

from the fin:

max,F

F

Fq

q (I.2120)

The maximum heat transfer rate occurs when the entire fin is at fin base temperature Tfb:

)TT(Ahq fbFomax,F (I.2121)

For an infinitely long fin (Cengel, 2003):

aL

1

ph

kA

L

1

)TT(Ah

)TT(pkAh

o

c

fbFo

fbco

F

(I.2122)


22/50

I2 - 22

For a fin with an insulated tip (Cengel, 2003):

aL

aLF

tanh (I.2123)

A short fin with high thermal conductivity would be nearly isothermal, thus 1F .

C. Fin Ef fectiveness

The performance of fins can be expressed in terms of the fin effectiveness, which gives the

ratio of the heat transfer rate from the fin of base area Afb to the heat transfer rate from the

same area of bare surface:

)TT(Ah

q

q

q

fbfbo

F

B

FF

(I.2124)

1F fin does not affect heat transfer from surface

1F fin retards heat transfer from surface

1F fin enhances heat transfer from surface

The following equation relates fin effectiveness to fin efficiency:

F

B

FF

A

A (I.2125)

D. Total H eat Transfer Area of F inned Sur face

The total surface area on the finned side of a heat exchanger tube consists of two parts: (1) thearea of the fins and (b) the area of the bare surface not covered by the base of each fin:

UFF AAA (I.2126)

where:A= total area of finned surface

AF = surface area of fins

F= fin efficiencyA

U= area of the unfinned portion of tube surface, A

U= A

OA

B ,where A

O is the

original surface area without fins and AB is the portion of the original area

covered by the base of the fins

E. Rate of H eat Transfer fr om F inned Surf ace

The total heat transfer rate for a finned surface qis the sum of the heat transfer rate at the fins,

Fq , and the heat transfer rate at the bare (unfinned) portion of the surface, Uq :


23/50

I2 - 23

)TT)(AA(hqqq o,wFFUoFU (I.2127)

The convective resistance outside a finned tube or wall is

)(

1

FFUo

o

AAh

R

(I.2128)

The overall effectivenessfor a finned surface is defined as the ratio of the total heat transferrate from the finned surface to the heat transfer rate from the same surface if there were no

fins:

)TT(Ah

)TT)(AA(h

q

q

o,wBSo

o,wT,FFT,Uo

BS

FSFS

(I.2129)

whereABSis the area of the bare surface when there are no fins, T,FA is the total surface area

of all the fins on the surface, and T,UA is the total area of the unfinned portion of the surface.

F. Heat Sinks

Heat sinks are specially designed finned surfaces that are used in the cooling of electronic

equipment. They lower the thermal resistance by increasing the heat transfer area. Their heat

transfer performance is expressed in terms of their thermal resistance (Cengel, 2003):

)TT(AhR

TTq fbFFo

fb

F

(I.2130)

I2.5. HEAT TRANSFER CORRELATIONS

The individual heat transfer coefficients are estimated from appropriate correlations of

experimental heat transfer data.

I2.5.1. Dimensionless Numbers

Parameters used in heat transfer correlations are conveniently combined into dimensionlessnumbers for which all dimensions cancel and the numerical value is independent of the units

used, provided they are consistent.A. Reynolds Number

The Reynolds numbergives the ratio of the inertial forces to the viscous forces in a fluid. It

governs the flow regime in forced convection:

cc VLVLN Re (I.2131)


24/50

I2 - 24

where Lcis a characteristic dimension. Fluid flow is laminar at low Reynolds numbers and

turbulent at high Reynolds numbers.

B. Nusselt Number

The Nusselt numbergives the ratio of the convective heat flux to the conductive heat fluxthrough a fluid layer. It represents the enhancement of heat transfer as a result of convectionrelative to conduction across the same fluid layer (Cengel, 2003):

k

hL

LTk

Th

q

qN

conduction

convection

Nu

/ (I.2132)

C. Prandtl Number

The Prandtl numberis the ratio of the momentum diffusivity to the thermal diffusivity of a

fluid. It also gives the ratio of the thickness of the hydrodynamic boundary layer to thethickness of the thermal boundary layer:

k

c

ckN

p

p

)/(

)/(Pr (I.2133)

For most liquids, NPr > 1, i.e., the hydrodynamic boundary layer is thicker than the thermal

boundary layer.

D. Grashof Number

The Grashof number gives the ratio of the buoyant forces to the viscous forces acting on afluid. It represents the natural convection effects and governs the flow regime in naturalconvection.

2

3

csGr

L)TT(gN

(I.2134)

where:g= gravitational acceleration

= coefficient of volume expansionTs= temperature of heat transfer surface

T = temperature of fluid sufficiently far from heat transfer surface= kinematic viscosity of fluid

E. Graetz Number

The Graetz numberis defined as


25/50

I2 - 25

kL

cmN

p

Gz

(I.2135)

I2.5.2. Equivalent Diameter for Fluid Flow

For non-circular conduits, the diameter Din both Reynolds number and Nusselt number are

replaced by an equivalent diameter(or hydraulic diameter),De , which is defined as

w

c

Hep

ArD 4

4 (I.2136)

where De = equivalent or hydraulic diameter

Ac= cross sectional area of the fluid streampw= wetted perimeter

I2.5.3. Temperature Dependence of Fluid Properties

Generally, the temperature, and hence the properties of a fluid stream vary from point to point

along the heat exchange surface. Thus, the local value of the film coefficient also varies. To

obtain an average value of the heat transfer coefficient, the fluid properties such as cp, , and kmust be evaluated at some average temperature. The average film coefficient thus obtained is

used in calculating the overall heat transfer coefficient.

A. Sensibl e Heat Transfer

1. External Flow

For heating or cooling of a fluid in immersed flow, the temperature, and hence the properties

of the fluid in the thermal boundary layer vary from the wall to the outer edge of the boundarylayer. To account for the variation of the properties with position across the boundary layer,

the fluid properties are evaluated at the film temperature, which is the arithmetic average of

the wall surface temperature and the free-stream fluid temperature:

)(21

TTT wf (I.2137)

2. Internal Flow

For heating or cooling of a fluid flowing inside a tube, the average value of the heat transfercoefficient is computed by evaluating the fluid properties at the mean bulk fluid temperature,

which is the arithmetic average of the bulk temperatures at the tube inlet and exit:

)(21

ba TTT (I.2138)

B. Latent H eat Transfer


26/50

I2 - 26

1. Condensation

The properties of the condensate are evaluated at the condensate film temperature given by:

4

)TT(3TT

4

3TT whhhf

(I.2139)

where: hT = temperature of the condensing vapor

wT = temperature of the tube wall surface

2. Boiling

The vapor properties are evaluated at the average temperature of the vapor film:

)(21

satwf TTT (I.2140)

The liquid properties, on the other hand, are evaluated at the saturation temperature, satT .

I2.5.4. Sieder-Tate Correction for Tubular Flow

When a viscous liquid stream is heated inside a tube, its velocity gradient increases near the

tube wall, giving a higher rate of heat transfer. This is caused by the lower viscosity near the

wall. Conversely, the velocity gradient at the wall decreases when it is cooled, giving a lower

rate of heat transfer.

The Sieder-Tate correction (Middleman, 1998) accounts for the effect of the temperature-dependent viscosity on the velocity profile, particularly when a large temperature drop is

involved:

w

v

(I.2141)

I2.5.5. Forced Convection Heat Transfer

A. Laminar Flow Forced Convection

1. Immersed Flow along a Heated Flat Plate )7.0,103( Pr5

Re NxN (Geankoplis, 1995):

Re3

Pr664.0 NNNNu (I.2142)

where: h= average heat transfer coefficient over the entire length of the plate

L= entire length of the plate

uo= free-stream velocity or velocity of fluid approaching the plate and beyond the

edge of the hydrodynamic boundary layer


27/50

I2 - 27

f

fo

f

ffp

f

Nu

LuN

k

cN

k

hLN

Re

,

Pr ,,

For gases, the equation can be used down to 7.0PrN .

2. Tubular Flow

14.03/12 vGzNu NN (I.2143)

where: = viscosity at the mean bulk temperature of the fluid

w =viscosity of the fluid evaluated at the tube wall temperature Tw

w

v

p

Gzi

NukL

cmN

k

DhN

,,

B. Turbulent F low Forced Convection

1. Immersed Flow Parallel to Flat Plate )7.0,103( Pr5

Re NxN (Geankoplis, 1995):

3/1

Pr

8.0

Re0366.0 NNNNu (I.2144)

where:f

fo

f

ffp

f

Nu

LuN

k

cN

k

hLN

Re

,

Pr ,,

2.Immersed Flow Outside a Tube

For heat transfer to a liquid flowing normal to a tube (McCabe, 2001):

52.0

Re

3.0

Pr 56.035.0 NNNNu

(I.2145)

where:f

o

f

ffp

f

oo

Nu

GDN

k

cN

k

DhN

Re

,

Pr ,,

The equation can also be used for gases from4

Re 101 N

3.Immersed Flow Past a Sphere of diameter DP )4006.0,1071( Pr4

Re NxN

(Geankoplis, 1995):

3/1

Pr

2/1

Re60.00.2 NNNNu (I.2146)

where:f

ffp

f

p

f

po

Nuk

cN

GDN

k

DhN

,

PrRe ,,


28/50

I2 - 28

4. Tubular Flow

The average heat transfer coefficient for fluid flowing in turbulent flow inside a tube isobtained from the Sieder-Tate equation:

For long tubes (L/D) >50:

vNu NNN 3/1

Pr

8.0

Re023.0 (I.2147)

where:k

cN

VDN

k

DhN

pi

Nu

PrRe ,,

For a short tube (L/D < 50) with uniform fluid velocity at the sharp-edged entrance (McCabe,2001):

7.0

1

L

D

h

hi (I.2148)

where: hi= average heat transfer coefficient over the short tube length

h = heat transfer coefficient for fully developed turbulent flow (long tube)

I2.5.6. Natural Convection Heat Transfer

When a hot surface transfers heat to an adjacent cold fluid, the density of the fluid parcel near

the surface decreases due to this heating process. Buoyancy forces cause it to rise in naturalconvection.

A. Simple Heating or Cooling

The general equation for natural convection from an isothermal surface (L


29/50

I2 - 29

Vertical planes and cylinders,

vertical heightL< 1 m

109

1.36

0.59

0.13

1/5

1/4

1/3

Horizontal cylinders, replaceL

by diameterD,D < 0.20 m

109

0.49

0.71

1.09

1.090.53

0.13

0

1/25

1/10

1/51/4

1/3

Horizontal plates

Upper surface of heated

plate or

lower surface of cooledplate

Lower surface of heated

plate orupper surface of cooled

plate

105-2 x 10

7

2 x 107-3 x 10

10

105-10

11

0.54

0.14

0.58

1/4

1/3

1/5

B. Condensation

In film-type condensation, the liquid condensate forms a film of liquid that flows over the

surface of the condenser tube under the action of gravity.

1. Condensation on Outside of Vertical Tube

oT

b

oTo

T

oo

T

TLTLD

m

TA

qh

(I.2150)

where:Tq = total rate of heat transfer

Tm = total rate of condensation

TL = total tube length

b = condensate loading at the bottom of tube, oTb D/m

ForNRe< 1200 (McCabe, 2001):

4/1

fo

2

f

3

f

3/1

fb

2

ff

LT

gk943.0

3

g

3

k4h

(I.2151)


30/50

I2 - 30

where:f

bRe

4N

(I.2152)

2. Condensation on Outside of Horizontal Tube

The flow of condensate over the outside surface of a horizontal tube is usually laminar. Theaverage coefficient is given by (McCabe, 2001):

4/1

0

23

729.0

fo

ff

DT

gkh

(I.2153)

ForNRe> 40, his multiplied by 1.2 to account for the effect of rippling (McCabe, 2001).

3. Condensation on Vertical Stack of Horizontal Tubes

The condensate falls cumulatively from tube to tube and the total condensate from the entirestack finally drops from the bottom tube. The average coefficient for condensation on the

outside ofNhorizontal tubes arranged in a vertical stack is given by (McCabe, 2001):

4/123

729.0

foo

ff

NDTN

gkh

(I.2154)

C. Fi lm Boil ing on Submerged Horizontal Cyli nder or Sphere

In film boiling, the hot surface is covered with a quiescent film of hot vapor, which offers

virtually all the resistance to heat transfer. There is slow and orderly formation of bubbles atthe interface between the liquid and the film of hot vapor. The bubbles detach themselves

from the interface and rise through the liquid. For film boiling on the outside of a horizontal

cylinder or a sphere of diameterD(Cengel, 2003):

)()(

)](4.0)[(4/1

3

sats

satsv

satspvvlvvTT

TTD

TTcgkC

A

q

(I.2155)

where:A= area of surface of cylinder or sphere in contact with fluidC= 0.62 for horizontal cylinder; C= 0.67 for sphere

= heat of vaporizationg= gravitational acceleration

LV , = density of liquid and vapor, respectively

VV k, = viscosity and thermal conductivity of vapor, respectively

Ts = surface temperature of hot cylinder or sphere

Tsat= boiling temperature at the specified pressure


31/50

I2 - 31

cpv= specific heat capacity of vapor

I2.5.7. Multipass and Cross-Flow Heat Exchangers

A. Shell -and-Tube Heat Exchanger

The Donohue Equation can be used to predict the shell-side heat transfer coefficient in ashell-and-tube heat exchanger:

14.0

v

33.0

Pr

6.0

ReNu NN2.0N (I.2156)

where:k

cN

GDN

k

DhN

peooo

Nu

PrRe ,,

Ge= weighted average mass velocity of fluid

cbe GGG (I.2157)

Gc= cross-flow mass velocity

c

cS

mG

(I.2158)

Gb= mass velocity parallel to tubes

b

bS

mG

(I.2159)

Sc= transverse flow area between tubes in row at or closest to exchanger centerline

p

D1PDS o

sc (I.2160)

Sb= free area for flow in baffle window

4

DN

4

DfS

2

o

b

2

s

bb

(I.2161)

Ds= inside diameter of shell

Do= outside diameter of tubes

Nb= number of tubes in baffle windowp = center-to-center distance between tubes

P =baffle pitch

fb= fraction of shell cross section occupied by baffle window, commonly 0.1955

(McCabe, 2001)

B. Cross-F low Heat Exchanger


32/50

I2 - 32

In a cross-flow heat exchanger, the shell-side heat-transfer coefficient can be estimated from

(McCabe, 2001):

a

33.0

Pr

61.0

ReNu FNN287.0N (I.2162)

where:k

cNGDN

kDhN poooNu

PrRe ,,

Fa= arrangement factorthat depends on Reynolds number and tube spacing

p= tube spacing in heat exchanger

G = mass velocity outside the tubes, based on minimum area for flow in any tube row

Typical values ofFaare given in the following table.

Table I22.Arrangement Factor for Cross-flow Heat Exchanger with Square Pitch(McCabe, 2001)

p/Do Fa

NRe= 2000 NRe= 8000 NRe= 20000 NRe= 400001.25 0.85 0.92 1.03 1.02

1.5 0.94 0.90 1.06 1.04

2.0 0.95 0.85 1.05 1.02

C. Plate Heat Exchanger

For the plate heat exchanger (McCabe, 2001):

33.0

Pr

67.0

Re37.0 NNNNu (I.2163)

I2.5.8. Scraped-Surface Heat Exchanger

Liquid-solid suspensions, viscous solutions, and juice concentrates are often cooled or heated

in a scraped-surface heat exchanger (Geankoplis, 1995). The viscous liquid passes at lowvelocity through a central tube. During the short time interval between passages of successive

scraper blades, heat is transferred to the liquid but penetrates only a small distance into the

stagnant liquid. The process is analogous to unsteady-state heat transfer to a semi-infinitesolid.

Time interval between passages of successive blades:

nB

1t (I.2164)

where: n= agitator speed, r/s

B= number of blades carried by shaft


33/50

I2 - 33

Total amount of heat transferred during time interval t (McCabe, 2001):

t)TT(k2

A

Qbw

T (I.2165)

Heat-transfer coefficient averaged over time interval t (McCabe, 2001):

nBck2

t

ck2

)TT(tA

Qh

pp

bw

Ti

(I.2166)

I2.5.9. Heat Transfer in Agitated Tanks

Many chemical and biological processes are often carried out in a cylindrical vessel agitated

by an impeller mounted on a shaft and driven by an electric motor. Often it is necessary to

cool or heat the contents of the vessel during the agitation. This is usually done by heat

transfer surfaces, which may be in the form of cooling or heating jackets in the wall of thevessel or coils of pipe immersed in the liquid (Geankoplis, 1995).

A. Agi tated Baff led Tank with Heating or Cooling Coil s

For heating or cooling liquids in an agitated baffled cylindrical tank equipped with a helical

coil, the heat transfer coefficient hcbetween the coil surface and the liquidcan be found from

(McCabe, 2001):

5.01.0

24.037.0

Pr

67.0

Re

t

c

t

a

vNu

D

D

D

DNKNN (I.2167)

where:w

v

paccNu

k

cN

nDN

k

DhN

,,, Pr

2

Re

n= impeller speed, r/s

Da= impeller diameter

Dt= tank diameter

Table I2-3. Values of K for Baffled Tank Heated or Cooled by Helical Coil

and With Various Agitation Devices (McCabe, 2001):

Agitation device Kturbine impeller 0.17

pitched turbine 0.1445

propeller 0.119

B. Jacketed Agitated Baff led Tank


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I2 - 34

The heat-transfer coefficient hjbetween the liquid and jacketed inner surface of an agitated

and baffled tank can be found from McCabe, 2001):

bva

Nu NNKN 3/1

PrRe (I.2168)

where:w

vpatj

Nuk

cNnDNkDhN

,,, Pr

2

Re

Table I2-4.Values of K, a, and b for Heating or Cooling Jacketed Baffled Tank With Various

Method of Agitation (McCabe, 2001).

Agitation device K a b

standard turbine 0.76 2/3 0.24

pitched turbine 0.684 2/3 0.24

propeller 0.456 2/3 0.24

anchor agitator, 10 < NRe< 300 1.0 1/2 0.18

anchor agitator, 300 < NRe< 40000 0.36 2/3 0.18

C. Transient Heat Transfer i n Agitated Tank

Consider the heating of a liquid of mass m and specific heat capacity cp in a well-agitated

vessel where at time t = 0, its temperature is Ta. The temperature of the liquid, bT , at any time

t> 0 can be found for the following two cases:

1.Heating by an isothermal medium at temperature Ts(McCabe, 2001):

pbs

as

mc

UAt

TT

TTln

(I.2169)

2.Heating by a flowing heat transfer medium (McCabe, 2001):

K

1K

mc

tcm

TT

TTln

p

phh

hab

haa

(I.2170)

where: Tha = temperature of the entering heat transfer mediumcp, cph = specific heat capacities of liquid being heated and heating medium,

respectively

phhcm

UAexpK

(I.2171)

I2.5.10. Heat Transfer in Packed Beds


35/50

I2 - 35

In a tubular reactor, the reactant gases pass through a bed of solid catalyst particles. In the

packed bed, the total resistance to heat transfer is the sum of the resistance in the region very

near the wall and the resistance in the rest of the packed bed. Thus, the heat transfer

coefficient is given by

wbedi hhh

111

(I.2172)

where: gk = thermal conductivity of the gas

4 ebed

kh

r (for bed with parabolic temperature profile) (I.2

173)

Re, Pr 5 0.1e

P

g

kN N

k (I.2174)

33.0

Pr

5.0

Re,94.1 NN

k

DhP

g

pw (I.2175)

I2.6. RADIATIVE HEAT TRANSFER

I2.6.1. Thermal Radiation

All objects with a temperature above absolute zero continuously emit electromagnetic

radiation. Such radiation, referred to as thermal radiation,is distributed over the 10-7

to 10-4

m wavelength range and results from changes in the electronic, vibrational, rotational and

translational states of electrons, atoms, and molecules.

Thermal radiation is a major energy-transfer mechanism in equipment where largetemperature differences are involved such as steam boilers, rotary kilns, and, blast furnaces.

I2.6.2. Absorption, Reflection and Transmission of Radiation

When matter appears in its path, electromagnetic radiation will be transmitted, reflected, or

absorbed. Radiation that is incident on opaque objects is usually absorbed within a fewmicrons from the surface (a surface phenomenon). The absorbed radiation is eventually

transformed into heat. Radiation may be reflected diffusely or specularly from a surface.Dust and other finely divided solid particles present in a gas medium deflect incident

radiation.

Irradiationis the rate at which thermal radiation from all directions is incident on a surface

per unit area of the surface. The absorptivity of a surface is the fraction of irradiation

absorbed by the surface; its reflectivity is the fraction of irradiation that it reflects; its


36/50

I2 - 36

transmissivity is the fraction of irradiation that is transmits. The sum of these fractionsmust be unity:

1 (I.2176)

A black body is a hypothetical object that absorbs all incident thermal radiation at all

temperatures regardless of wavelength and direction. Thus, for a blackbody,

0,0,1

I2.6.3. Absorption of Radiation by Opaque Solids

A black body, being an ideal absorber of all incident radiation of any wavelength ( 1 ) isused as a standard with which real surfaces are compared. Another idealized body, the gray

body does not absorb all incident radiation ( 1 ) but reflects some fraction of it. However,its absorptivity is independent of the wavelength of radiation. For a real opaque body, 0

thus 1 .

A real opaque body is anything other than gray or black. Its absorptivity is practically

independent of its temperature but is strongly dependent on the temperature of the radiation

source and can vary considerably with the wavelengthof the incident radiation.

A. Gray Body:

1

)(

f

B. Blackbody:

1

)(

f

C. Real Body:

1

)(

f

I2.6.4. Emission of Thermal Radiation

A. Plancks Equation

When a black body is heated, photons having a distribution of energy are emitted from its

surface. The amount of radiation energy emitted by a blackbody at an absolute temperature T


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I2 - 37

per unit time, per unit surface area, and per unit wavelength about the wavelength is givenby:

)1( /52

1

2

TCben

CE

(I.2177)

where: bE spectral blackbody emissive power, W/m2-m

C1= 3.742 x 108W-m

4/m

2

C2= 1.439 x 104m-K

n = index of refraction of radiation medium

B. Wiens Displacement Law

For a given absolute temperature T, the wavelength at which the emissive power of a

blackbody is a maximum is given by:

KmxT 3max 108978.2 (I.2178)

C. Stefan-Boltzmann L aw

The total blackbody emissive power at an absolute temperature T is obtained by integrating

Plancks law over the entire wavelength spectrum:

4

0TdE

A

qb

(I.2179)

where:

q/A= blackbody emissive power or total thermal energy (sum of radiation over allwavelengths) emitted by a blackbody per unit time and per unit surface area

q= total rate of energy emission from the blackbody

A = area of radiating blackbody surface

= Stefan-Boltzmann constant, 5.676 x 10-8

W/m2-K

4

T= absolute temperature of blackbody

The black body is a diffuseemitter, i.e., it emits thermal radiation uniformly in all directions

per unit area normal to the direction of emission.

Real bodies such as glossy surfaces or polished metal plates emit less radiation than a black

body. The emissivity of a real surface relates its radiation to that of the black body at a

given temperature:

4)(

)(

AT

q

blackbodyofq

bodyrealofq

(I.2180)

Thus, the radiation emitted by a real body is:

4ATq (I.2181)


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D. Kirchoffs Law

An object that is in thermal equilibrium with its surroundings is absorbing and emittingthermal radiation at equal rates (no net heat flux) and its temperature remains constant. Thus,

the emissivity and absorptivity of any solid surface in thermal equilibrium with its

surroundings are equal:

(I.2182)

I2.6.5. Radiation Exchange Between Surfaces

The net radiant energy gained or lost by an opaque body is the difference between the energythat it emits and the incident radiation that it absorbs. The exchange of radiation between two

emitting surfaces depends on the emissivity, absorptivity, size, shape, and relative orientation

of these two surfaces.

A. Radiation Shape Factor

In general, not all the radiation leaving one surface will reach another surface; some will belost to the surroundings (i.e., to other surfaces). Theradiation shape factor,F12,also known

as configuration factor, direct view factor, or angle factor, represents the fraction of

diffuse radiation leaving surface 1 that is directly intercepted by surface 2 (or the fraction

of surface 1 directly seen by surface 2).

If surfaceA1sees a number of other surfaces and if its entire hemispherical angle of vision is

filled by these surfaces, then

F11+F12+ F13+ = 1 (summation rule) (I.2183)

For a small body of surface areaA2, having no concavities, and surrounded by a large body ofsurface areaA1:

(a) none of the radiation leaving 2 is intercepted by itself: F22= 0

(b) all radiation leaving 2 is intercepted by 1: F21 = 1

(c) some radiation from 1 is intercepted by itself; some by 2: F11+ F12= 1

B. Dir ect Radiati on Between Parall el B lack Bodies of F in ite Size

Consider the exchange of thermal radiation between two black bodies of finite size that are infull view of each other and with no absorbing medium between them. Some of the radiation

from surface 1 does not strike surface 2, and vice versa. Thus, some of the radiation is

lost to the surroundings. The net rate of energy transfer from body 1 to body 2 is givenby the equation

)()( 424

1212

4

2

4

112112 TTFATTFAq (I.2184)


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I2 - 39

where: A1F12= A2F21 (reciprocity relation) (I.2185)

C. Radiation Between Parallel Black Bodies Connected by Adiabatic Wall s

Anadiabatic wallis one that is in thermal equilibrium with its surroundings. A reradiating

wall is an adiabatic wall whose backside is well insulated; the net heat transfer though it iszero. In the absence of convection effects and at steady state conditions, its heat transfer sideradiates back all the radiation that it receives.

For direct radiation between two black body surfaces 1 and 2 that are connected byadiabatic walls formed from line elements perpendicular to these two surfaces, the fraction of

the radiation from A1 that is intercepted by A2 would be larger than in the absence of the

adiabatic walls. In this case, the net rate of energy transfer from blackbody 1 to blackbody

2 is given by the equation

)()( 424

1212

4

2

4

112112 TTFATTFAq (I.2186)

where 2112 ,FF are referred to asinterchange factors.

When there are no adiabatic walls: 1212 FF

(I.2187)

D. Radiat ion Exchange Between Two Real Surfaces

To simplify the analysis for direct radiation between two real surfaces 1 and 2, theradiating surfaces are assumed to be opaque, diffuse emitters and reflectors (radiation

properties are independent of direction) and gray (radiation properties are independent ofwavelength). The net rate of energy transfer from real surface 1 to real surface 2 is givenby:

)()( 424

1212

4

2

4

112112 TTATTAq (I.2188)

where 12 is the overall interchange factorbetween any two opaque, diffuse, gray surfaces,which in general can be determined approximately from:

1

11

11

1

22

1

112

12

A

A

F

(I.2189)

where1 and 2 are the emissivities of radiation source 1 and radiation sink 2,

respectively.


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I2 - 40

These equations are exact for radiation exchange between two opaque surfaces that are

parallel infinite planes, concentric infinite cylinders, or concentric spheres, and no other

surfaces are in view. The equations also become exact as the emissivities of all surfaces in

the system approach unity.

1. For Parallel Infinite Planes (A1 = A2, F12 = 1.0):

1/1/1

)(

21

4

2

4

1

TT

A

q (I.2190)

2. For Concentric Infinite Cylinders (F12 = 1.0):

)1/1)((/1

)(

2211

4

2

4

11

AA

TTAq (I.2191)

3. For Convex Surface 1 in Large Enclosure 2( F12 = 1.0, A1/A2 = 0):

)( 424

111 TTAq (I.2192)

I2.6.6. Radiation Effect on Temperature Measurement

Consider a temperature sensor being dipped into a fluid flowing through a large channel

whose walls are at a different temperature than the fluid. The signal output from the sensor

will stabilize when its heat gain by convection equals its heat loss by radiation (or vice versa).

The actual fluid temperature is given by

h

TTTT wttt

)( 44

(I.2193)

where: T = actual fluid temperature

Tt= temperature measured by the thermometer

Tw= temperature of the surrounding surface

To reduce the radiation effect, the sensor is coated with a highly reflective material and for

outdoor measurements, protected from direct sunlight. Another way to reduce the radiationeffect is to insert a thin, highly reflective sheet between them aradiationshield.

I2.6.7. Radiation Shield

A radiation shield introduces additional resistance in the heat flow path between two

radiating surfaces, thus reducing the overall radiation heat transfer rate without adding orremoving any heat from the overall system. Multilayer radiation shields are used in cryogenic

and space applications.


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I2 - 41

For radiation heat transfer between two large parallel plates separated byNradiation shields:

111

...111

111

)(

11

44

,

BNANBABA

BANAB

TTAq

(I.2194)

where: AN = emissivity of surface of shieldNfacing plate A

BN = emissivity of surface of shieldNfacing plate B

I2.6.8. Radiation Exchange with Absorbing-Emitting Medium

In thin layers, most liquids and some solids absorb only a fraction of the incident radiation

and transmit the rest. Layers of water more than a few milllimeters thick are transparent to

visible light but absorb virtually all radiant energy with wavelengths greater than 1.5 m(McCabe, 2001).

Monatomic and symmetrical diatomic gases such as argon, oxygen, hydrogen, and nitrogenare almost completely transparent to thermal radiation because their nuclei and electrons

cannot be energized by photons of the available energy levels. At moderate temperatures,

polyatomic gases with asymmetric molecules (such as H2O, CO2, CO, SO2, and CnHn) mayparticipate in the radiation process by absorption, and at high temperatures, such as in a

furnace or combustion chamber, by both absorption and emission.

Gases emit and absorb radiation at a number of narrow wavelength bands (in contrast, solids

emit and absorb thermal radiation over the entire radiation spectrum). The emission andabsorption characteristics of a gas mixture also depend on its temperature, pressure, and

composition. Scattering caused by the gas molecules themselves (Rayleigh scattering) hasnegligible effect on radiation heat transfer but it can be affected by the presence of aerosols

(dust, ice particles, liquid droplets, soot) that scatter radiation.

A. The Greenhouse Eff ect

Ordinary glass is transparent to radiation of short wavelengths but opaque to radiation of

longer wavelengths. Thus, the glass walls of a greenhouse will readily transmit thermal

radiation from the sun, which is chiefly of short wavelengths, but will trap thermal radiationof longer wavelengths emitted by objects inside the greenhouse. Thus, there will be a net

radiation exchange into the greenhouse. The temperature inside the greenhouse will rise until

convective heat loss from the enclosure equals the input of radiant energy.

On a larger scale, the combustion gases in the earths atmosphere transmit the bulk of the

solar radiation but absorb the infrared radiation emitted from the surface of the earth,

producing a greenhouse effect. Environmentalists are concerned that the energy trapped on


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I2 - 42

earth will cause global warming that can lead to drastic changes in the global weather

patterns.

B. Beers Law

The decrease in the intensity of radiation dIas it passes through a medium of thickness dxisproportional to the intensity itself and the thickness:

dxIdI (I.2195)

where is the spectral absorption coefficient of the medium which is a function of the

wavelength of the incident radiation as well as the temperature, pressure, and composition

of absorbing medium.

The spectral transmissivity, , of the medium of thickness Lis the ratio of the intensity of

radiation leaving the medium to that entering the medium. It is obtained by separation ofvariables and integration of Beers law from distance x = 0 to x = L in the medium and

assuming a constant absorption coefficient

L

0,

L,e

I

I

(I.2196)

The ability of a medium to absorb radiation of wavelength is given by its absorption length

(or optical path length),L, which is defined as the distance of penetration into the mediumat which the incident radiation has been attenuated an amount equal to 1/ewhere eis the base

of natural logarithms (McCabe, 2001). Thus,

1L (I.2197)

A material whose thickness L is many times larger than L is considered to be opaque to

radiation of that wavelength. IfLis less than a few multiples of L, the material is considered

to be transparent.

The spectral absorptivityof a nonscattering (and thus nonreflecting) medium of thickness L

is

Le

11 (I.2198)

From Kirchoffs law, the spectral emissivityof a participating medium is

Le

1 (I.2199)

A participating medium with a large value of L is considered to be optically thick and will

emit like a black body at the given wavelength.


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I2 - 43

C. Emission of Radiati on by CO2and H2O in a Gas M ixtu re

For CO2 and H2O gases existing together in a mixture with nonparticipating gases, the

effective emissivity is given by

wcg (I.2200)

where the emissivity correction factor accounts for the fact that each gas is somewhatopaque to radiation from the other (McCabe, 2001).

D. Absorpti on of Radiation by CO2and H2O in a Gas M ixture

The absorptivity of a gas mixture containing CO2and H2O for radiation emitted by a source at

temperature Tsis

wcg (I.2201)

where and is determined at the temperature of the radiation source Ts.

The net rate of radiation heat transfer between the emitting-absorbing gas and a black

body surface surrounding it is

)TT(Aq 4sg4

ggs (I.2202)

When the enclosure is not a black body but with s > 0.7 (such as wall surface of furnace and

combustion chamber):

)TT(A2

1q 4sg

4

ggs

s

(I.2203)

E. Simul taneous Heat Transfer M echanisms

In many engineering applications, heat transfer occurs by a combination of the three

fundamental heat transfer mechanisms. However, not all three mechanisms can existsimultaneously in a heat transfer medium.

Table I2-5.Heat transfer mechanisms in various media.

System Heat transfer mechanism

Opaque solid Conduction


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I2 - 44

Semitransparent solid Conduction Radiation

Still fluid Conduction Radiation

Flowing fluid Convection Radiation

Vacuum Radiation

Convection and radiation can occur in parallel when a surface exchanges heat with its

surroundings. The total heat transfer rate between a small surface and its black bodysurroundings is:

)TT(A)TT(hAqqq 44sssssrc (I.2204)

For simplicity and convenience, a combined heat transfer coefficient, hcombined , that includesthe effects of both convection and radiation may be used:

)TT(Ah)TT(A)hh(q sscombinedssrc (I.2205)

where hc is the convective heat transfer coefficient and hr is the radiative heat transfercoefficient defined as:

)TT(A

qh

ss

r

r

(I.2206)

When the temperature difference is small,

3

ssr T4h (I.2207)

When the temperature difference is more than a few degrees but less than 20 % of Ts , thearithmetic average of Tsand Tcan be used to improve the accuracy of the preceding equation

(McCabe, 2001).

In film boiling at a very hot surface (>300oC), thermal radiation from the hot surface passing

across the vapor film to the liquid becomes significant and must be added to heat transfer by

convection. The convective heat-transfer coefficient can be predicted by iteration from(McCabe, 2001):

3/1

rc

o,c

o,cchh

hhh

(I.2208)

where hc,ois the convective heat-transfer coefficient in the absence of radiation.

The total heat transfer rate during film boiling, for the case when the radiative heat transferrate is less than the convective heat transfer rate, is (Cengel, 2003):

r43

c qqq (I.2209)


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I2 - 45


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I2 - 46

NOMENCLATURE

Symbol Definition SI Units

A heat transfer surface area perpendicular to the positivex

direction for heat transfer; also area of radiating surface

m

B number of blades carried by agitator shaft [ - ]k thermal conductivity W-m

--K

-

h convective heat transfer coefficient or film coefficient W-m-

-K-

C heat capacity rate W-K-

c heat capacity ratio [ - ]

cp specific constant-pressure heat capacity of heat transfer

medium

J-kg-

-K-

fb fraction of shell cross section occupied by baffle window [ - ]Fa arrangement factor for cross-flow heat exchanger [ - ]

FG correction factor for LMTD in multipass heat exchanger [ - ]

L length mLc characteristic length of the geometry m

m mass flow rate, rate of condensation kg-s-

QT total amount of heat absorbed or rejected by heat transfer

medium

J

q

q/A

rate of heat transfer; also rate of energy emission from

radiating surface

heat flux through a heat transfer medium; also emissive

power of radiating surface

W

W-m-2

U overall heat transfer coefficient W-m-

-K-

R thermal resistance K-W-

Rf fouling factor K-W-

-m

r radius of cylinder or sphere m

S area for fluid flow m

D diameter of cylinder or sphere m

g gravitational acceleration m-s-

s half-thickness of flat wall mFG correction factor that accounts for complex flow and

geometry of heat exchanger

[ - ]

P baffle pitch mp center-to-center distance between tubes m

p perimeter m

t time of heating or cooling sT absolute temperature K

T bulk temperature of heat transfer medium K

dxdT temperature gradient along the positive x direction K-m-

temperature drop along heat transfer path K

x thickness of plane wall m


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I2 - 47

xp thermal penetration distance m

V volume m

emissivity of radiating surface [ - ]

heat transfer effectiveness [ - ]

F fin effectiveness [ - ]

G mass velocity kg-s-

-m-

coefficient of volume expansion K-

NH number of heat transfer units [ - ]N dimensionless number [ - ]

number of layers in a composite wall; number of heat

shields; number of tubes in a vertical stack

[ - ]

n rotational speed r-s-

I intensity of radiation

thermal diffusivity (in convective heat transfer) m -s

absorptivity (in radiative heat transfer) [ - ]

absorption coefficient of medium m-

transmissivity of a body [ - ] reflectivity of a body [ - ]

density of heat transfer medium kg-m-

kinematic viscosity of fluid m -s-

dynamic viscosity of fluid kg-m-

-s-

v correction factor for effect of heating or cooling onvelocity gradient of fluid in internal flow

[ - ]

Z ratio of fall in temperature of hot fluid to rise in

temperature of cold fluid

[ - ]

heating effectiveness of heat exchanger (ratio of actual

temperature rise of cold fluid to maximum possible

temperature rise obtainable if the warm end approach werezero based on countercurrent flow)

[ - ]

F fin efficiency [ - ]

b condensate loading or mass rate per unit length of tubeperiphery

kg-s-

-m-

Stefan-Boltzmann constant, 5.67 x 10-

W-m-

-K-

latent heat of vaporization J-kg-

wavelength of electromagnetic radiation m

F radiation shape factor, also known as configuration factor,

direct view factor, or angle factor

[ - ]

12F interchange factor for radiation between two opaque,

diffuse, and gray surfaces

[ - ]

12 overall interchange factor [ - ]

Subscripts


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I2 - 48

i inside of heat transfer tube

o outside of heat transfer tube

a entrance terminal in the heat exchangera impeller

b exit terminal in the heat exchanger

b black bodyb baffle windowe equivalent

f heat transferfluid medium

f fluidfilmfb fin base

spectral or monochromaticj heating or cooling jacket

c heating or cooling coil

c interfacial contact

c convective

r radiativel liquid

v vapor

t tank

p particlesat vapor-liquid equilibrium (saturated) condition

s heat transfer surface

w tube wall

w,i inside surface of heat exchanger tube or wallw,o outside surface of heat exchanger tube or wall

B bare heat transfer surface

F finU unfinned portion of heat exchange surface

T total

L logarithmic

G geometrich hot fluid

c cold fluid

cr criticalmin minimum

max maximum free stream fluid

H hydraulicRe Reynolds

Nu Nusselt

Bi Biot

Fo Fourier

Pr Prandtl

Gr Grashof

Gz Graetz


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I2 - 49

REFERENCES

1. Bird, Stewart and Lightfoot, Transport Phenomena, John Wiley, 1960.

2. Brown, Unit Operations, John Wiley, 1950.

3.

Cengel,Heat Transfer, 2nded., McGraw-Hill, 2003.4. Foust,Principles of Unit Operations, 2

nded., John Wiley, 1980.

5. Geankoplis, Transport Processes and Unit Operations, 3rd

ed., Prentice Hall, 1995.

6. Hagen,Heat Transfer with Applications, Prentice-Hall, 1999.

7. Holman,Heat Transfer, 9th

ed., McGraw-Hill, 2002.8. McAdams,Heat Transmission, 3

rded., McGraw-Hill, 1954.

9. McCabe, Smith and Harriott, Unit Operations of Chemical Engineering, 6th

ed.,

McGraw-Hill, 2001.

10.Middleman,An Introduction to Mass and Heat Transfer: Principles of Analysis andDesign, John Wiley, 1998.

11.Perry and Green,Perrys Chemical Engineers Handbook, 7th

ed., McGraw-Hill,

1997.

Heat Transfer, I2 - 1

convective heat transfer, I2 - 12heat exchangers, design equations, I2 - 13

heat exchangers, extended surface, I2 - 21

heat exchanger, analysis, I2 - 18

heat exchanger, equipment, I2 - 12

Newton's law for convective heat transfer, I2 - 12correlations of film coefficients, I2 - 24

correction for heating and cooling effect, I2 - 28

dimensionless numbers in heat transfer correlations, I2 - 25

forced convection heat transfer, I2 - 28


50/50

heat transfer, agitated vessels, I2 - 35

heat transfer, packed beds, I2 - 37

hydraulic diameter, I2 - 26heat transfer , natural convection, I2 - 30

heat exchangers, scraped-surface, I2 - 34

shell side transfer coefficients, I2 - 33

heat exchangers, shell-and-tube, I2 - 33

heat transfer mechanisms, I2 - 1

radiative heat transfer, I2 - 37

absorption of radiation by opaque solids, I2 - 39absorption, reflection, transmission, I2 - 38

thermal radiation emission, I2 - 39

radiation effect on temperature measurement, I2 - 43

radiation exchange between surfaces, I2 - 40radiation shield, I2 - 43

thermal radiation, I2 - 37

heat transfer, steady-state conductive , I2 - 1

conduction through a multilayer wall, I2 - 4

conduction through a solid, I2 - 2Fourier's law of heat conduction, I2 - 1

thermal conductivity, I2 - 2

thermal contact resistance, I2 - 6

thermal insulators, I2 - 6

unsteady-state thermal conduction, I2 - 7dimensionless parameters, I2 - 7transient heat conduction in semi-infinite solids, I2 - 10

transient heat conduction in slabs, cylinders, and spheres, I2 - 9

Documents

I020 Heat Transfer