3. Macroscopic transport processes: Heat and mass transfer...

Heat and Mass Transfer

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3. Macroscopic transport processes: Heat and mass transfer in presence of convection

3.1 Heat transfer in technical appliances

In section 1 we have identified transport of heat by convection and conduction/diffusion. The superposition of

these two transport processes forms the basis of heat transfer in technical appliances.

Example: Cooling / heating of a chemical reactor.

Within the cooler /reactor there is flow of the coolant

/ reactant and thereby convective transport of heat.

Due to the temperature difference between the reactor

fluid and the coolant there is also heat transfer from

the reactant fluid to the coolant.

Fig.. 3.1-1: Transport / Transfer of heat in

technical heat exchanger. T

x

Kü

hlm

an

tel

Rea

kto

rwa

nd

Tmi

Tma x

x+Dx

JQkonv(x)

JQkonv(x+Dx)

JQüber(x)

3.1/1 - 1.2012


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The heat flux from the reactant to the coolant occurs

via conduction of heat through the reactor wall and the

adjacent fluid boundary layers with u = 0 at the wall.

Therefore, heat transfer in any case is based on

heat conduction! Convection only modifies the boun-

dary conditions. Heat transfer is a boundary value

problem of convective / diffusive transport of heat.

3.2 General model for heat transfer

From the discussion in section 2 we have learned to

write any heat transfer problem as:

As heat transfer occurs by heat conduction through the

wall boundary layer we can write for the heat flux:

From that we obtain for the heat transfer coefficient:

For few problems in section 2 the temperature gradient

at the wall could be accessed analytically.

Fig. 3.2-1: General model for heat transfer.

)12.3(111

)( ai

m am iQ

s

kTTk

j

)32.3(0

W imi

y

iTT

y

T

In most technical appliances velocity- and temperature

fields cannot be calculated in a simple way (as in the

examples in section 2.) so that the temperature

gradient (dT/dy)y=0 has to be approximated.

In these cases a linearization of the temperature near

the wall (system boundary) is applied (Taylor-series of

temperature, see figure 3.2-1):

With DT = T0 – TW we obtain:

)42.3(..... D

y

dy

dTTT W

)52.3(or0 D

D

yy

TT

dy

dT W

3.2/1 - 1.2012

)22.3()(0

y

W im iiQdy

dTTT j

Fig. 3.1-1: Heat transfer as heat conduction through

a layered geometry.

Med

ium

2

Med

ium

1

Radius

Tem

per

atu

r

Tmi

TWi

TWa

Ri Ra

Tma


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According to the model the heat transfer coefficient

is the heat conductivity divided by the thickness

Dy of a fictitious temperature boundary layer which

results from the linearization of the temperature.

The problem of the unknown heat transfer coefficient

is now shifted to the fictitious boundary layer

thickness Dy. The determination of this is done mostly

by experiments resulting in correlations for the heat

transfer coefficients for the various arrangements.

Correlations for heat transfer coefficients. Correla-

tions for heat transfer coefficients are generally given

in the form:

For simple problems (heat transfer from a sphere into

a fluid at rest, cold bridge, non stationary heating etc. )

correlations of the kind of equation (3.2-6) can be

accessed analytically. For the more complicated

technical cases the correlations have to be developed

from experimental investigations.

Fig. 3.2-2: Nusselt number for laminar flow in tubes.

)62.3(with ...)P r,(Re,

λ

LαN u

L

DfN u ch

)72.3(

PrReRe293,062,166,3

3

1

2

1

33

L

d

L

dNu

Equation (3.2-7) exhibits that the (lenght averaged)

Nusselt number asymptotically approaches a value of

3,66 for long tubes (thermal upgrowth).

Laminar flow in tubes.

3.2/2 - 1.2012

a

Lu ch

Pr,Re


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Fig.. 3.2-3 a, b: Nusselt number for turbulent flow in tubes.

Turbulent flow in tubes.

with

)82.3(

1Pr8

7,121

11Pr1000Re

83

2

3

2

L

dNu

The dependence of Nu on Re in turbulent flow is considerably larger than in laminar flow; Nu decreases with

increasing lenght of tube; influence of Prandtl number stronger than in laminar flow.

)92.3(6 4,1R elo g8 2,12

3.2/3 - 1.2012


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Tabl. 3.2-1: Correlations for the Nusselt number for different scenarios (further information

in literature (e.g. VDI-Wärmeatlas).

3.2/4 - 1.2012


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3.3 Heat transfer in technical heat exchangers

3.3.1 Over all heat transfer coefficient

In section 3.2 we have discussed heat transfer from a fluid to a wall (to the system boundary). In technical heat

exchangers we find a) transfer of heat from a fluid to the wall, b) transfer of heat through the wall and c)

transfer of heat from the wall to a fluid. In principle, we have a series of heat resistances formed by a boundary

layer, the wall and a second boundary layer. The problem of heat resistances in series has been discussed in

detail in section 2.1.4. This has to be transferred now to heat transfer in technical heat exchangers, see figure

3.3.1-1.

Fig. 3.3.1-1: Example for a

technical heat exchanger.

Med

ium

2

Med

ium

1

In figure 3.1-1 we have a technical heat exchanger. In

the inner tubes a hot gas flow from the bottom to the

top is established (e.g. hot reaction products from a

high temperature reaction). In the annuli around the

inner tubes coolant is coflowing (e.g. water which is

vaporized when flowing from the bottom to the top).

Due to the temperure differences between the hot gas

flow (medium 1) and the coolant (medium 2) we have

temperature gradients perpendicular to the direction of

the flow. This temperature gradient is the driving force

for heat transfer from the hot to the cool (from

medium 1 to medium 2).

The temperature profile across a tube in the heat

exchanger is enlarged in Fig. 3.3.1-2

3.3.1/1 - 1.2012


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Fig. 3.3.1-2: Temperature profile across a tube in

the heat exhanger.

Med

ium

2

Med

ium

1

The temperature profile in figure 3.3.1-2 across a tube

in the heat exchanger resembles the temperature

profile in a layered material, see section 2.1.4, when

replacing the two outer layers by fluid boundary

layers. This problem has been treated as a heat

conduction problem through heat conduction

resistances in series, see fig. 3.3.1-3.

For this problem we have derived:

with:

Radius

Tem

per

atu

r

Tmi

TWi

TWa

Ri Ra

For the over all heat transfer coefficient we have:

ki:heat conductivities per length, 1/ki: resistance.

The reciprocals of the heat transfer coefficients are

lenght-specific resistances wich add for the heat

transfer through all layers.

)31.3.3(11

ikk

)21.3.3(difference re temperatudriving

area

fluxheat k

)11.3.3( D TAkQJ

Tma

Fig. 3.3.1-3: Heat transfer in a layered material

by heat conduction.

JQ JQ

T1 T1/2

T3

T2/3

A

s1 T

x 1

s2

2 3

s2

JQ

3.3.1/2 - 1.2012

3

3

1

kR

2

2

1

kR

1

1

1

kR


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For the heat exchanger indicated in figure 3.3.1-2 the

single outer layers are replaced by the temperature

boundary layers (see figure 3.3.1-4). From this we

obtain:

Internal side:

Wall:

External side

Resolving equations (3.5.1-4) to (3.5.1-6) for the

temperature differences and adding we obtain:

Transformation with Aa~AW~Ai=A gives:

ikk

11

)41.3.3()( D

i

iiWimiiiQ

yTTA

J

)51.3.3()( WaWiiW

Q TTAs

J

)61.3.3()( D

a

aamaWaaaQ

yTTA

J

)71.3.3()()()(

111

maWaWaWiWimi

aaW

Wii

Q

TTTTTT

AA

s

A J

)81.3.3(111

1

1

1and

with

DD

aWii

mamiQ

s

k

k

TTTTAk

J

According to equation (3.3.1-8) the heat flux by

heat transfer is proportional to the area and the

total temperature difference. Proportionality factor

is the over all heat transfer coefficient. (All

quantities are local quantities, i.e. dependent on the

coordinate in flow direction.)

Fig. 3.3.1-4: Temperature profile perpendicular to the

fluid flow direction in the heat exhanger.

Med

ium

2

Med

ium

1

Radius

Tem

per

atu

r

Tmi

TWi

TWa

Ri Ra

Tma

3.3.1/3 - 1.2012


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In the previous section we have derived the basic

equation for heat transfer

with

This equation will be used in the following to treat

some simple heat transfer problems from process

engineering.

Temperature profile in the coolant for a well stirred

tank reactor. In the reactor an exothermic chemical

reaction is performed at a constant temperature Tmi. To

keep temperature at constant the heat of reaction has to

be transferred to the coolant. The temperature inside

the reactor is homogeneous, i.e. constant everywhere

inside the reactor. This is achieved by intensively

stirring the reactor. We look for the temperature profile

in the coolant, the effectivity of the heat exchanger, the

transferred heat per time…

3.3.2 Simple heat transfer problems from process engineering

)22.3.3(1111

aWii

s

kk

)12.3.3( D TAkQJ

Fig. 3.3.2-1: Heat transfer in a chemical reactor

(well stirred tank reactor).

3.3.2/1 - 1.2012


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Temperature profile. For the determination of the

temperature within the coolant we use a simple heat

balance for a differential balance volume as indicated

in figure 3.3.2-2.

The heat balance according to fig. 3.3.2-2 gives:

with (T=Tma), convective fluxes according to page

1.3.1/1 we obtain:

Balance:

Linearization of temperature (taylor series) gives:

Using this in the heat balance and separation of

variables and using dT = d(Tmi-T) results in:

Fig. 3.3.2-2: Heat balance for calculation of

the temperature profile in the coolant (heat

conduction in flow direction neglected!).

)52.3.3()()( DD

xxTmcxx pkonvQJ

)32.3.3()(=)()( D xxxxkonvQüberQkonvQ JJJ

)42.3.3()()(=)(

xTmcxTFucx ppkonvQ J

)62.3.3()(=)( D xTTxUkx miüberQJ

)72.3.3( D

x)T(xmcT(x)TΔxUkT(x)mc pmip

)82.3.3()( DD xdx

dTxTx)T(x

)92.3.3()(

dA

cm

kdxU

cm

k

TT

TTd

ppmi

mi

Solution of this ordinary differential equation:

Boundary condition: T=T0 for A=0, then

C = ln(Tmi-T0).

)102.3.3(ln 0

C

cm

kdA

TT

p

A

mi

T

x

Kü

hlm

an

tel

Rea

kto

rwa

nd

Tmi

Tma x

x+Dx

JQkonv(x)

JQkonv(x+Dx)

JQüber(x)

3.3.2/2 - 1.2012


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Further:

The solution then is:

According to equation (3.3.2-12) the temperature

difference Tma-T exponentially approaches zero.

Generally the temperature profile in heat exchangers is

written in the form:

Effectivity. g is the effectivity of the heat exchanger.

g describes the approach to thermal equilibrium. For

the heat exchanger under consideration for g = 1 the

maximum possible heat is transferred.

Transformation of equation (3.3.2-12) gives:

)122.3.3(0

p

m

cm

Ak

mi

mi eTT

TT

)132.3.3(0

0

TT

TT

mi

g

)142.3.3(10

0

p

m

cm

Ak

mi

eTT

TTg

Fig. 3.3.2-3: Temperature profile in the coolant. 3.3.2/3 - 1.2012

T

x

Kü

hlm

an

tel

Rea

kto

rwa

nd

Tmi

Tma x

x+Dx

JQkonv(x)

JQkonv(x+Dx)

JQüber(x)

)112.3.3(1

0

A

m kdAA

k


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The ratio

is called number of transfer units. NTU can be

interpreted as the ratio of heat flux per Kelvin by heat

transfer to the heat flux per Kelvin by convection. The

larger NTU the more effective the heat exchanger

works.

From equation (3.3.2-14) for NTU = 1 follows:

One tranfser unit causes a 63% approach to thermal

equilibrium for this kind of heat exchanger, compare

figure 3.3.2-3.

The number of transfer units can be tuned by the over

all heat transfer coefficient, the area of the heat

exchanger, the mass flux and the specific heat of the

coolant!

)152.3.3(

NTU

cm

Ak

p

)162.3.3(63,03678,01/111 1 eeg

1-1/e=0,63

Total heat flux for the heat exchanger. The total heat

flux of the heat exchanger can be calculated easily

with the help of the basic equation for heat transfer:

In equation (3.3.2-17) Tm is a temperature averaged

over the area of the heat exchanger that causes the

identical total heat flus as the factual temperature

profile in the coolant.

)1 72.3.3()( D mm immQ TTAkTAkJ

3.3.2/4 - 1.2012

Fig. 3.3.2-3: Temperature profile in the coolant.


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The averaged temperature Tm is calculated with the

help of a total heat balance for the heat exchanger:

This heat flux has to be absorbed from the coolant:

Comparing equations (3.3.2-18) and (3.3.2-19) we

obtain:

Transformation gives:

Taking NTU from (3.3.2-12) yields:

Fig. 3.3.2-3: Temperature profile in the coolant.

)1 82.3.3()( mm imQ TTAkJ

)1 92.3.3()( 0

TTcm epQJ

)2 02.3.3()()( 0

mm imep TTAkTTcm

)212.3.3()()()( 00

NTU

TTTT

cm

Ak

TTTT emimi

p

m

emmi

)222.3.3(

lnln

)()(

0

0

0

0

D

D

DD

D

e

e

emi

mi

emimimmmi

T

T

TT

TT

TT

TTTTTTT

For this kind of heat exchanger the total heat flux can

be easily calculated using equation (3.3.2-18) using an

averaged temperature difference which is given by the

„logarithmic mean“ according to equation (3.3.2-22).

This logarithmic mean is smaller than the arithmetic

mean due to the exponential decay of the temperature

difference, see figure 3.3.2-3.

3.3.2/5 - 1.2012


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Temperature profiles in a heat exchanger with

coflow. We treat a simple heat exchanger with

coflowing coolant and hot flow, compare figure 3.3.2-

4. In the inner tubes a hot flow is flowing from the

bottom to the top. In the annuli around the inner tubes

a coolant is co-flowing. We calculate the temperatures

in the hot and cold flow, the efficiency and the total

heat flux.

Temperatuer profiles. For the determination of the

temperatures we perform a simple heat balance for a

differential balance volume of the heat exchanger,

compare figure 3.3.2-4.

Heat balance for the coolant:

With convective heat fluxes according to section 1.3.1

and the basic heat transfer equation we obtain:

Fig. 3.3.2-4: Heat balance for calculation of

the temperature profiles in a coflowing heat

exchanger.

)252.3.3()(

)(u=)( a

D

DD

xxTmc

xxTAcxx

maapa

maapakonvQ J

)232.3.3()(=)()( D xxxxkonvQüberQkonvQ JJJ

)242.3.3()(

)(u=)( a

xTmc

xTAcx

maapa

maapakonvQ J

)262.3.3()()(=)( D xTxTxUkx mamiüberQJ

)272.3.3(

)(

D

x)(xTmc

(x)TxTΔxUk(x)Tmc

mamapa

mamimaapa

x

T

x

Außenrohr

Innenrohr

Tma

x x+Dx

JQkonv(x) JQkonv

(x+Dx)

JQüber(x)

JQkonv(x) JQkonv

(x+Dx)

Tmi

Balance:

The balance contains Tma and DT= Tmi-Tma, therefore

determination of DT(x) First!

3.3.2/6 - 1.2012


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We obtain finally a differential equation for the

temperature difference DT:

This differential equation can be solved easily after

separation of variables.

Fig. 3.3.2-5: Heat balance for calculation of the

temperature profiles in a coflowing heat

exchanger.

Expanding temperature into a Taylor-series and

linearization:

Introducing into the heat balance gives:

Analogously we obtain the balance for the hot flow:

Solving equations (3.3.2-29) and (3.3.2-31) for the

temperature gradients and subtracting gives:

Noting the definition of the temperature difference

)282.3.3()( DD xdx

dTxTx)(xT ma

mama

)292.3.3( D

TkTTkdA

dTcm mami

mapaa

)302.3.3()()()( D xxxxkonvQüberQkonvQ JJJ

)312.3.3( D

TkTTkdA

dTcm mami

mipii

)322.3.3(11

D

Tk

cmcmdA

dT

dA

dT

paapii

mami

)332.3.3(, DD mamimami dTdTTdTTT

)342.3.3(11

D

D

Tk

cmcmdA

Td

paapii

x

T

x

Außenrohr

Innenrohr

Tma

x x+Dx

JQkonv(x) JQkonv

(x+Dx)

JQüber(x)

JQkonv(x) JQkonv

(x+Dx)

Tmi

3.3.2/7- 1.2012


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Equation (3.2.2-38) can be transformed using dA =

Agesd into:

With that solution for the temperature difference the

temperatures in the coolant and the hot flow can be

calculated. For that we use equations (3.3.2-29) and

(3.3.2-31) after transformation:

Using DT from equation (3.3.2-39) we obtain:

Integration gives:

If the remaining integral on the right hand side is

replaced with the help of the averaged total heat

transfer coefficient we obtain:

Resolving the logaritm we obtain:

The temperature difference in the heat exchanger

exponentially decreases!

)352.3.3(11

D

D

dAk

cmcmT

Td

paapii

)362.3.3(11

ln00

D

D

A

paapii

dAk

cmcmT

T

)372.3.3()(

ln0

D

D

ai

paa

m

pii

m

NTUNTU

cm

Ak

cm

Ak

T

T

)382.3.3()(

0 DD ai NTUNTU

eTT

)392.3.3()(

0 DD gesagesi NTUNTU

eTT

)402.3.3( D TNTUd

dTgesa

ma

)412.3.3( D TNTUd

dTgesi

mi

)422.3.3()(

0 D

gesagesi NTUNTU

gesama eTNTU

d

dT

)432.3.3()(

0 D

gesagesi NTUNTU

gesimi eTNTU

d

dT

3.3.2/8- 1.2012


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Solution for the temperature in the hot flow:

Solution for the temperature in the coolant:

The solution for the temperature in the coolant

according to equation (3.3.2-45) is the effectivity g of

the heat exchanger in coflow!

Equation (3.3.2-44) can be rewritten as:

)452.3.3(

1)(

00

0

gesagesi NTUNTU

gesagesi

gesa

mami

mama eNTUNTU

NTU

TT

TT

Fig. 3.3.2-6: Effectivity of a coflow heat exchanger

according to equation (3.3.2-45), (NTUi = 5).

Fig. 3.3.2-7: Temperatur profiles in the coflow heat

exchanger according to equation (3.3.2-44a), (NTUi =5).

)442.3.3(

1)(

00

0

gesagesi NTUNTU

gesagesi

gesi

mami

mimi eNTUNTU

NTU

TT

TT

)442.3.3(

)(

00

0

a

NTUNTU

eNTUNTU

TT

TT

gesagesi

NTUNTU

gesigesa

mami

mami

gesagesi

3.3.2/9- 1.2012


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Total heat flux in coflow heat exchanger. Equation

(3.3.2-34) can be transformed by separating variables:

or

Integrating and replacing the term in brackets of

equation (3.3.2-46) by (NTUi + NTUa) from equation

(3.3.2-37) and using dJQ=kDTdA, we obtain:

Comparing this with the general equation for heat

transfer

)342.3.3(11

D

D

Tk

cmcmdA

Td

paapii

)462.3.3(11

D

D

dATk

cmcm

Td

paapii

)472.3.3(1

D Qai

m

dNTUNTUAk

Td J

)482.3.3(

ln 0

0

D

D

DD

e

emQ

T

T

TTAkJ

)4 92.3.3( D mmQ TAkJ

)502.3.3(

ln 0

0

D

D

DDD

e

em

T

T

TTT

we obviously and advantageously introduce again a

„logarithmic mean“ for calculation of the total heat

flux in this kind of heat exchanger.

Fig. 3.3.2-7: Temperature profiles in

the coflow heat exchanger.

3.3.2/10- 1.2012


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Only difference to the co-flowing heat exchanger: sign

in the heat balance !

Temperature profiles in a counter-flow heat

exchanger. We treat a simple heat exchanger with

counter flowing coolant and hot flow, compare figure

3.3.2-8. In the inner tubes a hot flow is flowing from

the right to the left. In the outer flow a coolant is

counter–flowing from left to right. We calculate the

temperatures in the hot and cold flow, the efficiency

and the total heat flux.

Temperature profiles. For the determination of the

temperatures we perform again a simple heat balance

for a differential balance volume of the heat

exchanger, compare figure 3.3.2-8.

Heat balance for the coolant (same as for the co-

flowing heat exchanger):

Balance equation again contains Tma and DT= Tmi-Tma,

therefore, first calculation of DT(x)!

Heat balance for the hot medium:

Same procedure as for the co-flowing heat exhanger

(expansion of temperature into a Taylor series etc.)

gives:


temperature profiles in a counter flowing heat

exchanger.

)512.3.3( D

TkTTkdA

dTcm mami

mapaa

)522.3.3()()()( D xxxxkonvQüberQkonvQ JJJ

)532.3.3( D

TkTTkdA

dTcm mami

mipii

x

T

x

Außenrohr

Innenrohr

Tma

x x+Dx

JQkonv(x) JQkonv

(x+Dx)

JQüber(x)

JQkonv(x) JQkonv

(x+Dx

)

Tmi

3.3.2/11 - 1.2008


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Solving equations (3.3.2-52) and (3.3.2-53) for the

temperature gradients and subtracting gives:

Noting the definition of the temperature difference, we

obtain finally a differential equation for DT:

Solution after separation of variables gives:

Factoring out (-1) and removing the logarithm gives:

)542.3.3(11

D

Tk

cmcmdA

dT

dA

dT

paapii

mami

)552.3.3( DD mamimami dTdTTdTTT

)562.3.3(11

D

D

Tk

cmcmdA

Td

paapii

The temperature difference in the counter-flowing

heat exchanger equally well decreases

exponentially! However, in this case much slower !

Equation (3.3.2-58) with the help of dA = Agesd can

be rewritten as:

)572.3.3()(

ln0

D

D

ai

paa

m

pii

m

NTUNTU

cm

Ak

cm

Ak

T

T

)582.3.3()(

0 DD ia NTUNTU

eTT)592.3.3(

)(

0 DD gesigesa NTUNTU

eTT

3.3.2/12 - 1.2008

x

T

x

Außenrohr

Innenrohr

Tma

x x+Dx

JQkonv(x) JQkonv

(x+Dx)

JQüber(x)

JQkonv(x) JQkonv

(x+Dx

)

Tmi


temperature profiles in a counter flowing heat

exchanger.


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With the solution for the total temperature difference

DT the temperature profiles in the hot and cold flow

can be calculated. For this we use equantion (3.3.2-51)

and (3.3.2-53) which can be written as:

Using DT from the equation (3.3.2-59) gives:

Then the temperature in the hot flow using the

boundary condition Tmi = Tmie for =0 and

DT0=Tmie-Tma0:

)602.3.3( D TNTUd

dTgesa

ma

)612.3.3( D TNTUd

dTgesi

mi

)622.3.3()(

0 D

gesigesa NTUNTU

gesama eTNTU

d

dT

)632.3.3()(

0 D

gesigesa NTUNTU

gesimi eTNTU

d

dT

)642.3.3(

1)(

0

gesigesa NTUNTU

gesigesa

gesi

mamie

miemi eNTUNTU

NTU

TT

TT

For the temperature in the cold flow we obtain using

the boundary condition Tma = Tmao for = 0:

In the equations for the temperatures in the hot and

counter-flowing cold flow the yet unknown

temperature Tmie is contained, which must be

calculated first.

For that we use equation (3.3.2-64) for =1 and add

(Tma0-Tma0). Transformation then results in:

If this result is substituted in equation (3.3.2-65) we

obtain an expression for the effectivity g of the counter

flowing heat exchanger.

)652.3.3(

1)(

0

0

gesigesa NTUNTU

gesigesa

gesa

mamie

mama eNTUNTU

NTU

TT

TT

)662.3.3(

11)(

000

gesigesa NTUNTU

gesigesa

gesi

mamimamie

eNTUNTU

NTU

TTTT

3.3.2/13 - 1.2008


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Using equation (3.3.2-66) in equation (3.3.2-64) we

obtain after addition of (Tmi0-Tmi0) and some

transformations:

Equation (3.3.2-68) can be rewritten as :

The temperature profiles according to (3.3.2-68a) is

given in figure 3.3.2-10.

)682.3.3(

1)(

)()(

00

0

gesigesa

gesigesagesigesa

NTUNTU

gesi

gesigesa

NTUNTUNTUNTU

mami

mimi

eNTU

NTUNTU

ee

TT

TT

Fig. 3.3.2-9: Effectivity of a counter-flowing

heat exchanger (NTUi = 5). 3.3.2/14 - 1.2008

)672.3.3(

1

1

)(

)(

00

0

gesigesa

gesigesa

NTUNTU

gesa

gesi

NTUNTU

mami

mama

eNTU

NTU

e

TT

TT

g

)682.3.3(

1

1

)(

)(

00

0 a

eNTU

NTU

eNTU

NTU

TT

TT

gesigesa

gesigesa

NTUNTU

gesi

gesa

NTUNTU

gesi

gesa

mami

mami


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Abb. 3.3.2-10: Temperature profile in a counter-

flowing heat exchanger according to equation

(3.5.2-68a) (NTUi = 5).

3.3.2/15 - 1.2008


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Heat and Mass Transfer 3.3.2/16- 1.2012

Summary of section 3.1 to 3.3

• A general model for heat transfer has been discussed.

• Correlations for heat transfer coefficients have been introduced.

• Heat transfer and over all heat transfer have been introduced.

• Some simple types of heat exchangers have been treated.

• Temperature profiles in heat exchangers, effectivity of heat exchangers and total heat

fluxes in heat exhangers can be calculated.


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3.4 Mass transfer in technical appliances

In section 1 we have identified transport of mass by convection and diffusion. The superposition of these two

transport processes forms the basis of mass transfer in technical appliances.

Example: Mass transfer in an absorption tower.

Within the absorption tower there is flow of the

solvent from top to bottom and of the gas from bottom

to top and thereby convective transport of mass. Due

to the concentration difference of the transferred

chemical species between the gas and the solvent there

is also mass transfer from the gas to the solvent.

3.4/1 - 1.2012

Fig. 3.4-1: Mass transfer in a technical

absorption tower.

x

x

x+Dx

Jnikonv(x)

Jnüber(x)

Jnikonv(x+Dx)

Jnakonv(x)

Jnakonv(x+Dx)


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The mass flux from the gas to the solvent occurs via

diffusion of mass through the phase boundary and the

adjacent boundary layers. Therefore, mass transfer

in any case is based on diffusion! Convection only

modifies the boundary layer conditions. Mass transfer

is a boundary value problem of convective /

diffusive transport of mass.

3.4.1 General model for mass transfer

From the discussion in section 2.5 we have learned to

write any mass transfer problem as:

As mass transfer occurs by diffusion through the phase

boundary we can write for the mass (molar) flux:

Comparing eq. (3.4.1-1) and (3.4.1-2) we obtain for

the mass transfer coefficient:

For few problems in section 2 the concentration

gradient could be accessed analytically.

Fig. 3.4.1-1: General model for mass transfer.

)11.4.3()( 0 D kk Wü bkn ccc j

)31.4.3(0

D

k

y

kk

c

y

cD

In most technical appliances velocity- and con-

centration fields cannot be calculated in a simple way

(as in the examples in section 2.) so that the concen-

tration gradient (dck/dy)y=0 has to be approximated.

In these cases a linearization of the concentration near

the phase boundary is applied (Taylor-series of

concentration, see figure 3.4.1-1):

With Dck = ckW – c k0 we obtain:

)41.4.3(..... D

y

dy

dccc k

kWk

)52.3(or0 D

D

y

D

y

cc

dy

dc kkkWk

3.4.1/1 - 1.2012

)21.4.3(0

y

kkübkn

dy

dcDj

y

ck

ck0

dD

ckW

Dy

y

Dk

D


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According to the model the mass transfer

coefficient is the diffusion coefficient Dk divided

by the thickness Dy of a fictitious concentration

boundary layer which results from the linearization

of the concentration profile.

The problem of the unknown mass transfer coefficient

is now shifted to the fictitious boundary layer

thickness Dy. The determination of this is done mostly

by experiments resulting in correlations for the mass

transfer coefficients for the various arrangements.

Correlations for mass transfer coefficients. Correla-

tions for mass transfer coefficients are generally given

in the form:

For simple problems (mass transfer from a sphere into

a fluid at rest, diffusion into the pores of a catalyst

etc.) correlations of the kind of equation (3.4.1-6) can

be accessed analytically. For the more complicated

technical cases the correlations have to be developed

from experimental investigations.

)61.4.3(with ...),(Re,

k

ch

D

LSh

L

DScfSh

Laminar flow in around spheres. For mass transfer

from spheres into a fluid at rest the Sherwood number

is, similar as the Nusselt Number for heat transfer

from a sphere, Sh=2. If there is laminar flow around

the sphere the appropriate correlation is given by:

Correlations for other cases relevant for process

engineering are given in table 3.4.1-1. From the table

the similarity of heat and mass transfer is obvious.

3.4.1/2 - 1.2012

k

ch

DSc

Lu

,Re

)71.4.3(R e6,00,2 3,05,0 S cS h


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Table. 3.4.1-1: Correlations for the Sherwood number for mass transfer in arrangements important

in provess engineeerimng (more information e.g. M. Baerns et al.: Technische Chemie, M. Jischa:

Konvektiver Impuls-, Wärme- und Stoffaustausch, E.U. Schlünder: Einführung in die

Stoffübertragung, ).

3.4.1/3 - 1.2012


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3.4.2 Over all mass transfer coefficient for mass transfer in technical appliances

In section 3.4.12 we have discussed mass transfer from a fluid to a phase boundary. In technical mass

exchangers we find a) transfer of mass from a fluid to the phase boundary, b) transfer of mass through the phase

boundary and c) transfer of mass from the phase boundary to a second fluid. In principle, we have a series of

mass transfer resistances formed by a boundary layer, the phase boundary and a second boundary layer. The

problem of mass transfer resistances in series can be treated analogously to the problem of heat transfer

resistances in series. This will be applied now to mass transfer in technical mass exchangers, see figure 3.4.2-1.

Fig. 3.4.2-1: Example of a technical mass

exchanger (film apsorption apparatus).

3.4.2/1 - 1.2012

In figure 3.4.2-1 we have a technical film absorption

apparatus. The solvent enters at the top and moves via the

different stages to the bottom. A gas phase is counter

flowing from bottom to the top. A chemical component k,

contained in the gas flow is transferred into the solvent.

Due to the concentration differences of the component k

between the gas flow (medium i) and the solvent (medium

a) we have concentration gradients perpendicular to the

direction of the flow. This concentration gradients are the

driving forces for mass transfer from the gas to the

solvent (from medium i to medium a).

The concentration profile across a stage in the mass

exchanger is enlarged in Fig. 3.4.2-2


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Solvent

The molar flux from the bulk of the gas phase to the

phase boundary is given according to the general mass

transfer equation by:

Fig. 3.4.2-1: Example of a technical mass

exchanger (film apsorption apparatus) and

concentration profiles.

Gas

x

ck

y

Dak

aD

y

Dik

iD

cki

ckPhi

ckPha

cka

The molar flux from the phase boundary to the bulk of

the solvent phase similarly is given by:

If there is no resistance for the transfer through the

phase boundary then the concentrations left and right

to the phase boundary ckPhi and ckPha are in phase

equilibrium (for absorption Henry‘s law applies).

3.4.2/2 - 1.2012

)12.4.3()( Phiik kkin ccAJ

)22.4.3()( ak kkPhaan ccAJ


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Applying Henry‘s law we can write:

and also

Here c*ki is the concentration that would be in phase

equilibrium with cka, compare figure 3.4.2-2.

Assuming stationary conditions the molar fluxes

according to equations (3.4.2-1) and (3.4.2-2) are

equivalent. Resolving these equations for the

concentration differences we obtain:

Multiplying equation (3.4.2-6) by k*k, we obtain:

or by using equations (3.4.2-3) and (3.4.2-4)

)52.4.3(

i

n

kkβA

cc k

Phii

J

)62.4.3(

a

n

kkβA

cc k

aPha

J

)92.4.3(1 *

*

a

k

i

nkkkkβA

k

βAcccc

kiPhiPhiiJ

)72.4.3(

*

**

a

kn

kkkkβA

kckck k

aPha

J

)82.4.3(

*

*

a

kn

kkβA

kcc k

iPhi

J

Fig. 3.4.2-2: Concentration profiles for

mass transfer from gas solvent.

Solvent

Gas

x

ck

y

Dak

aD

y

Dik

iD

cki

ckPhi

ckPha

cka

cki*

Adding equations (3.4.2-5) and (3.4.2-8) gives:

and finally

)1 02.4.3(* iik kkin ccAkJ

)112.4.3(1

1*

a

k

i

i

β

k

β

k

)32.4.3(0or ** PhaPhiPhaPhi kkkkkk ckcckc

)42.4.3(0or **** aiai kkkkkk ckcckc

3.4.2/3 - 1.2012


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We have now described the mass transfer analogously

to heat transfer: The molar flux is proportional to the

area, the over all mass transfer coefficient and a total

concentration difference. The driving concentration

difference has to be calculated somewhat differently

compared to heat transfer. This difference is due to the

concentration jump in the phase boundary.

with

In equations (3.4.2-10) and (3.4.2-11) the mass

transfer has been formulated with the concentration in

the gas phase. The same is possible with the

concentrations in the liquid phase. From the analogous

considerations we obtain equations (3.4.2-12) and

(3.4.2-13). Fig. 3.4.2-3: Concentration profiles for

mass transfer from gas solvent.

Solvent

Gas

x

ck

y

Dak

aD

y

Dik

iD

cki

ckPhi

ckPha

cka

cka*

)1 02.4.3(* iik kkin ccAkJ

)112.4.3(1

1*

a

k

i

i

β

k

β

k

)1 22.4.3(* k akan ccAkak

J

)132.4.3(11

1

*

aik

a

ββk

k

cki*

3.4.2/4 - 1.2012


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3.4.3 Simple mass transfer problems from process engineering

In the preceeding section the basic equations for mass

transfer have been derived as:

These relations now are to be used for the treatment of

simple problems of mass transfer in process

engineering.

Concentration profiles in a technical mass transfer

apparatus (absorption tower). In a technical

absorption tower (compare figure 3.4.3-1) a chemical

compound k in a gas mixture is transferred to a

solvent. The gas mixture containing the component k

enters the tower at the bottom and flows in upward

direction. The solvent is added at the top of the tower

and flows downwards. Requested is the concentration

profile of the component k in the gas phase and the

liquid phase.

Fig. 3.4.3-1: Mass transfer within an

absorption tower.

)12.5.4(* iik kkin ccAkJ

)22.5.4(1

1*

a

k

i

i

β

k

β

k

)32.5.4(* kakan ccAkak

J)42.5.4(

11

1

*

aik

a

ββk

k

4.5.2/1 - 2.2008


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For calculation of the concentration profiles within the

absorption tower a simple mass balance for the component k for

a differential volume of the absorption tower is formulated, see

figure 3.4.3-2.

Balance of molar fluxes for the gas phase in the balance

volume:

Inflow = Outflow + Transfer

For the molar fluxes we have:

Balance:

Epansion of the concentration into a Taylor-series and

linearisation gives:

Fig. 3.4.3-2: Material balance for a technical

absorption tower and concentration profiles

(schematically).

x

x

x+Dx

Jnikonv(x)

Jnüber(x)

Jnikonv(x+Dx)

Jnakonv(x)

Jnakonv(x+Dx)

x

ck

cki

ckie

ckio

cka

ckie

cka0

)52.5.4()()()( D xxxxkonvniübernkonvni JJJ

)62.5.4()(1

)(

xcmx kiikonvni

J

)82.5.4(=)( * kikiiübern ccAkxJ

)92.5.4()(1

)(1 * D

xxcmccAkxcm kiikikiikii

)72.5.4()(1

)( DD

xxcmxx kiikonvni

J

)102.5.4()( DD xdx

dcxcx)(xc i

ii

k

kk

4.5.2/2 - 2.2008


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Fig. 3.4.3-2: Material balance for a

technical absorption tower.

Inserting into the material balance gives:

Here we have used : A = a . Ageo. Dx.

The balance for the liquid phase gives analogously:

Expansion of the concentration into a Taylor-series,

linearisation and insertion into the balance equation

gives:

By use of

We obtain from equation (4.5.2-15):

x

x

x+Dx

Jnikonv(x)

Jnüber(x)

Jnikonv(x+Dx)

Jnakonv(x)

Jnakonv(x+Dx)

)112.5.4(*

ii

i

kk

i

geoikcc

m

Aak

dx

dc

)142.5.4()( DD xdx

dcxcx)(xc a

aa

k

kk

)122.5.4()()()( D xxxxkonvnakonvnaübern JJJ

)132.5.4(

)(1

)(1*

D

xcmxxcmccAk kaakaakikii

)152.5.4(*

ii

a

kk

a

geoikcc

m

Aak

dx

dc

)162.5.4(or **** aiai kkkkkk dckdcckc

)172.5.4(***

*

ii

ai

kk

a

geoi

k

k

k

kcc

m

Aakk

dx

dck

dx

dc

Adding equation (4.5.2-11) and (4.5.2-17) gives:

From the definition of the over all mass transfer

coefficient from equation (4.5.2-2) and (4.5.2-4)

follows:

)183.5.4(**

*

D

ii

iii

kk

a

ik

i

igeo

kkk

cc

m

kk

m

kAa

dx

cd

dx

dc

dx

dc

)192.5.4(or*

* k

aiika

k

kkkkk

4.5.2/3 - 2.2008


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Finally after transformation and separation of variables

we have:

Integration gives:

For the integral on the right hand side one can write

analogously as in the calculation of the temperature

profiles in heat exchangers:

With this we obtain analogously to the profile of the

temperature differences in counter-flow heat

exchangers:

After removing the logarithm follows:

x

x

x+Dx

Jnikonv(x)

Jnüber(x)

Jnikonv(x+Dx)

Jnakonv(x)

Jnakonv(x+Dx)

)202.5.4(

*

D

dxa

m

k

m

kA

cc

cd

a

a

i

igeo

kk

k

ii

i

)212.5.4(11

ln000

D

D

H

a

a

H

i

i

geo

k

kdxka

m

dxka

m

Ac

c

i

ie

)222.5.4(1

)(1

)(00

H

ama

H

imi dxkaH

kadxkaH

ka

)232.5.4()()(

ln

0

D

D

H

m

ka

m

kaA

c

c

a

ma

i

migeo

k

k

i

ie

)242.5.4(

)()(

0DD

x

m

ka

m

kaA

kka

ma

i

migeo

iiecc

The concentration difference between the

concentration of the component k in the gas phase and

the concentration in the liquid phase decreases

exponentially (compare counter-flow heat exchanger).

A similar result will be obtained, if the mass transfer is

formulated with the concentration of the liquid phase

(equations 4.5.2-3 and 4.5.2-4).

(In the above derivation density of fluid and gas are

treated to be equal!)

4.5.2/4 - 2.2008

Fig. 3.4.3-2: Material balance for a

technical absorption tower.


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With the solution for the concentration difference it is

possible to determine the concentration profiles in the

liquid phase and the gas phase. For that we use

equation (4.5.2-11) and equation (4.5.2-15),

respectively. Inserting the solution from equation (4.3-

24) and integrating gives:

Similarly we obtain for the liquid phase (equation

(4.5.2-15)):

The concentration ck*

io is the concentration which is in

thermodynamic equilibrium with the concentration

ckae, compare equation (4.5.2-16). This concentration

is not yet known and must be determined (compare

counter-flow heat exchanger).

For this equation (4.5.2-26) is formulated for the total

tower height and resolved for the requested

concentration difference. After some transformations

we obtain:

The concentration profiles of the component k in the

gas phase and the liquid phase as well as the

concentration difference are plotted in figures 3.4.3-3

and 3.4.3-4.

)112.5.4(*

ii

i

kk

i

geoikcc

m

Aak

dx

dc

)252.5.4(1

)()(

*

00

0

x

m

ka

m

kaA

i

ai

kk

kk a

ma

i

migeo

ii

ii e

m

mm

cc

cc

)262.5.4(1

)()(

*

00

0

x

m

ka

m

kaA

a

ai

kk

kk a

ma

i

migeo

ii

aa e

m

mm

cc

cc )272.5.4(

)()(

)()()()(

00

0

H

m

ka

m

kaA

i

a

H

m

ka

m

kaAx

m

ka

m

kaA

kk

kk

a

ma

i

migeo

a

ma

i

migeo

a

ma

i

migeo

ai

ii

e

m

m

ee

cc

cc

4.5.2/5 - 2.2008


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Fig. 3.4.3-3: Concentration difference

in an absorption tower. Fig. 3.4.3-4: Concentration profiles

in an absorption tower.

4.5.2/6 - 2.2008


© P

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Summary of section 3.4

• A general model for mass transfer has been discussed.

• Correlations for mass transfer coefficients have been introduced.

• Mass transfer and over all mass transfer have been introduced.

• Similarity between heat and mass transfer has been stressed.

• Mass transfer devices can be calculated analogously to heat transfer.

3.4.2/5 - 1.2012

3. Macroscopic transport processes: Heat and mass transfer...

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