3. macroscopic transport processes: heat and mass transfer...
TRANSCRIPT
Heat and Mass Transfer
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Heat and Mass Transfer
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
Heat and Mass Transfer
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Heat and Mass Transfer
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
Heat and Mass Transfer
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Heat and Mass Transfer
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
Heat and Mass Transfer
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Heat and Mass Transfer
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
Heat and Mass Transfer
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Heat and Mass Transfer
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
Heat and Mass Transfer
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Heat and Mass Transfer
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
Heat and Mass Transfer
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Heat and Mass Transfer
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
Heat and Mass Transfer
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Heat and Mass Transfer
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
Heat and Mass Transfer
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Heat and Mass Transfer
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
Heat and Mass Transfer
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Heat and Mass Transfer
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
Heat and Mass Transfer
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Heat and Mass Transfer
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
Heat and Mass Transfer
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Heat and Mass Transfer
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.
Heat and Mass Transfer
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Heat and Mass Transfer
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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Heat and Mass Transfer
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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Heat and Mass Transfer
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
Heat and Mass Transfer
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Heat and Mass Transfer
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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Heat and Mass Transfer
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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Heat and Mass Transfer
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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Heat and Mass Transfer
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:
Fig. 3.3.2-8: Heat balance for calculation of the
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
Heat and Mass Transfer
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Heat and Mass Transfer
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
Fig. 3.3.2-8: Heat balance for calculation of the
temperature profiles in a counter flowing heat
exchanger.
Heat and Mass Transfer
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Heat and Mass Transfer
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
Heat and Mass Transfer
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Heat and Mass Transfer
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
Heat and Mass Transfer
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Heat and Mass Transfer
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
Heat and Mass Transfer
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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.
Heat and Mass Transfer
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Heat and Mass Transfer
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)
Heat and Mass Transfer
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Heat and Mass Transfer
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
Heat and Mass Transfer
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Heat and Mass Transfer
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
Heat and Mass Transfer
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Heat and Mass Transfer
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
Heat and Mass Transfer
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Heat and Mass Transfer
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
Heat and Mass Transfer
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Heat and Mass Transfer
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
Heat and Mass Transfer
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Heat and Mass Transfer
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
Heat and Mass Transfer
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Heat and Mass Transfer
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
Heat and Mass Transfer
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Heat and Mass Transfer
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
Heat and Mass Transfer
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Heat and Mass Transfer
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
Heat and Mass Transfer
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Heat and Mass Transfer
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
Heat and Mass Transfer
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Heat and Mass Transfer
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.
Heat and Mass Transfer
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Heat and Mass Transfer
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
Heat and Mass Transfer
© P
rof.
Dr.
-Ing.
H.
Bo
ckho
rn,
Kar
lsru
he
20
12
Heat and Mass Transfer
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
Heat and Mass Transfer
© P
rof.
Dr.
-Ing.
H.
Bo
ckho
rn,
Kar
lsru
he
20
12
Heat and Mass Transfer
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