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MULTIPLE IMPELLER GAS-LIQUID CONTACTORS138

7 Heat Transfer

Controlling the system temperature within a narrow limit is a common process

requirement in many industrial practices, and heat transfer in a process vessel can be greatly

improved by applying a mechanical agitation. Although there are numerous experimental

correlations available for predicting the values of heat transfer coefficients, the range of

application is usually limited for the specific cases. Because heat transfer coefficient is

affected by so many factors, such as fluid properties, hydrodynamics generated by the

impeller and arrangements of coil and baffles. Therefore, it is still difficult to have a general

correlation to estimate heat transfer correctly for all purposes.

This chapter will be devoted to give a brief review of the current researches in heat

transfer in gassed multiple impeller stirred vessels along with our own studies on both local

and overall heat transfer coefficients in stirred vessels to illustrate how gassing will affect the

heat transfer in a mechanically agitated vessel.

7.1 Heat transfer coefficient

The amount of heat transferred from process fluid to heat transfer medium through heat

transfer surface can be estimated by:

Q=UA∆Tm (7.1-1)

where U=overall heat transfer coefficient (w Km°/ )

A=Heat transfer area ( 2m )

Tm =log. mean temperature difference between process fluid and heat

transfer medium ( K° )

The reciprocal value of U is a sum of a series of heat transfer resistances and can be

written as:

c2

1

2m1

RAA

h1

kh1

U1

+++=l (7.1-2)

here, h1 denotes process fluid side heat transfer film coefficient, h2 is film coefficient at heat

transfer medium side, l/km is conductive thermal resistance of heat transfer surface material

and Rc is the resistance due to fouled dirt. In many cases h1 is the limiting resistance, however,

Heat Transfer 139

for the low viscosity process fluids, and flow velocity of the heat transfer medium in jacket or

coil is very low, h2 will often become the limiting resistance of the system.

Since the film heat transfer coefficient expresses thermal conductivity of the boundary

layer clinging on the heat transfer surface, the thickness of this boundary layer directly affects

the rate of heat transfer through this layer. Thus, prior to grasp the performance of heat

transfer of a stirred vessel, it needs to understand the hydrodynamics of process fluid within

the system. Figure 7.1-1 depicts the different flow patterns in mechanical agitated vessel

generated by 45° pitched paddle and Rushton turbine impellers under various arrangements of

coils and baffles. These different flows against heat transfer surface naturally result in

different local heat transfer characteristics for the process vessels.

Rushton

Turbine

Pitched

Blade paddle

helical coil helical coil hair pin coil baffles

without baffles with baffles

Fig. 7.1-1 Flow patterns of the disk turbine and the pitch paddle impeller systems withvarious internal structures.

7.2 Local heat transfer coefficient

Figure 7.2-1 gives an example of how the value of local heat transfer coefficient will

vary with the angle with respects to liquid flow direction as fluid flows over a cylindrical pipe.

(Knudsen and Katz, 1955) Two peaks are found near at 0° and 180°, this fact implies that the

value of heat transfer coefficient will be heavily affected by flow characteristics of the


impeller and arrangement of the heat transfer surface. Due to the complex characteristic of the

flow pattern existed in stirred vessels, there have been only a few studies devoted to local

process side heat transfer.

Fig.7.2-1 Variation of heat transfer coefficient with respect to flow direction.

Using an electrochemical technique, Mann (1985) analogized the results of local

variation of the mass transfer coefficient to heat transfer. Karcz and Strek (1985) used the

same technique in an aerated stirred vessel with a dual turbine system. Using a micro foil

sensor, Fasano et al. (1991) and Haam and Brodkey (1992) also determined local heat transfer

coefficients for side wall and bottom surface. Instead of correlating the dimensionless local

heat transfer coefficient Nu as a function of power drawn by an impeller, they correlated Nu

in terms of the dimensionless superficial velocity of gas.

In order to examine the effect of the flow field on the rate of heat transfer in such an

aerated stirred vessel, Lu and Lan (1995) measured local value of heat transfer coefficient at

points which were located at different heights along both the tank wall (3.8°,45° and 86.2°)

and each side of the baffle plate as shown in Fig. 7.2-2.

Figures 7.2-3a~7.2-3c show how the value of local heat transfer coefficient varied with

height and the speed of impeller rotation at three different θ locations under ungassed

condition. Since the rate of heat transfer heavily depends on the velocity of the fluid sweeping

along the surface, the highest value of the local heat transfer coefficient is always found at the

impeller discharge region where fluid impinges on the wall at the maximum velocity.

Heat Transfer 141

Fig. 7.2-2 Locations where local heat transfer coefficient measured.

Fig. 7.2-3a~c Variation of local heat transfer coefficient with axial height under non aaeration condition at different θ locations.

For a given θ location and operating conditions, the value of local heat transfer film

coefficient, h, had the maximum at impeller plane, and it decreased on both sides along the

change in the height; however, the change of h along the height was different for each θ

location. For the case of baffle surface in front of flow (θ=86.2°), the change of h against axial

height was more gentle while the decrease of h along the height at a location just behind the

baffle (θ=3.8°) was rather steeper and became almost flat as the height exceeded a distance of

one diameter of the impeller (i. e. Z / H≧ 0.3).


This fact can be explained from difference in flow patterns (as shown in Fig. 7.2-4) in

each location. At the location behind the baffle plate (θ=3.8°), the flow shows a swirling or

vortices type flow near the impeller discharge region (Ede, 1961), and the impeller

characteristic had less effect on the upper zone of this location. It is interesting to examine the

change of the values of h at this location (θ=3.8°) under any operating conditions; the h

measures at Z/H = 0 always had the highest value if it was compared with the value of h at the

other θ locations. This is due to the fact that the strong swirling flow can be seen at the

location just behind the baffle plate, but the strength of this swirling flow decays sharply

along the increase in the height, (as shown in Fig. 7.2-4) which causes a steep decrease of h vs.

Z/H and also results in a flat variation of in the upper region.

(a) θ= 3.8º (b) θ= 45º (d) θ= 86.2º

Fig. 7.2-4 Schematic diagram of flow pattern near the wall in stirred tank.

Table 7.2-1 Local heat transfer coefficients at both sides of the baffle plates under non-aerated condition (N=6.67 rps).

Location b1 b0 θ=45∘wall f1 f0

Z/H=0

Z/H=0.625

average

6548.7

2692.7

4281.9

6514.0

3435.4

4647.7

5497.2

3498.6

4651.8

7055.8

4283.7

4959.8

6252.2

3229.5

4940.1

Unit:(W/m2‧K), Locations b1, b0, f1, f0 as shown in Fig. 7.2-2

In Table 7.2-1, the variation of local heat transfer coefficients and the average values for

both sides of the baffle plate are listed for two different heights. The values at the 45° mid-

plane at the wall are also presented here for comparison. The front point near the impeller

Heat Transfer 143

always has the highest local heat transfer coefficient and also the highest mean value. On the

backside of the baffle plate, the value of local h at the impeller discharge region shows to be

quite high, but it becomes almost the lowest value in the region near the free surface. It is

noticed that the average value of the mean heat transfer coefficient for the whole only

approximately 10% higher than that obtained at the 45° mid-plane.

7.3 Effect of gassing rates on heat transfer

Since the boundary layer thickness is directly affected by flow which sweeps over the

surface, local heat transfer coefficient in gassed stirred vessels can be related to flow patterns

of both liquid and bubbles. At low sparged rates, the impeller performs nearly equal to non-

gassed condition that pumps liquids and disperses gas radially. This action results a maximum

value of h in impeller plane. As gassing rate increases, the dispersed bubbles will be trapped

in the vicinity of the impeller. This weakens the pumping capacity and reduces the value of

local heat transfer coefficient.

Figure 7.3-1 shows a plot of h vs. VS for a jacket vessel obtained by Kurpiers et. al.

(1985). For a given rotational speed of the impeller, hj reduces first as the VS increases due to

its reduction in pumping capacity. As the value of h reaches to a minimum value, the heat

transfer will shift from agitation limiting to gassing controlling situation and the plot will

merge with the results obtained from bubble columns.

Fig. 7.3-1 Influence of aeration intensity on heat transfer. (Kurpiers et.al.,1985).

In Figures 7.3-2a~7.3-2c, how the value of h varies with height and rotational speeds for

a considerable high gas flow rate of Vs = 0.00534 m/s (equivalent to 1.11 VVM) at three

different θ locations are shown. Since the location of maximum liquid velocity shifts upward


due to aeration, the point of maximum h also shifts to the height Z/H = 0.1 except for the

points behind the baffle (i.e. θ =3.8°). The curves of θ=3.8° and θ=45° shown in Figure 7.3-3

show a flat variation of h at locations Z/H > 0.4 where the flow field in the upper region is not

affected by the performance of the impeller while at the points in front of the baffle, h decays

monotonously with the increase of height.To examine how the gas flow rate will affect the

values of local heat transfer coefficients, the plots of h vs. Z/H for several different gas flow

rates are shown for the location θ=45° in Figures 7.3-3a~c. Before the status of gas dispersion

in the system reaches flood stage, the trends of these plots are very similar to the results seen

in the cases of ungassed conditions except for the shift of the highest h point from Z/H=0 to

Z/H = 0.1.

Fig. 7.3-2a~c Variation of local heattransfer Coefficient with axial heightunder VS=5.34x10-3 m/s at different θlocations.

Fig. 7.3-3a~cEffect of gas flow rate onlocal heat transfer coefficient at θ= 45°.

Heat Transfer 145

If the gas flow rate continuously increases, or the rotational speed of the impeller

decreases to have a flooding situation, there occurs a sharp decrease in the value of h in the

place near the impeller due to insufficient pumping of the impeller.

For a given rotational speed, no significant change in the upper region of the tank is

observed because the flow of fluid in this region is not much affected by the pumping

capability of impeller, but the up rising gas flow rates.

7.4 Average Heat Transfer Coefficients

To show how averaged value of local heat transfer coefficient will differ from the local

values, the averaged local heat transfer coefficients of nine different heights at three different

azimuthal positions are shown in Fig. 7.4-1.

Fig. 7.4-1 Variation of mean heat transfer coefficient under non-aeration condition forvarious N and θ.

For a given rotational speed, the mean value of h at the plane in front of the baffle

(θ=86.2° ) is always the highest and the difference between it and mean h at the 45° middle

plane is approximately 5-7%, which is higher than 2% given by Bourne et al. (1985) for mass

transfer rates. If the mean Nu at the 45° mid-plane can be correlated in the form of140

b

w330r

ae PcRNu

..

−

⎟⎟⎠

⎞⎜⎜⎝

⎛µµ

= (7.4-1)

it gives a=0.66, c=0.729, i.e.140

b

w330r

660e PR7290Nu

....

−

⎟⎟⎠

⎞⎜⎜⎝

⎛µµ

= (7.4-2)

where c=0.729 is taken after the existing correlation (Chilton, 1994) of Nu for jacket wall,

which has c = 0.74. For all of the physical properties of fluid for calculation, the mean values

of data at TS and Tf are taken.


7.5 Correlation of Nu with Dimensionless Variables for Aerated System

Loiseau and Charpentier (1977) indicated that Michel and Millers experimental equation

(Michel and Miller, 1962)

46.056.0

g

3o

g )QNDP

(83.0P = (7.5-1)

can be used to estimate the gassed power drawn by a Rushton turbine impeller m its standard

geometry, with a gas flow rate of 0.0005m/s≦Vs≦0.09m/s and 1<PoND3/Qg0.56<107. For an

ungassed system, Calderbank and Moo-Young (1961) deduced from local isotropic turbulent

and proposed that3/14/1

v/p PrReNu = (7.5-2)

to correlate the heat transfer coefficient with the power drawn by the impeller for a stirred

vessel, where Rep/v is a modified Reynolds number and is defined as

3

24

v/pT)V/P(Re

µρ

= (7.5-3)

Modifying Eq. 7.5-2 by inserting (µw /µb)0.14 to include the effect of viscosity, it can be written

in the form of

14.0

b

w3/1r

aV/P P)(RecNu

−

⎟⎟⎠

⎞⎜⎜⎝

⎛µµ

= (7.5-4)

Since in most operations, the status of gas dispersion is either at loading or complete

dispersion, which satisfies the condition Pg>>Pa (aeration power) or P/V≒Pg/V, it follows that324

gv/p /T)V/P(Re µρ≈ (7.5-5)

Figure 7.5-1 shows a plot of Nu at 45° mid-plane for air-water system, and they are

compared with the data of Steiff et al.(1985). These results can be correlated as14.0

b

w33.0r

16.0V/P PRe913.2Nu

−

⎟⎟⎠

⎞⎜⎜⎝

⎛µµ

= (7.5-6)

for the range of 7.6 x 1014 < Rep/v < 3 × 1016.

Table 7.5-1 lists the values of constants for the same correlation for local and mean heat

transfer coefficients measured at both sides of the baffle plates. For the whole baffle, the

average value of Nu for air-water system can be given as14.0

b

w33.0r

129.0V/P PRe8.9Nu

−

⎟⎟⎠

⎞⎜⎜⎝

⎛µµ

= (7.5-7)

Heat Transfer 147

Fig. 7.5-1Correlations of the mean heat transfer coefficient against the modified

Reynolds number of the system.

Table 7.5-1 Correlation of local heat transfer coefficient at both sides of baffle platesunder aerated condition.

Position Z/H c abi 0 1.466 0.189b0 0 1.305 0.192fi 0 0.325 0.232f0 0 2.259 0.178bi ave.of all Z/H 2.841 0.160b0 ave.of all Z/H 4.113 0.152fi ave.of all Z/H 2.803 0.173f0 ave.of all Z/H 1.829 0.177

In Figure 7.5-2, the values hc /ρCP with and without aeration for the helical coil within

the system are plotted versus [(Pµ/ρ2V)1/4(CPµ/k)2/3]. This plot demonstrates that for or a given

value of net power consumption in agitation, there is no difference between the data obtained

under conditions with aeration and without aeration, except in the range of

[(Pµ/ρ2V)1/4(CPµ/k)2/3]<10-1 where the flooding phenomenon obviously has significant effect

on heat transfer coefficients. Thus, in the range of [(Pµ/ρ2V)1/4(CPµ/k)2/3]<10-1, hc can be

correlated with the net power consumption P and other variables as:


])k/C/()V/P[(1054.2Ch 3/2

p4/121

p

c µρµ×=ρ

− ( 7.5-8 )

Fig. 7.5-2 Heat transfer coefficient of coil in agitated vessel with dimenlsionlessvariables.

Fig. 7.5-3 hj in agitated vessel with and without aeration correlated in term of the netimpeller power consumption.

A similar plot of the heat transfer coefficient from jacket surface to the liquid vs. the

power consumed in agitation with and without gassing is summarized in Figure 7.5-3. Here P

again denotes net power drawn by the impeller either with or without gassing. The experiment

values in this figure show that for the same power consumption, there is no distinction

Heat Transfer 149

between hj, with aeration and without aeration. If the net power drawn by the impeller is

considered, and the results can be given as

71.03/2p

4/121

p

j ])k/C/()V/P[(1091.8Ch

µρµ×=ρ

− (7.5-9)

for the range [(Pµ/ρ2V)1/4(CPµ/k)2/3]>10-1. It is clear that in both cases of agitation with and

without gassing, heat transfer coefficient increases as the net power consumed in agitation

increases.

7.6 Type of heat transfer surfaces

In a mechanically agitated vessel, heat generated or needed for the process is mostly

removed or supplied through a jacketed wall or internal coils. Figure 7.6-1 depicts various

types of jacket structure commonly used. The jacket shown in Fig. 7.6-1(a) is normally used

in lower pressure steam heating, or smaller scale vessels, low pressure coolant cooling

systems. In order to increase external side film coefficient, it is often to install a spiral flow

path for external side heat transfer fluid to increase its flow rate as shown in Fig. 7.6-1(b). To

overcome the effect of external pressure exerting on vessel wall, hemi-cut pipe jacket as

shown in Fig. 7.6-1(c) or dimple jacket as shown in Fig. 7.6-1(d) is often adopted to reduce

the required wall thickness of the vessel.

Figure 7.6-2 shows different types of internal coil used in mechanically agitated vessels

for heat transfer purposes. Helical coil are the most commonly adopted in pilot or plant scale

vessels while plate while hair-pin type baffle coils are used in larger vessels or in the system

which generates large amount of heat. External heat exchanger as shown in Fig. 7.6-3 is

sometimes adopted where a large amount of process heat and no contaminating problem

existence.

(a) (b) (c) (d)

Fig. 7.6-1 Various type of jacket structures (a)plain jacket (b)jacket with internalbaffles (c)hemi-cut pipe jacket (d)dimple jacket.


(a) helical coil (b) hair pinbaffle coils

Fig. 7.6-2 Internal coils.

Fig. 7.6-3 Agitated vessel with externalheat exchanger.

7.7 Heat Transfer in Multiple Impeller Systems

The study on the heat transfer in multiple impeller systems is scare and limited to dual

impeller systems. Fig. 7.7-1 shows a plot of local heat transfer coefficient vs. height for

various gassing rate obtained by Karcz (1999), the largest amount of the heat transfer occurs

at the impeller discharge planes and the heat transfer coefficient decreases with the increase of

aeration rate.

Fig. 7.7-1 Variation of local of local heat transfer rate along the axial location in theaxial location in a dual impeller stirred vessel under various gassing rate.

The plots also shown that the higher heat transfer coefficient was found at the upper

impeller region than it was seen at the bottom impeller region. This is due to less air flow

through the upper impeller. Kurpiers et al. (1986) has explained the heat transfer in stirred

vessels by adopting Higbie’s penertration theory. Without examining the net power drawn by

each impeller, they concluded that it was difficult to estimate heat transfer rate in dual

Heat Transfer 151

impeller system as in the single impeller system. A correlation has been presented by Karcz

for estimation of average heat transfer coefficient for a dual impeller system as14.0

33.067.0 PrRe769.0−

⎟⎟⎠

⎞⎜⎜⎝

⎛=

b

wNuµµ

(7.7-1)

This correlation will give approximately 30% higher value of heat transfer coefficient for a

dual impeller system than a single impeller system.

Since the amount of energy dissipated by each impeller it different depended on amount

of bubbles surrounded the impeller and it can be related to the transport characteristics of each

impeller region, therefore, it is suggested that one may estimate power drawn by each

impeller (as described in the previous chapter), then the value of average heat transfer

coefficient for each impeller region can be estimated by using the Eq. 7.7-1.

NOTATION

Cp Specific heat [J/kg‧k]

d Internal diameter of circular conduit [m]

D Impeller diameter [m]

g Gravitation factor [m/s2]

h Local heat transfer coefficient [W/m2‧K]

h Mean heat transfer coefficient [W/m2‧K]

H Height of non-aeration liquid from bottom of stirred tank [m]

k Thermal conductivity [W/m‧K]

N Impeller rotational speed [1/s]

Nu Nusselt number [-]

P Power consumption [W]

Pg Power consumption with aeration [W]

Po Power consumption without aeration [W]

Pr Prandtl number [-]

qconv Convective heat flow rate [W]

qloss Heat loss of probe [W]

Qs Sparged gas flow rate [m3/s]

Rep.v Modified Reynolds number(3

24

µρT

VP⋅

) [-]

t Time [s]

T Temperature [K or℃]

V Volume of non-aerated liquid [m3]

Vs Superficial gas velocity [m/s]

w Impeller blade width [m]

Z Axial coordinate [m]


<Greeks Letters>α Thermal diffusivity [m2/s]

θ Shaft angular position [degree]

μ Fluid viscosity [Pa‧s]

μb Fluid viscosity at bulk fluid temperature [Pa‧s]

μw Fluid viscosity at wall temperature [Pa‧s]

ν Kinematic viscosity [m2/s]

ρ Fluid density [kg/m3]

ω Angular velocity [rad/s]

<Subscripts>c Coil

co Correlated

G Gas

j Jacket

L Liquid

<Dimensionless groups>Nu Local Nusselt number LkTh /⋅

Nu Mean Nusselt numberLkTh /⋅

Pr Prandtl number α/vRe Reynolds number vDN /2⋅Rep/v Modified Reynolds number 324 /)/( µρ⋅⋅TVP

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