metodo de wilson

Journal of Applied Fluid Mechanics, Vol. 4, No. 2, Issue 1, pp. 43-50, 2011.Available online at www.jafmonline.net, ISSN 1735-3572, EISSN 1735-3645.

An Investigation of Heat Transfer in a Mechanically AgitatedVessel

A. Debab1, N. Chergui1, K. Bekrentchir1 and J. Bertrand2

1Chemical Engineering Dept., Faculty of Science, University of Science and Technology (USTO), Oran, Algeria2 Chemical Engineering Laboratory, INP-ENSIACET, Toulouse, France

Corresponding Author Email: [email protected]

(Received January 8, 2009; accepted May 24, 2010)

ABSTRACT

The objective of this study is to optimize experimental conditions of agitating a non-Newtonian liquid usingexperimental design methodology. The measurements of the temperatures have been carried out in a jacketed vesselequipped with Turbine impellers. The rheological properties of aqueous solutions of carboxymethylcellulose sodiumsalt had been studied using shear stress/shear rate data. The results of the experimental studies, concerning the effectof the diameter of the impeller, the impeller speed and baffled or unbaffled vessel on the overall heat transfercoefficient have been approximated in the form of equations. Based on the optimization criterion, an agitated vesselequipped with Flat Blade Disc Turbine (FBDT) of diameter ratio d/D = 0.6 and baffles is proposed as the mostadvantageous for heat transfer processes.

Keywords: Stirred vessel, Non-Newtonian liquid, Experimental design, Heat transfer coefficient, Wilson plot.

NOMENCLATURE

A Heat transfer surface area (m2)Cp Specific heat (J kg-1K-1)d Diameter of the agitator (m)D Inner diameter of the agitated vessel (m)dt Time step (s)h Film heat transfer coefficient (Wm-2K-1)k Fluid consistency (Pa s-n)n Fluid index (-)Np Power number (Np=P. ?-1.N-3.d-5)Nu Nusselt number (Nu =h0.D.?-1)Pr Prandtl number (Pr =?.Cp. ?-1)N Speed of agitation (rpm)P Power consumption (W)q Volumetric flow rate (m3s-1)T Temperature (C)t Time (s)

U overall heat transfer coefficient (Wm-2K-1)V Volume of fluid (m3)? Thermal conductivity of agitated liquid

(W m-1 K-1)? Dynamic viscosity (kg m-1 s-1)?a Apparent viscosity (kg m-1 s-1)? Fluid density (kg m-3)? Shear stress (Pa)?? Shear rate (s-1)subscripts0 at the surface internal sidei inletj jacketr vessel

1. INTRODUCTIONMixing has found wide application in chemical andbiochemical processing. Many stirred tank bioreactorsand chemical reactors require precise control of bothmixing and heat transfer to achieve optimumproductivity (Oldshue 1983; Harnby et al. 1983; S.Nagata 1975). Heat transfer in agitated vessels is one ofthe most significant factors for controlling the outcomeof biochemical processes. The temperature offermentation generally must be maintained within verynarrow limits (Brian et al. 1989; Yce et al. 1999).Usually, agitated vessels have a heat transfer surface, in

the form of a jacket or internal coils, for addition orremoval of the heat.The intensity of heat transfer during mixing of fluidsdepends on the type of the agitator, the design of thevessel and conditions of the process. When designingan agitated vessel, the impeller, vessel geometry andbaffles should give the degree of mixing the processdemands, but it is impractical to specify the agitator togive a specific heat transfer coefficient. The main factorwhen selecting an agitator is the nature of the fluid.Large diameter agitators operating at low speedsnormally give excellent blending and heat transfercharacteristics with high viscosity fluids. Small

A. Debab et al. / JAFM, Vol. 4, No. 2, Issue 1, pp. 43-50, 2011.

44

impellers operating at high speed are more suited forlow viscosity fluids. Turbine impellers are normallyused for high speed, low viscosity applications.The normal design of turbine impeller has either four orsix flat blades on a central disc. The ratio of turbinediameter to vessel diameter d/D is usually in the rangefrom 1/3 to 2/3. Baffles are essential in stirred batchvessels to provide good mixing patterns throughout thevessel. They act to reduce tangential flow and promoteaxial motion. As baffling increases turbulence, it affectsthe heat transfer rates.Because heat transfer in agitated vessels is complex, anempirical approach based on dimensionless analysis hasbeen used to predict the average heat transfercoefficients at the jacketed wall. Hence, the results ofmany heat transfer studies are frequently correlatedusing a dimensionless equation.

?? ? ??? ??????????????? (1)Where ?10, ?20, ?30 and ?40 are found by fitting Eq. (1) toexperimental data. A review of many such correlationshas been presented by Mohan et al. (1992).

Extensive characterization work has been carried out byauthors for different agitator types. The values of theconstant ?10 reported are given in Table 1.

Table 1 Exponent for different impellers and Reregimes

Type of agitator Re range ?10Flat blade disc turbineUnbaffled vesselBaffled vessel

Re400

0.540.74

Propeller with threeblades and baffled vessel

5500 to 37000 0.64

The typical values for ?20, ?30 and ?40 are respectively2/3, 1/3 and 0.14 (Chapman et al. 1965; Strek et al.1967; Fletcher P. 1987). The constant ?10 which is usedto multiply the whole equation have been reported torange from 0.33 to 1.0, mainly varying due to systemgeometry and type of impeller.

Attempts were made to study batch heat transfer in anagitated jacketed vessel, with a view to developing adesign equation to determine film heat transfercoefficient, h0, in such vessels. But as the variablesinfluencing the heat transfer coefficient are quite largein number, complete study covering all the variablescould not be made. It is essential to have completeinformation on the effects of possible variables on therate of heat transfer and also a general correlation validover wide ranges of operating conditions for heattransfer coefficients for design and efficient working ofthe process. The aim of the present work was to studythe effect of the impeller speed N, impeller diameter dand vessel with or without baffles on heat transfercoefficient using an experimental method in which themeasurements are mathematically planned.

1.1 Vessel modelingFigure 1 shows the configuration for the agitated vesselused in the experiments.

Fig. 1. A typical arrangement of a jacketed vessel withlabels.

An energy balance in the vessel, considering the vesselvolume and the heat capacity of the vessel contents tobe constant, reads:(????? ????) ????? = ???????? ?(?)???? ?(?)? ? ??? ? ??? (2)Where ??? and ??? are the power terms associated withthe energy transfer from the stirrer and the heat ofreaction, respectively.

On the other hand the energy balance in the jacket canbe expressed as:(????? ????? ????? = ????(?? ?(?)????? ?(?)????????? ???? ?(?)? ??? ?(?)? (3)It is assumed that the heat loss of the usually insulatedjacket is negligible.

When the test begins, the temperatures in the vessel andin the jacket are respectively Tr(0) and Tj(0). Then, thevessel temperature is forced to go up or down when thejacket temperature is changed, suddenly or slowly, to anew set point. The jacket inlet temperature Tji(t) isrecorded, and so is the time to go from Tr(0) to Tr(t).

With known jacketed surfaces area, flow rates, heatcapacities and densities, the time-temperature responseis applied to the pertinent mathematical model.

The initial conditions are Tj(t) = Tj(0) and Tr(t) = Tr(0).In Eq. (2), since there is no reaction the heat of reactionand kinetic energy transferred from the impeller to thevessel fluid are neglected. Equations (2) and (3) can berewritten in terms of dimensional groups, as describedbelow. Tr(t) is isolated in Eq. (3) as a function of dTj/dt,Tj0(t) and Tji(t). The outcome is introduced into Eq. (2)to obtain dTr/dt as a function of the same variables.Then, the obtained equations are differentiated withrespect to time and solved by Laplace transform to yieldthe expression for Tr(t):

? ? ? ?? ?1 2j 1 1 2 2j

1( ) . .t tr j j jT t Ce C e? ?? ? ? ? ? ??? ? ? ? ? ?

(4)

Where

? ? ? ?21/)0()0( ???? ??? rjjnn TTC n= 1, 2? ?21 1 1 21 42 z z z? ? ? ? ?

Tr

M

Tji

Tjo, ?j, Vj,C

Tj

A. Debab

2 1 1 2?

1z

j?

j?

1.2Transfer

Overall ifrom

??

A plot of time versus temperature such as shown inFigaccuracy for design purposes.

If the time incrementconstant time intervaland initial temperatures respectively of the liquid beingheated, then we can write

??

Where

????

2.1The laboratory equipment is shownexperimentsvessel ofTurbineagitatorpositionedequal to the half of the height of the liquid medium.speed of therpm

The agitated vessel is equipped with a straight jacket,which has a circulating liquid in a clo

T

T

T

A. Debab

2 1 1 212

? ? ? ? ?

1 ??

jU A? ?

j q?

1.2 The determination of theransfer

Overall ifrom Eq.

=?

A plot of time versus temperature such as shown inFig.2accuracy for design purposes.

Fig.

If the time incrementconstant time intervaland initial temperatures respectively of the liquid beingheated, then we can write=

?

Where

???? ?

??

2

2.1 EquipmentThe laboratory equipment is shownexperimentsvessel ofTurbineagitatorpositionedequal to the half of the height of the liquid medium.speed of therpm.


Tri

Tr

Trf

A. Debab

?2 1 1 212 z z z? ? ? ?j ??i iU A?

j Vq

The determination of theransfer

Overall inside heat transfer coefficientEq. (

?? ?

A plot of time versus temperature such as shown incan be used to evaluate U

accuracy for design purposes.

Fig. 2


?? ?

Where

? ????

2. M

EquipmentThe laboratory equipment is shownexperimentsvessel ofTurbine agitatoragitator diameterpositionedequal to the half of the height of the liquid medium.speed of the


A. Debab et

2 1 1 212

z z z? ? ? ?

j??

?i i j j jU A

Cp V?

jV

The determination of theransfer Coefficient

nside heat transfer coefficient(2), it can therefore be rewritten to give:

??

????(A plot of time versus temperature such as shown in

can be used to evaluate Uaccuracy for design purposes.

2. Temperature plot for unsteady


??

????(??? ~MATERIAL

EquipmentThe laboratory equipment is shownexperiments wevessel of 2 liters

agitatordiameter

positioned in the vessel at a height from the bottomequal to the half of the height of the liquid medium.speed of the agitator


t

al.

22 1 1 2z z z? ? ? ?

and

j j jCp V?

The determination of theoefficient

nside heat transfer coefficient, it can therefore be rewritten to give:

?????

? (?)?A plot of time versus temperature such as shown in


Temperature plot for unsteady


?????

? (?)?? ~ ??

??

ATERIAL

EquipmentThe laboratory equipment is shown

were conducted in a cylindrical liters

agitatordiameter

in the vessel at a height from the bottomequal to the half of the height of the liquid medium.

agitator


ti

/ JAFM

22 1 1 24z z z? ? ? ?

and z2

?j j jCp V



???( )? ??A plot of time versus temperature such as shown in




???( )? ??? ???

??

ATERIAL

EquipmentThe laboratory equipment is shown

re conducted in a cylindrical liters equipped

agitator (6diameters were used:

in the vessel at a height from the bottomequal to the half of the height of the liquid medium.

agitator


JAFM

?2 1 1 24z z z? ? ? ??2

j j jCp Vand



) ?(?)?A plot of time versus temperature such as shown in


Temperature plot for unsteadytransfer

If the time incrementconstant time interval ?t, andand initial temperatures respectively of the liquid beingheated, then we can write

) ?(?)?

ATERIAL AND

The laboratory equipment is shownre conducted in a cylindrical

equipped6FBDT)

s were used:in the vessel at a height from the bottom

equal to the half of the height of the liquid medium.agitator was varied from


JAFM, Vol

2 1 1 24z z z

r??and ??

The determination of the


( )? ? ??A plot of time versus temperature such as shown in


Temperature plot for unsteadytransfer

If the time increment dt is evaluated as a specific?t, and

and initial temperatures respectively of the liquid beingheated, then we can write Eq

( )? ? ?

AND M


equippedFBDT)


equal to the half of the height of the liquid medium.was varied from


Vol.

jr?? = (

The determination of the


)? ?????A plot of time versus temperature such as shown in


Temperature plot for unsteadytransfer.

is evaluated as a specific?t, and T

and initial temperatures respectively of the liquid beingEq. (5)

)? ?????

AND METHOD


equipped withas seen in




tf

. 4,

(???The determination of the O


A plot of time versus temperature such as shown incan be used to evaluate U


is evaluated as a specificTrf and

and initial temperatures respectively of the liquid being) as:

? ? ?

??

ETHOD


with aas seen in

s were used: 81in the vessel at a height from the bottom



, No.

( ??)?Overall


A plot of time versus temperature such as shown incan be used to evaluate Ui


is evaluated as a specificand

and initial temperatures respectively of the liquid beingas:

????

ETHOD


a 6-Flat Blade Discas seen in

81mm andin the vessel at a height from the bottom


The agitated vessel is equipped with a straight jacket,which has a circulating liquid in a closed loop. Heat is

. 2,

) (???verall

nside heat transfer coefficient Ui, it can therefore be rewritten to give:

A plot of time versus temperature such as shown inwith sufficient

Temperature plot for unsteady-state heat

is evaluated as a specificand Tri

and initial temperatures respectively of the liquid being

?

The laboratory equipment is shown inre conducted in a cylindrical

Flat Blade Discas seen in

m andin the vessel at a height from the bottom

equal to the half of the height of the liquid medium.was varied from 260

The agitated vessel is equipped with a straight jacket,sed loop. Heat is

, Issue

) ( ????verall Heat

i is, it can therefore be rewritten to give:


state heat

is evaluated as a specificare the final


?

in Figre conducted in a cylindrical

Flat Blade Discas seen in Fig

m andin the vessel at a height from the bottom

equal to the half of the height of the liquid medium.260 rpm to


Issue 1

???)Heat

obtained, it can therefore be rewritten to give:


state heat



Fig.3; thejacketed

Flat Blade DiscFig.4.

m and 45in the vessel at a height from the bottom

equal to the half of the height of the liquid medium. rpm to


1, pp

)

obtained

(5)A plot of time versus temperature such as shown in

with sufficient

state heat



(6

; thejacketed

Flat Blade DiscTwo

45mm,in the vessel at a height from the bottom

equal to the half of the height of the liquid medium. The rpm to 850


pp. 43

45

)

obtained

)A plot of time versus temperature such as shown in

with sufficient



6)

; thejacketed

Flat Blade DiscTwo

m,in the vessel at a height from the bottom

The850


43-50

45

provided from this fluid by means of an electric heatexchanger and a thermostatic bath, as indicated inFigsensors; one is positioned in the high part of theagitated fluid (TWater circulate in the jacket and his temperature ismeasured with two sensors positioned at the inlet (Tand the outlet side (Tspaced, flat wall baffles were used each with a widthequal

The Temperature controller controls the jackettemperature and allows tracking a desired set point ofthe vessel temperature.

The jacketthe flow of the circulating heat transfer fluid throughthe jacket; it was calculated for theflow rateexperimentsWerner

2.2The commonly used method to determine the heattransfer coefficient is the Wilson plot method (1915parameters defining the dimensions of the equipment(the heat transfer surface), the fluidscircuit, the mass flow rates, and the averagetemperatures in theheat transfer coefficient on each side of the heat transfersurface depends on the flow regime. In order to

1

50, 2011

provided from this fluid by means of an electric heatexchanger and a thermostatic bath, as indicated inFig.sensors; one is positioned in the high part of theagitated fluid (TWater circulate in the jacket and his temperature ismeasured with two sensors positioned at the inlet (Tand the outlet side (Tspaced, flat wall baffles were used each with a widthequal


Fig


2.2The commonly used method to determine the heattransfer coefficient is the Wilson plot method (1915parameters defining the dimensions of the equipment(the heat transfer surface), the fluidscircuit, the mass flow rates, and the averagetemperatures in theheat transfer coefficient on each side of the heat transfersurface depends on the flow regime. In order to

1

Trb

2011

provided from this fluid by means of an electric heatexchanger and a thermostatic bath, as indicated in

. 3. It is instrumented by four PTsensors; one is positioned in the high part of theagitated fluid (TWater circulate in the jacket and his temperature ismeasured with two sensors positioned at the inlet (Tand the outlet side (Tspaced, flat wall baffles were used each with a widthequal to


Fig.equipment4-Pump6-Thermostatic bath


Wilson plot metThe commonly used method to determine the heattransfer coefficient is the Wilson plot method (1915). The input data for this method are the physicalparameters defining the dimensions of the equipment(the heat transfer surface), the fluidscircuit, the mass flow rates, and the averagetemperatures in theheat transfer coefficient on each side of the heat transfersurface depends on the flow regime. In order to

b

2

M

2011.


. It is instrumented by four PTsensors; one is positioned in the high part of theagitated fluid (TWater circulate in the jacket and his temperature ismeasured with two sensors positioned at the inlet (Tand the outlet side (Tspaced, flat wall baffles were used each with a width

to 1/10


. 3. Schematic representation of the laboratoryequipment

PumpThermostatic bath

Fig.

The jacket-the flow of the circulating heat transfer fluid throughthe jacket; it was calculated for theflow rate, which was keptexperimentsWerner et al.

Wilson plot metThe commonly used method to determine the heattransfer coefficient is the Wilson plot method (

). The input data for this method are the physicalparameters defining the dimensions of the equipment(the heat transfer surface), the fluidscircuit, the mass flow rates, and the averagetemperatures in theheat transfer coefficient on each side of the heat transfersurface depends on the flow regime. In order to

T

2

M


. It is instrumented by four PTsensors; one is positioned in the high part of theagitated fluid (TWater circulate in the jacket and his temperature ismeasured with two sensors positioned at the inlet (Tand the outlet side (Tspaced, flat wall baffles were used each with a width

1/10 of the inner diameter of the vessel


. Schematic representation of the laboratoryequipment:

Pump, 5-Thermostatic bath

Fig. 4.Agitator used in the investigation

-side heat transfer coefficient hthe flow of the circulating heat transfer fluid throughthe jacket; it was calculated for the

, which was keptexperiments according to the correlation given

et al. (1986



Trh

8


. It is instrumented by four PTsensors; one is positioned in the high part of theagitated fluid (Trh) andWater circulate in the jacket and his temperature ismeasured with two sensors positioned at the inlet (Tand the outlet side (Tspaced, flat wall baffles were used each with a width

of the inner diameter of the vessel


. Schematic representation of the laboratory: 1-Vessel-Temperature controller,

Thermostatic bath

Agitator used in the investigation

side heat transfer coefficient hthe flow of the circulating heat transfer fluid throughthe jacket; it was calculated for the

, which was keptaccording to the correlation given1986



Tjo

Tj


. It is instrumented by four PTsensors; one is positioned in the high part of the

) andWater circulate in the jacket and his temperature ismeasured with two sensors positioned at the inlet (Tand the outlet side (Tspaced, flat wall baffles were used each with a width



. Schematic representation of the laboratoryVessel

Temperature controller,Thermostatic bath



, which was keptaccording to the correlation given1986).


). The input data for this method are the physicalparameters defining the dimensions of the equipment(the heat transfer surface), the fluidscircuit, the mass flow rates, and the averagetemperatures in the inlet and outlet connector pipes. Theheat transfer coefficient on each side of the heat transfersurface depends on the flow regime. In order to



) and the other in the low part (TWater circulate in the jacket and his temperature ismeasured with two sensors positioned at the inlet (Tand the outlet side (Tjo) of the jacket. Four equallyspaced, flat wall baffles were used each with a width



. Schematic representation of the laboratoryVessel, 2

Temperature controller,Thermostatic bath, 7



, which was keptaccording to the correlation given

Wilson plot methodThe commonly used method to determine the heattransfer coefficient is the Wilson plot method (

). The input data for this method are the physicalparameters defining the dimensions of the equipment(the heat transfer surface), the fluidscircuit, the mass flow rates, and the average

inlet and outlet connector pipes. Theheat transfer coefficient on each side of the heat transfersurface depends on the flow regime. In order to



the other in the low part (TWater circulate in the jacket and his temperature ismeasured with two sensors positioned at the inlet (T

) of the jacket. Four equallyspaced, flat wall baffles were used each with a width


The Temperature controller controls the jackettemperature and allows tracking a desired set point of

. Schematic representation of the laboratory2-Agitator,

Temperature controller,7-Electric heater



, which was keptaccording to the correlation given

hodThe commonly used method to determine the heattransfer coefficient is the Wilson plot method (



Digital Computer

5

Data Acquisition

3

7







. Schematic representation of the laboratoryAgitator,

Temperature controller,Electric heater



constant during all theaccording to the correlation given

The commonly used method to determine the heattransfer coefficient is the Wilson plot method (



Digital Computer

5

Data Acquisition

7







. Schematic representation of the laboratoryAgitator,








Digital Computer

Data Acquisition

4


. It is instrumented by four PT-100sensors; one is positioned in the high part of the





. Schematic representation of the laboratoryAgitator, 3-Flow meter,



side heat transfer coefficient hthe flow of the circulating heat transfer fluid throughthe jacket; it was calculated for the maximum mass



). The input data for this method are the physicalparameters defining the dimensions of the equipment(the heat transfer surface), the fluids flowing in eachcircuit, the mass flow rates, and the average


Digital Computer

4


100 temperaturesensors; one is positioned in the high part of the





. Schematic representation of the laboratoryFlow meter,

Electric heater, 8-


side heat transfer coefficient hj depends onthe flow of the circulating heat transfer fluid through

maximum massconstant during all the

according to the correlation given


). The input data for this method are the physicalparameters defining the dimensions of the equipment

flowing in eachcircuit, the mass flow rates, and the average


6

8CoolingWater


temperaturesensors; one is positioned in the high part of the



of the inner diameter of the vessel.



-Baffles.


depends onthe flow of the circulating heat transfer fluid through







M

8CoolingWater







Baffles.




The commonly used method to determine the heattransfer coefficient is the Wilson plot method (Wilson




M

Cooling



the other in the low part (Trb).Water circulate in the jacket and his temperature ismeasured with two sensors positioned at the inlet (Tji)

) of the jacket. Four equally-spaced, flat wall baffles were used each with a width


. Schematic representation of the laboratory

Baffles.



according to the correlation given by

The commonly used method to determine the heatWilson






).Water circulate in the jacket and his temperature is

)

spaced, flat wall baffles were used each with a width




by

The commonly used method to determine the heat





46

experimentally determine this parameter in givenequipment, varied is the mass flow rate in one side, andthe mass flow rate in the other side is kept constant.Temperature and mass flow rate are the inputparameters varying during the experiment.

The original Wilson plot method was modified in avariety of ways. The modification mainly consisted inchanging the number of determined parameters. Theoriginal version required knowledge of the exponent atthe Reynolds number. Briggs et al. (1969) proposed aniterative method to determine both heat transfercoefficients without knowing of the exponent at theReynolds number. Further modifications allowed fordetermining of up to five parameters can be found inFernandez-Seara et al. (2007).

The overall heat transfer resistance through the wall ofthe agitated vessel is described by the followingapproximation formula:

?

??= ???? ?? + ????? (7)

Where: Rw is the wall resistance. It should be noticedthat every liquid can produce a layer or deposit ofextraneous materials on the heat transfer surface, whichwill provoke a time-modification of both film heattransfer coefficients. In these cases, the heat transferresistance increases considerably due to the fact thatthis type of materials normally has a lower thermalconductivity. This effect is referred to as the fouling ordirt factor, and should be avoided or, if this is notpossible, taken into account as a further resistance toheat transfer.

Substituting to Eq. (1) definitions of the similaritynumbers, the heat transfer coefficient hi can beexpressed in the form:

. 200?? Nhi ? (8)

Where, ?0 is a constant.

According to Eq. (8), only the stirrer speed can modifythe value of the internal film heat transfer coefficientfor a given vessel, with the same liquid and temperatureconditions inside it. The value of ?0 is described by Eq.(9).

.dViPr20

4030

2

100

???

????? ???

????

D (9)

Where ?10, ?20, ?30, ?40 are constant for every system.

Substituting Eq. (8) to Eq. (7) we arrive at the followingequation, used in the calculations:1??

= 1??????? ? ?? + ????? (10)

When the power generated in the vessel is small, theviscosity number (Vi) can be neglected, if temperaturedifference between the wall of the vessel and the fluidtemperature is not so large.

Assuming that ?0 has a constant value under theconditions previously mentioned, it is possible todetermine experimentally the value of hi, using theWilson plot.

Substituting the new variables:

? = ???

(11)

? ? ????? (12)

? = ???

(13)

? ? ?? + ????? (14)Equation (10) can be linearized:

? ? ? ? ? ? ? (15)

A straight line can be plotted in Fig. 5 using Eq. (15).

Fig. 5. Original Wilson plot for hi experimentaldetermination

The slope of the straight line is the reciprocal of ?0.When the stirrer speed has no effects on the value of hi(N ???), 1/U gives the parameter b.

2.3 Planned experimental methodThe influence of different geometrical parameters of theagitated vessel on the heat transfer coefficient can bedetermined using experimental method in which themeasurements are mathematically planned (Goupy1996). As detected in previous study (Chergui 2005)the most influential factors on the heat transfercoefficient are the impeller diameter (X1), the baffles(X2) and the impeller speed (X3). In order to evaluatethe effects and interactions of these three factors, a 23factorial design is used. The experimental conditionsrequired by this design are defined in Table 2.

Table 2 Levels of the factors

d (m)X1

BafflesX2

N (rpm)X3

Level + 0.081 With

baffles

850

Level 0.045 Withoutbaffles

260

The levels chosen for each of the three parameters arealso presented in this table, e.g., a low level istransformed to the number -1 and a high level to thenumber +1.

y = a.x+b

bx

y


47

The mathematical model associated with this type ofplanned experiments is a linear regression which takesinto account the effects of the factors as well as theeffects of their interactions.It is written as:

0 1 1 2 2 3 3 12 1 2

13 1 3 23 2 3 123 1 2 3

Y b b X b X b X b X Xb X X b X X b X X X? ? ? ? ? ?

? ? (16)

Where Y represents the estimated response, it representsthe overall heat transfer coefficient hi and Xi areindependent variables in code. The constant b0, is theaverage experimental response, the coefficients b1, b2and b3 are the estimated effects of the factorsconsidered and the extent to which these terms affectthe performance of the method is called main effect.The coefficients b12, b13, b23 and b123 are called theinteraction terms. The coefficients bij of the model areestimated from experimental responses. Following themodel fitted for each response, we represent graphicallyisoresponses surfaces which are three-dimensionalmodels of the relationship between the responses andtwo factors. This response surfaces methodology allowsexperimental responses behavior to be described asprecisely as possible as a function of factor variationand optimal conditions of the factors to be determinedfor each experimental response. We can see from Eq.(16) that the factorial design provides information aboutthe importance of interactions between the factors. Thismeans that sometimes the level in which some factorsmust be set is influenced by their interaction withothers, so that we can ensure a better expectedexperimental response.

The statistical significance of each term, in Eq. (16),was verified using the t-Student.

3. RESULTS AND DISCUSSION

3.1 Rheological Behavior of Aqueous Solutionof Carboxymethyl Cellulose Sodium SaltA 2% dilute concentration of aqueous polymer solutionof sodium carboxymethyl cellulose sodium salt (CMC)was chosen as a test fluid. To study the effect oftemperature on rheology, the measurements werecarried out at temperatures between 30C to 60C andshear rates ranging between 6.45 S-1 to 645 S-1. Theexperimental set up to perform these experiments werea variable speed, coaxial cylinder thermostatedrheometer (VT550). A plot of the flow curves, shearstress versus shear rate at four different temperaturesare shown in Fig. 6.

0

10

20

30

40

50

60

70

80

90

0 200 400 600 800

?(P

a)

?? (s-1)

T=30CT=40CT=50CT=60C

Fig. 6. The dependence of shear stress on the shearrate.

We observed that the experimented fluid exhibits arheofluidifiant behavior governed by the Ostwald deWaele power-law model:

nk ?? ?.? (17)Where k is the consistence index and n, is the flowbehavior index.

Table 3 provides some sample numerical values of kand n.

Table 3 Values of the index of flow and consistency forvarious values of temperatures

T,C 2% C.M.C

N k , Pa.sn

30 0.7272 0.714240 0.761 0.466150 0.8153 0.254460 0.8483 0.1646

The consistency index, k, decreases with increasingtemperature T. It is correlated with temperature asfollows:

bTeak .? (18)Where a and b are constants.

The fluid behavior index, n, showed no noticeablevariation with temperature change.

In Fig. 7 we could observe the influence of thetemperature on the apparent viscosity. A decrease inviscosity was produced when the temperature increases.

0,01

0,1

1

1 10 100 1000

a(P

a.s)

??(1/s)

T=30C

T=40C

T=50C

T=60C

Fig. 7. The dependence of apparent viscosityon the shear rate.

The apparent viscosity is computed from the followingequation:

1na k? ? ?? ? (19)

3.2 Influence of Non-Newtonian FlowBehavior on Mixing Process CharacteristicsIn an agitated vessel, the rheological properties of thematerial to be processed are of main importanceregarding the parameters which have to be consideredfor the configuration. The viscosity influences theformation of the flow field inside the tank and,therefore, the mixing and power behavior.


48

Thus, the viscosity is the most important characteristicvalue. While Newtonian fluids are characterized by aconstant viscosity, the processing of non-Newtonianmaterials demands the consideration of a changingviscosity through a viscosity function.

For the configuration of agitators for the processing ofnon-Newtonian materials, the changing viscosity needsto be considered, where the Reynolds number is thedecisive characteristic value. The main problem is thedetermination of the correct viscosity at a set rotationrate; as a consequence the Reynolds number cannot becalculated. This means that also the mixing time andpower characteristics only can be included asapproximations in the configuration stage. For thisreason, several methods, which enabled theconstruction of a stirring apparatus for the processing ofnon-Newtonian fluids by relating speed and viscosity toeach other, were developed in the past. For power lawfluids, a linear correlation between rotation rate of thestirrer and mean shear rate ?? in the agitated vessel canbe described by a linear correlation according to themethod of Metzner and Otto (1957).

NK s??? (20)Where the constant Ks in pseudo-plastic fluids has thevalue 11.5, when a flat blade disc turbine is used; forthe pitched blade turbine (?=45) the value is 13 and forthe propeller it is 10 (Reinhold et al. 1980).

3.3 Heat Transfer Calculation3.3.1 Experimental Determination of Heat TransferCoefficients by Wilson Plot

A computer programme was used to analyze the datataken from the experiments in this heat transfer study.The first step of the calculation deals with theestimation of an averaged Overall heat transfercoefficient Ui from Eq. (6) and the experimental time-temperature results presented in Fig.8

The values calculated from Eq. (6) are presented inTable 4.

Table 4 Values of N, Re and Ui

FBDT(d=81mm) FBDT(d=45mm)

N(rpm) Re Ui Re Ui

260 142.80 129.30 140.78 83.95

450 241.95 188.68 265.07 116.22

650 353.75 240.82 373.49 146.68

850 430.91 276.19 573.08 189.72

From Table 4, it can be seen that the heat transfercoefficient is calculated in the transition regime.

In the second part of the steady, the time response ofthe theoretical vessel temperature Tv(t) using Eq. (4) ispresented in Fig. 8. From these plots it could be clearly

seen that the agreement between the experimental dataand the theoretical values calculated using Eq. (4) issatisfactory. Similar plots prepared for the agitator of45mm diameter also indicated satisfactory agreement.

Fig. 8. Time-Temperature Plots for FBDT d= 81mm and N=850 rpm.

The third step deals with the determination of the partialheat transfer coefficient hi using Wilson plot method.

The calculated values of the local heat transfercoefficient hi from the Wilson graphs have been plottedagainst the speed of the impeller for each of theimpellers. From this plots it has been deducted thecorrelations given in Table 5.

Table 5 Values of hi

FBDT(d=81mm) FBDT(d=45mm)

hi= 3.697 N2/3 hi = 2.158 N2/3

3.3.2 Prediction of the Film Heat TransferCoefficient by the Planned Experimental Method

The mathematical model derived from the experimentalresults is presented by the following equation:

1 2 3

1 2 1 3 2 3

1 2 3

428,37 113,53 21,29 90,4517,05 23,84 15,19,23

Y X X XX X X X X X

X X X

? ? ? ? ?? ? ? (21)

Where018,0

063,0dX1?? and

5,2665,526NX3

??

A second test was done in order to check the test ofFischer (Goupy 1996), this test shows that themathematical model is representative. To be able torepresent graphically this equation; it is necessary tostay constant one of the parameters, for example thebaffles referred by the parameter (X2,), then we obtainthe following equations:

For X2 = +1 (vessel with baffles)

? = 449.66 + 130.58?? + 105.55?? +33.07???? (22)

295

300

305

310

315

320

325

330

335

0 500 1000 1500 2000 2500 3000

Tr(K

)

time (s)

Tr-model

Tr-exp

A. Debab

For X

? =14TheEq.andmoreX3transfer coefficientthe diameter of the agitator increases from0.081loops generated by each impeller.

The applicability of the models for the prediction of theoverall heat transfer coefficientused in this study is demonstrated inexperimental data obtained for various operatingconditions are compared with those predicted from(6)

Fig.

A. Debab

For X= 40714.61?The mathematical modelEq. 23and Fig.more weight than the third, which is the impeller speed

3 and the diameter of the agitator Xtransfer coefficientthe diameter of the agitator increases from0.081mloops generated by each impeller.

The applicability of the models for the prediction of theoverall heat transfer coefficientused in this study is demonstrated inexperimental data obtained for various operatingconditions are compared with those predicted from

).

Fig

Fig

Fig.

A. Debab

For X2=-1407.06???

mathematical model23, reported asFig. 10

weight than the third, which is the impeller speedand the diameter of the agitator X

transfer coefficientthe diameter of the agitator increases from

m, this is due to flow pattern and the circulationloops generated by each impeller.


Fig.

Fig. 10

Fig. 11.

A. Debab et

1 (vessel without baffles)06 +??

mathematical modelreported as10 respectively



this is due to flow pattern and the circulationloops generated by each impeller.


. 9. Response surface contours

10. Response surface

. Comparisonthose

al.

(vessel without baffles)+ 96mathematical model

reported asrespectively





Response surface contours

Response surface

Comparisonthose

/ JAFM

(vessel without baffles)96.48mathematical model

reported as responserespectively


transfer coefficient hithe diameter of the agitator increases from




Response surface

Comparisonthose calculated from

JAFM

(vessel without baffles)48?? +mathematical model

responserespectively. We note that two factors have


i (represented bythe diameter of the agitator increases from




Response surface

Comparison ofcalculated from

JAFM, Vol

(vessel without baffles)+ 75mathematical models

response. We note that two factors have


(represented bythe diameter of the agitator increases from




Response surface

of the experimental values andcalculated from

Vol.

(vessel without baffles)75.35described by

response surface contours. We note that two factors have


(represented bythe diameter of the agitator increases from




Response surface contours for

the experimental values andcalculated from

. 4,

(vessel without baffles)35??described bysurface contours

. We note that two factors haveweight than the third, which is the impeller speed

and the diameter of the agitator X(represented by

the diameter of the agitator increases fromthis is due to flow pattern and the circulation

loops generated by each impeller.

The applicability of the models for the prediction of theoverall heat transfer coefficient in the agitated vesselused in this study is demonstrated inexperimental data obtained for various operatingconditions are compared with those predicted from


contours for

the experimental values andcalculated from Eq.

, No.

(vessel without baffles) +described bysurface contours


and the diameter of the agitator X1(represented by Y) increases when


The applicability of the models for the prediction of thein the agitated vessel

used in this study is demonstrated in Figexperimental data obtained for various operatingconditions are compared with those predicted from


contours for

the experimental values andEq. (22

. 2,

described bysurface contours


1. The film heatY) increases when



Fig. 11experimental data obtained for various operatingconditions are compared with those predicted from

for X

contours for

the experimental values and22).

, Issue

described by Eq.surface contours


The film heat) increases when

the diameter of the agitator increases from 0.045this is due to flow pattern and the circulation


11, where allexperimental data obtained for various operatingconditions are compared with those predicted from

for X2=

X2=+

the experimental values and.

Issue 1

22surface contours in Fig



0.045this is due to flow pattern and the circulation


, where allexperimental data obtained for various operatingconditions are compared with those predicted from

=-1.

=+1.

the experimental values and

1, pp

(23

22 andFig.



0.045m tothis is due to flow pattern and the circulation


, where allexperimental data obtained for various operatingconditions are compared with those predicted from Eq.

.


pp. 43

49

23)

and9



m tothis is due to flow pattern and the circulation


, where allexperimental data obtained for various operating

Eq.


43-50

49

In this work, experimental design methodology hasbeen used in order to determine the effects ofimpeller diameter,baffles on the overall heat transfer coefficientmechanically stirrResponse surface contoursoptimal parametetransfer coefficiof the impeller so that

The rheologicalof CMC employed in this study allows concluding thatthe concentration ofNewtonianThe fluid behavior index, n, showed no noticeablevariation with temperature changeindex, k, decreases with increasing temperature.

The prthe Wilson method to prediccoefficientthe calculated values of the mean overall heat transfercoefficient

Brain, T.J.S. and

Briggs, D.E. and

Chapman, F.S. and F

Chergui

Debab

Fernandez

Fletcher

Goupy

Harnby, N., F. Edwards and A

Metzner, A

50, 2011



The prthe Wilson method to prediccoefficientthe calculated values of the mean overall heat transfercoefficient

Brain, T.J.S. and

Briggs, D.E. and

Chapman, F.S. and F

Chergui

Debab

Fernandez

Fletcher

Goupy

Harnby, N., F. Edwards and A

Metzner, A

2011

4.In this work, experimental design methodology hasbeen used in order to determine the effects ofimpeller diameter,baffles on the overall heat transfer coefficientmechanically stirrResponse surface contoursoptimal parametetransfer coefficiof the impeller so that


The present analysis demonstrated the feasibility to usethe Wilson method to prediccoefficientthe calculated values of the mean overall heat transfercoefficient

RBrain, T.J.S. and

tank bioreactors,

Briggs, D.E. andplot techniques for obtaining heat transfercorrelations for shell and tube heat exchangers,ChemSeries,

Chapman, F.S. and Ftransfer correlations for agitated liquids in processvessels,

CherguiVesselTechnology of Oran (USTO), Algeria.

Debab, AJacketed VInstitute

Fernandez(2007method and its modifications to determineconvection coefficients in heat exchange devices,Applied Thermal Engineering,

Fletcherbatch reactor design,33-

Goupy,Ed.

Harnby, N., F. Edwards and Ain the Process I

Metzner, ANewtonian fluids,

2011.

CONCLUSION



esent analysis demonstrated the feasibility to usethe Wilson method to prediccoefficient in an agitated vesselthe calculated values of the mean overall heat transfercoefficient compare favorably

REFERENCESBrain, T.J.S. and

tank bioreactors,

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, J. (1996d. Dunod, Paris.


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The rheologicalof CMC employed in this study allows concluding thatthe concentration of

behaviorThe fluid behavior index, n, showed no noticeablevariation with temperature changeindex, k, decreases with increasing temperature.

esent analysis demonstrated the feasibility to usethe Wilson method to predic

in an agitated vesselthe calculated values of the mean overall heat transfer

compare favorably

EFERENCES

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icalSeries, 92(

Chapman, F.S. and Ftransfer correlations for agitated liquids in processvessels, Chem

. (2005. Master t


. (2002Jacketed VInstitute (ENSIACET), Toulouse, France.

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. (1987batch reactor design,

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CONCLUSION

In this work, experimental design methodology hasbeen used in order to determine the effects ofimpeller diameter,baffles on the overall heat transfer coefficientmechanically stirrResponse surface contoursoptimal parameters to be determined. The filmtransfer coefficientof the impeller so that

The rheological characterization of the aqueous solutionof CMC employed in this study allows concluding thatthe concentration of

behaviorThe fluid behavior index, n, showed no noticeablevariation with temperature changeindex, k, decreases with increasing temperature.



compare favorably

EFERENCES

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Briggs, D.E. and Hplot techniques for obtaining heat transfercorrelations for shell and tube heat exchangers,

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2005).Master t


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CONCLUSION

In this work, experimental design methodology hasbeen used in order to determine the effects ofimpeller diameter, the speed of the impeller and thebaffles on the overall heat transfer coefficientmechanically stirred tank equippedResponse surface contours

rs to be determined. The filment is strongly dependent on the speed

of the impeller so that it is

characterization of the aqueous solutionof CMC employed in this study allows concluding thatthe concentration of

behavior in theThe fluid behavior index, n, showed no noticeablevariation with temperature changeindex, k, decreases with increasing temperature.



compare favorably

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and R. OttoNewtonian fluids,

CONCLUSION

In this work, experimental design methodology hasbeen used in order to determine the effects of

the speed of the impeller and thebaffles on the overall heat transfer coefficient

ed tank equippedResponse surface contours

rs to be determined. The filmis strongly dependent on the speed

it is

characterization of the aqueous solutionof CMC employed in this study allows concluding thatthe concentration of 2%

in theThe fluid behavior index, n, showed no noticeablevariation with temperature changeindex, k, decreases with increasing temperature.



compare favorably

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Youngplot techniques for obtaining heat transfercorrelations for shell and tube heat exchangers,

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ical Engineering

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Heat transfer coefficients for stirbatch reactor design,

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and R. OttoNewtonian fluids, AIChE



ed tank equippedResponse surface contours methodology allowed the


it is proportional to N

characterization of the aqueous solutionof CMC employed in this study allows concluding that

2% mass represent a nonin the range of

The fluid behavior index, n, showed no noticeablevariation with temperature changeindex, k, decreases with increasing temperature.



compare favorably.

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ical Engineering

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Harnby, N., F. Edwards and Andustries.

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proportional to N


mass represent a nonrange of

The fluid behavior index, n, showed no noticeablevariation with temperature changeindex, k, decreases with increasing temperature.


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.

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proportional to N


mass represent a nonrange of shear rates studied.

The fluid behavior index, n, showed no noticeablevariation with temperature change but the consistencyindex, k, decreases with increasing temperature.

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metodo de wilson

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