Online in situ viscosity determination in stirred tank reactors by measurement of the heat transfer capacity

Chemical Engineering Science 152 (2016) 116–126

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Online in situ viscosity determination in stirred tank reactors by measurement of the heat transfer capacity Martin Wunderlich a, Patrick P. Trampnau a, Emanuel F. Lopes b, Jochen Büchs a,n, Lars Regestein a a b

RWTH Aachen University, AVT – Aachener Verfahrenstechnik, Biochemical Engineering, Worringer Weg 1, 52074 Aachen, Germany Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

H I G H L I G H T S







Online and in situ viscosity determination in turbulent stirred tank reactor. Viscosity determination is based on the measurement of the heat transfer capacity. Precise measurement of heat transfer capacity by means of a calibration heater. Viscosity determination showed to be precise and accurate.

art ic l e i nf o

a b s t r a c t

Article history: Received 11 February 2016 Received in revised form 17 May 2016 Accepted 1 June 2016 Available online 3 June 2016

Viscosity plays an important role in a variety of biotechnological and chemical processes, such as in the production of biopolymers and in fermentations with filamentous microorganisms as well as in some dissolution, crystallization, and hydrogenation processes. Most of the established online methods for measuring the viscosity, however, struggle with the complexity of multiphase liquids like aerated liquids or suspensions and fermentation broths. This work presents a method to consider the viscosity of the whole reactor content regardless of its composition by means of calorimetric measurement of the heat transfer capacity (UA) and the use of a heat transfer model. Measurements were carried out with polyvinylpyrrolidone (PVP) model solutions (0–110 g/L) in a 50 L pilot scale stirred tank reactor with different viscosities (0.001–0.12 Pa  s), mechanical power input (0.04–27 kW/m³), and aeration rates (0– 2 vvm). The heat transfer capacity (UA) measurement by means of a calibration heater (1.25–12.5 kW/m³) was found to be very precise (o 1.5% standard deviation) and the online in situ determination of the viscosity fairly accurate (9.8% arithmetic mean error) in comparison to the offline measured viscosity. This suggests that the presented method is suited for online in situ viscosity determination in stirred tank reactors. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Online measurement Viscosity Stirred tank reactor Nusselt number

1. Introduction Viscosity is a key parameter in various fermentation processes like in the production of extracellular polysaccharides like xanthan, alginate, pullulan, or polyglutamic acid as well as in fermentations with filamentous biomass growth, as observed for many fungi and Streptomyces sp., e. g. during the production of antibiotics and organic acids (Henzler, 2007). In the chemical industry, unit operations as hydrogenation (e.g. of vegetable oils), crystallization, and dissolution (Nienow, 2000) may show notable n

Corresponding author. E-mail address: [email protected] (J. Büchs).

http://dx.doi.org/10.1016/j.ces.2016.06.003 0009-2509/& 2016 Elsevier Ltd. All rights reserved.

viscosity changes. Therefore, the viscosity is often directly linked to product formation and biomass growth, respectively, and has direct impact on the mixing performance and generally on mass transfer (Henzler, 2007). Consequently, the ability to measure the viscosity inside a stirred tank reactor (STR) is of special interest for online monitoring and process control. 1.1. Methods for measuring the viscosity Until some 25 years ago, viscosity measurement was commonly carried out with offline samples using, for instance, a rheometer for a detailed analysis of the rheological behavior of the sample. Though offline measurement provides detailed

M. Wunderlich et al. / Chemical Engineering Science 152 (2016) 116–126

Nomenclature Latin letters Heat exchange area (0.424 m2) Empirical exponent of Reynolds number, Eq. (8). Rheology factor, Eq. (21). Empirical exponent of Prandtl number, Eq. (8). Constant accounting for the geometry of reactor and stirrer, Eq. (8). cp,r Heat capacity at constant pressure of vessel liquid (30 °C)* (4180 kJ kg  1 K  1) cp,j(Tj) Heat capacity at constant pressure of jacket liquid** [kJ kg  1 K  1] DO Inner diameter of jacket inlet (0.025 m) DR Inner diameter of reactor vessel (0.29 m) Dstr Stirrer diameter (0.12 m) d Characteristic dimension [m] ds Characteristic dimension of jacket, ds ¼ (8/3)0.5  δ [m] e Empirical exponent of viscosity number, Eq. (8). Fgas Volume specific aeration rate (at 0 °;C, 101325 Pa) [m3gas mL 3 min  1] Fr Froude number, Fr ¼ n2  Dstr/g [-] g Gravitational acceleration of the earth (9.81 m s  2) Ga Galileo number, Ga ¼ DR2  g  (ηn/ρL)  2 [-] hL,0 Height of unaerated liquid in vessel (0.625 m) he Distance between reactor bottom and sparger (0.002 m) hj Height of jacket cylinder (0.468 m) K Consistency factor [Pa  sm] L Adjustment parameter for shear rate calculation, see (Henzler, 2007), Eqs. (20), (22). lj Mean path length through jacket (1.21 m) m Flow behavior index [-] ṁj Mass flow rate through jacket [kg s  1] Mln(T1,T2) Logarithmic mean value of T1 and T2 [K] n Stirring rate [s  1] Nu Nusselt number [-] Pstr Mechanical power input of stirrer [kW] PoG Gassed power number, PoG ¼ Pstr,G/(ρL  n3  DR 5) [-] Pr Prandtl number [-] Calibration heat flow [W] Q̇ cal qcal Volume specific calibration heat flow [kW m  3] R² Coefficient of determination [-] Rf Thermal resistance due to fouling [m2 K W  1] Re Reynolds number [-] ReG Gas Reynolds number [-] s Wall thickness of inner reactor wall (0.005 m) Tr Reactor temperature [K] Tj,in Jacket inlet temperature [K] Tj,out Jacket outlet temperature [K] A a B b Cr

information about the rheology of a sample, the sample itself may not be representative for certain processes: When separating a sample from the bulk, conformity changes may occur due to gelation, thixotropic behavior, phase separation of emulsions and suspensions, or mechanical damage due to the sampling itself as for example because of high shear in a sampling valve (Chhabra and Richardson, 2008). Furthermore, for control of the product quality and of the production process, for process automation and for cost reduction, online viscometry was established (Chhabra and Richardson, 2008).

ua ub uj uo U UA UAmeas Vr,L Vj,L

117

Water velocity in annular space of jacket [m s  1] Water velocity in jacket due to buoyancy [m s  1] Mean water velocity in jacket [m s  1] Water velocity in inlet tube of jacket [m s  1] Overall heat transfer coefficient [W m  2 K  1] Heat transfer capacity [W K  1] Measured heat transfer capacity [W K  1] Liquid volume of reactor (0.04 m3) Liquid volume of jacket (0.00377 m3)

Greek letters

α βj,L(Tj) γṙ , n

δ ηr,n ηj,n λr,L λj,L(Tj) λsteel ξ ρr,L ρj,L(Tj) τ *

**/***

Convective heat transfer coefficient [W K  1] Coefficient of volumetric thermal expansion** [W K  1] Shear rate in vessel bulk [s  1] Reactor annulus width (0.00795 m) Apparent dynamic viscosity of vessel bulk [Pa  s] Apparent dynamic viscosity of jacketbulk*** [Pa  s] Thermal conductivity of reactor liquid* (0.614 W m  1 K  1) Thermal conductivity of jacket liquid** (W m  1 K  1) Thermal conductivity of stainless steel 1.4435 (15 W m  1 K  1) Energy dissipation parameter of stirrer (0.285, interpolated; Kurpiers, 1985) Mass density of reactor liquid (30 °C)* (995.65 kg m  3) Mass density of jacket liquid** [kg m  3] Residence time of circuit water in jacket [s] Exact values are unknown and, therefore, approximated with the values of water at 30 °C given by (Wagner and Kretzschmar, 2010). Exact values for water were calculated with the logarithmic mean temperature of the jacket (Tj) with a polynomic regression function. The Polynomial were of 2nd (**) or 3rd (***) degree and fitted to the data between 10 °C and 40 °C given by Wagner and Kretzschmar (2010).

Indices i j L max n off on r w

Counter for sum Jacket side Liquid Maximum In bulk When calibration heater is off When calibration heater is on Reactor side In wall boundary layer

For online viscosity measurement there are different approaches: In stirred tank reactors, the mechanical power input of the stirrer can be measured for determining the viscosity (impeller viscometer) (Doran, 2013; Morawski et al., 1999). However, this method is limited to the laminar flow regime or to non-baffled STRs when using standard (fast rotating) impellers such as Rushton or pitched blade turbines, or marine propellers. Only under these conditions the power number changes with the Reynolds number and, therefore, with the viscosity. But, in order to provide homogeneous mixing and a sufficient oxygen supply for aerobic

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fermentations, most STRs are baffled and operated at turbulent or, at least, transitional flow regime. Under these conditions the power number is either completely independent of the viscosity of the bulk liquid or very insensitive. A different, very common approach is to measure the viscosity atline with established technologies: For instance, a rotational or a capillary viscometer can be connected to the reactor via bypass (Chhabra and Richardson, 2008; Pörtner et al., 1989). Also, the ultrasound Doppler velocimetry was found to be suitable for atline viscosity measurement (Köseli et al., 2006). However, this technique struggles with bubbly liquids. Other techniques use magnetic resonance imaging (MRI) and laser Doppler anemometer (LDA), which both are sophisticated, yet quite expensive measuring methods (Chhabra and Richardson, 2008). All these atline techniques have in common that a small fraction of the liquid is separated from the bulk liquid of the reactor and analyzed at conditions that in terms of aeration and shear rate – similar to the aforementioned offline samples – may differ from the ones within the STR, influencing the shear-dependent apparent viscosity of non-Newtonian liquids. This leads to results not being representative for the entire reactor content. In contrast to the inline and atline techniques, a real in situ measurement is provided by vibrational rheometers, which are based on the measurement of the energy absorption of vibrating rods, spheres, or blades and are widely used in process industries (Barnes, 2000; Chhabra and Richardson, 2008). Many of the commercial “inline viscometers” from diverse manufacturers are based on the principle of such a vibrating element. Probes like these usually measure at distinctive spots in the reactor, which are not necessarily representative for the whole vessel bulk: Due to an inhomogeneous shear rate distribution in the STR (Cutter, 1966), especially for shear sensitive, i. e. non-Newtonian, liquids, a local viscosity measurement may not provide conclusive data for the whole reactor content. Furthermore, the vibrating probes are less precise when used in bubbly fluids. Finally, the probes with their usually high vibrating frequencies (ultrasound) apply additional force on the liquid, which again may result in a change of the measured viscosity of a non-Newtonian liquid (Barnes, 2000; Chhabra and Richardson, 2008). The method presented in this work considers the overall apparent viscosity of the entire content of an STR, based on the measurement of the heat transfer capacity (UA). It is commonly known that viscosity has an impact on the heat transfer in STRs (Chhabra and Richardson, 2008; Gaddis, 2010). This is considered when calculating the required heat transfer areas during design and scale up of STRs (Radež et al., 1991). However, no in-depth research has been carried out, yet, to investigate if an accurate online measurement of the heat transfer capacity (UA) of a reactor allows conclusions about the bulk viscosity inside an STR. Therefore, this dependence of the bulk viscosity on UA is shown in this work. 1.2. Theory of viscosity determination by means of measuring the heat transfer capacity

temperature of the jacket circuit water (Tj) has to drop. Hence, the temperature difference between reactor interior and jacket (ΔTr|j) increases. With the jacket temperature being measured at the inlet (Tj,in) and outlet (Tj,out) of the jacket (see chapter 2), the mean temperature difference between reactor interior and jacket can be calculated:

ΔTr j = Mln ( Tr − Tj,out, Tr − Tj,in ) =

Q̇ = U⋅A⋅ΔT

(1)

When the state of the calibration heater is changed (e. g. from off to on), an additional heat flow (Q̇ cal ) is applied to the vessel liquid. To remove this additional heat flow while maintaining a constant reactor temperature (Tr), with U and A being constant, the

⎛ Tr −Tj,in ⎞ ln ⎜ T −T ⎟ ⎝ r j,out ⎠

(2)

and in case of Tj,in = Tj,out or ( Tr − Tj,in )⋅( Tr − Tj,out ) ≤ 0:

ΔTr | j = Mln ( Tr − Tj,out, Tr − Tj,in ) =

( Tr − Tj,in ) + ( Tr − Tj,out ) 2

(3)

The term ( Tr − Tj,in )⋅( Tr − Tj,out ) might become negative (or zero on very rare occasions) in two situations and would lead to a nondefined mathematical operation (logarithm of a negative number or zero or division by zero). In order to prevent data processing errors due to non-defined operations, in those cases the arithmetic mean is used instead of the logarithmic mean. The two situations in question are: First, during the transition time when intentionally switching from heating to cooling operations and vice versa, for example when changing the power state of the calibration heater while no other significant heat sinks or sources are present, the term might become negative. Second, when all measured temperatures are almost of the same value and, hence, neither significant cooling nor heating is required, the term occasionally might become negative due to frequently alternating heating and cooling operations (micro fluctuations of the liquid temperatures), and/or signal noise. The change of the calibration heat power (ΔQ̇ cal ) divided by the above mentioned change of the reactor/jacket temperature difference results in the heat transfer capacity:

UA =

ΔQ̇ cal ⎡ Δon off ⎣ Mln ( Tr − Tj,out, Tr − Tj,in ) ⎤⎦

(4)

Summarizing, the measured signal to determine UA is the temperature change in the jacket induced by changing the calibration heat power and is not influenced by heating or cooling due to chemical or biological reactions, as implicitly shown by Regestein et al. (2013a, 2013b). For processes in steady state or for slowly changing processes, half cycles of measurement, i.e. considering only one state change of the calibration heater, are sufficient and provide more frequently measuring points. For processes with fast changing heat generation (or consumption) on the other hand, as for example during the advanced feed phase of fed-batch fermentations, full cycles of measurement have to be considered. I.e., for a full measurement cycle the state of the calibration heater changes (state “1A” (e.g. “off”)-state “2” (“on”)) and later returns to the previous state (state “2” (“on”)state “1B” (“off”)). This allows accounting for process based changes of the jacket temperature by extending Eq. (4) to:

UA = Basis for the online viscosity determination is the periodic measurement of the heat transfer capacity (UA) using a calibration heater according to Regestein et al. (2013a, 2013b). The equation for heat transfer is:

( Tr − Tj,in ) − ( Tr − Tj,out )

ΔQ̇ cal

( ΔTr|j,1A + ΔTr j,1B )/2 − ΔTr j,2

(5)

With ΔTr | j = Mln ( Tr − Tj,out, Tr − Tj,in ). For exponentially accelerated processes (e.g. auto-catalytic reactions such as fermentations), the logarithmic mean (cp. Eq. (2)) instead of the arithmetic mean ((ΔTr|j,1 þ ΔTr|j,2)/2) provides more accurate results. Alternatively, in all cases mathematical methods such as trend lines can be used as shown in (Regestein et al., 2013a). In an STR, UA depends on the heat transfer resistance of the vessel wall (s/λsteel) and the convective heat transfer coefficients of the reactor side (αr) and of the jacket side (αj) as well as on the

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Fig. 1. Schematic representation of the heat transfer resistances and the temperature profile in a stirred tank reactor. While the bulk liquid in both the reactor interior and the vessel jacket are considered to show a homogeneous temperature distribution, the heat transfer between both is impaired by the reactor wall (s/λ) and its boundary layers (αr and αj), leading to a temperature drop in each of the three compartments. The single heat transfer resistances are summarized in the overall heat transfer coefficient (U) (cp. Eq. (6)).

total heat exchange area (A) (Fig. 1, Eq. (6)). Potential fouling of the inner reactor wall and in the jacket is considered with the fouling resistances (Rf,r and Rf,j) and will be discussed in Section 3.1.

UA =

1 1 αr

+ Rf,r +

s λsteel

+ Rf,j +

1 αj

Nu⋅λL d

(6)

(7)

With d being the characteristic dimension: d¼ DR in case of the reactor interior and d ¼dS in case of the jacket. When assuming a homogenous distribution of heat and concentrations (discussed in Section 3.1), by measuring UA, the apparent viscosity inside the vessel can be calculated. Model equations for these calculations are given within Section 1.3. Because the heat exchange area (A) is implicit part of the heat transfer capacity (UA) of a system, it is required that A is either known (e. g. by means of level sensors) or kept constant throughout a fermentation/ chemical process. The latter is possible if the filling volume level of the reactor outreaches the water level of the jacket circuit. In this case an increase of the reactor filling volume, e. g. due to changes in aeration and gas hold-up or due to feeding in a fed-batch process, does not affect the heat transfer area. 1.3. Heat transfer from the inside of a stirred tank reactor to the inner reactor wall The most commonly cited equation to calculate the Nusselt number (Nur) – and with it the heat transfer to and from reactor walls – is given in the form of Eq. (8) for fast rotating impellers like Ruston/ pitched blade turbines and marine propellers (Chhabra, 2003; Gaddis, 2010; Mohan et al., 1992):

⎛ η ⎞e r,n ⎟⎟ Nur = Cr⋅Re ar ⋅Pr br ⋅⎜⎜ ⎝ ηr,w ⎠

Rer =

⋅A

where α (Eq. (7)) is a function of the Nusselt number (Nu), which again is a function of the bulk viscosity.

α=

the apparent viscosity of the bulk liquid (ηr,n):

(8)

For the reactor side heat transfer, Cr is a constant accounting for the geometry of the reactor and of the stirrer and usually amounts to values between 0.3 and 1.5 (Chhabra, 2003). The exponents of the Reynolds number (Rer), Prandtl number (Prr), and of the viscosity number (ηr,n/ηr,w) for heating and cooling conditions were empirically determined and typically amount to a E2/3, bE1/3, eE0.14 (Gaddis, 2010; Mohan et al., 1992). Both Reynolds number (Eq. (9), given for the reactor side heat transfer (Gaddis, 2010)) and Prandtl number (Eq. (10)) depend on

Prr =

2 Dstr ⋅ρr,L ⋅n

ηr,n

(9)

ηr,n⋅cp,r λr,L

(10)

When calculating Nur with Eqs. (9) and (10) introduced into Eq. (8), the influence of the apparent viscosity (ηr,n) is partially, yet not completely compensated due to the different exponents of Reynolds and Prandtl number. With the given standard values for the exponents of Rer and Prr, Nur (and hence UA) is influenced by the reactor side apparent viscosity according to the proportionality (see also Table 1):

⎛ η ⎞0.14 −1/3 ⎜ r,n ⎟ Nur ~ ηr,n ⋅⎜ ⎟ ⎝ ηr,w ⎠

(11)

In case of no intense heating or cooling (cp. Section 3.1), as for many fermentation processes, this proportionality can be simplified to: −1/3 Nur ~ ηr,n

(12)

A limitation of Eq. (8) is that it neither considers the influence of aeration on the heat transfer nor the different flow conditions of multiple stirrers on the agitator shaft compared to the flow conditions of a single stage stirrer. Aeration results in gas hold-up and hence in a diminished mechanical power input by the stirrer due to the formation of gas cavities behind the stirrer blades, which reduces the flow resistance of the bulk liquid. A detailed insight in cavity formation and its influence on the mechanical power input is given by Sieblist et al. (2011). Air bubbles might be effective as insulators, additionally impairing the heat transfer. On the other hand, rising gas bubbles induce liquid movement, which might improve the mixing process (represented by the gas Reynolds number) (Kurpiers, 1985). Kurpiers, (1985) extended the basic model (Eq. (8)) considering the influence of aeration on the Nusselt number:

⎧ ⎛ D ⎞ ⎛ D3 ⎞ Nur = 0.1⋅⎨ ξ⋅⎜ R ⎟⋅⎜ R ⎟⋅PoG⋅Re 3r ⎩ ⎝ Dstr ⎠ ⎝ Vr, L ⎠ ⎪

⎪

0.25 ⎡ ⎛ D ⎞ ⎛ Dstr ⎞ 0.5 ⎤ ⎫ ⎪ + Ga⋅ReG⋅⎢ 1 − 2⋅⎜ R ⎟⋅⎜ ⋅Pr 0.4 ⎟⋅PoG ⋅Fr ⎥ ⎬ r ⋅ ⎪ ⎝ Dstr ⎠ ⎝ hL,0 − he ⎠ ⎦⎭ ⎣

⎛ η ⎞0.23 ⎜⎜ r,n ⎟⎟ ⎝ ηr,w ⎠

(13)

This equation, originally derived for a one stage stirrer and

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explained in detail in Kurpiers (1985), Kurpiers et al. (1985), was fitted by Kurpiers to the experimental data of a two stage stirrer, leading to Eq. (14). Kurpiers stated that fitting was necessary because the flow conditions in an STR change in a way not yet completely understood when using multiple stirrer stages and, therefore, no theoretic derivation was possible (Kurpiers, 1985; Kurpiers et al., 1985).

⎫0.238 ⎧ ⎛ D ⎞ ⎛ D3 ⎞ Nur = 0.1⋅⎨ 11.58⋅⎜ R ⎟⋅⎜ R ⎟⋅PoG⋅Re 3r + Ga⋅ReG ⎬ ⋅Pr 0.362 ⋅ r ⎝ Dstr ⎠ ⎝ Vr, L ⎠ ⎭ ⎩ ⎪

⎪

⎪

⎪

⎛ η ⎞0.23 ⎜⎜ r,n ⎟⎟ ⎝ ηr,w ⎠

(14)

For both aerated and unaerated systems, all dimensionless numbers in Eq. (14) are calculated with the values of the physical properties of the unaerated liquid (Gaddis, 2010; Kurpiers, 1985). Eq. (14) was used in this study as basis for the calculations of Nur and, thus, of the convective heat transfer coefficient of the reactor interior (αr).

1.4. Heat transfer from the inner reactor wall to the jacket liquid Lehrer (Lehrer, 1970) stated that “the heat transfer in jacketed vessels is generally limited by the vessel side resistance to a greater extent than by the jacket side resistance, particularly when […] the vessel contains a liquid with relatively high viscosity”, making this method robust to variations on the jacket side. Additionally, the heat transfer of the jacket side is widely independent of the viscosity of the reactor side. Yet, knowledge about the heat transfer of the jacket side is mandatory for the calculation of the vessel side apparent viscosity (ηn,r) from UA with the help of Eq. (6). The jacket side amounts to about 25% of the total heat transfer resistance for typical operation conditions and up to 50% in unfavorable operating conditions, according to own measurements and calculations. For calculations of the heat transfer on the jacket side of the vessel, i.e. from the inner vessel wall to the circuit water in the jacket, three different approaches were considered: those of Lehrer (1970), (1981), Baker (Baker and Walter, 1979), and Stein and Schmidt (Stein and Schmidt, 1993). Though the model of Stein and Schmidt is the most recent and most sophisticated one, it is only well applicable to plain cylindrical jackets. The equation for the jacket side Nusselt number of Lehrer is given in two different forms, a more complex and a simplified one, with the latter one being almost identical to the model used by Baker (Baker and Walter, 1979), which was used in this work:

⎛ 2⋅δ ⎞ 0.794 1/3 Nuj = 0.019⋅⎜ 1 + 3.5⋅ ⋅Pr j ⎟⋅Re j ⎝ DR ⎠

(15)

with

Rej =

uj⋅dS⋅ρj,L (Tj ) ηj,n (Tj )

(16)

The main difference between the models of Lehrer and Baker is the way of estimating the mean water flow velocity (uj) for the calculation of the Reynolds number (Rej). Baker introduced equations for calculating the water flow for jackets with hemisphericallike jacket bottoms whereas the derivation of Lehrer considers cylindrical jackets. However, when correctly applied, both the Lehrer and Baker model provide very similar results (Lehrer, 1981).

2. Materials and methods 2.1. Stirred tank reactor Experiments were conducted in a 50 L stirred tank reactor (Fig. 2, LP 351, Bioengineering AG, Wald, Switzerland), which has been described in detail previously (Knabben et al., 2010; Knoll et al., 2007; Maier et al., 2001; Regestein et al., 2013a). The tempering system was modified as described in the following paragraph. The reactor (inner diameter: 0.29 m) was equipped with three (installation heights: 0.105 m, 0.315 m, and 0.525 m) six blade Rushton turbines (diameter: 0.12 m, height: 0.04 m), four baffles (each 30 mm width) and a ring sparger (diameter: 93 mm) for aeration. A hollow shaft of the stirrer allows the direct measurement of the torque and, hence, the mechanical power input of the stirrer into the vessel bulk. This measuring technique is not affected by loses due to bearings and mechanical sealings. The vessel temperature was controlled by a manually optimized cascaded proportional (P) controller for the outer loop (temperature of jacket inlet) and a proportional-integral (PI) controller for the inner loop (temperature of vessel interior, see Fig. 2; IMAGO 500, JUMO GmbH & Co. KG, Fulda, Germany). The actuators of the P/PI temperature controller are continuous pneumatic control valves (RC 200 ½” with preinstalled SRI990 analog positioner, Badger Meter Europa GmbH, Neuffen, Germany), which were newly installed, replacing the original standard on/off valves of the reactor. The mass flow rate of the circuit water was adjusted by a Magna 25–60 circulator pump (Grundfos GmbH, Erkrath, Germany) and measured by a coriolis mass flow meter (Proline Promass 80 M ½” 6500 kg/h, Endress & Hauser Flowtec AG, Reinach, Switzerland). All Pt100 temperature sensors were calibrated simultaneously in stirred water at 20 °C and 30 °C. The jacket of the here investigated vessel has a widely irregular shape. Therefore, it was divided into two parts according to Supplementary Fig. 1 in order to allow a simplified calculation of the water flow and the mean Reynolds number: The first part consists of the torispherical bottom and the lower cylindrical jacket (A) with an estimated tangential flow. The second part consists of the upper cylindrical vessel with cut-outs for probes and for the sight glass and is considered to generate mainly axial flow (B). The mean water velocity in the annular space of jacket (ua) is calculated by division of the mean path length (lj) through the residence time of the circuit water in the jacket:

ua = lj⋅

ṁ j ρj,L (Tj,L )⋅Vj,L

(17)

The characteristic velocity in the annular space of a jacket (uj) can be calculated as described in detail by Gaddis (2010):

uj =

ua ⋅uO + ub

(18)

2.2. Measurement of the heat transfer capacity The heat transfer capacity (UA) of the STR filled with polyvinylpyrrolidone (PVP) solutions was measured using a calibration heater (Supplementary Fig. 2) in an in-house built steel holder according to Regestein et al. (2013a). The calibration heat flow ( Q̇ cal ) was manually set between 500 W and 50 W (≙ 12.5 and 1.25 kW/m3 at 40 L reactor filling volume, cp. Section 2.4) in dependence of the viscosity in order to prevent overheating at higher viscosities. It was adjusted with an open loop control with pulse width modulation through an output regulator with a solid state switching device (3RF2920-0HA13 and 3RF2120-1AA02, Siemens AG, Munich, Germany) operated at nominal 230 V and 50 Hz. The line voltage and line frequency as well as the current were

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121

Fig. 2. Scheme of reactor setup including the jacket circuit with a cascaded P/PI temperature controller (TC) and continuous control valves, supply streams, and sensors. The calibration heater (dark gray, see Supplementary Fig. 1) was plugged into the reactor via a standard Ingold connector. The mechanical power input of the stirrer (Pstr) was determined via torque measurement without the losses of bearings and mechanical sealings.

monitored with a HM8115-2 power meter and a HZ815 power adapter (both HAMEG Instruments GmbH, Mainhausen, Germany). Calibration and sleep phases were altered every 30–60 min after all temperatures were stable at least for 15 min.

power input and the rheological properties of the liquid: m−1

ηr,n

⎛ P /V ⎞ m + 1 = ⎜⎜ L2⋅ str 2 L ⎟⎟ ⎝ K1 − m ⎠

(22)

2.3. Data processing and modeling For calculating and modeling the reactor side and jacket side Nusselt numbers, the equations of Kurpiers et al. (1985) and Baker and Walter (1979) were used. The following measured parameters were incorporated into the model equations: ηr,n, Tj,in Tj,out, ṁj, Pstr, n, Fgas (cp. Fig. 2). All invariable parameters and constants used for modeling are listed in the nomenclature list. The regression of proportionality factors and exponents was accomplished by the least square method with normalized values: min

⎛ UA ⎞2 meas, i − f ηr,n, i , Tj,in, i, Tj,out, i, ṁ j, i , Pstr, i, ni , Fgas, i ⎟ ⎟ UAmeas, i ⎝ ⎠

∑ ⎜⎜ i

(

)

1

(20)

with L being a function of m (Henzler, 2007):

L = B⋅exp ( 2.7⋅m)

Nearly Newtonian polyvinylpyrrolidone solutions (PVP – Luviskol K90 powder, BASF AG, Ludwigshafen, Germany) of different concentrations were prepared in the reactor with deionized water. The PVP powder was dosed stepwise into 35 L of stirred water and dissolved for at least 2 h at 60 °C and 500 rpm. Finally, the reactor content was tempered to 30 °C and filled up to 40 L with deionized water so all three stirrers were equally covered with liquid. The filling volume level of the reactor outreached the circuit water level (cp. Section 1.2).

(19)

For parity plots of online and offline measured viscosities, the average shear rate inside the STR was applied as the criterion for comparison. It was calculated according to the power concept of Henzler Henzler (2007): 2 ⎛ Pstr/Vr,L ⎞ m + 1 γr,n ̇ = L m + 1⋅⎜ ⎟ ⎝ K ⎠

2.4. Model liquid

(21)

with B ¼1.8 for near-Newtonian solutions. Eq. (20) is introduced into the power law equation (Ostwald-de Waele relationship) to directly calculate the apparent viscosity from the mechanical

2.5. Offline viscosity measurement The viscosity of the model liquid was determined offline as duplicates: 0.425 mL sample volume were analyzed at 30 °C with a Modular Compact Rheometer (MCR 301, Anton Paar GmbH, Graz, Austria), using a cone/ plate system (CP50, 0.467°) with a gap width of 0.054 mm. The shear rates were varied according to a logarithmic shear rate profile ranging from 12.4/s to 24000/s with 10 measuring points per order of magnitude. Consistency factors (K) and flow indices (m) where calculated according to the powerlaw equation (Ostwald-de Waele relationship) as a regression with normalized least squares in the shear rate range of interest over one order of magnitude.

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3. Results and discussion The influence of the viscosity of near-Newtonian polyvinylpyrrolidone (PVP) solutions inside a 50 L stirred tank reactor (STR) on the heat transfer capacity (UA) was investigated at different reactor operating conditions. Heat transfer models according to Kurpiers (1985) (vessel side heat transfer) as well as according to Lehrer (1970, 1981), Baker and Walter (1979), and Stein and Schmidt (1993) (jacket side heat transfer) were tested for the description of the experimental results. Thereby, the coefficients of the model equations for both the vessel side and jacket side Nusselt number (Nur and Nuj) were determined simultaneously with the complete data set, using the normalized least square method according to Eq. (19). 3.1. Influence of stirring rate and aeration rate on the vessel side heat transfer Under typical operation conditions the Reynolds number of the reactor interior usually was between 103 and 3  106) and only dropped down to 220 for very high PVP concentrations and very low stirring rates (Z100 g/L PVP at r 250 rpm and Z80 g/L PVP at 125 rpm). Considering that laminar flow in an STR occurs at Reo10, ReZ103 in the vessel means advanced transitional to turbulent flow, with fully turbulent conditions beginning at Re4104. Because of the measuring time of 30–60 min for one state of the calibration heater, sufficiently homogenous bulk liquid can be assumed. The impact of the apparent viscosity in the vessel bulk (ηr,n) in conjunction with the stirring rate and the aeration rate on the heat transfer capacity (UA) of an STR was investigated and is shown in Fig. 3 (filled symbols). In both cases there is a significant dependence of UA on ηr,n: With increasing viscosity, UA decreases whereas the stirring rate (n) shows a positive correlation (Fig. 3A): Maximum UA values of 4400 W/K were reached for water-like viscosities and high stirring rates of 1000 rpm, whereas for solutions with high PVP concentrations (Z100 g/L) and low stirring rates (125 rpm) UA dropped below 150 W/K. A change of the stirring rate within the upper range results in a lower change of UA compared to changes within the lower stirring rate range. This behavior is in accordance with the theoretical trend of decreasing heat transfer capacity with increasing viscosity and/or decreasing stirring rate derived from the original equation of Kurpiers (Eq. (13)). Radež et al. (1991) found this correlation to be U  η  0.34 . However, the presented study cannot confirm a formal r correlation between the heat transfer capacity or heat transfer coefficient (U), respectively, and the viscosity. Rather, it was found that the correlation is better (but not perfectly) described with logarithmic curves. This is due to the thermal resistance of the reactor wall (λsteel/s) and especially due to the variable thermal resistance of the jacket side boundary layer (1/αj), both of which also affect the heat transfer coefficient. Instead, the given proportionality was expected and found for the Nusselt number (Nur), i.e Nur  ηr  1/3 (theoretically derived, see Eq. (12)) and Nur  η  0.358 (experimentally determined, cp. Table 1). r At a constant stirring rate of 500 rpm, the aeration rate (Fgas) has a minor impact on UA and shows a negative correlation (Fig. 3B): The higher the aeration rate, the lower the heat transfer capacity. Similar to the influence of the stirring rate, the change of the aeration rate in the upper range results in a lower change of UA compared to changes of the aeration rate in the lower range. Although at elevated viscosities (4 0.04 Pa  s) the influence of the aeration rate diminishes, there is a visible difference in UA between the aerated and the unaerated system.

Fig. 3. Influence of the stirring rate (n) (Fig. 3A) and influence of the aeration rate (Fgas) (Fig. 3B) as a function of the apparent viscosity (ηr,n) on the heat transfer capacity (UA). The model of UA (Eqs. (6), (7), (23), and (24); lines with open circles) was fitted to the experimentally determined values (filled symbols). All UA measurements were carried out in a 50 L stirred tank reactor and repeated 5 – 20 times, with standard deviations of 0.4 – 2.6%. Reactor operating conditions were: Vr,L ¼40 L of 0–110 g/L PVP solution, Tr ¼ 30 °C, Fgas ¼ 0 vvm (Fig. 3A), n¼ 500 rpm (Fig. 3B), ṁj ¼1800 kg/h (τ¼ 7.5 s), qcal ¼ 1.25–12.5 kW/m³ (Fig. 3A), qcal ¼ 5 kW/m³ (Fig. 3B). Offline viscosity measurements were carried out in duplicates, with ηr,n determined according to Eq. (22).

3.1.1. Model simplification Although developed for a reactor with two stirrer stages, the model of Kurpiers (Eq. (14)) could be fitted well (Fig. 3, lines with open circles) to the experimental data (filled symbols) with Nur being a function of (ηr,n)  0.358 (see Eq. (23), cp. Table 1). In an attempt to simplify the model of Kurpiers, the impact of its individual terms on the calculated Nusselt number (Nur) was determined as follows: It was found that under the investigated operating conditions, the Galileo number (Ga) and gas Reynolds number (ReG) had a minor impact on the reactor side Nusselt number (Nur). The influence of aeration on the heat transfer was

Table 1 Comparison of model parameters for the calculation of the reactor side Nusselt number Nur when using 1, 2, and 3 stirrer stages (Eqs. (13), (14), and (23)) and their b−a impact on the proportionality with the reactor side apparent viscosity Nur ~ηr,n

(cp. Eq. (11)). 2 stirrer stages (Kurpiers et al., 1985)

3 stirrer stages (this work)

1 stirrer stage (Gaddis, 2010; Mohan et al., 1992) unaerated, empiric

1 stirrer stage (Kurpiers et al., 1985)

Exponent a of Rer

0.666

3  0.250 ¼ 0.750

3  0.238 ¼ 0.714

Exponent b of Prr ba

0.333

0.400

0.362

1.958  0.238 ¼ 0.466 0.108

 0.333

 0.350

 0.352

 0.358

aerated and unaerated, experimental

M. Wunderlich et al. / Chemical Engineering Science 152 (2016) 116–126

found to be very well represented by considering the mechanical power input of the stirrer during aeration (PoG), which decreases with increasing aeration rate, thus resulting in a lower heat transfer coefficient (Fig. 3B). For systems in which the mechanical power input cannot be measured, it can be calculated according to the equations given by Henzler (1982) or Möckel et al. (1990), which were found to provide reasonable results for the diminished mechanical power input of the stirrer caused by aeration (data not shown). The gas Reynolds number (ReG) on the other hand is of special importance for reactors which either do not have a stirrer (bubble columns) or in which the stirrers are flooded by excess aeration. In those systems, heat transfer is mainly driven by the rising gas bubbles, represented by a high gas Reynolds number (ReG). A flooded stirrer, however, heavily impairs the mixing process and gas dispersion, thus being an undesired operational condition to be prevented. Because of the missing relevance and due to the fact that in the investigated reactor system the product of Galileo number and gas Reynolds number (ReG) always was one to two orders of magnitude lower than the other term, it was neglected in all shown calculations and figures. The model of Kurpiers (Eq. (14)) could be simplified further by omitting the viscosity number (ηn/ηw), which accounts for intense heating and cooling processes. However, in this study neither intensive heating nor cooling was required: The temperature differences between the reactor side bulk liquid and the jacket side bulk liquid always were o 5 K (typically between 1–3 K), meaning that the temperature difference between the bulk liquid (Tn) and the boundary layer (Tw) is even considerably lower. While a temperature drop from 30 °C to 25 °C would lead to an increase in viscosity by E20% for 80 g/L PVP solution and E11% for water, the correction factor by means of the viscosity number results in (1/1.2)0.23 ¼0.96. Thus, the influence on the Nusselt number is only 4% even in the unlikely case of 5 K of temperature difference between bulk and boundary layer. Hence, in the considered processes the viscosity number was neglected for simplification. 3.1.2. Fouling Further simplifications consider the fouling terms in Eq. (6): The reactor side surface of the vessel wall (stainless steel) is regularly cleaned before and after experiments and fouling layers never have been observed. Therefore, fouling on the reactor side vessel wall is considered to be negligible in this study. However, fouling may apply for other processes or reactor sizes: In case of the production of (bio-)polymers, molecule polymerization also occurs close to the reactor wall, which might lead to reaction fouling if the polymerization product is not continuously removed. As for biofouling, reaction times in (fed-)batch processes typically are too short for microorganisms to grow in considerable amount on the reactor wall. But for fermentations with certain types of filamentous organisms, such as fungi, or in continuous processes biofouling may occur. In all cases, there is an increased likelihood of fouling when the reactor is operated without thorough mixing/ at low Reynolds numbers (Re). Considering vessel sizes, with increasing reactor scale the Reynolds number, and with it the Nusselt number (Nu), typically increase faster than the characteristic dimension (dR) of the convective heat transfer coefficient (α) (cp. Eqs. (7) to (9)). As a consequence the thermal resistance of the reactor-side boundary layer, which is equal to (1/αr), is low in large scale reactors whereas the thermal resistance of the reactor wall (s/λsteel) increases due to the greater wall thickness (s). Hence, the influence of viscosity on the heat transfer capacity (UA) is reduced and would be further diminished if fouling occurs (cp. Eq. (6)). However, due to the increased Reynolds number for large reactors potential fouling is less likely. The jacket side surface of the reactor wall, on the other hand, is not easy to inspect visually. Yet, the potential impact of different

123

types of fouling is considered to be negligible, too, because of the following reasons: Chemical reaction fouling, solidification fouling, and macro fouling (such as the settlement of objects or growth of higher organisms) does not apply in a closed loop, water based tempering system as used in this study. Also corrosion fouling can widely be excluded due to the material of the reactor wall, which is stainless steel. Biofouling is counteracted by regular sterilization (121 °C for 20 min, minimum every four weeks) of both the reactor interior and the tempering system. Precipitation fouling, e.g. the crystallization of the common components such as CaCO3 or CaSO4 can be widely excluded because the tempering water in use is soft (water hardnessE1.4 mmol L  1). 3.1.3. Simplified and fitted model Including the above mentioned simplifications and assumptions, the model fitting to the experimental data by adjusting the proportionality factor and the exponents of Reynolds and Prandtl number lead to the following equation:

⎫0.238 ⎧ ⎛ D ⎞ ⎛ D3 ⎞ ⎬ ⋅Pr 0.108 Nur = 4.216⋅⎨ ⎜ R ⎟⋅⎜ R ⎟⋅PoG⋅Re1.958 r r ⎭ ⎩ ⎝ Dstr ⎠ ⎝ Vr,L ⎠ ⎪

⎪

⎪

⎪

(23)

The comparison of the equations given by Kurpiers for calculating the Nusselt numbers for one stirrer stage (Eq. (13)) and two stirrer stages (Eq. (14)) shows that both the exponents of Reynolds number and Prandtl number are lower in the equation for two stirrer stages than for one stirrer stage. This trend was found to be continued for three stirrer stages in the present study (Eq. (23)) as shown in Table 1. At the moment it is not possible to deduce the reasons for this trend, since no advances have been made since Kurpiers stated that the flow conditions in an STR with multiple stirrer stages are not yet well understood (Kurpiers, 1985; Kurpiers et al., 1985). However, although the fitted exponents of Rer and Prr differ from the standard empiric values for unaerated STRs (cp. text to Eq. (8)) and from the values found by Kurpiers (1985), Kurpiers et al. (1985), it is remarkable that the difference of the exponents of Reynolds number (a) and Prandtl number (b), and, hence, the sensitivity of the heat transfer to the viscosity, is not significantly altered. Thus, the reactor side Nusselt number remains a function of about ηr,n  1/3 (Eq. (11)). 3.2. Influence of the circuit water mass flow rate on the jacket side heat transfer The experimental data (filled symbols) of the influence of the mass flow rate of the circuit water (ṁj) on the heat transfer capacity (UA) are shown in Fig. 4 at two distinct apparent viscosities of the vessel bulk (ηr,n) and at a constant stirring rate of 500 rpm without aeration. High mass flow rates of the circuit water result in high heat transfer capacities whereas a high viscosity of the vessel bulk, as already described in Section 3.1, results in a lower heat transfer capacity. The model equations of Lehrer (1970) (simplified), and Baker and Walter (1979) could be fitted similarly well to the experimental data (not shown) by adjusting the proportionality factor and the exponent of the Reynolds number. Since the Prandtl number of the jacket (Prj) was not significantly varied, its influence on the jacket side Nusselt number (Nuj) could not be investigated and, hence, its exponent was held constant. However, in the model of Lehrer it is mathematically possible to diminish the influence of Prj by increasing the proportionality factors of both numerator and denominator of the equation. This frequently happened when solving the minimization problem, so either more coefficients need to be held constant or the equation will be reduced to the simpler form by the minimization algorithm. Therefore, and due to slightly better fitting results, the model of Baker was chosen for

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Fig. 4. Influence of the circuit water mass flow rate in the vessel jacket (ṁj) on the heat transfer capacity (UA) at two different apparent dynamic viscosities inside the vessel. The model of UA (Eqs. (6), (7), (23), and (24); lines with open circles) was fitted to the experimentally determined values (filled symbols). All UA measurements were carried out in a 50 L stirred tank reactor and repeated 7–17 times, with standard deviations of 0.5–1.3%. Reactor operating conditions were: Vr,L ¼ 40 L of 0 g/L and 90 g/L PVP solution, Tr ¼30 °C, n¼ 500 rpm, Fgas ¼ 0 vvm, qcal ¼12.5 kW/m³ and 5 kW/m³. Offline viscosity measurements were carried out in duplicates, with ηr,n determined according to Eq. (22).

the calculation of the jacket side Nusselt number and fitted (Fig. 4, lines with open circles) to the experimental data:

⎛ 2⋅δ ⎞ 0.749 1/3 Nuj = 0.0288⋅⎜ 1 + 3.5⋅ ⎟⋅Re j ⋅Pr j ⎝ DR ⎠

(24)

3.3. Precision and sensitivity of the measurement of the heat transfer capacity Repetitive measurements of UA were carried out in order to determine the precision of the measurement. It was found that in a broad variety of reactor operating conditions and vessel viscosities the UA measurement showed very good repeatability (Fig. 5). Standard deviations were always o1.5% with absolute deviations as low as 3% in 13 repetitions. In order to determine the lowest volumetric calibration heat power which still provides a reliable UA measurement, the heat power was reduced stepwise from 12.5 kW/m³ down to 0.05 kW/ m³. Fig. 6 shows that in the range of 1.25–12.5 kW/m³ the UA measurement provides precise results with o 3% standard deviation. While a calibration heat power of 2.5–5 kW/m³ may be favored for a precise measurement with E1.5% standard deviation, further increase of the calibration heat power does not provide useful improvements of the measurement precision but bears the

Fig. 5. Precision of the measurement of the heat transfer capacity (UA) at different reactor operating conditions: All UA measurements show a standard deviation of E1% with maximum deviations of o3%. Measurements were carried out in a 50 L stirred tank reactor. Reactor operating conditions were (if not stated differently in the diagram): Vr,L ¼40 L of 0–110 g/L PVP solution, Tr ¼ 30 °C, n¼ 500 rpm, Fgas ¼ 0 vvm, ṁj ¼ 1500 kg/h (τ ¼ 9.0 s), qcal ¼2.5–12.5 kW/m³.

Fig. 6. Sensitivity of the measurement of the heat transfer capacity (UA): Dependence of the standard deviation of the UA measurement on the volume specific calibration heat flow (qcal). A heat flow of qcal Z 1 kW/m³ provides acceptable UA measurements with o 3% standard deviation. At qcal Z 2.5 kW/m³ standard deviations of UA further decrease to o 1.5%. All UA measurements were carried out in a 50 L stirred tank reactor and repeated 5–20 times. Reactor operating conditions were: Vr,L ¼ 40 L of 0–110 g/L PVP solution, Tr ¼ 30 °C, n ¼500 rpm (250 rpm at 0.001 Pa s), Fgas ¼ 0 vvm, ṁj ¼1800 kg/h (τ ¼ 7.5 s).

risk of overheating the calibration heater or burning the liquid close to the heater in case of insufficient heat removal, e. g. at high viscosities and/or low stirring rates. Furthermore, a low calibration heat power may be desirable because of the reduced energy consumption, especially for larger reactors. However, if the calibration heat power is too low, the ratio of the measurement signal to signal noise decreases. Here, the measurement signal is the mean temperature drop (or rise) of the circuit water when changing the state of the calibration heater. Signal noise appears as temperature fluctuations in both the reactor interior and the circuit water. These may appear from the noise of the temperature probes or may be caused by an insufficient homogenization of the bulk liquid inside the reactor and/or jacket, or by fluctuations due to the temperature control system: Precisely, a very fast operation of heating and cooling valves as well as a quick alternation of heating and cooling may result in frequent, noise-like temperature changes in the jacket circuit. Therefore, a well-adjusted temperature controller with continuous control valves is mandatory for a fast and precise UA measurement (Voisard et al., 2002), especially if the signal/noise ratio is low. 3.4. Comparison of online and offline measured viscosities Fig. 7 shows the accuracy of the online viscosity determination in comparison to the offline measured viscosities at a variety of reactor operating conditions. Offline viscosity was measured with a rotational shear rheometer and plotted against the online viscosity, which was calculated from the measured UA values with Eqs. (6), (22), (23), and (24). The viscosities were compared at the shear rate in the reactor, which was calculated according to Eq. (20). The parity plot shows a satisfying agreement of the online to the offline measurements, with a high coefficient of determination of R2 ¼0.97. The vast majority (75 of 84 measuring points) showed an error of o20%, with the arithmetic mean error of all measuring points being 9.8%. Deviations appear not to be systematic but random since no correlations with stirring rate, aeration rate, circuit water flow, heat transfer capacity, calibration heat power, standard deviation within a series of measurement, or with viscosity could be found. Although the offline viscosity measurement by a rheometer is very sophisticated and precise nowadays, the determination of the average shear rate in a stirred tank reactor is not. But for every liquid which does not show perfect Newtonian properties, the shear rate is an essential parameter due to its influence on the viscosity. Hence, the offline determined viscosity cannot be

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margin of UA of 7 3% (cp. Section 3.3), the highest errors (relative to the determined viscosity) have to be expected at high UA values, i.e at water-like viscosities at high stirring rates. With decreasing UA the propagated relative error decreases, too. The margin of UA 73% turns into an estimated error of approx. 725% for typical operation conditions (250 W/K oUAo300 W/K) and reaches approx. 715% for high viscosities and/or low stirring rates (UAo 250 W/K, cp. Fig. 3). Above 350 W/K the viscosity determination becomes less reliable with estimated errors of at least 740%. 3.5. Discussion on transferability and scalability

Fig. 7. Accuracy of the online viscosity determination. Online viscosity was calculated from measured UA values via Eqs. (6), (7), (23), and (24). Offline viscosity was measured with a rotational cone/plate shear rheometer in duplicates, with ηr,n determined according to Eq. (22). All UA measurements were carried out in a 50 L stirred tank reactor and repeated 5–20 times, with standard deviations of 0.4–2.6%. Reactor operating conditions were: Vr,L ¼ 40 L of 0 – 110 g/L PVP solution, Tr ¼ 30 °C, n¼ 125–1000 rpm, Fgas ¼ 0–2 vvm, ṁj ¼600–1800 kg/h (τ ¼ 22.5–7.5 s), qcal ¼ 1.25– 12.5 kW/m³.

Fig. 8. Bland–Altman plot for the comparison of the online (stirred tank reactor) and offline (rheometer) viscosity measurement derived from Fig. 7. The plot shows in average a slight underestimation (  3.7% mean) of the online method compared to the offline method. Applying a confidence level of 95%, the confidence interval is found between þ22.5% and  29.8% of difference between both methods.

considered as gold standard but as subjected to errors, too. Therefore, for statistical analysis a plot according to Bland and Altman (2003) was chosen (Fig. 8). It depicts the difference of the offline and online measurements in dependence of the average viscosity (arithmetic mean of offline and online measurement) as best guess. The plot reinforces the statement of a satisfactory agreement of both methods and that no correlation between viscosity and deviation could be found. The average difference amounts to 3.7%, which indicates that the online method slightly underestimates the viscosity in comparison to the offline measurement. The standard error (SE) of the differences between on- and offline measurements amounts to 13.3%. Assuming a normal distribution of the errors, the confidence interval for a 95% confidence level is set up by the mean value 71.96  SE. Hence, 95% of all measurements are expected to be found in the interval of 29.8% to þ 22.5% deviation between on- and offline measurement. An analysis of the error propagation shows that the error of the heat transfer capacity (UA) has by far (at least 5 to 10 times of other influences) the strongest impact on the online viscosity determination (data not shown). When assuming a linear error

While the fitted model Eqs. (23) and (24) for the online viscosity determination are most probably reactor specific and not necessarily transferable to other reactors, the use of various dimensionless numbers within the model equations suggests that the model itself can be adapted to geometrically similar systems. Even for not yet calibrated reactor systems, the UA measurement will immediately allow a qualitative monitoring of viscosity changes. It may also be possible to correlate usual offline viscosity measurements with simultaneously measured, reactor specific UA values to establish the online viscosity determination in an already existing STR. For large scale STRs, the method to measure UA for the viscosity determination, as introduced in this study, will be limited by high energy consumption by both the calibration heater itself and the necessary counter cooling. However, it is conceivable that instead of a calibration heater, a calibration cooler can be used in the very same way for UA measurement, thereby reducing the additional energy costs and possibly allowing the application of this method to larger reactors. A calibration cooler could be designed as cooling probe/finger for smaller reactor systems or as an additional cooling coil in large scale reactors. An alternative way to determine UA, applicable especially for large scale reactors, is the installation of a mass flow meter for the circuit water system in order to calculate UA from heat balance calorimetry (also referred to as heat capacity method) as suggested by Regestein (Regestein et al., 2013a) and conducted for an external plate heat exchanger by Türker (Türker, 2003, 2004). Because almost all STRs have heat exchangers for temperature control, such as a jacket, coils, and/or (external) plates, and because probably every liquid based heat exchange system is sensitive to viscosity, UA measurement can be retrofitted with moderate effort to a variety of heat exchange systems, e. g. by installing a calibration heater or a mass flow meter and additional temperature sensors into the tempering system (Regestein et al., 2013a).

4. Conclusion In this work a new approach for the online determination of the apparent viscosity in stirred tank reactors is suggested and evaluated in regards of precision, accuracy, sensitivity and repeatability. It was shown that the measurement of the heat transfer capacity in conjunction with (semi-)empirical heat transfer models for stirred tank reactors is well suited for the online determination of the viscosity of different PVP solutions inside the reactor. As heat transfer models, simplified equations of Kurpiers (1985) and Baker and Walter (1979) (Eqs. (14) and (15)) were used and fitted to the results from the investigated reactor system (Eqs. (23) and (24)), showing almost exactly the expected correlation of Nur  ηr  1/3 after fitting (Table 1). Combining and rearranging Eqs. (6), (7), (9), (10), (16), (23), and (24) leads to the explicit formula of the apparent viscosity in dependence of the measured parameters:

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⎤0.238 ⎡⎛ ⎛ 3⎞ ⎛ c ⎞0.108 λr,L Pstr, G ⎥ D ⎞ D 2 ⋅ρ ⋅n 0.466 ⋅⎜ p,r ⎟ ηn0.358 = 4.216⋅⎢ ⎜ R ⎟⋅⎜⎜ R ⎟⎟⋅ ⋅ Dstr ⋅ ⋅ r,L ⎢ ⎝ Dstr ⎠ Vr,L ρ ⋅n3⋅D 5 ⎥ DR ⎝ λr,L ⎠ ⎠ ⎝ r,L ⎣ R⎦ ⎧ ⎤−1⎫ ⎡ 1/3 0.846 ⎛ ⎪ 1 λ j,H2 O ⎥ ⎪ ⎛ ηj,n⋅cp,j ⎞ ⎢ s 2⋅δ ⎞ ⎛⎜ uj⋅d S⋅ρ j, L ⎞⎟ ⎟ ⋅ ⎨ ⋅A − − ⎢ 0.0288⋅⎜ 1 + 3.5⋅ ⋅⎜⎜ ⎟⋅⎜ ⎟ ⎥ ⎬ ⎟ ⎝ λ DR ⎠ ⎝ dS ⎥ ⎪ ηj,n ⎠ ⎪ UA ⎝ λ j,H2 O ⎠ ⎦ ⎭ ⎣⎢ ⎩

(

)

(25)

With a sufficiently high calibration heat power ( 42.5 kW/m3), the measurement of the most important parameter, the heat transfer capacity (UA), was shown to be very precise (E 1.5% standard deviation) and in general accurate (9.8% as arithmetic mean error of measurement). The error propagation shows that a margin of a maximum error of UA of 73% turns into an error of η of approx. 725% for typical operation conditions (250 W/ Ko UAo300 W/K). Whereas the heat transfer of single stirrer tank reactors can be considered basic knowledge, less research had been carried out for reactors with two, and even less for reactors with three stirrer stages, which are commonly used for fermentation processes. The presented work closes the lack of in depth investigation of the heat transfer capacity in dependence of viscosity and reactor operating conditions. For the first time heat transfer was investigated for a stirred tank reactor with three stirrer stages. The results show that standard parameters of the heat transfer equations might not provide sufficiently accurate estimations for all stirred tank reactors. As so, more research with different vessel sizes is necessary to propagate scalability of these refitted dimensionless equations. This study also paves the way for the online viscosity determination of liquids with shear thinning properties, as encountered in almost all biopolymer solutions and with filamentous microorganisms. Currently, the influence of non-Newtonian model solutions on the heat transfer capacity and, hence, on the presented online viscosity determination is under investigation in our laboratory.

Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Appendix A. Supplementary material Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.ces.2016.06.003.

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