Heat transfer investigations in a liquid that is mixed by means of a multi-ribbon mixer


 The objective of this paper is to present the investigations of the heat transfer process carried out by means of the multi-ribbon mixer. It is shown that the heat transfer process for the synergic effect of the mixing process and the flowing liquid through the mixer has significantly higher values of the heat transfer coefficients than the mixer with motionless impellers. The empirical correlations between the heat transfer coefficient and the operational parameters obtained in this work can provide guidance for the design and operation of an apparatus equipped with the multi-ribbon impeller. These empirical correlations can be used to predict the heat transfer coefficient for the multi-ribbon mixer.


INTRODUCTION
Owing to their importance in the fi eld of many industrial applications such as manufacturing of chemicals, drugs, and food, mixing operation of liquid have been the subject of many investigations. The mixing process of viscosity or non-Newtonian liquids is often carried out by using the mixer equipped with the Rushton turbine 1 or in the hybrid mixing system composed of the helical ribbon and the Rushton turbine 2 . The relevant reports have been focused on the studies of the application of helical ribbon agitator 3-10 as well on the numerical analysis 11- 12 .
A growing interest in the investigation of the effect of ribbon mixer on the heat transfer process has been noticed in the 13-18 . Delaplace et al. 19 analyzed the data of the heat transfer process in the atypical helical ribbon impeller supported by two vertical arms and concluded that the heat fl ux sensors can monitor the thermal boundary layer thickness. It should be noticed that the rate of heat transfer from the vessel wall to the mixing liquid is dependent on the thickness of the thermal boundary layer 20 . These authors reported a new and convenient synthetic procedure to obtain the heat transfer effi ciency for ribbon impellers operating in a laminar regime. Nzihou et al. 21 studied the effects of a rheo-reactor (the vessel with a helical-ribbon impeller) on the heat transfer coeffi cient. One study by Delaplace et al. 22 examined the heat transfer for a jacketed vessel fi tted with an atypical helical ribbon. In this work, numerical simulations of heat transfer phenomena to highly viscous Newtonian at unsteady states in this mixer were also attempted using CFD fi nite volume software. Rai et al. 23 found that the fl at-bottomed vessel equipped with a helical ribbon agitator can be applied for the heat transfer enhancement for some Newtonian and non-Newtonian fl uids. As noted by these authors the heat transfer coeffi cient did not increase substantially when aeration was coupled with agitation as compared to the only aeration. Saraceno et al. 24 discuss the heat transfer effi ciency in the mixing system equipped with an agitator type helical ribbon. Gammoudi et al. 25 analyzed the hydrodynamic and thermal behaviors of yield stress fl uids within the vessel equipped with simple helical ribbon stirrers by using the numerical simulation approach.
It may be noted that there is a lack of systematic studies with the heat transfer studies in the multi-ribbon mixer. Previous research has indicated that the heat transfer coeffi cient at the wall of the agitated vessel depended on many factors, such as the type and geometry of the vessel and agitator 26 . To obtain an accurate correlation of heat transfer coeffi cient and to predict the heat transfer effi ciency in the mixed liquid, it is necessary to clarify the relation between heat transfer and fl uid fl ow in the mixer. Therefore, in the present study, some analyses on the heat transfer performance of the multi-ribbon mixer are presented. This study aims to investigate the heat transfer process in the multi-ribbon mixer for a Newtonian fl uid. It should be noticed that the two cases were considered: i) multi-ribbon mixer with motionless impellers acting as a heat exchanger (tap water as the working liquid fl owed through the mixer); ii) multi-ribbon mixer with working impellers and fl uid fl ow through the mixer (the effect of mixing process on heat transfer when the water fl owed through the mixer). Due to practical constraints, this paper cannot provide comprehensive experimental work with the highly viscous Newtonian and non-Newtonian fl uid. Therefore, the obtained results may be treated as preliminary research which can be the basis for further work with the usage of the multi-ribbon mixer to intensify the heat transfer process.

Theoretical
The mathematical description of heat transfer operations may be defi ned by using the governing equation of the temperature fi eld (the Fourier-Kirchhoff equation) 27 . This equation for p = const and Φ V = 0 (where p is pressure and Φ V is viscous dissipation function) is given as follows where: c p -specifi c heat, J · kg -1 · K -1 ; T -temperature, K; w -velocity vector, m · s -1 ; ρ -liquid density, kg · m -3 ; λ -thermal conductivity, J · m -1 · K -1 · s -1 ; τ -time, s. Equation (1) may be rewritten in the symbolic form which is useful for the dimensional analysis. The introduction of non-dimensional quantities denoted by an asterisk (*) into this relationship yields (2) The non-dimensional form of Eq. (2) may be scaled against the convective term (λ Tl -2 ). The result of this operation is given as follows (3) Considering the following non-dimensional groups (4) (5) we obtain the following relation (6) where: a -thermal diffusivity, m 2 · s; l -characteristic length, m; Fo -dimensionless Fourier number; Pe -dimensionless Péclet number.
The heat transfer process between the wall and the fl uid can be described by means of the following relationship (7) where: T f -temperature of fl uid, K; T w -temperature of wall, K; α -heat transfer coeffi cient, J · m -2 · K -1 · s -1 .
The Eq. (7) may be also expressed in the following non-dimensional form (8) The consequence of this operation is that the heat transfer process may be characterized by using the dimensionless Nusselt number (9) From the above equations, it follows that the heat transfer process for the steady conditions may be defi ned by using the equation It should be noticed that the dimensionless Péclet number may be expressed as follows (11) where: Pr -dimensionless Prandtl number; Re -dimensionless Reynolds number; η -dynamic viscosity, kg · m -1 · s -1 ; ν -kinematic viscosity, m 2 · s -1 .
In the case of the application of the experimental methods, the heat transfer coeffi cient is correlated by means of the following equation 20 (12) where: a, C -parameters; Vi -viscosity simplex defi ned as the ratio dynamic viscosity of the fl uid and the dynamic viscosity of the fl uid at the wall temperature.

Experimental set-up
Experimental investigations were carried out using the set-up with the multi-ribbon mixer shown schematically in Fig. 1. The mixer is constructed in the form of two partly penetrating horizontal stainless steel cylinders. The main geometrical parameters of the mixer are as follows: diameter of each cylinder: 0.186 m, length of the cylinder: 0.34 m, total volume: 0.025 m 3 . The tested mixer is equipped with the jacket and adequate piping for the delivery of the heating medium (water steam) into the jacket of the mixer. The heating jacket was supplied with stabilized steam at a pressure of 0.11 MPa and temperature approximately 102 o C. A manometer and thermometer were used to control these operational parameters.
The mixing process was performed using two co--operating helical multi-ribbon impellers. In the case of this experimental work, we tested three geometrical confi gurations of impellers (see Fig. 2).
The geometrical parameters of the tested impellers are collected in Table 1.

Experimental procedure
In the present investigations, the averaged heat transfer coeffi cient from the heating jacket to the liquid was measured by the stabilized heat fl ow method. All experiments were carried out for the steady-state conditions, in which heat supplied to the tank from steam condensing in the jacket was received by the circulating liquid. A detailed description of the measurement of heat transfer coeffi cients on the tank wall was presented by Niedzielska and Kuncewicz 20 . It should be noticed that the heat passing into the vessel through the wall and the heat absorbed by the mixed liquid are essentially equal.
The heat transfer rate exchanged between the heating medium (condensing steam) and the mixed liquid (water; the inlet temperature of this liquid was about 5°C) fl owing through the mixer is defi ned by the following equation (13) where: w G  -mass fl ow rate, kg · s -1 ; c p -mixing liquid specifi c heat, J · kg -1 · deg -1 ; T in -temperature of the liquid at the mixer inlet, o C; T out -temperature of the liquid at the mixer outlet, o C; Q  -heat transfer rate, J · s -1 .
The heat transfer coeffi cient from heating wall to mixed liquid is described by the following relation (14) where: F HT -heat transfer area, m 2 ; T w -temperature of the vessel wall, o C; Figure 2. The views of the tested two co-operating helical multi-ribbon impellers Table 1. The list of the geometrical parameters of the tested impellers T bulk -temperature of the mixed liquid, o C; α -heat transfer coeffi cient, J · m -2 · K -1 · s -1 . In the case of this experimental work, the temperatures were measured with thermal sensors (Pt100A) and a measuring instrument that controlled the temperature of the set-up and supervised the real-time acquisition of all experimental data coming from the sensors. According to PN-EN 60751:1997+A2, the temperature measurement errors with this sensor at 5°C and 100°C are equal to 0.17°C and 0.35°C, respectively. The location of temperature sensors is shown in Fig. 3.
It should be noticed that the temperature of the vessel wall T w was calculated as the average of temperatures from T 1 to T 5 . η w -dynamic viscosity of the fl uid at wall temperature, kg · m -1 · s -1 ; λ -thermal conductivity of liquid, W · m -1 · deg -1 ; ν -kinematic viscosity of the fluid at mixing temperature, m 2 · s -1 . The liquid velocity fl owing through the mixer is defi ned as follows (16) where: F fl ow -cross-sectional area of the mixer, m 2 ; ρ -density of the liquid, kg · m -3 . The diameter of the mixer, D mixer , was equal to 0.26 m. This is equivalent diameter calculated as follows (17) where: F m -cross-sectional area of the mixer which is composed -of two cylinders, m 2 ; F c -cross-sectional area of the mixer cylinder, m 2 ; D -diameter of cylinder, m.
The mass fl ow ratio of water was varied between 0.0278 and 0.2222 kg · s -1 . The liquid velocity w fl ow was changed between 0.00052 and 0.0042 m · s -1 . The dimensionless Prandtl number and the viscosity simplex were changed in the range between 4.54-13.41 and 1.02-5.24, respectively.
The generalization of the effect of mixing on the heat transfer when fl uid fl ows through the vessel is given in the following form (18) where: N -rotational speed of impeller, s -1 ; d mix -diameter of impeller, m.
The rotational speed of impeller was varied from 20 rpm to 104 rpm.

RESULTS AND DISCUSSION
In the case of this experimental work, the enhancement of the heat transfer process was realized by applying the tested multi-ribbon mixer with impellers (see Fig. 2). As it was mentioned above, the infl uence of motionless impellers on this process was discussed. In this case, the tested mixer may be treated as the heat exchanger.
In the present report, we consider that the heat transfer in the multi-ribbon mixer acted as the heat exchanger may be described by the proposed relationship between the dimensionless Nusselt number and the dimensionless Reynolds number for the fl uid fl ow through the mixer (Eq. (15)). To establish the effect of the hydrodynamic conditions on the heat transfer process, the experimental

Calculation of dimensionless numbers
The investigations were carried out for two cases. Firstly, the infl uence of the motionless impellers (see Fig. 2) on the heat transfer process was analyzed. In this case, the tested vessel equipped with the multi-ribbon impeller may be treated as the heat exchanger. Secondly, the effect of various types of impellers on heat transfer performance was discussed.
In the fi rst case, the heat transfer process may be conveniently correlated using the following relation (15) where: D mixer -diameter of the mixer, m; w fl ow -liquid velocity fl owing through the mixer, m · s -1 ; η -dynamic viscosity of the fl uid at mixing temperature, kg · m -1 · s -1 ; data obtained in this work are graphically illustrated in the log-log system in Fig. 4.
less insert in the mixer. The enhancement of the heat transfer process can be attributed to the breakdown of the thermal boundary layer especially near the wall of the mixer. It should be noticed that the enhancement of heat transfer is provided by the mixer in which the clearance between the ribbons is larger. This leads to an increase in the turbulence in the fl owing liquid. Thus, the improvement of the heat transfer ratio may be obtained.
To establish the effect of the tested mixing system and the applied hydrodynamic conditions connected with the fl owing fl uid through the mixer, the experimental data obtained in this work are described by the following correlation     The constants and exponents of Eq. (20) for the tested impellers are collected in Table 3. The applied mixing system has a signifi cant effect on the heat transfer coeffi cient expressed in the form of the term Nu Pr -0.33 Vi -0.14 . An increase in the dimensionless Reynolds number for the mixing system, Re mix , increased the values of the mentioned term. This trend can be better observed from the plot shown in Fig. 5. Table 4 presents some of the main heat transfer effects associated with the application of the tested mixing systems in the multi-ribbon mixer.
The current study found that the application of the tested multi-ribbon impeller and the mixing system has a signifi cant effect on the heat transfer ratio. Figure 5 shows that the values of this coeffi cient for the tested mixer increase with Re mix . In the case of impeller A_1 the values of term Nu Pr -0.33 Vi -0.14 decrease with Re fl ow . The experimental results reveal that, for the mixing system with the impeller A_2 and A_3, the values of heat transfer coeffi cients increase with Re fl ow . This indicates that the geometrical confi guration of the impeller has    The experimental results shown in Fig. 4 suggest that the heat transfer process may be analytically described by the following function (based on Eq. (15) The constants and exponents were computed by means of the Matlab software and the principle of least squares and the obtained values are collected in Table 2.
It can be seen from Fig. 4 that the experimental points for the various impellers may be described by the same type of relation (Eq. (19)) using the various values of the parameters p 1 and p 2 . It is clear that the heat transfer process is dependent on the geometrical confi guration of the multi-ribbon impeller that acted as the motion-A_3). In the case of impeller confi guration A_1, this parameter decreased along with Re fl ow and increase along with Re mix . Experiments related that the term Nu Pr -0.33 Vi -0.14 in the system with the mixed liquid using the impeller is higher than the values obtained in the mixing system with the motionless impellers. a signifi cant effect on the tested process. From the analysis of Fig. 5, it follows that the synergic effect of the mixing process and the fl owing liquid is also effective. It is observed that the heat transfer coeffi cient is highest for the confi guration A_3 (see Fig. 2c).

CONCLUSIONS
This study sought to determine the heat transfer process for a mixing system equipped with the multi--ribbon impeller. The experimental results showed that the synergic effect of the mixing process and the fl owing liquid provided higher values of the heat transfer coeffi cients. This parameter increased along with Re mix and with Re fl ow (for impeller confi gurations of A_2 and