Acta Polytechnica

This paper presents results obtained between 2010 and 2014 in the field of fan aerodynamics at the Department of Composite Technology at the VZLÚ aerospace research and experimental institute in Prague – Letňany. The need for rapid and accurate methods for the preliminary design of blade machinery led to the creation of a mathematical model based on the basic laws of turbomachine aerodynamics. The mathematical model, the derivation of which is briefly described below, has been encoded in a computer programme, which enables the theoretical characteristics of a fan of the designed geometry to be determined rapidly. The validity of the mathematical model is assessed continuously by measuring model fans in the measuring unit, which was developed and manufactured specifically for this purpose. The paper also presents a comparison between measured characteristics and characteristics determined by the mathematical model as the basis for a discussion on possible causes of measured deviations and calculation deviations.


Introduction
One of the most common operations in chemical, food and pharmaceutic industries is a mixing of liquid materials.This operation frequently requires cooling or heating the agitated liquid.The required size of the transfer surface depends mainly on the intensity of the heat transfer between the agitated liquid and the heat transfer surface.The heat transfer rate Q depends on the size of heat transfer surface S, heat transfer coefficient α, and mean temperature difference ∆T .Q = αS∆T. ( Therefore, the knowledge of the heat transfer coefficient α at the heat transfer surface (for example, at the vessel bottom, vessel walls, baffles, inserts, etc.) is very important for an optimum design of the mixing equipment.
Experimental methods based on measuring various thermal quantities (heat fluxes, heat flow rates, temperatures) are the ones most frequently used for determining values of heat transfer coefficients.Besides that, methods based on an analogy between various transport phenomena (heat and mass transport, for example) can also be used.Nowadays, numerical methods implemented in many CFD packages solving momentum and heat transport equations are also frequently used to determine heat transfer coefficients in laminar as well as turbulent flow regimes.In principle, the methods based on measuring thermal quantities use (1).With the liquid and wall temperatures along with the heat flow rate transferred between the wall and liquid determined, the heat transfer coefficient can be expressed from this equation (assuming we know the size of the heat transfer surface).The heat flow rate can be measured by various techniques.One possibility is using thin film heat flux sensors, which measure small temperature differences arising from the heat flow rate through a thin film element made of material with known properties.Transient methods determine the heat flow rate, for example, on the basis of a monitored temperature profile of the liquid accumulating the supplied heat, see [1,2] for a description of this approach.
In general, any experimental method where we can derive some relationship between the heat transfer coefficient and directly measured physical quantities can be used to determine heat transfer coefficients.Wandelt and Roetzel [8] found an analytical solution of the case where a plate (representing the heat transfer surface) is heated from one side by an oscillating (sine) heat flux.This solution depends on properties of the plate and heat transfer coefficients on both sides.By measuring the time dependency of the plate surface temperature and on the basis of their mathematical model, it is then possible to indirectly evaluate the heat transfer coefficient on the other side of the plate.
Wandelt and Roetzel's temperature oscillation infrared thermography method (TOIRT) has been extensively studied by Freund [9] in his dissertation thesis.Freund studied the theoretical background of this method and focused on some well-known cases like a pipe flow or a jet impinging a plate perpendicularly.Freund and Kabelac [10]  Freund et al. [11] also used the TOIRT method in a study of heat transfer in spray cooling systems.
Impinging jets are widely used in industrial applications for local cooling and heating.Many papers dealing with heat transfer in various geometries can be found in the literature and impinging jets represent frequent reference cases for testing the results of heat transfer measurements or numerical simulations.In studies of impinging jets, most of the authors focus on cases with the nozzle-plate spacing related to the nozzle diameter greater then 1, only few authors deal with relative distances h/d less than 1 or around 1. Lytle and Webb [3] measured local values of a heat transfer coefficient for 0.1 < h/d < 6 and the Reynolds number within the range of 3600 < Re < 27600.They presented the radial dependency of the local Nusselt number, correlations for the Nusselt number in the stagnation point and the mean Nusselt numbers averaged over the heat transfer surface.
Katti and Prabhu [4] presented results of the radial dependency of the local Nusselt number based on experimental data for 0.5 < h/d < 8 and 12000 < Re < 28000.They presented expressions describing the local Nusselt number in the stagnant, transition, and wall jet regions, and they also proposed a correlation for the stagnant Nusselt number.Persoons et al. [12] studied local heat transfer in a pulsating impinging jet for the nozzle-to-surface spacing 1 ≤ h/d ≤ 6, Reynolds numbers 6000 ≤ Re ≤ 14000, and pulsation frequency 0 ≤ f ≤ 55 Hz).Many works also focused on numerical modelling of impinging jets, for example, Draksler et al. [13] or Jensen and Walther [14].Zuckerman and Lior [15] published an extensive review paper on impinging jets summarizing empirical correlations and numerical simulation techniques.
Another very frequent field of study is the heat transfer in agitated vessels.Many papers can be found about heat transfer in the whole vessel (Wichterle [16], Mohan et al. [17], Akse et al. [5], Karcz [7]) but only a few deal with local heat transfer coefficients at the bottom of the agitated vessel or at the vessel wall (Karcz and Cudak [18], Engeskaug et al. [19], Mohan et al. [17] and Akse et al. [5]).
Mohan et al. [17] presented a review of the heat transfer in agitated vessels.This work focused on heat transfer in single-phase and two-phase (gas-liquid) agitated systems equipped with either jacket or heat transfer coils.They also presented axial dependencies of local heat transfer coefficients in an air-water agitated system.The heat transfer in a jacketed vessel agitated by the standard Rushton turbine was described by Akse et al. [5].Edwards and Wilkinson [6] also presented a review of the heat transfer in agitated vessels with various types of agitators and heat transfer surfaces.Typical correlation parameters (see equation 4) for the average Nusselt number in a vessel agitated by a six-blade disc impeller (Rushton turbine) are summarized in Table 1.Typical values of parameters for the average Nusselt number on the wall of the vessel agitated by the six-blade impeller with a pitch angle of 45°published by Karcz [7] are also mentioned in this table.
The local convective heat transfer coefficients are usually evaluated by experiments or numerical models.Two most frequently used experimental methods are Thermochromic Liquid Method (TLC), see Vejrazka and Marty [20] for an example, and electro-diffusion method (EDD) (Karcz and Cudak [18]).Usually, the measured values of heat transfer coefficients are expressed in terms of correlations for the dimensionless Nusselt number where The Prandtl number is usually defined as Transport and thermophysical properties, in the correlations for the Nusselt number and corresponding dimensionless numbers (see equations above), are related to fluid (liquid).

Theoretical background of TOIRT
The TOIRT method by Wandelt and Roetzel [8] is based on measuring the wall temperature with an IR camera (see Figure 1).This temperature depends on two main factors: heat transfer coefficients on both sides of the wall with thickness δ and modulated heat flux q.The heat flux is modulated by a sine function q(t) = q sin(ωt) and it is applied to one side of the wall.The temperature field monitored at the same side by the IR camera then shows a sinusoidal dependency.According to the TOIRT method, the phase shift between the sine-modulated heat flux and wall temperature response can be used to indirectly evaluate the heat transfer coefficient α 0 on the other side of the wall.The time and spatial dependency of the temperature T in a homogeneous wall (with a thermal diffusivity a w ) is described by Fourier's equation by neglecting the lateral heat conduction in the wall (the first two partial derivatives on the right-hand side of the previous equation), it is possible to find an analytical solution in the case of the periodically The temperature oscillation method for heat transfer measurement by Wandelt and Roetzel [8].
oscillation heat flux.The boundary conditions of the third kind can be written as where α 0 is the heat transfer coefficient we are looking for, α δ is the heat transfer coefficient on the other side of the wall, ω is the angular frequency of the oscillating heat flux and q is its amplitude.The authors [8] used the Laplace transformation to solve this system of equations and found solution in the following form where the phase shift ϕ(z) at the surface with periodically oscillating heat flux can be described as Dimensionless parameters r, ψ 0 , ξ and c 0,1,2,3 in the previous equation are defined as The big advantage of the equation ( 12) when determining the heat transfer coefficient α 0 is no dependency on the amplitude of the heat flux.However, the amplitude size influences the accuracy of the measured surface temperature, which then has an indirect impact on the accuracy of the key quantity, that is the phase shift between the oscillating heat flux and the measured temperature response.Therefore, the correct and accurate evaluation of the heat transfer coefficient on the base of the experimental data depends mainly on the accuracy of this phase shift.Figure 2 at the top illustrates a real temperature response measured by an IR camera at one point of the heat transfer surface.It is obvious that it does not oscillates around a constant average value as it would imply from boundary conditions ( 9) and ( 10) but it slowly increases with time.This increase is caused by the accumulation of heat in the measured volume because the oscillating heat flux does not represent a heat source and sink with zero average value.The "slow component" of the temperature increase must be removed and the measured data can be transformed into the form depicted in the middle of Figure 2, that is oscillations around zero average value.This transformation can be accomplished by many algorithms.The simplest one is subtracting a function describing the "slow component" (approximated by a piece-wise linear function, for example) so that the resulting average value is zero.Freund [9] mentioned that removing the exponential part of the time dependency should have no significant impact on the results.
On the basis of the transformed measured data, see Figure 2, it is then possible to find the phase shift ϕ.We used two methods in our case (their results are compared to detect possible discrepancies).The slower method uses a classic non-linear regression to find parameters of a model function describing the transformed experimental data.Equation (11) represents a suitable model function for approximating ∆T .Then, parameters A and ϕ can be determined with the least-squares method based on the minimizing function where k is the index representing the sum over experimental data ∆T k at times t k (see Figure 2).In practical implementation of this algorithm, we used the function nlinfit in system MATLAB ® .The second method for evaluating the phase shift is based on the discrete Fourier transformation (DFT) [21] which seems to be very suitable with respect to the periodic behaviour of the signal.Coefficients x n of the Fourier series describing original signal f can be determined as It is sufficient to analyse only the first harmonic component, in our case, this means that we can evaluate only the first coefficient in (16) and express the value of the phase shift as The DFT requires N 2 of operations for the transformation of the whole spectrum where N is the number of discrete points, therefore, it is substantially faster than the non-linear regression mentioned above.

Application of the method
The temperature oscillation method generates the sine-modulated heat flux and monitors the temperature fields in precise time intervals.To satisfy these requirements, we used the two-channel function generator BK Precision 4052.The first channel generates the sine wave in a voltage range of 3-10 V.This signal is connected with 1.6 kW power supply and halogen lamps.We used 500 W linear halogen lamps because of their bad efficiency, therefore, most of the electric power is transformed to the heat flux.The second channel of the BK 4052 triggers the thermoIMAGER TIM 160 IR camera with a spatial resolution of 160 × 120 pixels.To get better results and constant emissivity ε, the Phase shift ( target surfaces were painted by black matte spray colour with emissivity ε = 0.96.The experimental equipment and wiring schematics are depicted in Figure 3. The phase-shift between the heat flux and wall temperature response must be measured precisely.Therefore, it is important to synchronize the heat flux and the IR camera trigger signal.The dynamic characteristic of the whole system, i.e. the voltage supplyhalogen lamps, has to be taken into account, otherwise, the experimental results would be affected by a systematic error.We used two methods to determine the dynamic characteristic.The first one measures the temperature response at one side of the heat transfer surface (wall), which is insulated on the other side (zero-heat method).The phase shift is then determined from the shift between the signal controlling the heat flux and the temperature response signal.The second method measures the transition characteristic of the voltage supply -the halogen lamp system on the basis of a step change of the voltage supply control input and the corresponding heat flux measurement (step-voltage method).Assuming the system is of the first order, the phase shift can be determined as the time constant of the measured response.The dynamic characteristic of the system, i.e., time constant, depends only on the properties of the system and not on the period of the oscillating signal.Nevertheless, several periods were tested (2, 5, 10 and 20 s, with corresponding frequencies 0.5, 0.2, 0.1, and 0.05 Hz) and same results were obtained.The comparison of both methods is in Table 2.

Validation of the method: Pipe flow
The TOIRT experimental method was validated on two cases: a flow of water in a pipe, and a jet of water impinging a plate perpendicularly.The mean temperature of the liquid (water) was maintained within the range of 16 ± 1.5 °C during the pipe-flow experiment.Corresponding liquid properties at this temperature are: density 999 kg m −3 , thermal conductivity 0.598 W m −1 K −1 , dynamic viscosity 1.1 mPa s, and Prandtl number 7.36.Temperature of the flowing liquid was measured at the inlet and outlet of the measuring section.The maximum temperature increase of the outer pipe surface was 5 °C during the experiments.Neglecting the thermal resistance of the wall and the heat accumulation, we could assume the same liquid temperature at the inner pipe surface.The dynamic viscosity for this temperature is 0.98 mPa s.Taking into account the temperature dependency of the viscosity as described by Sieder-Tate's correction, its influence would be (0.98/1.1) 0.14 = 0.98, that is 2 %.This influence was neglected.Water was pumped from a water tank by a centrifugal pump and it flowed through a copper pipe (density 8941 kg m −3 , thermal conductivity 392 W m −1 K −1 , thermal diffusivity 1.136 × 10 −4 m 2 s −1 ) with an inner diameter of d = 24 mm.The measurement section (100 mm long, approx.4d) started at the distance approximately L = 80d from the inlet, therefore, we assumed a fully developed velocity profile there.The volumetric flow rate was measured by the induction flow meter (Krohne IFC010D) and it was used in the calculation of the mean velocity u in the pipe and consequently the Reynolds number according to (5).The experimental setup is in with various parameters: the period of the sine wave 10 or 20 s, scan frequency of the IR camera 10 or 20 Hz and the power modulation between 30 and 100 %.The heat transfer coefficients were evaluated at the inner side of the pipe wall and the results were described as the Nusselt number dependency on the Reynolds number Nu = C Re m Pr 0.4 where Pr 0.4 is the Prandtl number with a commonly used exponent for heating.Our evaluated dependency (Re from 10 4 to 5 × 10 4 ) was Nu = 0.029 Re 0.8 Pr 0.4 (19) which is very close to the well-known Dittus-Boelter correlation Nu = 0.023 Re 0.8 Pr 0.4 , ( valid for Re > 10 4 and Pr from 0.6 to 160.

Validation of the method: Impinging jet flow
Heat transfer in an impinging jet was investigated at a plane stainless steel plate (density 7800 kg m −3 , thermal conductivity 14.6 W m −1 K −1 and thermal diffusivity 3.736 × 10 −6 m 2 s −1 ) of a diameter of 392 mm.The stainless plate formed a flat bottom of a cylindrical unbaffled vessel and it was impinged by a liquid stream coming from the pipe of an inner diameter of 22 mm, see Figure 5.The jet outlet nozzle was located at h/d = 1 above the vessel bottom.The centrifugal pump controlled by an electric frequency inverter formed a closed loop with the water in the vessel, hence the water level was constant during the experiment.The volumetric flow was measured by the induction flow meter (Krohne OptiFlux 5300) and it was used in calculating the mean velocity Solnař [22] with literature data by Persoons et al. [12], and Katti and Prabhu [4].
in the pipe and Reynolds number according to (5).
The mean temperature of the liquid was maintained within a range of 20.5 ± 1.5 • C during the experiment.
According to the distance of the IR camera from the bottom (see Figure 5), the size of the measured point was 3.3 mm/px.Figure 6 compares the experimental results [22] with literature data [4,12] for two different Reynolds numbers.

Experimental setup
Schematic drawings of the agitated vessel (stainless steel, inner diameter D = 300 mm) with two types of impellers (axial and radial impellers) and four baffles (angle 90°) are depicted in Figure 7.
The axial down-pumping impeller (6PBT45) with six pitched blades (pitch angle = 45°) and diameter d = 100 mm (ratio D/d = 3) was used.The radial impeller was a Rushton turbine with the same diameter as the axial-flow impeller.The thickness of the vessel wall was 1.03 mm and it was coated by a black matte colour at the outer side of the vessel in the area between baffles (the measurement area).Thermophysical properties of the vessel wall were: thermal conductivity 14.6 W m −1 K −1 , density 7800 kg m −3 and specific heat capacity 501 J kg −1 K −1 .The water level in the vessel was H = D.The height of impellers h above the bottom and water level H were adjusted with an accuracy of 1 mm.The IR camera monitored the temperature field along the height of the vessel wall with a spatial resolution of about 2.5 mm/px.The sensitivity of the IR camera was 0.08 K.The temperature of the liquid in the vessel was measured by a Pt1000 temperature sensor.It was within a range of 22.0 ± 0.7 °C, and corresponding properties of water are: density 997.8 kg m −3 , thermal conductivity 0.605 W m −1 K −1 , dynamic viscosity 0.99 mPa s, and Prandtl number 6.92.Both impellers were situated at h/d = 1 above the vessel bottom.Radial baffles of a width B = 0.1 D were used in both cases.The Reynolds number in a mixing system is defined by (6) with the impeller diameter as the characteristic dimension d.The Reynolds number, in the case of 6PBT45, was within the range 9 × 10 3 -9 × 10 4 , and in the case of Rushton turbine, the range was 9 × 10 3 -6 × 10 4 .
The following parameters were used in each experiment: the period of the sine wave 10 or 20 s, the number of periods 5 or 10, and the sampling frequency of the IR camera was always set to 10 Hz.

Experimental results
Local values of the Nusselt numbers can be calculated according to (2), and it is possible to express the mean Nusselt number dependency by a commonly used correlation (18).
Two forms of experimental data are presented in this paper.The first one describes graphical dependencies of the local Nusselt number on dimensionless height z/H for various Reynolds numbers (Re from 10 4 to 5 × 10 4 ).The second form describes the mean Nusselt number along the wall as a function of the Reynolds number.The Nusselt number profiles for various Reynolds numbers and the Rushton turbine are in Figure 8.There is a significant peak of the Nusselt number at a dimensionless height z/H = 0.33, which corresponds to the impeller location.Above z/H > 0.4, the measured data show an exponential dependency.
Figure 9 illustrates the Nusselt number profiles for various Reynolds numbers and the 6PBT45 axial-flow impeller.The Nusselt numbers are slightly larger in the lower part of the vessel, from z/H = 0 to z/H = 0.5, but there is no visible peak in the Nusselt number dependency as in the case of the Rushton turbine.The axial-flow impeller was located at z/H = 0.33.The local values of the Nusselt number in Figures 8  and 9 were evaluated as average values over the width of the monitored surface at a specific dimensionless distance from the bottom z/H.
Mean values of the Nusselt number along the vessel wall and their dependencies on the Reynolds number  are displayed in Figure 10.The dotted lines represent 95 % confidence bands, which are no more than ±15 % wide in our case.Evaluated parameters of the correlation (18) are summarized in Table 3 for both cases, the Rushton turbine and the 6PBT45 axial-flow impeller.Contours of the heat transfer coefficients at the wall are depicted in Figure 11.The part of the tank wall monitored by the IR camera is represented by 120x15 pixels here, that is 300x37.5 mm according to the camera resolution 2.5 mm/px.This means that the wall temperature was monitored along the whole height of the liquid in the vessel (H = 300 mm).

Discussion
Figure 12 shows a good agreement of our experimental results for the Rushton turbine with Karcz and Cudak [18] measured by the EDD experimental method.A bigger difference (approx.50 %) is visible within the range z/H = 0.25 − 0.5 where we have a clearly smoother distribution of experimental data.Vlček et al. [23] studied velocity fields in a vessel agitated by the 6PBT45 impeller using LDA experiments and LES-based simulations in ANSYS CFD software.

C m
Rushton turbine 8.75 ± 33 % 0.336 ± 9.5 % 6PBT45 5.98 ± 34 % 0.328 ± 9.6 % it is possible to calculate the Nusselt number as 395.Using ( 6) and rotation speed N = 300 min −1 , the Reynolds number in the agitated vessel Re M is 50 000.Substituting to correlation (18) for the 6PBT45 axial impeller and a same Prandtl number, we can evaluate the Nusselt number Nu = 404, which is quite close to the value based on the case of the flow along a plate (error 2.3 %).
In the case of the vessel agitated by the 6PBT45 impeller, it is possible to find regions with some similarities between the heat transfer at the vessel wall and the heat transfer in a flow along a flat plate.Such regions with similar results cannot be identified in the case of the Rushton turbine (radial-flow impeller).
The evaluated experimental data do not show similarities with the flow along the flat plate for the entire height of the vessel wall.In general, such similarities can be found only in regions where the flow is aligned with the vessel wall.Such regions are larger in configurations with an impeller placed in a draft tube [25], or in cases with axial-flow impellers and larger H/D ratio.

Conclusion
The temperature oscillation infra-red thermography method was used for measuring local heat transfer coefficients at the wall of an agitated vessel with four baffles.The Rushton turbine and a 6PBT45 axial-flow impeller located at dimensionless distance z/H = 0.33 from the vessel bottom were used in the experiments.
We have validated the TOIRT method on the pipe flow and impinging jet experiments.The experimental results of the heat transfer in the agitated vessel are presented as graphical dependencies of the Nusselt number with respect to a dimensionless coordinate z/H.In the case of the Rushton turbine, a significant peak of the Nusselt number at the vessel wall can be observed around z/H = 0.33 (the impeller location).
No such peak can be identified in the case of the axial-flow impeller 6PBT45.
Parameters of the commonly used correlation for the mean Nusselt number were evaluated in both cases, the Rushton turbine and 6PBT45 axial-flow impeller.Similarities between the heat transfer at the wall of the vessel agitated by the 6PBT45 impeller and the flow along a flat plate can be identified in some regions of the wall, but they cannot be recognized for the entire height of the vessel wall or the Rushton turbine impeller.
The disadvantages of the TOIRT method consist in the necessity of an accurate measuring of the wall temperature response.The temperature variations are relatively small, therefore, relatively expensive IR cameras with good sensitivity are required, otherwise, the key parameter of this method, the phase shift, is determined inaccurately and the resulting value of the heat transfer coefficient is inaccurate as well.Another problem lies in light reflections from the surface irradiated by the halogen lamps, which could easily distort the image scanned by the IR camera.Nevertheless, we think that the advantages of the TOIRT method prevails and they can be briefly summarized as (1) contactless, (2) relatively fast for a heat transfer measurement, (3) with an option to easily change the spatial resolution.

Figure 2 .
Figure 2. Illustration of processing the temperature time responses at the heat transfer surface measured by IR camera.(top) Original measured temperature response.(middle) Transformed temperature response oscillating around zero value, that is with removed accumulation (exponential) component.(bottom) Comparison of normalized values of the generated heat flux q * with first harmonics of the preprocessed temperature response ∆T * in order to find out the phase shift ϕ.

Figure 3 .
Figure 3. Wiring schematic and experimental equipment for the temperature oscillation method.

Figure 10 .
Figure 10.Mean values of the Nusselt numbers along the wall of the agitated vessel and the nonlinear regression lines based on the model function (18).Dotted lines represent 95 % confidence bands of the fitted functions.The width of these bands is no more than ±15 %.

Figure 12 .Figure 13 .
Figure 12.Comparison of the experimental results for the Rushton turbine with Karcz and Cudak[18]

a
Abbreviations6BD six-blade disc impeller (Rushton turbine) 6PBT45 six-blade impeller with a pitched angle of 45°C FD Computational Fluid Dynamics DFT Discrete Fourier Transform EDD Electro-Diffusion method IR Infra Red LES Large Eddy Simulation LDA Laser Doppler Anemometry TLC Thermochromic Liquid Crystals TOIRT Temperature Oscillation Infra-Red Thermography px pixel

Table 1 .
(2)sented another study based on the TOIRT method where heat transfer coefficients in a plate heat exchanger were measured.Parameters m, n, and C in correlation (4) for various cases in the literature.L char represents the characteristic length in equation(2).Subscript 0 refers to the stagnant point in the impinging jet.6BD refers to six-blade disc impeller (Rushton turbine), and 6PBT45 to six-pitched-blade turbine impeller (pitched angle 45°) in common geometry configuration, i.e.H/D = 1, d/D = 1/3.

Table 2 .
Time-frame synchronization results, the phase-shift results are for sine wave with period 10 s (frequency 0.1 Hz).

Table 3 .
(18)uated parameters of(18)for mean values of the Nusselt number Nu.