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Journal of Food Process Engineering ISSN 1745-4530

DIMENSIONLESS NUMBERS OF STRUCTURAL AND PROCESS SIMILITUDE OF A WHIRLPOOL HOT TRUB SEPARATOR JAROSLAW DIAKUN and MAREK JAKUBOWSKI1 Department of Mechanical Engineering, Koszalin University of Technology, Koszalin, Poland

1

Corresponding author. Koszalin University of Technology, Department of Mechanical Engineering, Raclawicka 15-17, Koszalin PL75620, Poland. TEL: +48-94-3478457; FAX: +48-94-3426753; EMAIL: [email protected] Received for Publication April 9, 2013 Accepted for Publication August 13, 2013 doi:10.1111/jfpe.12043

ABSTRACT The paper presents an application of a dimension similitude theory in an analysis of the flow in a hot trub separator. It presents a procedure for derivation of dimensionless numbers based on a comprehensive analysis of all the forces occurring during fluid flow in a whirlpool. A combination of quotients of elementary forces has allowed to attain the dimensionless numbers. Among them, there are well-known numbers such as the Reynolds and Strouhal numbers, and there is also one original number. Modifications of these numbers have been derived to describe process conditions characteristic for the flow in the whirlpool. Dimensionless numbers relating to the time and the secondary flow parameters occurring at the bottom of the tank have been highlighted. General forms of functions of the dimensionless numbers have been proposed for calculating the time of the whirling motion and the parameters of the flow that form the sediment cone.

PRACTICAL APPLICATIONS To this point, designing and assembling whirlpools has not incorporated the knowledge coming from analysis of flows inside the tank occurring during the process of separation. All the information introduced by this publication may contribute to the improvement in designing tanks such as whirlpool. Presented similarity numbers allow for scaling flow phenomena occurring in a cycling vat.

INTRODUCTION A centrifugal-settling vat (called whirlpool) is a separator used in the brewing industry to remove so-called “hot trub” from the wort after its is boiled (Kunze 2010). The efficiency of this process has a significant impact on the quality of wort (chemical composition, physical characteristics, micronutrients availability) and consequently on the performance of the fermentation process (Poreda et al. 2009). Whirlpool is a cylindrical tank in which a sediment cone is formed due to the natural gravitational sedimentation and the secondary flow syndrome caused by the whirling motion of a mixture of fluid (the wort) and a protein suspension (hot trub). Forming of a sediment cone is considered to be a paradox because the centrifugal force does not move the sediment particles outside but the opposite occurrence is observed – the sediment is accumulated in the central zone of the tank bottom. Albert Einstein called this 748

phenomenon “a cup of tea effect” (Einstein 1926). Whirlpool is an effective device to clarify sedimenting suspensions. Compared to centrifugal clarifiers, it requires less energy to perform clarifying process and when matched up to classic settling vats, operating on gravitational sedimentation only, it proves much better efficiency of separation process in a shorter time. Fluid whirl motion in a whirlpool and formation of secondary flows are very complex occurrences in terms of mathematical description. A range of equations, flow continuity equation, momentum equation and clamping equations of the turbulent energy dissipation, which describe processes occurring in the three-dimensional space of the tank and in time, are not possible to solve directly. However, it is possible to carry out the analysis using computer simulations and calculations conducted by the authors (Jakubowski and Diakun 2007; Diakun and Jakubowski 2010). Dimensionless numbers are used to create simplified Journal of Food Process Engineering 36 (2013) 748–752 © 2013 Wiley Periodicals, Inc.

J. DIAKUN and M. JAKUBOWSKI

descriptions of usually complex (in terms of the mathematical description) fluid flow issues and occurrences related to them (Krantz 2007). They allow creating formulas and functions that generally describe structural and process relations and which are particularized by analyzing the results of experimental research. There are already existing and published studies with dimensionless numbers used in describing the specific fluid motion in a whirlpool, in which authors have attempted to describe the turbulent flow of fluid by applying the Reynolds number (Dürholt 1988). These attempts apply to only one of many dimensionless numbers. Therefore, our approach is based on combination of quotients of elemental forces defining the flow in a whirlpool.

MATERIALS AND METHODS

DIMENSIONLESS NUMBERS FOR THE WHIRLPOOL FLOW

h

,

u H D

ub

hb

The Purpose of the Analysis The purpose of the analysis is to derive and interpret a compilation of dimensionless similitude numbers describing structure and occurrences of a fluid flow in a whirlpool settling-turbulent vat. The analysis considers only a singlephase flow of fluid motion, which means that it does not include the influence of dispersed phase (in this case of hot trub). A general form of the function of related dimensionless numbers has been proposed.

Parametric Object Identification A simplified diagram of the fluid flow in a whirlpool tank with size indications, by means of which constructional and process relations can be described, is shown in Fig. 1. Separated mixture is poured into a whirlpool through the inlet located tangentially to the casing of the tank. This way of pouring allows to obtain whirling, turbulent motion of the mixture. The method determines the formation of a primary turbulent flow. After filling a whirlpool, the fluid motion is at first slowed down and then totally stopped due to a retardation effect at the bottom and by the walls of the tank and also due to the energy dissipation. In brewery engineering, the time between the filling of the tank and the stopping of the mixture’s motion is called whirlpool’s break time. Retardation and fluid’s internal motions during the whirling time create local secondary flows. The most important for a whirlpool operation is a turbulent secondary flow formed in the area near the bottom of the tank, in so-called “Ekman boundary layer” (Shore 2007). This flow has a cross-sectional shape of a flattened torus. This flow is responsible for the accumulation of hot trub in the central zone of the tank’s bottom. For the proper operation of the separator such as whirlpool, it is important that this flow is of proper shape, and its velocity is as high as possible. Journal of Food Process Engineering 36 (2013) 748–752 © 2013 Wiley Periodicals, Inc.

FIG. 1. A DIAGRAM AND GEOMETRICAL PARAMETERS OF A WHIRLPOOL VAT

The construction of the separator’s tank and its basic process parameters are described by the numerical values of dimensional size of the tank: its diameter – D, and basic parameter for the process, that is the discharge head after the stabilization of the free surface of liquid – H. The properties of fluid in motion for single-phase flow are determined by its density – ρ and its dynamic viscosity – μ. External interaction takes place due to gravity – g. In this case, the process parameter is the whirling time – t. The primary flow in a whirlpool can be formulated using the basic value, which is the tangential velocity of the flow – u. The velocity of the liquid flowing out of the filling hole of the tank can be used for the calculation of the dimensionless similitude numbers. The value of the whirling velocity is lower than the maximum speed assumed for the analysis due to energy dissipation effect. Based on experimental research PIV (particle image velocimetry), it is possible to determine the potential divergence of the value in the form of a coefficient. The research concerning this area is carried out at the moment as a part of the research project. The analysis of the secondary flow issues, which are a consequence of the impact of the primary flow, allows to define the basic parameters characterizing the secondary flow that occurs at the bottom of the tank. These parameters are specified in Fig. 1: height – hb, and maximum velocity of the flow – ub. The height of the meniscus of a free surface’s cross-section of the rotational motion, Δh, can be expressed as a function of the diameter D, velocity u and fluid’s density ρ. It is therefore a dependent value and will not be included in the scope of further consideration as the base value. 749



There is a set of nine values identified as primary ones. They can be expressed by using the three independent values: the linear dimension, the mass of rotating liquid and time. According to the fundamental expression on dimensional analysis presented by Buckingham (called the Π theorem), there are six dimensionless numbers (9 − 3 = 6), which are power multiplication products and which can be used for the analysis (Yarin 2012). The dimensionless numbers of similitude have been derived using the method of proportions (quotients) of the values of the same kind.

Criteria of Dimension and Velocity Relationship’s Similitude

(1)

• the number of the height of the vortex at the bottom of the vat:

hb , D

(2)

• the number of the velocity of the vortex at the bottom of the vat:

K ub =

ub . u

(3)

To derive the remaining three numbers of the process similitude, there have been assumed the proportions of forces occurring during the rotating flow and during the formation of the sediment cone at the bottom of the tank. Following are the forces that can be expressed as a function of the basic values: • inertial force of the initial vortex (F1) resulting from velocity changes (deceleration) of a rotating liquid. In differential notation:

dF1 =

du dm. dt

(4.1)

Integrating this formula, taking into account the velocity and mass radial distribution, and then having differentiation of velocity over time, we can obtain the formula for the force: 750

All the constant values resulting from the course of the transformation of this formula have been included as a numerical value L1. Because in the generated dimensionless numbers the relations of forces occur and there is significant impact of variable values and factors, the numerical number becomes irrelevant. So the formula can be noted:

F1 ≈

u 2 D Hρ t

F2 = c N u2 A ρ,

There are three similitude numbers that are quotients of base values and they use the vessel diameter (D) and the maximum initial velocity of the primary flow (u) as reference: • the number of dimensional proportions of the vat:

K hb =

(4.2)

(4.3)

• the force of the velocity breaking effect near the walls, the same as the resisting force of the liquid medium, existing due to the transfer of momentum between the adjacent particles of the liquid:

RESULTS AND DISCUSSION

H KH = , D

u F1 = L1 D 2 H ρ. t

(5.1)

where cN is a resistance coefficient, and A is the surface. Thus, the force of the velocity break effect of the cylindrical tank walls:

F21 ≈ u2 D H ρ,

(5.2)

and the force of the velocity breaking effect of the tank’s bottom:

F22 ≈ u2 D 2 ρ.

(5.3)

In the formulas (5.2) and (5.3), the (4.3) constants are omitted. Eqs. (5.2) and (5.3) can also be derived from the relation that includes in the initial notation tangential stress of the fluid in the boundary layer of the wall and the bottom of the whirlpool. Applied derivation and the one from the tangential stress of the fluid have the same physical meaning – fluid flow breaking near of the whirlpool’s wall: • the force of viscous liquid’s dissipation due to the velocity gradient – viscous deformation:

F3 =

du μ A. dx

(6.1)

Taking into account the nature of the impact of the factors and omitting the constant values, we can obtain formulas for the dissipation force of the primary flow of viscous liquid:

F31 ≈ u μ H ,

(6.2)

and the dissipation force of viscous liquid flow at the bottom: D2 F32 ≈ ub μ . (6.3) hb Journal of Food Process Engineering 36 (2013) 748–752 © 2013 Wiley Periodicals, Inc.



The centrifugal force of rotation of the primary flow has the same form as the retardation force of the wall of the cylinder (Eq. 5.2), and the force of hydrostatic pressure has the same form as the retardation force of the bottom of the vat (Eq. 5.3).

Criteria for Force Relations It is possible to create a combination of reciprocal relations (quotients) of particular forces. Similitude number resulting from the relation of inertial forces to the retardation force of the wall (the first approach will take into account the wall of the tank):

K F1

F 21

u 2 ρ F1 t D H = = 2 , F21 u D H ρ

(7.1)

and after reduction:

K F1

F 21

=

D . ut

(7.2)

In this way, we have obtained formula (7.2), which is known as the Strouhal number, or the Thomson number (Kuneš 2012), and which characterizes the unsteady flow. In the whirlpool, two retardation forces occur – the one of the wall and another at the bottom. Summing up these two forces, we get:

K

=

F1 F 21 + F 22

1 D H (D + H ) = ⎛1 + ⎞ , ut ut ⎝ D⎠

(8.1)

and by introducing Eq. (1), we get:

K

F1 F 21 + F 22

=

1 (1 + K H ). ut

(8.2)

The resulting expression defines the criterion of rotating time in the whirlpool. In the brewing industry, it is commonly named as a break time (Kunze 2010). We will apply its inverse:

Kt =

1 ut . D (1 + K H )

(8.3)

Similitude criteria result from the ratio of inertial force to viscous dissipation force for the dissipation of the initial flow:

K F1

F S1

u 2 ρ D2 ρ F1 t D H = = = . FS1 uμ H tμ

(9)

This form of a similitude number does not occur in the literature. Journal of Food Process Engineering 36 (2013) 748–752 © 2013 Wiley Periodicals, Inc.

The formula for the dissipation force of the flow in the Ekman boundary layer is

K F1

FS 2

u 2 D Hρ wHh ρ F1 b = = t = . D2 FS 2 ub t μ ub μ hb

(10.1)

After introducing Eqs. (2) and (3) that describe the flow at the bottom and after writing a fraction in the reverse form, we finally obtain:

Kb =

tμ K ub = . D 2 ρ K H K hb

(10.2)

In this formula, there are similitude numbers that characterize secondary flow that occurs at the bottom of the tank; therefore, it can be defined as the similitude number of the cone forming flow. Subsequently, we can determine a similitude number resulting from the relation of the retardation force of the wall to the viscous dissipation force. The first approach takes into account the tank wall and the dissipation rate of the primary flow:

K F21

F S1

=

F21 u2 D H ρ u D ρ = = . μ FS1 uμ H

(11)

This expression is the standard notation of the Reynolds number. In the case of the whirlpool vessel, it is important to consider not only the retardation effect of the wall but also the cylinder’s bottom. Because the maximum velocity of the secondary flow at the bottom is lower than the order of magnitude of the primary flow velocity, plus the nature and impact of the secondary flow are local, the dissipation force of the flow can be ignored. Taking it into account, the adjusted Reynolds number for whirlpool tank can be noted:

Rew = K F21 + F22 = F S1

uD ρ ⎛ 1 ⎞ ⎜⎝ 1 + ⎟, μ KH ⎠

(12)

The form of the above formula shows that the retardation effects of the bottom of the tank increase especially for vessels with small ratio KH value of discharge head. A secondary flow at the bottom of the tank is formed with a significant impact of the tank’s bottom retardation and the hydrostatic pressure, which is relative to the height of the rotation meniscus – Δh. These two effects are described by the same force F22. For a description of the secondary flow at the whirlpool’s bottom, it is important to examine the relation between the retardation force of the wall and the viscous dissipation of the flow at the bottom. When we take into consideration the entries of the secondary flow (2) and (3), we get the corrected Reynolds number appropriate for this flow: 751


Reb = K F22

FS 2

=

uD ρ K hb . μ K ub


(13)

Equations for Dimensional Numbers of Similitude of a Whirlpool Vat The important process value for a whirlpool vat is a whirling time (also known as standing time) of the mixture in the separator. It depends on the inertial mass of the rotating wort and the forces that cause retardation. In a whirlpool, the retardation of the clarified wort is a consequence of walls and bottom influence plus energy dissipation originating from the liquid viscous deformation. The latter component is smaller by several orders than retardation effect of the walls and the bottom. This is reflected, among other things, by the value of the Reynolds number, which binds quantitatively the ratio of these two components of resistance force. Similitude number, which is a relation of inertia to resistance force of the tank’s walls and the bottom Kt (8.3), is therefore a dimensionless discriminate of rotation time, which can be the basis for calculating the break time:

t = Kt =

D (1 + K H ). u

(14)

By analyzing this group of similitude numbers, we can conclude that dimensionless number of time is a function of the Reynolds number for a whirlpool vat (Eq. 12):

K t = f ( Rew ) ,

(15)

Defining the explicit function requires experimental research. Such studies are carried at this point. In the research PIV, measurements are used. The parameters of the flow at the bottom (the discharge head and the velocity value) are expressed by the formulas (2) and (3). They are present in derivation of dimensionless similitude numbers in formulas (10.2) and (13). The function of these two dimensionless quantities determines the process dependence of the parameters of the silting flow:

f ( K b, Reb ) = 0.

(16)

The explicit function also requires experimental research (similar to Eq. 14). In both expressions of this function, there are two parameters of the secondary flow and in this sense, this function is implicit.

CONCLUSION The combination of the quotients of the basic forces has allowed to derive the dimensionless numbers. Among them, there are well-known numbers such as the Reynolds number and the Strouhal (Thomson) number, and there is

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also the original one derived by the authors. Modifications of these numbers have been derived to describe the turbulent flow in a whirlpool vat. There have been distinguished dimensionless numbers of time and parameters of the secondary flow, which is accountable for phenomenon of sediment con creation in a whirlpool. Two functions of dimensionless numbers have been proposed, one of them allows to specify the rotation time, and the other formulates the parameters of the flow responsible for the secondary flow of the sediment cone in the Ekman boundary layer. The derived relations can be helpful in designing whirlpool vats and also in sizing/scaling the flow occurring in them.

ACKNOWLEDGMENT Analyses were made within the frames of the research financed from funds of Polish Ministry of Science and Higher Education in 2010–2013 (research project N N313 429639). REFERENCES DIAKUN, J. and JAKUBOWSKI, M. 2010. Analysis of the secondary flow velocities forming a settling cone in a whirlpool vat. Chem. Process Eng. 31, 477–488. DÜRHOLT, A. 1988. Experimental investigation of the unsteady rotational flow in the settling vessel “whirlpool.” Fortschr.-Ber. VDI 14(38), 24–28. VDI-Verlag Düsseldorf (in German). EINSTEIN, A. 1926. The cause of the formation of meanders in the courses of rivers and of the so-called Baer’s law. Naturwissenschaften. 14(2), 223–224. (in German). JAKUBOWSKI, M. and DIAKUN, J. 2007. Simulation investigations of the effects of whirlpool dimensional ratios on the state of secondary whirls. J. Food Eng. 1(83), 107–110. KRANTZ, W.B. 2007. Scaling Analysis in Modeling Transport and Reaction Processes: A Systematic Approach to Model Building and the Art of Approximation, pp. 32–38, John Wiley & Sons, Inc, Hoboken, NJ. KUNEŠ, J. 2012. Dimensionless Physical Quantities in Science and Engineering, pp. 87–90, Elsevier Inc., Waltham, MA. KUNZE, W. 2010. Technology Brewing and Malting, pp. 387–391, VLB, Berlin, Germany. ´ SKI, T. and POREDA, A., ANTKIEWICZ, P., TUSZYN MAKAREWICZ, M. 2009. Accumulation and release of metal ions by Brewer’s yeast during successive fermentations. J. Inst. Brew. 15(1), 78–83. SHORE, S.N. 2007. Astrophysical Hydrodynamics: An Introduction, pp. 80–81, Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, Germany. YARIN, L.P. 2012. The Pi-Theorem: Applications to Fluid Mechanics and Heat and Mass Transfer, pp. 3–7, Springer-Verlag Berlin Heidelberg, Germany.

Journal of Food Process Engineering 36 (2013) 748–752 © 2013 Wiley Periodicals, Inc.