A

aflTvostQot©

K

1

oclosbee[cbgo

1d

Biochemical Engineering Journal 34 (2007) 200–208

Calculating liquid distribution in shake flasks on rotaryshakers at waterlike viscosities

Jochen Buchs ∗, Ulrike Maier, Stefan Lotter, Cyril P. PeterBiochemical Engineering, RWTH Aachen University, Worringerweg 1, 52056 Aachen, Germany

Received 19 September 2005; accepted 7 December 2006

bstract

Screening projects in biotechnological industry performed in shake flasks risk unwanted development if not failure, when operating conditionsre not suitable. Limited knowledge, however, is available for the mechanistical design of operating conditions in this type of bioreactor. Theundamental engineering variables are influenced by the geometry of the rotating bulk liquid: for momentum transfer, the contact area between theiquid and the flask inner wall is the friction area, and for mass transfer, the wetted wall exposed to the surrounding air is the mass exchange area.o assess the geometry of the rotating bulk liquid moving inside a shaken Erlenmeyer flask, with respect to the mentioned important engineeringariables mentioned, a mechanistical model for the liquid distribution in shake flasks is described in this work. The model is based on a superpositionf two individual movements: a circular translatoric movement and a rotation of the flask counteracting the first motion to keep the shake flasks’patial orientation. If the effect of viscosity is neglected, the liquid distribution results in an exactly symmetrical paraboloid. A comparison ofhe calculated liquid distribution with photographs shows very good qualitative agreement of the real liquid distributions by the model equations.

uantitative agreement has been demonstrated by comparison of the liquid height. Furthermore, model equations are presented for the calculationf the contact area between the liquid and the flask wall. This may eventually lead to a prediction of the volumetric power consumption. Similarly,he calculation of the mass transfer area (i.e. liquid surface area and wetted flask wall) is presented.

2006 Elsevier B.V. All rights reserved.

ellin

dEtaflew

wfliee

eywords: Fermentation; Shake flask; Power consumption; Mass transfer; Mod

. Introduction

Shake flasks are widely used in bioprocess development andptimization due to their ease of handling and low operatingosts. Especially, in early stages of process development, aarge number of parallel experiments is performed to determineptimal medium composition or to find an suitable microbialtrain. As very decisive and orienteering decisions are takenased on the experimental results, suitable and reproductivexperimental conditions are crucial [1]. Until today, importantngineering parameters like the volumetric power consumption2], the oxygen transfer capacity [3,4] or the hydromechani-al stress [5,6] have been quantified experimentally and may

e assessed through empirical or semi-empirical equations. Toain deeper understanding of the aforementioned mechanismsn a theoretical basis, the geometry, i.e. the contour and spatial

∗ Corresponding author. Tel.: +49 241 8025546; fax: +49 241 8022265.E-mail address: buechs@biovt.rwth-aachen.de (J. Buchs).

tetwflda

369-703X/$ – see front matter © 2006 Elsevier B.V. All rights reserved.oi:10.1016/j.bej.2006.12.005

g; Scale-down

istribution of the rotating liquid mass moving inside a shakenrlenmeyer flask is of crucial importance. The liquid distribu-

ion gives important information about the momentum transferrea, which is the contact area between the liquid mass and theask inner wall, and the mass transfer area, which is the surfacexposed to the surrounding air, including the film on the flaskall [3].This work aims at proving a mechanistical physical model,

hich may serve to calculate the liquid distribution in shakeasks to provide the geometrical information required to

nvestigate momentum and gas/liquid mass transfer. First, thequations, which describe the liquid distribution for commonlymployed shaking conditions [2,7] and with waterlike viscosi-ies will be established. This information will serve to derive thequations for the calculation of the momentum transfer area (i.e.he contact area between the liquid mass and the flask wall), as

ell as the mass transfer area (i.e. liquid surface area and wettedask wall) will be derived. The theoretically calculated liquidistribution is verified through comparison with photographsnd validated through liquid height measurements.

J. Buchs et al. / Biochemical Engineering Journal 34 (2007) 200–208 201

Nomenclature∑A total mass transfer area (m2)

AB area of wetted surface on the flask base, not cov-ered by the liquid sickle (m2)

�AB area of incremental ring of wetted surface on theflask base (m2)

ACB contact area between liquid sickle and flask bot-tom (m2)

�ACB area of incremental ring of contact area betweenliquid sickle and flask bottom (m2)

ACW contact area between liquid sickle and flask wall(m2)

�ACW area of incremental ring of contact area betweenliquid sickle and flask wall (m2)

AS inner surface area of liquid bulk facing the rota-tional paraboloid (m2)

�AS area of incremental slice of inner surface of liquidsickle facing the rotational paraboloid (m2)

AW area of wetted surface at flask wall (m2)�AW area of incremental slice of wetted surface at flask

wall (m2)d0 shaking diameter (m)F abbreviation term (m−1)g gravitational acceleration (m s−2)h0 height coordinate of lowest point of rotational

paraboloid (m)m slope of conen shaking frequency (s−1)�r width of incremental ring at flask base (m)r′ radius of incremental ring at flask base (m)r′1 inner radius of incremental ring at flask base (m)r′2 outer radius of incremental ring at flask base (m)rf radius of incremental slice of liquid (m)rg radius of flask base (m)rk maximum flask radius (m)rmin lower summation boundary for integration of r′

(m)rmax upper summation boundary for integration of r′

(m)rp radius of rotational paraboloid at specific height

z (m)rt radius of torus segment (m)V volume (m3)V1 volume of slice of liquid for Case 1 (Fig. 3) (m3)V2 volume of slice of liquid for Case 2 (Fig. 3) (m3)V3 volume of slice of liquid for Case 3 (Fig. 3) (m3)VL total filling volume (m3)V calc

L calculated filling volume (m3)x x coordinate (m)xmin lower boundary of x for summation (m)xmax upper boundary of x for summation (m)xs distance of the intersection point of the rotational

paraboloid and the flask wall from the flask center(m)

x′s distance of the intersection point of the incremen-

tal ring on the flask base from the flask center(m)

y y coordinatez z coordinate (m)�z thickness of slice of liquid (m)z1 lower height coordinate for slice of liquid (m)z2 upper height coordinate for slice of liquid (m)zc height coordinate along the conical part of the

shake flask (m)zmax maximum liquid height (m)zp height coordinate along the surface of the rota-

tional paraboloid (m)zt height coordinate along the torus part of the shake

flask (m)

Greek lettersα opening angle of sickle seen from flask center at

specific height zα′ angle between intersection points of radius r′ with

liquid sickle at the flask base seen from flask cen-ter

β opening angle of sickle seen from center of shak-ing motion at specific height z

ϕ angle describing wall section of torus at z1ψ angle describing wall section of torus at z2�ω1 angular velocity of shaking movement rotation

around shaft 1

2

2

msmkmilto

|

ceorbs

�ω2 angular velocity of shake flask rotation aroundshaft 2

. Model formulation

.1. Basic concepts

Most commonly shake flasks are agitated by orbital shakingachines. Fig. 1 illustrates the theoretical description of this

haking motion. The shake flask performs a circular translatoricovement with the radius equal to half the shaking diameter

eeping its orientation relative to the surrounding. This move-ent can theoretically be described as a superposition of two

ndividual movements: the angular velocity of the circular trans-atoric movement with the shaking radius around the center ofhe shaking motion (shaft 1) is defined by the shaking frequencyf the shaking machine n:

�ω1| = 2πn (1)

The second rotation around the flask center (shaft 2) mustounteract the first rotation to keep the shake flask’s spatial ori-ntation. Therefore: �ω2 = −�ω1. Driven by the centrifugal field

f the first rotation (�ω1), the liquid is distributed in the flask as aotational paraboloid if viscous effects are neglected. Enclosedy the flask walls, the liquid bulk forms a particular geometricalhape similar to a crescent, which is also termed as “sickle”. For

202 J. Buchs et al. / Biochemical Engineer

Ftl

orlpuvt

2

Frtft

Fd

tt

z

w

F

c

z

z

mtt

dotecdi

ig. 1. Theoretical partition of shake flask movement: superposition of circularranslatoric and rotational movement. Rotational paraboloid determining theiquid distribution.

ur model, the following simplifications are made: the fluid isegarded as an ideal frictionless liquid. Moreover, in reality aiquid film remains at the flask wall after the liquid sickle hasassed [3]. This film contains a small portion of the filling vol-me of the flask. This model, however, assumes that the liquidolume on the flask wall can be neglected for the calculation ofhe liquid distribution.

.2. Model equations for the liquid distribution

The geometric dimensions of a shake flask are shown inig. 2. The flask consists of three parts: a flat base of radiusg, a torus segment of radius rt and a cone segment described by

he slope m. The liquid distribution of a frictionless fluid resultsrom the surface coordinates of the rotational paraboloid and thehree parts of the shake flask.

ig. 2. Geometrical dimensions of a shake flask and the rotational paraboloidetermining the liquid distribution.

fisaaf

2

dodac

V

cap

ing Journal 34 (2007) 200–208

For the x- and y-coordinates as given in Fig. 2, the height alonghe surface of the rotational paraboloid (zp) may be calculatedhrough:

p = 1

2

(2πn)2

g

[(x+ d0

2

)2

+ y2

]+ h0

= F

[(x+ d0

2

)2

+ y2

]+ h0 (2)

ith the simplifying variable F:

= 1

2

(2πn)2

g(3)

The shape of the torus segment of the shake flask (zt) isalculated by:

t = rt −√r2

t − (√x2 + y2 − rg)

2(4)

The shape of the cone segment of the shake flask (zc) is:

c = rt +m(rk −√x2 + y2) (5)

The paraboloid and flask wall intersect at the point of theaximum liquid height zmax. This height results from combina-

ion of Eq. (2) with (4) or (5) (cf. Fig. 2), and may easily be usedo compare the calculations with measurements.

Eq. (2), however, contains one unknown, h0, which isepicted in Fig. 2. It is the distance between the lowest partf the rotational paraboloid and the base of shake flask alonghe axis of rotation of the shaker drive. For a given flask geom-try (flask type, flask nominal volume VK), and given operatingonditions (filling volume VL, shaking frequency n and shakingiameter d0), this parameter (h0) is fixed and may be assessedn an iterative way: assuming a value of h0 � 0, the theoreticallling volume (V calc

L ) is calculated according to the followingection and compared to the given filling volume VL. If thesere not equal, h0 is reduced by an incrementally defined valuend the procedure is repeated, until V calc

L = VL and thus h0 isound.

.3. Model equations for the liquid distribution

To calculate the theoretical filling volume, the liquid bulk isivided horizontally into thin incremental slices�z. The volumef each slice is calculated according to one of three cases asiscussed further below, and then all incremental volumes aredded over the whole liquid height zmax, which results fromombination of Eqs. (2) with (4) or (5) as shown above:

calcL =

zmax∑z=0

V (z) (6)

For the calculation of the incremental volume, three differentases have to be considered. Fig. 3 shows the liquid distributiont a low shaking frequency, which causes the lowest point of thearaboloid to be located above the flask bottom (h0 > 0). In the

J. Buchs et al. / Biochemical Engineering Journal 34 (2007) 200–208 203

Ft

ulCsa

C

C

Fig. 4. Geometrical dimensions for Case 3 in Fig. 3: incremental slice of liquidifl

C

(aE

ai

ig. 3. Location of the incremental slice of liquid in three cases when calculatinghe filling volume in a shake flask.

pper part of the picture, a cutaway view of the flask and theiquid is shown, whereas in the lower part the cutting planes forase 2 and for Case 3 are given. The volume of an incremental

lice of thickness �z, vertically located at z is then calculatedccording to:

ase 1: In the lowest part of the flask below the rotationalparaboloid (z ≤ h0), the flask slice is completely filledwith liquid. The volume V1 is therefore given by:

V1 = πr2f�z (7)

The radius rf, which describes the largest radius ofthe liquid slice, depends on its vertical location (z) aswell as the shape of the flask wall. When the slice islocated in the torus (z < rt):

rf = rg +√z(2rt − z) (8)

and if located in the cone (z ≥ rt):

rf = rk − z− rt

m(9)

ase 2: The slice considered is located above the lowest part ofthe rotational paraboloid (z > h0), which does not inter-sect with the flask wall (cf. Fig. 3). This slice is filledwith liquid except for the disc enclosed by the rotational

x

s situated above the lowest point of the paraboloid and intersecting with theask wall.

paraboloid. The volume V2 is then to be calculated by:

V2 = π�z(r2f − r2

p) (10)

with rf from Eq. (8) (slice located in torus) or Eq. (9)(slice located in cone) and

rp =√z− h0

F(11)

ase 3: The slice considered is located above the lowest pointof the rotational paraboloid (z > h0), and the rotationalparaboloid intersects with the flask wall (cf. Fig. 3).Fig. 4 shows the geometrical relations for this case morein detail. The volume V3 is hence defined by:

V3 = �z

2[r2

f (α− sinα) − r2p(β − sinβ)] (12)

with angles α and β (cf. Fig. 4):

α = 2 arccos

(xs

rf

)(13)

β = 2 arccos

(xs + (d0/2)

rp

)(14)

The radius of the liquid slice rf can be calculated with Eq.8) (slice located in torus) or Eq. (9) (slice located in cone)nd the radius rp of the paraboloid at the height z follows fromq. (11).

The distance xs of the intersecting point between flask wallnd rotational paraboloid in x-direction is—if the slice is locatedn the torus part of the shake flask (z < rt):

s = 1

d0

[z− h0

F−

(d0

2

)2

−(rg +

√2rtz− z2

)2]

(15)

2 ineer

a

x

owoitdsduaaa

2

firsuwtfl(

Fl

2

C

C

C

04 J. Buchs et al. / Biochemical Eng

nd if located in the cone (z ≥ rt):

s = 1

d0

[z− h0

F−

(d0

2

)2

−(rk − z− rt

m

)2]

(16)

In summary, Eqs. (6)–(16) allow the calculation of the the-retical filling volume at a given value of h0. By comparisonith the given filling volume, the position of the lowest partf the rotational paraboloid along the z-axis (h0) is determinedteratively. Eq. (2) then defines the geometry and shape ofhe liquid sickle in the shake flask at given operating con-itions by providing a functional relationship between thepatial coordinates x, y and zp. The following paragraphs willescribe the calculation of the contact area between the liq-id bulk and the glass wall to yield the momentum transferrea as well as the area that is exposed to the surroundingir by the liquid bulk to integrally yield the mass transferrea.

.4. Model equations for the momentum transfer area

During fluid motion, power is dissipated due to internalriction. In shake flasks, the compensating mechanical powers introduced over the flask wall, which can therefore beegarded as the power inducing element (comparable to thetirrer in an agitated tank). The contact area between the liq-id bulk and the flask is the liquid covered surface of theall. The total momentum transfer area therefore consists of

wo parts, which are depicted in Fig. 5: first, the area on theask bottom (ACB) and second, the area on the flask wallACW).

ig. 5. Geometrical dimensions for calculating different surface areas of theiquid sickle inside a shake flask.

2w

t

ing Journal 34 (2007) 200–208

.4.1. Contact area on flask base—ACB

ase 1: For z = 0 (i.e. the flask base, see Fig. 3) and h0 > 0, theflask base is fully coved with liquid and therefore:

ACB = πr2g (17)

ase 2: For z = 0, h0 < 0 (i.e. the flask base is not fully coveredby the liquid), and the paraboloid (rp) does not intersectwith the flask base (rg) (cf. Fig. 3), the contact areabetween the liquid sickle and the flask base dACB iscalculated by:

ACB = π(r2g − r2

p) (18)

ase 3: For z = 0, h0 < 0 (i.e. the flask base is not fully coveredby liquid), and the paraboloid (rp) intersects with theflask base (rg) (cf. Fig. 3), the contact area betweenthe liquid sickle and the flask base ACB is calculatedby summing up circular rings �ACB across the wholeflask base (Fig. 5b). The incremental area of one ringis expressed as:

�ACB = π(r′22 − r′21)α′

2π(19)

with

r′1 = r′ − �r

2and r′2 = r′ + �r

2.

Fig. 5b shows that the angle α′ denotes the frac-tion of the circular ring covered by liquid. It is givenanalogously to Eq. (13) by:

α′ = 2 arccos

(x′

s

r′

)(20)

with:

x′s = − 1

d0

[r′2 +

(d0

2

)2

+ h0

F

], (21)

where F is calculated according to Eq. (3) and h0results from the calculation of the liquid distribu-tion (cf. Section 2.2). The summation boundaries arexmin +�r/2 ≤ r′ ≤ rg −�r/2 with

xmin =√

−h0

F− d0

2(22)

Finally, ACB for Case 3 is calculated through:

ACB =r′=rmax∑r′=rmin

dACB (23)

.4.2. Contact area between the liquid sickle and the flaskall—ACW

To calculate the liquid covered area of the shake flask wall,he total area is divided into incremental slices of height�z, for

ineer

wt

�

iw

ϕ

w

ϕ

ψ

w

z

a

z

c

�

i

α

wp

α

2

tgsgwpsfib

2

sbfa

�

α

a

x

wlatzi

2p

t

�

wlEi

2p

t

�

Er

�

J. Buchs et al. / Biochemical Eng

hich the contact area �ACW is calculated. In the torus part ofhe shake flask (z < rt), �ACW is given by:

ACW = 2πrt[(rk − rt)(ϕ − ψ) +�z]α

2π. (24)

The angleα is calculated with Eqs. (13) and (8), and xs accord-ng to Eq. (15). Angles ϕ and ψ accounting for the wall inclineithin the torus segment are given by:

= arcsin

[rt − z1

rt

], (25)

hen z1 > 0. In all other cases

= π

2(26)

= arcsin

[rt − z2

rt

](27)

ith

1 = z− �z

2(28)

nd

2 = z+ �z

2(29)

The summation boundaries are �z/2 ≤ z ≤ rt −�z/2.In the cone part of the shake flask (z ≥ rt), the contact area is

alculated by

ACW = 2π�z

√1

1 +m2

(rk + rt − z

m

) ( α

2π

)(30)

When d0/2 + rp ≥ rf, with rf after Eq. (8) (i.e. the paraboloidntersects the flask wall),

= arcsin

(xs

rf

), (31)

ith xs according to Eq. (16). When d0/2 + rp < rf (i.e. thearaboloid does not intersect the flask wall),

= 2π (32)

The summation boundaries are rt ≤ z ≤ zmax.

.5. Model equations for the gas/liquid mass transfer area

The liquid sickle within a shake flask periodically moves onhe inside of the wall leaving a thin liquid film [3]. Hence, theas/liquid mass transfer area consists subsequently of the sickleurface itself and the surface of the wetted glass wall. The totalas/liquid mass transfer area therefore consists of three parts,hich are depicted in Fig. 5: first, the area on the flask bottom

eriodically covered by the liquid bulk (AB), second the innerurface area of the liquid bulk facing the paraboloid (AS) andnally the area of the film on the flask wall wetted periodicallyy the liquid bulk (AW).

∑

ing Journal 34 (2007) 200–208 205

.5.1. Surface area on flask base—AB

The total surface area on the flask base AB may be obtained byummation of the incremental circular rings�AB across the flaskase. At h0 < 0 and z = 0 (flask base, see Fig. 5b), the mass trans-er area of the circular ring at the flask base �AB is calculatednalogous to Eq. (19) by:

AB = π(r′22 − r′21)

[1 − α′

2π

](33)

The angle α′ is given similar to Eq. (13) by:

′ = 2 arccos

(x′

s

r′

)(34)

nd the distance x′s:

′s = − 1

d0

[r′2 +

(d0

2

)2

+ h0

F

](35)

ith F according to Eq. (3) and h0 resulting from the calcu-ation of the liquid distribution (cf. Section 2.2). When z = 0nd h0 ≥ 0 (i.e. the paraboloid does not intersect the flask base),he integration boundaries are d0/2 − rp ≤ r′ ≤ d0/2 + rp. When= 0 and h0 < 0 (i.e. the paraboloid intersects the flask base), the

ntegration boundaries are d0/2 + rp ≤ r′ ≤ rg.

.5.2. Inner surface area of the liquid bulk facing thearabolid—AS

For z < rt (torus) and z ≥ rt (cone), the mass transfer area ofhe sickle slice �AS (Fig. 5a) is given by:

AS = π

6F2 [1 + 4F (z2 − h0)]1.5 − [1 + 4F (z1 − h0)]1.5 β

2π,

(36)

ith the angle β according to Eq. (14), xs after Eq. (15) (sliceocated in torus) or Eq. (16) (slice located in cone), rp fromq. (11), z1 from Eq. (28) and z2 according to Eq. (29). The

ntegration boundaries are 0 ≤ z ≤ zmax.

.5.3. Surface area of the film on the flask wall wettederiodically by the liquid bulk—AW

The wall area wetted by the liquid sickle slice �AW in theorus (0 < z < rt) is given by:

AW = 2πrt[(rk − rt)(ϕ − ψ) +�z][1 − α

2π

](37)

The angle α is calculated with Eq. (13) and xs according toq. (15). Angles ϕ and ψ are defined by Eqs. (26) and (27),

espectively. The summation boundaries are 0 < z < rt.For z ≥ rt (cone), the wetted wall area is calculated by:

AW = 2π�z

√1

1 +m2

(rk + rt − z

m

) (1 − α

2π

)(38)

With summation boundaries rt ≤ z ≤ zmax.

Taken together, the total mass transfer area is obtained by:

A = AB + AS + AW =∑r′�AB +

∑z

�AS +∑z

�AW

(39)

206 J. Buchs et al. / Biochemical Engineer

Fc

3

u(cntssarflcttostdTai

4

4

fliteaaslboitt

varasri

dnstwiw

ptt(scata

orsTmalli

dscbc

iflbiawlwi[m

ig. 6. Experimental setup for taking pictures of a moving shake flask with aamera.

. Material and methods

The model for the calculation of the liquid distribution innbaffled shake flasks was implemented in Visual Basic 6.0Microsoft, Redmond, USA). To experimentally validate the cal-ulations of the liquid distribution, unbaffled shake flasks with aominal volume of 250 mL (Schott, Mainz, Germany) were agi-ated on orbital shakers (Adolf Kuhner AG, Switzerland). Fig. 6hows the experimental setup for taking pictures of the liquidickle at a fixed position relative to the direction of centrifugalcceleration. The shaker was operated in a completely darkenedoom. A photo camera (Canon, Japan) was aimed at the shakeask with the lens opened. When the shaker table was in theorrect position, a mechanism mounted on the shaking machineriggered a flashlight (PE 3057, National, Japan) illuminatinghe fluid in the shake flask. To prevent continuous triggeringf the flash, the trigger mechanism was activated by a manualwitch and deactivated by the first triggering event. To controlhe correct position of the shaker table at trigger time, a smallish containing a steel ball was mounted on the shaker table.he motion of the steel ball is considered frictionless and thuslways pointing in the direction of the centrifugal acceleration,ndicating the position of the shaker table.

. Results and discussion

.1. Calculation of the liquid distribution in shake flasks

Fig. 7 shows liquid distribution in a standard 250 mL shakeask for different shaking conditions. The left-hand pictures

n each subfigure show the calculated liquid distribution fromhree different views, while the right-hand pictures depict thexperimental situation. The top pictures of each subfigure showside view of the shake flask in motion, while the centrifugal

cceleration points to the right. The calculated distribution ishown as a cutaway view through the middle of the flask. Theightly shaded area is the actual cutting area through the liquidulk, whereas the heavily shaded area represents the plan view

f the sickle behind the cutting plane. The lower horizontal lines the division between the torus segment and the cone, whereashe upper horizontal line is the maximum liquid height (zmax) inhe flask. The second picture on the left-hand side shows a plane

actfl

ing Journal 34 (2007) 200–208

iew of the calculated liquid distribution with the centrifugalcceleration pointing away from the reader. The third pictureepresents a top view where the flask base radius rg is depicteds strong circle, and the smallest radius covered by the liquidickle rmin (Eq. (22)) as lighter circle. The second picture on theight in each subfigure is a combinational view as the photographs taken from a 45◦ angle.

Fig. 7A shows the calculated and experimental liquidistribution at common operating conditions of VL = 25 mL,= 250 rpm and d0 = 50 mm. The liquid forms the characteristic

ickle shape and the bulk is pointing into the direction of the cen-rifugal acceleration. The calculated distribution compares wellith the experimental one, i.e. the same maximum liquid height

s observed in both cases, and the flask base is partly coveredith liquid.For the discussion of the influence of different operating

arameters, in each of the following subfigures, one parame-er is varied while all others are held constant. Fig. 7B depictshe situation for a decreased shaking frequency of n = 150 rpmVL = 25 mL, d0 = 50 mm). While a pronounced sickle shape istill observed with the bulk pointing towards the direction ofentrifugal acceleration, the maximum liquid height is reducednd the smallest radius of the flask base, which is covered byhe liquid sickle is smaller, because of the decreased centrifugalcceleration.

Fig. 7C shows the situation for a reduced filling volumef VL = 10 mL (n = 250 rpm, d0 = 50 mm). The liquid height iseduced and the flask base is just not covered by the liquid as theickle’s smallest diameter rotates just on the flask base radius.he calculated maximum liquid height is larger than experi-entally seen. This is possibly caused by the assumption that the

mount of liquid in the film on the inside of the flask wall is neg-igible. As the filling volume investigated is relatively small, theiquid volume in the film reduces the amount of liquid remainingn the bulk relatively more than with larger filling volumes.

Fig. 7D shows the situation for a reduced shaking diameter of0 = 25 mm (VL = 25 mL, n = 250 rpm). The slope of the liquidickle towards the flask wall is much smaller and the liquidovers a larger part of the flask circumference. This may welle explained by the reduced distance between the axis of theentrifugal acceleration and the flask central axis.

Taken together, this systematic study on the influence of var-ous operating conditions on the liquid distribution in shakeasks has demonstrated that the liquid height, the covered flaskase area and the covered flask circumference vary as expectedntuitively with the change of centrifugal acceleration and themount of liquid. Moreover, the experimental findings are veryell predicted by the model calculations presented. The calcu-

ated distribution results in a completely symmetric liquid bulk,hich is due to the assumption of frictionless liquid. The exper-

mental findings show a more or less skewed distribution. Ref.3] has compared the maximum liquid height calculated by thisodel with the measured value and found a maximum devi-

tion of ±15%. Thus, the validity of the model presented isonfirmed, and it may therefore serve to calculate the momen-um transfer area and the mass transfer area for any set of shakeask operating conditions.

J. Buchs et al. / Biochemical Engineering Journal 34 (2007) 200–208 207

F k. Opn 5 mL,

5

a

ig. 7. Calculated and photographed liquid distribution in a 250 mL shake flas= 150 min−1, d0 = 5 cm. (C) VL = 10 mL, n = 250 min−1, d0 = 5 cm. (D) VL = 2

. Conclusion

To describe the liquid distribution in an orbitally shaken flask,physical model has been derived. The model calculations were

csbm

erating conditions: (A) VL = 25 mL, n = 250 min−1, d0 = 5 cm. (B) VL = 25 mL,n = 250 min−1, d0 = 2.5 cm.

ompared to photographs of the liquid sickle inside a movinghake flask. This comparison has shown very good agreementetween the model calculation and the reality. Therefore, aathematical model is now available to calculate the liquid

2 ineer

dsau

A

wd

R

[

[

[

[

[

08 J. Buchs et al. / Biochemical Eng

istribution in shake flasks. Important engineering parametersuch as the momentum transfer area and the mass transferrea may now be calculated to facilitate a deeper theoreticalnderstanding of the transport phenomena in shake flasks.

cknowledgements

The authors wish to thank Mr. Udo Kosfeld for his creativeork in assembling the trigger mechanism and Ms. Carina Kor-an for her patient effort in refining the artwork.

eferences

1] D. Freedman, The shaker in bioengineering, Proc. Biochem. 2 (1969) 35–40.

[

[

ing Journal 34 (2007) 200–208

2] J. Buchs, U. Maier, C. Milbradt, B. Zoels, Power consumption in shakingflasks on rotary shaking machines: I. Power consumption measurementsin unbaffled flasks at low liquid viscosity, Biotechnol. Bioeng. 68 (2000)589–593.

3] U. Maier, J. Buchs, Characterisation of the gas–liquid mass transfer in shak-ing bioreactors, Biochem. Eng. J. 7 (2) (2001) 99–107.

4] U. Maier, M. Losen, J. Buchs, Advances in understanding and modelingthe gas–liquid mass transfer in shake flasks, Biochem. Eng. J. 17 (3) (2004)155–167.

5] J. Buchs, B. Zoels, Evaluation of maximum to specific power consump-tion ratio in shaking bioreactors, J. Chem. Eng. J. 34 (5) (2001) 647–653.

6] C.P. Peter, Y. Suzuki, J. Buchs, Hydromechanical stress in shake flasks: Cor-relation for the maximum local energy dissipation rate, Biotechnol. Bioeng.93 (6) (2006) 1164–1176.

7] J. Buchs, Introduction to advantages and problems of shaken cultures,Biochem. Eng. J. 7 (2) (2001) 91–98.