Chapter 7,Scale-Up
8.1 Similitude
For the optimum design of a production-scale
fermentation system (prototype),we must
translate the data on a small scale (model) to
the large scale,The fundamental requirement
for scale-up is that the model and prototype
should be similar to each other.
Two kinds of conditions must be satisfied to
ensure similarity between model and prototype,
They are:
(1) Geometric similarity of the physical
boundaries:
The model and the prototype must be the same shape,and
all linear dimensions of the model must be related to the
corresponding dimensions of the prototype by a constant
scale factor.
(2) Dynamic similarity of the flow fields,The ratio of flow
velovities of corresponding fluid particles is the same in
model and prototype as well as the ratio of all forces acting
on corresponding fluid particles,When dynamic similarity
of two flow fields with geometrically similar flow patterns.
The first requirement is obvious and easy to accomplish,
but the second is difficult to understand and also to
accomplish and needs explanation,For example,if force
that may act on a fluid element in a fermenter during
agitation are the viscosity force FV,drag force on impeller
FD,and gravity force FG,each can be expressed with
characteristic quantities associated with the agitating
system,According to Newton’s equation of viscosity,
viscosity force is (9.1)
Where du/dy is velocity gradient and A is the area on
which the viscosity force acts,For the agitating system,
the fluid dynamics involved are too complex to
calculate a wide range of velocity gradients present.
However,it can be assumed that the average velocity
gradient is proportional to agitation speed N and the
area A is to D2I,which results.
(9.2)
The drag force FD can be characterized in an agitating
system as
A
dy
duF
V )(
2
IV NDF
(9.3)
Since gravity force FG is equal to mass m times gravity
constant g,(9.4)
The summation of all forces is equal to the inertial force
FI as,
= = (9.5)
Then dynamic similarity between a model(m) and a
prototype(p) is achieved if
= = = (9.6)
gDF IG 3
F?
GDV FFF 24 NDF II
PV
mV
F
F
)(
)(
PD
mD
F
F
)(
)(
PG
mG
F
F
)(
)(
PI
mI
F
F
)(
)(
ND
PF
I
mo
D?
Or in dimensionless forms:
=
= (9.7)
=
The ratio of inertial force to viscosity force is
= = = (9.8)
P
V
I
F
F )(
m
V
I
F
F )(
P
D
I
F
F )(
P
G
I
F
F )(
m
D
I
F
F )(
m
G
I
F
F )(
V
I
F
F
2
24
I
I
ND
ND
ND I2 iNRe
Which is the Reynolds number,Similarly,
= = = (9.9)
= = = (9.10)
Dynamic similarity is achieved when the values of the
nondimensional parameters are the same at geometrically
similar locations.
(NRei )P = (NRei )m
(NP )P = (NP )m (9.11)
(NFr )P = (NFr )m
D
I
F
F
NDP
ND
Imo
I
/
24?
mo
I
P
DN 53?
PN
1
G
I
F
F
gD
ND
I
I
3
24
g
ND I 2 FrN
Therefore,using dimensionless parameters for the
correlation of data has advantages not only for the
consistency of units,but also for the scale-up purposes.
However,it is difficult,if not impossible,to satisfy the
dynamic similarity when more than one dimensionless group
is involved in a system,which creates the needs of scale-up
criteria,The following example addresses this problem.
------------------------------------------------------
Example 9.1
The power consumption by an agitator in an unbaffled vessel
can be expressed as
53
I
mo
DN
P
),(
22
g
DNNDf II
Can you determine the power consumption and impeller
speed of a 1000 gallon fermenter based on the findings of
the optimum condition from a geometrically similar one-
gallon vessel? If you cannot,can you scale up by using a
different fluid system?
Solution:
Since Vp/Vm=1000,the scale ratio is,
= 1000 1/3 = 10 (9.12)
To achieve dynamic similarity,the three dimensionless
numbers for the prototype and the model must be equal,
as follows:
= (9.13)
mI
PI
D
D
)(
)(
P
I
mo
DN
P )(
53? m
I
mo
DN
P )(
53?
P
IND )(
2
= (9.14)
= (9.15)
If you use the same fluid for the model and the prototype,
ρ p=ρ m and μ p=μ m,Canceling out the same physical
properties and substituting Eq.(8.12) to Eq,(8.13) yields
(Pmo )P = 105( Pmo )m (9.16)
The equality of the Reynolds number requires
NP = 0.01Nm (9.17)
m
IND )(
2
P
I
g
DN )( 2
m
I
g
DN )( 2
3)(
m
P
N
N
On the other hand,the equality of the Froude number
requires
= (9.18)
Which is conflicting with the previous requirement for
the equality of the Reynolds number,Therefore,it is
impossible to satisfy the requirement of the dynamic
similarity unless you use different fluid systems.
Ifρp ρm and μp μm,to satisfy Eqs.(8.14) and (8.15),the
following relationship must hold.
= (9.19)
PN
mN10
1

m)(?
P)(
6.31
1
Therefore,if the kinematic viscosity of the prototype is
similar to that of water,the kinematic viscosity of the fluid
which needs to be employed for the model should be 1/31.6
of the kinetic viscosity of water,It is impossible to find the
fluid whose kinematic viscosity is that small,As a
conclusion,if all three dimensionless groups are important,
it is impossible to satisfy the dynamic similarity.
----------------------------------------------------------------
The previous example problem illustrates the difficulties
involved in the scale-up of the findings of small-scale results,
Therefore,we need to reduce the number of dimensionless
parameters involved to as few as possible,and we also need
to determine which is the most important parameter,so that
we may set this
we may set this parameter constant,However,even
though only one dimensionless parameter may be
involved,we may need to define the scale-up criteria.
As an example,for a fully baffled vessel when
NRei>10000,the power number is constant according to
Eq.(9.52),For a geometrically similar vessel,the
dynamic similarity will be satisfied by
= (9.20)
If the fluid employed for the prototype and the model
remains the same,the power consumption in the
prototype is
(9.21)
P
I
mo
DN
P )(
53? m
I
mo
DN
P )(
53?
PmoP )(
5
3
)(
)(
)()(?
mI
pI
m
p
mmo D
D
N
N
P
Where (DI)p/(DI)m is equal to the scale ratio,With a
known scale ratio and known operating conditions of a
model,we are still unable to predict the operating
conditions of a prototype because there are two unknown
variables,Pmo and N,Therefore,we need to have a certain
criteria which can be used as a basis.
8.2 Criteria of Scale-Up
Most often,power consumption per unit volume Pmo/v is
employed as a criterion for scale-up,In this case,to satisfy
the equality of power numbers of a model and a prototype,
(9.22)
2
3
3 )(
)(
)()(?
mI
pI
m
p
m
I
mo
D
D
N
N
D
P
p
I
mo
D
P )(
3
Note that Pmo/DI3 represents the power per volume
because the liquid volume is proportional to DI3for the
geometrically similar vessels,For the constant Pmo/DI3,
(9.23)
As a result,if we consider scale-up from a 20-gallon to a
2500-gallon agitated vessel,the scale ratio is equal to 5,
and the impeller speed of the prototype will be
(9.24)
2
3
)(
)(
)(
pI
mI
m
p
D
D
N
N
mm
pI
mI
p NND
D
N 34.0
)(
)(
3/2
Which shows that the impeller speed in a prototype vessel is
about one third of that in a model,for constant Pmo/v,the
Reynolds number and the impeller tip speed cannot be the
same,for the scale ratio of 5,
(9.25)
(9.26)
Table 8.1 shows the values of properties for a prototype
(2500-gallon) when those for a model (20-gallon) are
arbitrarily set as 1.0 (Oldshue,1966),The parameter values
of the prototype depend on the criteria used for the scale-up.
mP ii NN )(5.8)( ReRe?
mIPI NDND )(7.1)(?
Table 9.1
Properties of Agitator on Scale-Up
Property Model(20gal ) Prototype(2500gal)
Pmo 1.0 125 3125 25 0.2
Pmo/v 1.0 1.0 25 0.2 0.0016
N 1.0 0.34 1.0 0.2 0.04
DI 1.0 5.0 5.0 5.0 5.0
Q 1.0 42.5 125 25 5.0
Q/v 1.0 0.34 1.0 0.2 0.04
NDI 1.0 1.7 5.0 1.0 0.2
1.0 8.5 25 5.0 1.0 /2IND
When Pmo/v is set constant,The third column shows the
parameter values of the prototype,when Pmo/v is set
constant,The values in the third column seem to be more
reasonable than those in the fourth,fifth,and sixth
columns,which are calculated based on the constant value
of Q/v,NDI,and,respectively,For example when
the Reynolds number is set constant for the two scales,the
Pmo/v reduces to 0.16 percent of the model and actual
power consumption Pmo also reduces to 20 percent of the
model,which is totally unreasonable,
As a conclusion,there is no one scale-up rule that applies
to many different kinds of mixing operations,
Theoretically we can scale up based on geometrical and
dynamic similarities,but it has been shown that it is
possible for only a few limited cases,However,some
/2IND
principles for the scale-up are as follows (Oldshue,
1985):
1,It is important to identify which properties are
important for the optimum operation of a mixing
system,This can be mass transfer,pumping capacity,
shear rate,or others,once the important properties are
identified,the system can be scaled up so that those
properties can be maintained,which may result in,the
variation of the less important variables including the
geometrical similarity.
2.,The major differences between a big tank and a
small tank are that the big tank has a longer blend time,
a higher maximum impeller shear rate,and a low
average impeller shear rate.
3,For homogeneous chemical reactions,the power per
volume can be used as a scale-up criterion,As a rule of
thumb,the intensity of agitation can be classified based
on the power input per 1000 gallon as shown in Table
8.2.
Table 9.2
Criteria of Agitation Intensity
hp per 1000 gal Agitation Intensity
0.5～ 1 mild
2 ～ 3 vigorous
4 ～ 10 intense
4,For the scale-up of the gas-liquid contactor,the
volumetric mass transfer coefficient kLa can be used as a
scale-up criterion,In general,the volumetric mass-
transfer coefficient is approximately correlated to the
power per volume,Therefore,constant power per
volume can mean a constant kLa.
5,Typical impeller-to-tank diameter ratio DI/T for
fermenters is 0.33 to 0.44,By using a large impeller
adequate mixing can be provided at an agitation speed
which dose not damage living organisms,fermenters are
not usually operated for an optimum gas-liquid mass
transfer because of the shear sensitivity of cells,which
is discussed in the next section.
8.3 生物反应器放大的基本方法
1,常用的放大方法基础方法 主要为求解三传一反方程。
半基础方法 为求解简化的模型方程。
因次分析法（类比法） 使用相似原理，包括对流动及反应过程的分析。
经验法 即使凭经验，也仍需以一定的理论为依据。
试差法 多用于从实验室结果设计反应器原型。
实际上常常是以上多种方法混合使用，而因次分析法及经验法用得最多。目前仍是多种方法互相借鉴。
借助经验关联式进行判断，应用理论公式进行运算,
最后经实践检验。
“生物过程的反应与传递原理,课程模拟设计练习
1,阅读附件,并根据所给的条件对一个 气升式 生物反应器中培养液 -絮凝细胞颗粒 -空气三相系统的气含率和传质系数进行模拟研究,给出其关联式 。
2,根据上题所给的条件和您得到的关联式,保持 Pmo/v基本不变,将冷模实验的小反应器体积放大到 500倍,请设计一个中试气升式生物反应器,给出基本几何尺寸,并画出示意图 。