Which is the Reynolds number. Similarly F PD,N (9.9) F Pmo /D, 10 2 F PD;N D, N Fr (9.10) D g Dynamic similarity is achieved when the values of the nondimensional parameters are the same at geometrically similar locations x7/x7 Rei p l丿m (NPP=(N Pm (NErP(NFr)
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 P D N D N mo I I / 4 2 mo I P N D 3 5 NP 1 G I F F D g D N I I 3 4 2 g DI N 2 NFr
Therefore, using dimensionless parameters for the correlation of data has advantages not only for the consistency ofunits, but also for the scale-up purposes However, it is difficult, if not impossible, to satisfy the ynamIc 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 no PNDZ N2DI DND
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 = 3 5 I mo N D P ( , ) 2 2 g ND N D f I I
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 vv=1000. the scale ratio is (D) P=100013 =10 (9.12) (D) To achieve dynamic similarity, the three . ionless numbers for the prototype and the model must be equal as follows P VD)=( nmo nmo ON (9.13)
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 onegallon 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) I m I P D D ( ) ( ) P I mo N D P ( ) 3 5 m I mo N D P ( ) 3 5