Empirical correlations

The general form of the power characteristic describing the relationship between the power number ($N_e$) and the Reynolds number ($Re$) (see Nagata, 1975) is

$$N_e=\frac{A}{Re}+B \left(\frac{10^3+0.6 \cdot f \cdot Re^\alpha} {10^3+1.6 \cdot f \cdot Re^\alpha} \right)^p$$ (13)

where the first term represents power consumption in the laminar range and the second term represent power consumption in the turbulent range. Coefficients of the empirical equation $A$, $B$, $p$, $f$ and $\alpha$ are to be estimated from the geometrical characteristics of the CSTR. The following correlations are available for parameter estimation:

1. Correlation of the coefficient $A$
   

   $$A=14.+\frac{b}{D} \left[670. \cdot \left(\frac{d}{D}-0.6\right)^2+85.\right]$$ (14)

   
   
   where, $D$ is vessel diameter, $d$ is impeller diameter, $b$ is blade height.
2. Correlation of the coefficient $B$
   

   $$B=10.^{\left[1.3-4. \cdot \left(\frac{b}{D}-0.5\right)^2-1.14\frac{d}{D}\right]}$$ (15)

   
   

3. Correlation of the exponent $p$
   

   $$p=1.1+4.\cdot \frac{b}{D}-2.5 \left(\frac{d}{D}-0.5 \right)^5-7. \left( \frac{b}{D} \right)^4$$ (16)

   
   

4. Values for $f$ and $\alpha$: $f=2$, $\alpha=0.66$

Correlations are derived for a simple configuration (single paddle impeller, $H/D=1$, where $H$ is liquid height, vertical blades $-$ $\theta=90^o$). Corrections are needed for CSTRs with different configuration:

1. For different number of paddles and different type of impeller, an equivalent blade height is defined as:
   

   $$b_{eq}=b \cdot n_p$$ (17)

   
   
   where $n_p$ is the number of paddles.
2. Effect of the liquid depth is accounted for by a multiplicative factor $C$ for the term representing power consumption in the turbulent range. $C$ is given by:
   

   $$C=\left(\frac{H}{D}\right)^{0.35+\frac{b}{D}}$$ (18)

   
   

3. For paddles with blade inclination $\theta \neq 90^o$ an additional multiplicative factor $C_1$ is used for the turbulent term,
   

   $$C_1=(\sin \theta)^{1.2}$$ (19)

   
   

Power consumption increases if baffles are present in the vessel. Maximum power consumption is attained when the following relation holds for the number of baffles ($n_B$) and the baffle width ($B_w$) (``fully baffled'' conditions):

$$\left(\frac{B_w}{D}\right)^{1.2} \cdot n_B=0.35$$ (20)

In this case, the value of $N_{e,max}$ is given by

$$N_{e,max}=\frac{A}{Re}+B \cdot C$$ (21)

Power numbers $N_{e,B}$ under partially baffled conditions are estimated by the following equation:

$$\frac{N_{e,max}-N_{e,B}}{N_{e,max}-N_{e,\infty}}=\left[1-2.9 \left(\frac{B_w}{D}\right)^{1.2} n_B\right]^2$$ (22)

where $N_{e,\infty}$ is the power number obtained at $Re$ tending to infinity, i.e.

$$N_{e,\infty}=B\left(\frac{0.6}{1.6}\right)^p$$ (23)

These equations were used to estimate the power characteristic versus Reynolds number for the laboratory vessel and the industrial CSTR model CE12500.