By Gregory T. Benz

**Introduction**

Many agitator applications, such as blending of miscible liquids or suspending slowly settling solids such as starch and grain slurries used in ethanol production, may be classified as flow-controlled applications. That is to say, the process result depends on creating a certain characteristic velocity, V_{b}, which is calculated on a square-batch basis (liquid height equals tank diameter) by dividing the flow produced by the impeller (Q) by the cross sectional area of the tank (A):

1) V_{b }= Q/A

For other batch geometries, the calculations are done on the basis of a square batch with the same volume. The flow created by an impeller is equal to its pumping number, N_{Q}, times its shaft speed, N, times the impeller diameter cubed:

2) Q = N_{q}ND^{3}

The pumping number for a given impeller type is a function mainly of Reynolds number and D/T (impeller to tank diameter) ratio. Thus, the same amount of pumping can be created by a small impeller turning at a high speed or a larger impeller turning at a slower speed. As we shall see, the required power is very different as a function of impeller diameter.

Impeller power draw is equal to the impeller power number, N_{P}, times the density of the fluid, times the shaft speed cubed and the impeller diameter to the 5^{th} power:

3) P = N_{P}ρN^{3}D^{5}

Power number is also dependent on Reynolds number and D/T ratio. For the purposes of this article, turbulent flow will be assumed. Under such conditions, the pumping number and power number are both constant at a given D/T ratio.

For constant flow, power required decreases as D/T increases, though with limits. A D/T ratio greater than 1 does not seem to work well, for example. (More seriously, there are flow pattern effects at large D/T, as we will see later).

**Example Problem**

To keep it simple, we will use a square batch geometry. We choose a 20 foot diameter flat bottomed tank, with a 20 foot liquid level, giving us a volume of 47000 gallons or 178000 liters. The fluid in the tank could be a starch slurry, a dry grind corn slurry or other grain slurry common in the ethanol industry (but not cellulosic unless full hydrolysis has occurred), with a specific gravity of 1.2 and a viscosity of less than 50 cP, which will result in turbulent flow. By experience, it has been found that a characteristic velocity of 18 feet/minute or 0.091 m/s is adequate to maintain suspension of such slurries. Below we have calculated a table of results for D/T ratios ranging from 0.2 to 0.6. Sizing and impeller characteristics are based on a generic, 3-blade hydrofoil, typical of that produced by most agitator vendors. The impeller is located 4 feet off tank bottom.

Table 1 Grain slurry tank agitation parameters. | ||||||

D/T | Impeller diameter, feet | Power number, N_{P} |
Pumping Number, N_{Q} |
Required shaft speed, rpm | Power draw, Hp, at SG =1.2 | Estimated capital cost, $K |

0.2 | 4 | 0.35 | 0.68 | 130 | 15.4 | 55 |

0.3 |
6 | 0.3 | 0.53 | 49 | 5.5 | 42 |

0.4 | 8 | 0.28 | 0.44 | 25 | 2.8 | 44 |

0.5 | 10 | 0.26 | 0.38 | 15 | 1.7 | 47 |

0.6 | 12 | 0.28 | 0.36 | 9 | 1.0 | 67 |

**Discussion**

As expected, the power draw dramatically decreases as the D/T increases. The capital cost, estimated by the author, goes through a minimum. Larger impellers cost more than smaller ones. However, the shaft design is often critical speed limited at smaller D/T ratios, so a more expensive shaft system may be required. At really slow shaft speeds, a more expensive gear drive system may be required. Under normal competitive bidding situations, an agitator vendor will tend to quote the lowest capital cost design, which in this case would be about a D/T of 0.3. However, when the present worth of electric power is taken into account, a D/T ratio of 0.4 or 0.5 may be a better choice. It is up to the buyer to ask the vendor to quote energy saving designs. We included a D/T of 0.6 as an extreme case, but it actually creates a poor flow pattern, as will be seen below.

**D/T effects on flow pattern**

As axial-flow impellers become larger, the flow pattern produced becomes more radial. There is also a tendency to get dead spots in the bottom center area of the tank, and at the junction of the side wall and the tank bottom, particularly in flat or sloped bottom tanks. In such areas, solid fillets tend to build up, which is not a desirable result. The progression of flow pattern changes is illustrated in the below series of CFD plots.

Based on the above figures, a maximum D/T of 0.5 should be considered for this application. Build-up of solids in the tank center and at the bottom sidewall is likely with a D/T of 0.6.

**Conclusions**

For flow controlled applications, agitator power input does not define process results. Within reason, significant energy savings may be achieved by using larger impellers rather than smaller ones, up to a D/T ratio of about 0.5 in low viscosity, turbulent flow applications.

**Acknowledgment**

The CFD figures used in this article were provided by Chemineer, a brand of NOV, Inc.

**About the Author**

Gregory Benz is a member of Lee Enterprises Consulting, the world’s premier bioeconomy consulting group, with more than 100 consultants and experts worldwide who collaborate on interdisciplinary projects, including those requiring the technologies discussed in this article. The opinions expressed herein are those of the author, and do not necessarily express the views of Lee Enterprises Consulting. Mr. Benz is also President of Benz Technology International, Inc.