Power Consumption of a Stirred Nonaerated Tank A Simple Example

The first step is to find all parameters which are relevant to the question (Fig. 5.13). Will this parameter influence the power consumption in the stirred tank if changed?

Looking at Fig. 5.13, we will first mention all the geometric parameters:

D = diameter of the tank d = diameter of the stirrer H = height of the water h = distance of the stirrer from the bottom

The stirrer may be equipped with three plates with a given width and distance from the wall. Usually, the stirrer diameter d is selected as the reference and we can define three dimensionless geometric numbers:

Tanks of different sizes are considered to be geometrically similar if these values are constant for all tanks.

Furthermore, we will consider a group of tanks with different stirrers of diameter d. Let us now follow the method of Zlokarnik (1999). Fig. 5.13 Cylindrical tank with Simplex rotor, Type A, showing main dimensions.

The next step is to look for parameters which will influence the power input P. These parameters can be divided into:

• Geometric parameters: diameter of the stirrer d.

• Material parameters of the water: density p, kinematic viscosity v.

• Process parameters: stirrer speed n.

Therefore, the list of relevant variables is: {P; d; p; v; n}.

The next step is to formulate the dimensional matrix. Before doing this, we have to decide which parameters should occur only in one dimensionless characteristic number. It is most effective to select P and v. After making this decision, the dimensional matrix can be formed (Table 5.3), with each number in the matrix representing the power of the unit dimension (length, time, mass) occurring in the variable.

This matrix must be transformed to obtain a unit matrix in the first 3 x 3 matrix, that is, with a diagonal of ones. To obtain a 0 instead of -3 in the first column (p) and a 1 instead of -1 in the third column (n), the Gaussian method is applied by performing linear combinations of the rows. In this manner -3 times the elements of the first row are added to the second row. The exponents listed in the last row need only be multiplied by -1 to obtain a 1 in the proper location (Table 5.4).

The numbers in the remaining matrix to the right of the unit matrix below P and v give us the exponents of the first three variables which yield dimensionless num-

Table 5.3 Dimensional analysis of a stirrer, matrix of unit dimensions for relevant parameters with respect to selected P and v (Zlokarnik 1999).

P

d

n P11

V

Mass M Length L Time T

-3 0

0-

0 1 0 2 1 -3

0 2 -1

Core matrix

 P d n P11 V M 1 0 0 1 0 3 M + L 0 1 0 5 2 -T 0 0 1 3 1 Unit matrix Rest matrix

bers. The first dimensionless number n1 formed with P is known as the Newton number. The second, formed with v, is the Reynold's number:

p1 n3d5

and:

2 p0n1d2 nd2 There must be a function of the five parameters for which:

Another way to find Eq. (5.85) starting from the list of the five variables f (P, p, n, d, v) = 0 is described below.

Looking for a function, with a reduced number of dimensionless parameter, we have at first to write:

and working in the MLT-system (mass, length, time) for the needed dimensions, we introduce P [ML2 T-3], p [M L-3], n [T-1], d [L], v [L2 T-1]:

As can be demonstrated now, this problem can be described by using only two dimensionless numbers resulting from the fact, that we use five parameter with dimensions and three units (5-3 = 2). If we want to obtein two dimensionless numbers each with P and v, the exponents of p, n and d have to be replaced by the exponents of P and v. Writing n ((ML2 T-3)a (ML-3)13 (T-1)Y (L)5 (L2 T-1)E)

we obtain three equations:

and the solutions

3 = - a Y = - 3 a - e 5= - 5a- 2e which have to be introduced into ni (Pa, p3, nY, d5, ve). ni is only dimensionless, if

giving the numbers Ne and Re (see Eqs. 5.83 and 5.84).