B number

Bbnnagel formula B-numbers Boar urine odor Boat keels Bock beer Bodied oils... 

Fig. 25. McCabe-Thiele diagram, (a) Number of theoretical plates, 5 (b) number of actual plates, 8.

Since the solutions AT are related to one another by a multiphcative constant, there is only one hneady independent solution and hence only one independent B-number. Choose, for simplicity, = 1. Equation 17 can be rewritten as (eq. 18) ... 

The example demonstrates that not all the B-numbers of equation 5 are linearly independent. A set of linearly independent B-numbers is said to be complete if every B-number of Dis a product of powers of the B-numbers of the set. To determine the number of elements in a complete set of B-numbers, it is only necessary to determine the number of linearly independent solutions of equation 13. The solution to the latter is well known and can be found in any text on matrix algebra (see, for example, (39) and (40)). Thus the following theorems can be stated. 

Theorem 1. The number of products in a complete set of B-numbers associated with a physical phenomenon is equal to n — r, where n is the number of variables that are involved in the phenomenon and ris the rank of the associated dimensional matrix. 

This result was first discussed by Buckingham (8) and stated in its present form by Langhaar (23). It states in effect that an equation is dimensionally homogeneous if and only if it can be reduced to a relationship among a complete set of B-numbers. Buckingham s result (8) was originally stated as Theorem 2. 

In terms of linear vector space, Buckingham s theorem (Theorem 2) simply states that the null space of the dimensional matrix has a fixed dimension, and Van Driest s rule (Theorem 3) then specifies the nullity of the dimensional matrix. The problem of finding a complete set of B-numbers is equivalent to that of computing a fundamental system of solutions of equation 13 called a complete set of B-vectors. For simplicity, the matrix formed by a complete set of B-vectors will be called a complete B-matrix. It can also be demonstrated that the choice of reference dimensions does not affect the B-numbers (22). 

Theorem 4. The set of B-numbers associated with a physical phenomenon is invariant with respect to the choice of the reference dimensions provided that the reference dimensions are considered independent, and that the number of these reference dimensions is not altered. 

The implication of this theorem is important in that in computing a complete set of dimensionless products or B-numbers associated with a physical phenomenon, it does not matter which set of dimensions are chosen as the reference dimensions as long as they are independent and their number is not altered. 

Once the dimensional matrix has been set up and the number of products in a complete set of B-numbers determined, a complete set of B-vectors must be computed. In the following, a systematic procedure for this purpose is presented. 

This relation is not in the best form for the calculation of the drag since 8 appears in both products. Hence it is necessary to change the two independent B-numbers by requiting that 8 occur in only one of them. To this end, we let Af be a nonsingular submatrix of of order 2 containing the tow corresponding to 8. Thus, tow 1 and, eg, tow 5 of ate chosen to give... 

Thus, the transpose of is a complete B-matiis. Since there are seven variables involved in the phenomenon and the tank of Dis 4, from Theorem 1 there are three dimensionless products in a complete set of B-numbers, each of which corresponds to a row of... 

Hence, the right-hand side of equation 52 is a desired complete B-matrix. A functional relationship among the associated B-numbers can be obtained and is given by (eq. 53) ... 

The number of independent variables is reduced from the original nine to four. This is a great saving in terms of the number of experiments required to determine the desired function. For example, suppose that a decision is made to test only four values for each variable. Then it would require 4 = 262144 experiments to test aU. combinations of these values in the original equation. As a result of equation 59, only 4 = 256 tests are now required for four values each of the four B-numbers. 

Proton magnetic resonance (carbon tetrachloride) B (number of, protons, multiplicity) 3.63 (2, singlet), 3.48 (2, singlet), 3.27 (6, singlet). The infrared and mass spectra are also as reported. ... 

Enter the 9-digit number assigned by Dun and Bradstreet (D B) for your facility or each establishment within your facility. These numbers code the facility for financial purposes. This number may be available from your facility s treasurer or financial officer. You can also obtain the numbers from your local Dun and Bradstreet office (check the telephone book White Paoest. If none of your establishments has been assigned a D B number, enter not applicable, NA, in box a. If only some of your establishments have been assigned Dun and Bradstreet numbers, enter those numbers in Section 3.7. 

B = number of blades on impeller R = number of baffles = agitator diameter H = liquid height... 

Internal gear None Power input and heat generated increases with increasing viscosity For higher viscosities (a) Speed may be reduced (b) Number of gear teeth reduced (c) Lobe-shaped gears employed... 

B. Number of Propagation Centers in One-Component Catalysts 1. The Chromium-Oxide Catalysts... 

H.2 Indicate which of the following are conserved in a chemical reaction (a) mass (b) number of atoms (c) number of molecules (d) number of electrons. 

Figure 3. Carbon Monoxide at Cuitlahuac Station, (a) Number of 8-h moving periods in which the Standard is exceeded, (b) Number of hours in which 26 ppm is exceeded.

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