Buckingham Pi Theorem
The Buckingham Pi Theorem, also known as the π theorem, is a key concept in dimensional analysis. It states that if there are n variables and m fundamental dimensions, then there are n - m dimensionless π terms that can be formed. These dimensionless π terms can be used to express the relationships between the variables in a system, allowing for simplification and analysis.
4 Key excerpts on "Buckingham Pi Theorem"
eBook - ePub- Kaveh Azar(Author)
- 2020(Publication Date)
- CRC Press(Publisher)
...At present, however, the derivation of the specific π-terms probably does not seem to be any more direct than for the power-product method. 3.8 Identify independent q values intuitively: the Buckingham Pi Theorem Both White (1979) and Kline (1986) emphasize a more intuitive approach to obtaining the dimensionless products. This approach allows the engineer to more directly generate the πs that are important to them, rather than sorting through a very large number of sets of π hoping to stumble upon the best set for a particular use. This approach is based upon the Buckingham Pi Theorem which is stated in Kline as follows: In any given physical problem we have one or more dependent parameters, each of which is a function of some independent parameters. Let us denote any particular dependent parameter under scrutiny as qi. If the dependent parameters are m – 1 in number, then we may call them q 2, q 3, …, q m. And we may write in functional notation: q 1 = f 9 (q 1, q 2, q 3, …, q m) where f 9 is an unspecified function. Mathematically, this relation is entirely equivalent to the relation f 10 = (q 1, q 2, …, q m) = 0 (11) where f 10 is some other unspecified function. The Pi theorem states that given a relation among m parameters of the form of Equation 11, an equivalent relation expressed in terms of n nondimensional parameters can be found of the form f 11 = (π 1, π 2, …, π n) = 0 where the number n is given by the relation n = m − k where m is the number of q s in Equation 11 and k is the largest number of parameters contained in the original list of parameters q 1, q 2, …,. q m that will not combine into a nondimensional form. Following the usual practice, we shall refer to the nondimensional groups π 1, π 2, …, π n as pi’s. In order to proceed with a specific problem, select the maximum number of variables that do not form a π. Then proceed to form a set of π values by combining this set with each of the remaining variables in a power product...
eBook - ePub- William S. Janna(Author)
- 2020(Publication Date)
- CRC Press(Publisher)
...We would therefore expect 6 variables − 3 dimensions = 3 dimensionless groups 4.1.2 B uckingham PI M ethod Another approach to dimensional analysis is through the Buckingham pi method. In this method, the dimensionless ratios are called ∏ groups or ∏ parameters and could have been developed in a manner outlined by the following steps: 1. Select variables that are considered pertinent to the problem by using intuitive reasoning and write the relationship between the variables all on one side of the equation. 2. Choose repeating variables—that is, variables containing all m dimensions of the problem. It is convenient to select one variable that specifies scale (e.g., a length), another that specifies kinematic conditions (velocity), and one that is a characteristic of the fluid (such as density). For a system with three fundamental dimensions, ρ, V, and D are suitable. For a system with four fundamental dimensions, ρ, V, D, and g c are satisfactory. The repeating variables chosen should be independent of each other. 3. Write ∏ groups in terms of unknown exponents. 4. Write the equations relating the exponents for each group and solve them. 5. Obtain the appropriate dimensionless parameters by substitution into the ∏ groups. Solving a problem by following these five steps is illustrated in the next example. Example 4.2 Liquid flows horizontally through a circular tube filled with sand grains that are spherical and of the same diameter. As liquid flows through this sand bed, the liquid experiences a pressure drop. The pressure drop ∆ p is a function of average fluid velocity V, diameter D of the sand grains, spacing S between adjacent grains, liquid density ρ, and viscosity μ...
eBook - ePub- Keith J. Moss(Author)
- 2015(Publication Date)
- Routledge(Publisher)
...Buckingham’s Pi theorem of (n − m) = (6 − 3) = 3, therefore, agrees with this solution. You should now confirm that the three groups of terms are dimensionless. Example A2.5 Find the form of the equation for forced convection from a fluid transported in a straight pipe The heat transfer by forced convection from a fluid transported in a long straight tube is governed by the variables h, d, μ, ρ, k, C, θ and u such that the heat transfer coefficient for forced convection h = φ (d, μ, ρ, k, C, θ, u). Using dimensional analysis, determine the form of the equation and the dimensionless grouping. Solution If an exponential relationship is assumed in which B is the constant of proportionality and the indices a, b, c, e, f and g are numerical constants, Introducing dimensions to each of the terms. using Table A1.1 and including the indices as appropriate: Collecting the indices: For M, For L, For T, For θ, There are six unknown indices and four indicial equations. Evaluating in terms of the unknown indices c and f. From equation A2.7z, Substitute equation A2.7r into equation A2.7v, from which Substitute equations A2.7r and A2.7s into A2.7y, from which Substitute equations A2.7r, A2.7s and A2.7t into equation A2.7w from which Substituting the indicial equations for a, b, g and e into equation A2.7 The variables are now related to the unknown indices c and f and index 1.0. There are therefore three groups of variables. Thus Adopting Buckingham’s Pi theorem there are (n − m) = (7 − 4) = 3 dimensionless groups here, thus by rearranging the equation they are: The group (hd / k) is known as the Nusselt number Nu ; the group (dρu/μ) is the Reynolds number Re and the group (μC / k) is known as the Prandtl number Pr. The value of the numerical (dimensionless) constants B, c and f are found empirically (by experiment). It has been established that the values of B, c and f are constant for a very wide range of Re and Pr numbers and B = 0.023, c = 0.8 and f = 0.33...
- Amithirigala Widhanelage Jayawardena(Author)
- 2021(Publication Date)
- CRC Press(Publisher)
...Clearly the above four parameters satisfy these criteria. Therefore, it is possible to write F = f (D, V, μ, ρ) (8.1) If an experiment is carried out to establish the relationship of F with each of the parameters D, V, ρ, and, μ. 10 4 experiments would be needed assuming each relationship requires ten trials. For example, F vs. D : keeping V, ρ, μ constant require ten trials F vs. V : keeping D, ρ, μ constant require ten trials F vs. μ : keeping D, V, ρ constant require ten trials F vs. ρ : keeping D, V, μ constant require ten trials If each experiment takes 1 hour, the whole exercise would take 10,000 hours. Assuming 8 hours working day, this is equivalent to 1,250 days or 50 months assuming 25 working days per month or 4 years. This is both impractical in terms of time and the handling of the enormous amount of data. Meaningful results can be obtained with less effort by the use of dimensional analysis by expressing the relevant parameters in dimensionless form: F ρ V 2 D 2 = f (ρ V D μ) (8.2) The function “f” still needs to be determined experimentally. However, we need only ten experiments rather than 10,000 experiments. It is no longer necessary to select ten different sizes or ten different fluids. Instead, only ten different combinations of the basic parameters would be needed. 8.2 Buckingham’s pi () theorem Suppose the dependent parameter of a physical problem is expressed as a function of n − 1 independent parameters as q 1 = f (q 2, q 3, …, q n) (8.3) which may be written as g (q 1, q 2, …, q n) = 0 (8.4) where g is a function different from f Then, Buckingham’s π theorem states that the n parameters may be grouped into (n − m) independent dimensionless ratios or π parameters, expressed in the. form G (π 1, π 2, …, π n − m) = 0 (8.5a) or π 1 = G 1 (π 2, π, 3 …, π n − m) (8.5b) where m is usually, but not necessarily equal to the minimum number of independent dimensions required to specify all the parameters q 1, q 2, …, q n...
