(a) Explanation Write the functions for the distance. l = f ( h , d , V , γ , μ ) (I) Here, the particle specific weight is γ , the viscosity of the fluid is μ , the distance from the ground is l , the diameter of the particles d , the velocity of the fluid is V and the height is h . Write the dimension of the diameter in F L T unit system. d = [ L ] (II) Here, the dimensions for diameter is L . Write the dimension of the length in F L T unit system. l = [ F ] (III) Here, the dimensions for length is F . Write the dimension of the velocity in F L T unit system. V = [ L T − 1 ] (IV) Here, the dimension for distance is L and the time is T . Write the expression for absolute viscosity. μ = u ρ (V) Here, the kinematic viscosity is u and the density of the fluid is ρ . Substitute [ L 2 T − 1 ] for u and [ F L − 4 T 2 ] for ρ in Equation (V). μ = [ L 2 T − 1 ] × [ F L − 4 T 2 ] = [ F L − 2 T ] (VI) Here, the dimension for force is F , the dimensions for length is L and the dimension for time is T . Write the expression for the specific weight of the liquid. γ = m v (VII) Here, the weight of liquid is m , the volume of liquid is v and the specific weight of the liquid is γ . Substitute [ F ] for m and [ L 3 ] for v in Equation (VII). γ = [ F ] [ L 3 ] = [ F L − 3 ] (VIII) Here, the dimensions for force is F and the dimensions for length is L . Write the expression for the number of variable. n = k − r (IX) Here, the number of variable is k , the number of dimensions is r and the number of pi terms is n . Write the expression for first pi terms. Π 1 = l μ a γ b h c (VIII) Here, the constant are a , c and b . Write the expression for first pi terms. Π 2 = d μ a γ b h c (IX) Write the expression for first pi terms. Π 3 = V μ a γ b h c (X) Write the dimension for pi term. Π = L 0 T 0 (XI) Write the function for first pi terms. Π 1 = f ( Π 2 , Π 3 ) (XII) Write the expression for velocity for model fluid. V m = ( μ μ m ) ( h m h ) 2 ( γ m γ ) 2 × V (XIII) Here, the viscosity for prototype is μ , the height for the prototype is h , the velocity for the prototype is V and the specific weight for prototype is γ , the viscosity for model is μ m , the height for the model is h m , the velocity for the model is V m and the specific weight for model is γ m , Conclusion: The number of variable is 6 and the number of dimensions is 3 . Substitute 6 for k and 3 for r in Equation (IX). n = 6 − 3 = 3 Substitute [ L ] for h , [ F 0 L 0 T 0 ] for Π 1 , [ F L − 3 ] for γ , [ L ] for l and [ F L − 2 T ] for μ in Equation (VIII). [ F 0 L 0 T 0 ] = [ L ] [ F L − 2 T ] a [ F L − 3 ] b [ L ] c [ F 0 L 0 T 0 ] = [ L ] [ F a L − 2 a T a ] [ F b L − 3 b ] [ L c ] [ F 0 L 0 T 0 ] = [ F a + b L 1 − 2 a − 3 b + c T a ] (XIV) Compare the exponents of T in Equation (XIV). a = 0 (XV) Compare the exponents of F in Equation (XIV). a + b = 0 b = − a (XVI) Compare the exponents of L in Equation (XIV). 1 − 2 a − 3 b + c = 0 (XVII) Substitute 0 for a in Equation (XVI). b = 0 (XVIII) Substitute 0 for a and 0 for b in Equation (XVII) (b) To determine The predicted length for prototype.

8th Edition

ISBN: 9781119547990

Author: HOCHSTEIN

Publisher: JOHN WILEY+SONS INC.

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Chapter 7.8, Problem 54P

(a)

To determine

The velocity of the model.

(b)

To determine

The predicted length for prototype.

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Chapter 7.8, Problem 53P

Chapter 7.8, Problem 55P

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