The volume of liquid flowing per second is called the volume flow rate Q and has
ID: 1952666 • Letter: T
Question
The volume of liquid flowing per second is called the volume flow rate Q and has the dimensions of [L]3/[T]. The flow rate of a liquid through a hypodermic needle during an injection can be estimated with the following equation:Q = Rn(P2 - P1)/(8L)
The length and radius of the needle are L and R, respectively, both of which have the dimension [L]. The pressures at one end of the needle and the other end are P2 and P1, both of which have the dimensions of [M]/{[L][T]2}. The symbol represents the viscosity of the liquid and has the dimensions of [M]/{[L][T]}. The symbol stands for pi and, like the number 8 and the exponent n, has no dimensions. Using dimensional analysis, determine the value of n in the expressions for Q.
Explanation / Answer
Your statement of the question has some ambiguity. I assume that n is indeed an integer (why do you say decimal ?), and that on the bottom line, 8(n)(l), the character which you put as n is meant to be ?. With these assumptions : Subsititute the dimensions for each term in the equation ; . . . . p R^n P Q = . ------------- . . . . . 8 ? l (The subtraction P2 - P1 is only done to find the difference in pressure, and the result is just a pressure, so we only need to consider one set of dimensions for pressure). Doing the substitution : . . . . . . . . p L^n M L?¹ T?² L³ T?¹ . =. . --------------------- . . . . . . . . . 8 M L?¹ T?¹ (It is far simpler to change all the dimensions to index form as above than to deal with "something over something over something else" and so on) I put in the p and 8 for completeness above, but, being dimensionless, they have no effect on the rest, and we can drop them. Now, for further simplification, I move all the bottom line to the top, not forgetting to change the signs of the exponents : L³ T?¹ = (L^n M L?¹ T?²) (M?¹ L T) Now just add the exponents of the like terms L³ T?¹ = L^n M¹?¹ L¹?¹ T?¹ . . . . . . .(the T has power (-2 + 1) of course : can't do this with superscript) So L³ T?¹ = L^n T?¹ And comparing the two sides of the equation, it is obvious that n = 3.
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