Question

In turbulent flow, one can estimate the Nusselt number using the analogy between heat and momentum transfer (Colburn analogy). This analogy relates the Nusselt number to the coefficient of friction, \(C_{p}\) as (a) \(\mathrm{Nu}=0.5 C_{f} \operatorname{Re} \mathrm{Pr}^{1 / 3}\) (c) \(\mathrm{Nu}=C_{f} \operatorname{Re} \mathrm{Pr}^{1 / 3}\) (e) \(\mathrm{Nu}=C_{f} \operatorname{Re}^{1 / 2} \operatorname{Pr}^{1 / 3}\) (d) \(\mathrm{Nu}=C_{f} \operatorname{Re} P r^{2 / 3}\)

Short answer

Answer: (e) Nu=Cf Re^(1/2) Pr^(1/3)

Step-by-step solution

(Understand the given formulas)

First, let's break down the given formulas for Nusselt number estimation based on the Colburn Analogy. (a) \(\mathrm{Nu}=0.5 C_{f} \operatorname{Re} \mathrm{Pr}^{1 / 3}\) (c) \(\mathrm{Nu}=C_{f} \operatorname{Re} \mathrm{Pr}^{1 / 3}\) (e) \(\mathrm{Nu}=C_{f} \operatorname{Re}^{1 / 2} \operatorname{Pr}^{1 / 3}\) (d) \(\mathrm{Nu}=C_{f} \operatorname{Re} P r^{2 / 3}\) All of these formulas have the same variables: Nusselt number (Nu), coefficient of friction (\(C_f\)), Reynolds number (Re), and Prandtl number (Pr). However, they have different coefficients and exponents.

(Colburn Analogy)

The Colburn analogy is a useful method for estimating heat transfer in turbulent flows because it relates the Nusselt number to the coefficient of friction, Reynolds number, and Prandtl number. The correct formula for the Nusselt number using the Colburn analogy is: $$ \mathrm{Nu}=0.0296 C_{f} \operatorname{Re}^{4/5} \operatorname{Pr}^{1 / 3} $$ Now, compare the correct Colburn analogy formula with each of the given formulas to see which one is the most accurate representation.

(Comparing formulas)

Comparing each of the given formulas to the correct Colburn analogy formula: (a) \(\mathrm{Nu}=0.5 C_{f} \operatorname{Re} \mathrm{Pr}^{1 / 3}\) - This formula has different coefficients and Re exponents compared to the correct Colburn analogy formula. (c) \(\mathrm{Nu}=C_{f} \operatorname{Re} \mathrm{Pr}^{1 / 3}\) - This formula has different coefficients and Re exponents compared to the correct Colburn analogy formula. (e) \(\mathrm{Nu}=C_{f} \operatorname{Re}^{1 / 2} \operatorname{Pr}^{1 / 3}\) - This formula has the correct Pr exponent but a different coefficient and Re exponent compared to the correct Colburn analogy formula. (d) \(\mathrm{Nu}=C_{f} \operatorname{Re} P r^{2 / 3}\) - This formula has different coefficients and exponents for both Re and Pr compared to the correct Colburn analogy formula.

(Selecting the most accurate representation)

None of the given formulas exactly matches the correct Colburn analogy formula for estimating Nusselt number in turbulent flow. However, selecting the closest representation: $$ \textbf{(e) }\mathrm{Nu}=C_{f} \operatorname{Re}^{1 / 2} \operatorname{Pr}^{1 / 3} $$ Formula (e) has the correct Pr exponent and a relatively similar Reynolds number exponent compared to the correct Colburn analogy formula, making it the most accurate representation among the given choices.