Question

Consider steady, laminar, two-dimensional, incompressible flow with constant properties and a Prandtl number of unity. For a given geometry, is it correct to say that both the average friction and heat transfer coefficients depend on the Reynolds number only?

Short answer

Answer: Yes, for a given geometry under the specified conditions of steady, laminar, two-dimensional, incompressible flow with constant properties and a Prandtl number of unity, both the average friction (Cf) and heat transfer coefficients (h) depend on the Reynolds number only.

Step-by-step solution

Examine the dependence of the average friction coefficient on the Reynolds number

For laminar flows, the average friction coefficient, Cf, is generally given by the Blasius relation: Cf = 0.664 / sqrt(Re) Here, Re denotes the Reynolds number. The average friction coefficient is inversely proportional to the square root of the Reynolds number, so indeed, its value depends only on the Reynolds number for a given laminar flow.

Examine the dependence of the average heat transfer coefficient on the Reynolds number

For laminar flows with Prandtl number (Pr) equal to unity, the average Nusselt number (Nu) can be expressed as a function of the Reynolds number and Prandtl number. A common correlation is the Gnielinski correlation: Nu = (f/2) * Re * Pr / (1 + 12.7 * sqrt(f/2) * (Pr^(2/3) - 1)) Given that Pr = 1 for this case, we can simplify this correlation as: Nu = (f/2) * Re Here, f denotes a factor that depends only on the geometry and is constant for a given geometry. Nu is the ratio of the average heat transfer coefficient (h) to the thermal conductivity (k), and it depends on the product of the Reynolds number and the geometry factor f.

Draw conclusions on the dependence of the average friction and heat transfer coefficients on the Reynolds number

Based on the relationships derived in Steps 1 and 2, we can conclude that the average friction coefficient (Cf) depends solely on the Reynolds number for a steady, laminar, two-dimensional, incompressible flow with constant properties and a Prandtl number of unity. For the average heat transfer coefficient (h), its dependence on the Reynolds number is not immediate, as it is a part of the Nusselt number (Nu). However, for a given geometry where the factor f is constant, the Nusselt number essentially depends on the Reynolds number due to the simplified Gnielinski correlation. So, for a given geometry under the specified conditions of steady, laminar, two-dimensional, incompressible flow with constant properties and a Prandtl number of unity, it is correct to say that both the average friction and heat transfer coefficients depend on the Reynolds number only.