Obtain by dimensional analysis a functional relationship for the wall heat transfer coefficient for a fluid flowing through a straight pipe of circular cross-section. Assume that the effects of natural convection may be neglected in comparison with those of forced convection. It is found by

Question

Obtain by dimensional analysis a functional relationship for the wall heat transfer coefficient for a fluid flowing through a straight pipe of circular cross-section. Assume that the effects of natural convection may be neglected in comparison with those of forced convection.
It is found by experiment that, when the flow is turbulent, increasing the flowrate by a factor of 2 always results in a 50% increase in the coefficient. How would a 50% increase in density of the fluid be expected to affect the coefficient, all other variables remaining constant?

Accepted Answer

For heat transfer for a fluid flowing through a circular pipe, the dimensional analysis is detailed in Section 9.4.2 and, for forced convection, the heat transfer coefficient at the wall is given by equations 9.64 and 9.58 which may be written as:

[latex]h d / k=f\left(d u \rho / \mu, C_p \mu / k\right)[/latex]

or: [latex]h d / k=K(d u \rho / \mu)^n\left(C_p \mu / k\right)^m[/latex]

∴ [latex]h_2 / h_1=\left(u_2 / u_1\right)^n[/latex].

Increasing the flowrate by a factor of 2 results in a 50% increase in the coefficient, or:

[latex]1.5=2.0^n[/latex] and [latex]n=(\ln 1.5 / \ln 2.0)=0.585[/latex].

Also: [latex]h_2 / h_1=\left(\rho_2 / \rho_1\right)^{0.585}[/latex]

When [latex]\left(\rho_2 / \rho_1\right)=1.50, h_2 / h_1=(1.50)^{0.585}=1.27[/latex] and the coefficient is increased by [latex]\underline{\underline{27 \%}}[/latex]

Question 1.P.9: Obtain by dimensional analysis a functional relationship for...

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