# Laminar flow in a tube, heat transfer coefficient-sanity check

## Main Question or Discussion Point

Hi there,

Hopefully this is a very easy question and you all can just confirm this for me.

When calculating heat transfer into a fluid from a heated tube, is it correct to say that the heat transfer coefficient is *not* dependent on the tube diameter?

So, if we solve for T_{out}, we get:

Substituting h for K*N/D, which is fluid thermal conductivity K, Nusselt number (depends on flow conditions and location in flow), and D is diameter, we get:

Finally, for our circular duct, A = pi*D*dx, so we get:

So, is there no dependence on tube diameter? I know that Nusselt number is *weakly* dependent on diameter when the flow is still developing, but that seems like it. In other words, given a large enough furnace, I could put a tube of any size in that furnace, and the flow would heat just as quickly regardless of tube diameter. That feels wrong to me, is there something I am missing?

Thanks!

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Chestermiller
Mentor
For laminar flow in a tube (with constant wall temperature), what is the equation for the local Nussult number as a function of the Reynolds number, Prantdl number, and x/D in the thermal entrance region?

For laminar flow in a tube (with constant wall temperature), what is the equation for asymptotic Nussult number at large distances along the tube?

Chestermiller
Mentor
As best as I can tell, your assessment is correct. For laminar flow, Nu is virtually independent of D.

Chestermiller
Mentor
Dimensional analysis of the partial differential heat balance equation shows that the dimensionless temperature $\frac{T-T_0}{T_w-T_0}$ is a function only of the dimensionless axial position $\frac{kz}{WC_p}$ and the dimensionless radius r/R. The dimensionless axial position is independent of diameter.

Yup. After sleeping on it, I felt pretty correct about what I had written. But it is nice to have had someone else look it over. Thank you for your time!

Go Blue!

'06 School of Education