# How to size a heat exchanger for a shear thinning fluid?

To size a heat exchanger, I need to know (among other things) the Reynolds number (Re) as an indication of the flow conditions. The Re number depends on the viscosity. In a shear thinning fluid, I can't assume a constant viscosity, instead it will depend on the shear rate that is not constant throughout the pipe. I've asked on Physics SE about the shear rate in turbulent shear thinning fluid flow in a pipe, but received no helpful answer.

Here at Engineering.SE, I am not looking for an in-depth examination of the beauty of the Navier-Stokes equation, but for a practical approach to sizing a heat exchange for such a fluid. How to go about it?

• (1) Is there a reason you cannot use experiment? (2) Do you have a preferred style of heat exchanger e.g. shell and tube? Is your shear thinning fluid on the tube side? (3) This abstract suggests it is a hard problem, probably beyond the scope of a Q&A website: dx.doi.org/10.1111/j.1745-4530.2008.00321.x Jan 28 '15 at 10:16
• 1) no ressources (plus admittedly I can delegate the problem to a supplier but I'm curious) 2) shell an tube, yes 3) lemme have a look ...
– mart
Jan 28 '15 at 10:40
• Did you ever come to a conclusion about this practical approach?
– Air
Mar 17 '15 at 19:55
• If you don't mind, asking out of curiosity - what are you using that shear-thinning liquid for? I've read about them a little but I couldn't find anything on applications outside of food industry (stick sauces to food).
– SF.
Mar 31 '15 at 20:13
• My company build biogas plants, and the thick slurries we have are shear thinning. We only sometimes use HX, mostly we heat our plants with pipes in the tanks, if we use HX we let the suppliers do the sizing. I just want to see if I land in a similiar ballpark.
– mart
Apr 1 '15 at 5:09

First, as Arthur notes, even the best Nusselt number correlations are often as much as 20% off, so don't expect any analytical method to give results that are much better than approximate.

With that said, there are ways to compute the Reynolds number for shear thinning fluids. Rudman, Blackburn, et al suggest using the effective viscosity for the mean wall shear stress. If you have a fluid that can be modeled as a Herschel-Bulkley Fluid, then the Reynolds number equation takes the following form. $$Re = \frac{\rho \bar{U} D}{\eta_w}$$ $$\eta_w = K^{\frac{1}{n}} \frac{\tau_w}{(\tau_w - \tau_y)^{1/n}}$$ $$\tau_w \approx \frac{4}{D} \frac{\Delta p}{L}$$

The authors note that other formulations of the Reynolds number exist for shear thinning flows, and there is no "perfect" Reynolds number for shear thinning flows, but state that this formulation takes the dynamics near the walls of the pipe into account well.

Any heat exchanger has different values for the viscosity along its length, since it depends on the fluid's temperature. You should use an average value for the viscosity. Surely if it varies too much it is going to be harder to estimate this value, but it can be done. You can do it by knowing the shear rate in the different regions of your stream, and then prorate how much of your fluid goes though each region (in percentage).

You have to keep in mind that with the best Nusselt number correlation you will have at least a 20% error, so it is always a little uncertain