## 62 Thermophysical properties

The most important thermo-physical properties of non-Newtonian fluids are thermal conductivity, density, specific heat, surface tension and coefficient of thermal expansion. While the first three characteristics enter into virtually all heat transfer calculations, surface tension often exerts a strong influence on boiling heat transfer and bubble dynamics in non-Newtonian fluids, as seen in Chapter 5. Likewise, the coefficient of thermal expansion is important in heat transfer by free convection.

Only a very limited range of measurements of physical properties has been made, and for dilute and moderately concentrated aqueous solutions of commonly used polymers including carboxymethyl cellulose, polyethylene oxide, carbopol, polyacrylamide, density, specific heat, thermal conductivity, coefficient of thermal expansion and surface tension differ from the values for water by no more than 5-10% [Porter, 1971; Cho and Hartnett, 1982; Irvine, Jr. etal., 1987]. Thermal conductivity might be expected to be shear rate dependent, because both apparent viscosity and thermal conductivity are dependent on structure. Although limited measurements [Loulou etal., 1992] on carbopol solutions confirm this, the effect is small. For engineering design calculations, there will be little error in assuming that all the above physical properties of aqueous polymer solutions, except apparent viscosity, are equal to the values for water.

Some values of these properties of polymer melts are also available [Brandrup and Immergut, 1989; Domininghaus, 1993]. Unfortunately though, no simple predictive expressions are available for their estimation. Besides, values seem to be strongly dependent on the method of preparation of the polymer, the molecular weight distribution, etc., and therefore extrapolation from one system to another, even under nominally identical conditions, can lead to significant errors. For industrially important slurries and pastes exhibiting strong non-Newtonian behaviour, the thermo-physical properties (density, specific heat and thermal conductivity) can deviate significantly from those of its constituents. Early measurements [Orr and Dallavale, 1954] on aqueous suspensions of powdered copper, graphite, aluminium and glass beads suggest that both the density and the specific heat can be approximated by the weighted average of the individual constituents, i.e.

where 0 is the volume fraction of the solids, and the subscripts s and L refer to the values for the solid and the liquid medium respectively.

The thermal conductivity of these systems, on the other hand, seems generally to be well correlated by the following expression [Tareef, 1940; Orr and

Dallavale, 1954; Skelland, 1967]:

Thermal conductivities of suspensions up to 60% (by weight) in water and other suspending media are well represented by equation (6.3). It can readily be seen that even for a suspension of highly conducting particles (kL/ks ! 0), the thermal conductivity of a suspension can be increased by several folds. Furthermore, the resulting increase in apparent viscosity from such addition would more than offset the effects of increase in thermal conductivity on the rate of heat transfer.

For suspensions of mixed particle size, the following expression due to Bruggemann [1935] is found to be satisfactory:

The scant experimental data [Rajaiah etal., 1992] for suspensions (0 < 0.3) of alumina (0.5-0.8 mm) particles in a paraffin hydrocarbon are in line with the predictions of equation (6.4). An exhaustive review of the thermal conductivity of structured media including polymer solutions, filled and unfilled polymer melts, suspensions and foodstuffs has been published by Dutta and Mashelkar [1987]. Figure 6.1 shows the predictions of equations (6.3) and (6.4) for a range of values of (kL/ks); the two predictions are fairly close, except for the limiting value of kL/ks = 0.

### Example 6.1

Estimate the value of thermal conductivity at 20°C of 25% (by vol) for aqueous suspensions of (a) alumina, ks = 30W/mK (b) thorium oxide, ks = 14.2W/mK (c) glass beads, ks = 1.20 W/mK.

Solution

Here 0 = 0.25; thermal conductivity of water, kL = 0.60 W/mK.

The values of the thermal conductivity of various suspensions are calculated using equations (6.3) and (6.4) for the purposes of comparison.

Here 0 = 0.25; thermal conductivity of water, kL = 0.60 W/mK.

The values of the thermal conductivity of various suspensions are calculated using equations (6.3) and (6.4) for the purposes of comparison.

Suspension |
Value of ksus |
, W/mK |

equation (6.3) |
equation (6.4) | |

alumina |
1.92 |
2.2 |

thorium oxide |
1.85 |
2.06 |

glass beads |
1.29 |
1.20 |

The values obtained by the two methods are seen to be close and the difference, being 10%, is well within the limits of experimental error in such measurements.

Of all the physico-chemical properties, it is the rheology which shows the strongest temperature dependence. For instance, the decrease in apparent viscosity at a fixed shear rate is well represented by the Arrhenius-type exponential expression; the pre-exponential factor and the activation energy are then both functions of shear rate. It is thus customary to denote the temperature dependence using rheological constants such as the power-law consistency coefficient and flow behaviour index. It is now reasonably well established that the flow behaviour index, n, of suspensions, polymer melts and solutions is nearly independent of temperature, at least over a range of 40-50°C, whereas the consistency coefficient exhibits an exponential dependence on temperature, i.e.

where the coefficient m0 and E, the activation energy of viscous flow, are evaluated using experimental results for the temperature range of interest. Similarly, in the case of Bingham plastic model, both the plastic viscosity and the yield stress decrease with temperature in a similar fashion, but each with different values of the pre-exponential factors and the activation energies. Temperature dependencies of the other rheological characteristics such as the primary and the secondary normal stress differences, extensional viscosity, etc. have been discussed in detail by Ferry [1980].

## Post a comment