Viscous shear heating

When a very viscous molten polymer is forced to flow through a slit or capillary die, viscous shear heating can become significant above a certain critical value of y or ct. Under such situations, nonlinear profiles of wall normal stress in a slit or capillary die may be observed, as described in the preceding section. Therefore, continuous-flow capillary/slit rheometry is limited to y or a, below which viscous shear heating can be neglected. 

The upper limit of y or a above which the extent of viscous shear heating becomes significant can be predicted theoretically by solving the following equations of momentum and heat transfer in a slit die ... 

Figure 5.31 The range of shear stresses where a cone-and-plate rheometer can be used to obtain the rheological properties of polymeric liquids before flow instability sets in, and the range of shear stresses where a capiUary/slit rheometer can be used to obtain the rheological properties of polymeric liquids before significant viscous shear heating is encountered.

If the heat flux from friction or viscous shear is properly estimated, the surface temperature, which is of interest in most engineering problems, can be determined through integrating an analytical solution of temperature rise caused by a moving point heat source, without having to solve the energy equation. For two solid bodies with velocity u j and Ui in dry contacts, the temperature rises at the surfaces can be predicted by the formula presented in Ref. [22],... 

The power put into a fluid mixer produces pumping Q and a velocity head H. In fact all the power P which is proportional to QH appears as heat in the fluid and must be dissipated through the mechanism of viscous shear. The pumping capacity of the impeller has been measured for a wide variety of impellers. Correlations are available to predict, in a general way, the pumping capacity of the many impeller types in many types of configurations. The impeller pumping capacity is proportional to the impeller speed N and the cube of the impeller diameter D,... 

All of the power applied by a mixer to a fluid through the impeller appears as heat. The conversion of power to heat is through viscous shear and is approximately 2500 Btu/hr/hp. Viscous shear is present in turbulent flow only at the microscale level. As a result, the power per unit volume is a major component of the phenomena of microscale mixing. At a l-/zm level, in fact, it doesn t matter what specific impeller design is used to apply the power. 

Calculate the drag (viscous-friction) force on the plate in Prob. 5-21 under the conditions of no heat transfer. Do not use the analogy between fluid friction and heat transfer for this calculation i.e., calculate the drag directly by evaluating the viscous-shear stress at the wall. 

The heat-conduction equation describes the transport of energy, the viscous-shear equation describes the transport of momentum across fluid layers, and the diffusion law describes the transport of mass. 

The other situation is on the microscale particles. They are particles less than 100 microns and they see largely the energy dissipation which occurs through the mechanism of viscous shear rates and shear stresses and ultimately the scale at which all energy is transformed into heat. 

The situation is analogous to momentum flux, where the relative Importance of turbulent shear to viscous shear follows the same general pattern. Under certain ideal conditions, the correspondence between heat flow and momentum flow is exact, and at any specific value of rjr the ratio of heat transfer by conduction to that by turbulence equals the ratio of momentum flux by viscous forces to that by Reynolds stresses. In the general case, however, the correspondence is only approximate and may be greatly in error. The study of the relationship between heat and momentum flux for the entire spectrum of fluids leads to the so-called analogy theory, and the equations so derived are called analogy equations. A detailed treatment of the theory is beyond the scope of this book, but some of the more elementary relationships are considered. 

It is important to notice the similarity between Eqs. 1.1,1.6, and 1.20. The heat conduction equation, Eq. 1.1, describes the transport of energy the diffusion law, Eq. 1.6, describes the transport of mass and the viscous shear equation, Eq. 1.20, describes the transport of momentum across fluid layers. We note also that the kinematic viscosity v, the thermal diffusivity a, and the diffusion coefficient D all have the same dimensions L2/f. As shown in Table 1.10, a dimensionless number can be formed from the ratio of any two of these quantities, which will give relative speeds at which momentum, energy, and mass diffuse through the medium. 

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