Effect of Viscous Energy Dissipation

Experimental and numerical analyses were performed on the heat transfer characteristics of water flowing through triangular silicon micro-channels with hydraulic diameter of 160 pm in the range of Reynolds number Re = 3.2—84 (Tiselj et al. 2004). It was shown that dissipation effects can be neglected and the heat transfer may be described by conventional Navier-Stokes and energy equations as a common basis. Experiments carried out by Hetsroni et al. (2004) in a pipe of inner diameter of 1.07 mm also did not show effect of the Brinkman number on the Nusselt number in the range Re = 10—100. 

Hetsroni et al. (2005) evaluated the effect of inlet temperature, channel size and fluid properties on energy dissipation in the flow of a viscous fluid. For fully developed laminar flow in circular micro-channels, they obtained an equation for the adiabatic increase of the fluid temperature due to viscous dissipation  

For an incompressible fluid, the density variation with temperature is negligible compared to the viscosity variation. Hence, the viscosity variation is a function of temperature only and can be a cause of radical transformation of flow and transition from stable flow to the oscillatory regime. The critical Reynolds number also depends significantly on the specific heat, Prandtl number and micro-channel radius. For flow of high-viscosity fluids in micro-channels of tq 10 m the critical Reynolds number is less than 2,300. In this case the oscillatory regime occurs at values of Re 2,300. 

We can estimate the values of the Brinkman number, at which the viscous dissipation becomes important. Assuming that the physical properties of the fluid are constant, the energy equation for fully developed flow in a circular tube at 7(v = const, is  

A detailed study of the influence of viscous heating on the temperature field in micro-channels of different geometries (rectangular, trapezoidal, double-trapezoidal) has been performed by Morini (2005). The momentum and energy conservation equations for flow of an incompressible Newtonian fluid were used to estimate 

For fully developed isothermal flow, the velocity gradient is obtained by differentiating the expression for velocity (equation 3.7) to give  

Evidently, for the limiting case of a shear-thinning fluid (Newtonian fluid n = 1), the velocity gradient is a maximum and hence, the viscous heat generation is also maximmn. The effect of this process on the developing temperatme profile can be illustrated by considering the situation depicted in Figme 6.7. 

Viscous dissipation can be quantified in terms of the so-called Brinkman number, Br, which is defined as the ratio of the heat generated by viscous action to that dissipated by conduction. Thus for streamline flow in a circular mbe (on the basis of per imit volmne of fluid)  

Clearly, the viscous dissipation effects will be significant whenever the Brinkman munber is much greater than imity. 

While the rigorous solutions of the thermal energy equation are quite complex, some useful insights can be gained by quahtative considerations of the results of Forrest and Wilkinson [1973, 1974] and Dinh and Armstrong [1982] amongst others or from the review papers [Winter, 1977 Lawal and Mujumdar, 1987]. 

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Under certain conditions the energy dissipation may lead to an oscillatory regime of laminar flow in micro-channels. The relation of hydraulic diameter to channel length and the Reynolds number are important factors that determine the effect of viscous energy dissipation on flow parameters. The oscillatory flow regime occurs in micro-channels at Reynolds numbers less than Recr- In this case the existence of velocity fluctuations does not indicate change from laminar to turbulent flow. 

The subject of this chapter is single-phase heat transfer in micro-channels. Several aspects of the problem are considered in the frame of a continuum model, corresponding to small Knudsen number. A number of special problems of the theory of heat transfer in micro-channels, such as the effect of viscous energy dissipation, axial heat conduction, heat transfer characteristics of gaseous flows in microchannels, and electro-osmotic heat transfer in micro-channels, are also discussed in this chapter. 

There are a number of additional physical phenomena, such as wall slip, electric effects and viscous energy dissipation, which may need to be taken into account. Generally applicable models are not available for some of these effects, particularly wall slip. 

Celata et al. (2005) evaluated the effect of viscous heating on friction factor for flow of an incompressible fluid in a micro-channel. By integrating the energy equation over the micro-channel length, a criterion that determines conditions when viscous dissipation effect is signiflcant was obtained ... 

The relation of hydraulic diameter to channel length and the Reynolds number are important factors that determine the effect of the viscous energy dissipation on flow parameters. 

If the dominating domain is selected correctly, the error induced by the simplification will be no more than about 20%. However, even well into the viscous dissipation domain, the effects of the surface tension are still significant, while in the surface tension domain, the effects of viscous dissipation disappear far more rapidly as one moves away from the borderline. In other words, the viscous energy dissipation contribution to the spread factor rapidly declines within the surface tension-dominated domain, while significant residual surface tension effects extend well into the viscous energy dissipation domain. 

Neglect the effects of viscous work (i.e., viscous dissipation), and compare with the steady-state form of the thermal-energy equation for an incompressible fluid. Are they the same equation If so, why If not, why ... 

To consider the effect of internal energy generation and viscous dissipation, the following formula obtained by Tyagi [6] is recommended ... 

Thermally Developing Flow. Wibulswas [160] and Aparecido and Cotta [161] have solved the thermal entrance problem for rectangular ducts with the thermal boundary condition of uniform temperature and uniform heat flux at four walls. However, the effects of viscous dissipation, fluid axial conduction, and thermal energy sources in the fluid are neglected in their analyses. The local and mean Nusselt numbers Nu j, Num T, and Nu hi and Num Hi obtained by Wibulswas [160] are presented in Tables 5.32 and 5.33. 

Other examples of the effect of viscous dissipation on capillary data are given by Cox and Macosko (1974b) and Warren (1988). These workers and Winter (1975, 1977) indicate how to correct data affected by shear heating to true viscosity values. This requires numerical solution of the momentum and energy equations, a capability available in many standard fluid mechanics software packages. However, note that for typical capillary dies the stress level at which viscous dissipation becomes important is near the region for polymer melt fracture, tu, lO Pa. As already pointed out, it is not possible to get true viscosity data aft - the onset of melt h cture. 

The component with the lower viscosity tends to encapsulate the more viscous (or more elastic) component (207) during mixing, because this reduces the rate of energy dissipation. Thus the viscosities may be used to offset the effect of the proportions of the components to control which phase is continuous (2,209). Frequently, there is an intermediate situation where a cocontinuous or interpenetrating network of phases can be generated by careflil control of composition, microrheology, and processing conditions. Rubbery thermoplastic blends have been produced by this route (212). 

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