The Friction Heating Term

The amount of frictional work per unit mass in typical fluid flow problems is generally less than in the examples cited above. I 

The heat capacity of liquids is generally greater than that of solids. For example, the amount of heat required to raise the temperature of 1 Ibm of water by 1°F will raise the temperature of 1 Ibm of steel by about 8 F. 

Solving this equation for the friction heating per unit mass, we see that it is given by the Am — dQ/dm term on the right of Eq. 5.3. 

however, there were friction in the heater, then Am — dQldm would be a positive number, whose value would be exactly equal to the amount of friction heating per unit mass. 

The increased infernal energy produced by friction heating usually is useless for industrial purposes, so that friction heating is often referred to as friction loss. Energy does not disappear in this case. Rather, energy of a valuable form is converted to energy of a useless form, hence the loss of energy. 

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The most interesting applications of Bernoulli s equation include the effects of friction. Before we can solve these, we must learn how to evaluate the term, which we do in Chap. 6. However, in many flow problems the friction heating terms are small compared with the other terms and can be neglected. We can solve these by means of Bernoulli s equation without the friction heating term. A good example of this type of problem is the tank-draining problem, which leads to Torricelli s equation. 

It has been found experimentally that the friction heating term in Eq. 5.26 is normally less than 1 percent of the total it may be ignored, giving... 

It is apparent that the frictional heating term, Ff, will have to be dealt if the Bernoulli Equation is to be applied. This subject will be deferred until the next chapter. However, it should be recognized that there are instances where fluid mechanics situations can be treated by the Bernoulli Equation if P/, is taken to be zero. Some of these will be discussed in the following sections. 

The calculation to determine the expansion factor can be completed once y and the frictional loss terms 2 Kf are specified. This computation can be done once and for all with the results shown in Figures 4-13 and 4-14. As shown in Figure 4-13, the pressure ratio ( f - P2)/Pi is a weak function of the heat capacity ratio y. The expansion factor Yg has little dependence on y, with the value of Yg varying by less than 1 % from the value at y = 1.4 over the range from y = 1.2 to y = 1.67. Figure 4-14 shows the expansion factor for y = 1.4. 

Figure 2 shows the thickness of the lubricating layer, / , as a function of distance along the dynamic contact zone. The upstream boundary value of h is the depth of the quasi-liquid layer based on the ice surface temperature." The quasi-liquid layer serves as the lubricant over the first 0.1% of the contact zone. The upper curve represents the frictional melting term (first term on the rhs of Eq (5)). The effects of adding squeeze flow and heat conduction into the ice are shown by the two intermediate curves. The lowest curve combines all factors, and shows... 

Here we use to avoid confusion with F for force. Most civil engineering texts call this quantity ghf or where g is the acceleration of gravity and hf or stands for friction head loss (Sec. 5.4). Some thermodynamics textbooks introduce the idea of the lost work in explaining the second law of thermodynamics. For a constant-density fluid at the heat reservoir temperature, the friction heating per unit mass is exactly equal to the lost work per unit mass, so some texts call this term LW. Other texts call it (-AP/p),since for the most common pipe friction problem, steady flow in horizontal, constant-area pipes. S - -tsPIp. 

For constant-kensity fluids, the term Au - dQldm in Bernoulli s equation represents the friction heating per unit mass. 

This equation rests on the plausible assumption, that the friction heating is proportional to the length of the pipe and on the more questionable assumption that it is proportional to VlD. Do these assumptions agree with the experimental data Yes and no. To save writing, we now define a new term, the friction factor /, which is equal to twice the proportionality constant in Eq. 6.15 ... 

The behavior will be described by Bernoulli s equation, with the laminar-flow friction heating term given by Eq. 12.13 ... 

The method that is used is to consider an incompressible fluid (good approximation for most liquids and also for gases under certain conditions), and we can equate the internal energy and heat combination to a friction heating term ... 

Correlations for Convective Heat Transfer. In the design or sizing of a heat exchanger, the heat-transfer coefficients on the inner and outer walls of the tube and the friction coefficient in the tube must be calculated. Summaries of the various correlations for convective heat-transfer coefficients for internal and external flows are given in Tables 3 and 4, respectively, in terms of the Nusselt number. In addition, the friction coefficient is given for the deterrnination of the pumping requirement. 

In the macroscopic heat-transfer term of equation 9, the first group in brackets represents the usual Dittus-Boelter equation for heat-transfer coefficients. The second bracket is the ratio of frictional pressure drop per unit length for two-phase flow to that for Hquid phase alone. The Prandd-number function is an empirical correction term. The final bracket is the ratio of the binary macroscopic heat-transfer coefficient to the heat-transfer coefficient that would be calculated for a pure fluid with properties identical to those of the fluid mixture. This term is built on the postulate that mass transfer does not affect the boiling mechanism itself but does affect the driving force. 

It is often convenient to correlate heat-transfer data in terms of a heat transfer factor, which is similar to the friction factor used for pressure drop (see Volume 1, Chapters 3 and 9). The heat-transfer factor is defined by ... 

Critical phenomena of gels have been studied mainly by dynamic light scattering technique, which is one of the most well-established methods to study these phenomena [18-20]. Recently, the critical phenomena of gels were also studied by friction measurement [85, 86] and by calorimetry [55, 56]. In the case of these methods, the divergence of the specific heat or dissipation of the friction coefficient could be monitored as a function of an external intensive variable, such as temperature. These phenomena might be more plausible to some readers than the divergence of the scattered intensity since they can observe the critical phenomena in terms of a macroscopic physical parameter. 

The force exerted by the substance within the cylinder on the lower force of the piston under these conditions is the product of the pressure exerted by the substance on the surface of the piston and the area of the piston. Moreover, the product of the area and the differential displacement of the piston is equal to the differential change of volume. The integral J F dh is then equal to P dV. This relation is the only change that is made in Equation (2.15) or a similar equation for quasistatic processes. The frictional effects or the collisions result in a temperature increase either of the surroundings, or of both the system and surroundings as the case may be, or the effects may be interpreted in terms of heat, as discussed above. 

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