Loss term

In practice, the loss term AF is usually not deterrnined by detailed examination of the flow field. Instead, the momentum and mass balances are employed to determine the pressure and velocity changes these are substituted into the mechanical energy equation and AFis deterrnined by difference. Eor the sudden expansion of a turbulent fluid depicted in Eigure 21b, which deflvers no work to the surroundings, appHcation of equations 49, 60, and 68 yields... 

Lasing occurs whenever the gain arising from stimulated emission exceeds the cavity losses. Internal losses, a, result from absorption and scattering of light. The reflectivity, R, of the mirror facet must be <1 and this contributes a loss term of (1/L)ln(l/E), where Eis the cavity length. At threshold, the gain, is equal to losses and... 

The first term on the right side of Eq. (5-179) is so nearly dominant for most furnaces that consideration of the main features of chamber performance is clarified by ignoring the loss terms and Lr or by assuming that they and have a constant mean value. The relation of a modified chamber efficiency T g(1 o) lo modified firing density D/(l — and to the normahzed sink temperature T = T-[/Tp is shown in Fig. 5-23, which is based on Eq. (5-178), with the radiative and convective transfer terms (GSi)/ja(TG — T ) -i- hiAijTc Ti) replaced by a combined radiation/conduction term (GS,) ,a(T - T ). where (GS])/ = (GS])/ + /jiA]/4oTgi Tg is adequately approximated by the arithmetic mean of Tg and T. ... 

The viscous or frictional loss term in the mechanical energy balance for most cases is obtained experimentally. For many common fittings found in piping systems, such as expansions, contrac tions, elbows and valves, data are available to estimate the losses. Substitution into the energy balance then allows calculation of pressure drop. A common error is to assume that pressure drop and frictional losses are equivalent. Equation (6-16) shows that in addition to fric tional losses, other factors such as shaft work and velocity or elevation change influence pressure drop. 

The coupling of Fg with the other loss terms ilr (cf. Eq. 10) means that even a modest absolute increase in Fg may lead to a much larger increase in fracture energy F. 

Thus there are three modifications to the a/s efficiency analysis, involving (i) the specific heats ( and n ), (ii) the fuel-air ratio / and the increased turbine mass flow (I +/), and (iii) the pressure loss term S. The second of these is small for most gas turbines which have large air-fuel ratios and / is of the order of l/IOO. The third, which can be significant, can also be allowed for a modification of the a/s turbine efficiency, as given in Hawthorne and Davis [I]. (However, this is not very convenient as the isentropic efficiency tjt then varies with r and jc, leading to substantial modifications of the Hawthome-Davis chart.)... 

Expression (7.12a) for overall efficiency is similar to that for the combined doubly cyclic plant the term itl[//ps corresponds to the heat loss term of Section 13. 

Now consider the loss term C[f] in equation 9.29. The probability of a loss is proportional to four terms (1) the number of type-1 hard-spheres already in the volume (= /i) (2) the number of type-2 hard-spheres entering the volume (= /2 d V2) (3) the probability, P[ ff ] = P vi,V2, V, V2), that two hard-spheres with velocities V, V2 will actually collide and make transitions to states with velocities hi, V2 t (4) the total volume of the allowed outgoing velocity-space (= d v dPv2)-Integrating over all possible momenta, we find that the loss term is given by... 

Our final form for the Boltzman equation is then obtained by substituting these last two expressions for the gain and loss terms into equation 9.29 and adding the term Vijf x, v, t) to the LHS for the case where there is an external force F Vjj is the gradient operator with respect to v and m is the hard-sphere mass) ... 

Finally, after all their internal checks and balance (the company was indeed fastidious in matters of product liability), they agreed they had gotten to understand the Flyback topology far better now. So I quickly incorporated the new loss term into the Mathcad and Excel... 

The last resort for reducing the clamp losses is to reduce the switching frequency, since the clamp loss term is purely a switching loss term and is therefore proportional to the switching frequency. 

For highly conductive liquids and solids the loss term not only results from a single relaxation term, as given by Eq. (8), but also from term resulting from ionic conductivity, cr, as described by Eq. (14) ... 

Fis the net frictional loss term (length force/mass),... 

The frictional loss term F in Equation 4-28 represents the loss of mechanical energy resulting from friction and includes losses resulting from flow through lengths of pipe fittings such as valves, elbows, orifices and pipe entrances and exits. For each frictional device a loss term of the following form is used ... 

For fluids flowing through pipes the excess head loss term Kt is given by... 

Determine the excess head loss terms for the pipe (using Equation 4-30), for the fittings (using Equation 4-38), and for any entrance and exit effects (using Equation 4-39). Sum the head loss terms, and compute the net frictional loss term using Equation 4-29. Use the velocity at point 2. 

Because the friction factor/and the frictional loss term Fare functions of the Reynolds number and velocity, the solution is found by trial and error. The trial and error solution is shown in the following table ... 

An important part of the frictional loss term is the assumption of a constant Fanning friction factor/across the length of the pipe. This assumption is valid only at high Reynolds numbers. 

K are the excess head loss terms, including pipe entrances and exits, pipe lengths, and fittings (unitless). 

The excess head loss terms 2 Kt are found using the 2-K method presented earlier in section 4-4. For most accidental discharges of gases the flow is fully developed turbulent flow. This means that for pipes the friction factor is independent of the Reynolds number and that for fittings Kf = and the solution is direct. 

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. 

Assume fully developed turbulent flow to determine the friction factor for the pipe and the excess head loss terms for the fittings and pipe entrances and exits. The Reynolds number can be calculated at the completion of the calculation to check this assumption. Sum the individual excess head loss terms to get 2 Kf. 

Gartner s equation can be derived by calculating that part of the photocurrent which comes from the bulk. The concentration p(x) of holes obeys the following equation, which combines the familiar diffusion equation with a source and a loss term ... 

Figure 4.3 tries to show the behavior of these terms in which <2r is only approximately represented for the Arrhenius dependence in temperature. For a given fuel and its associated kinetics, <2r is a unique function of temperature. However, the heat loss term depends on the surface area of the vessel. In Figure 4.3, we see the curves for increasing... 

The net heat flux is taken here to represent radiative heating in an environment at Tcx, with an initial temperature T,yj as well. From Equation (7.20) a more general form can apply if the flame heat flux is taken as constant. This nonlinear problem cannot yield an analytical solution. To circumvent this difficulty, the radiative loss term is approximated by a linearized relationship using an effective coefficient, hr ... 

Again as in the thermally thin case, the heat loss term is not included, suggesting that this only holds where qe is large. 

It should be pointed out that Equation (8.6), and its counterpart for thermally thick materials, will hold only for Ts > 7 smm, a minimum surface temperature for spread. Even if we include the heat loss term in Equation (8.4) by a mean-value approximation for the integrand,... 

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