Backward characteristic

Starting at a set of locations along the pipeline, i(to), = 1,2,... at time, to, equation (19.26) defines a set of forward trajectories or characteristic curves in distance and time, while equation (19.28) defines a set of backward characteristic curves starting at the same point ... 

Equations (19.32) and (19.33) apply to all interior points within the pipeline, so that the pressures and flows may be calculated for i = 1,2,3,. AT — 1. However, at the upstream boundary, when i = 0, no forward characteristic is present, as may be seen from inspecting Figure 19.1. At this location only the backward characteristic is available, so that only equation (19.31) is valid, which is a single equation containing the two unknowns of pressure and mass flow. A solution is possible, however, if the upstream conditions allow us to specify one of the following at i = 0 ... 

To find the flow at the pipe inlet at time j, Wq.j, the backward characteristic, equation (19.31) may Im applied with < = I. Since the fluid is assumed to be a liquid at constant temperature, we will assign a constant value to specific volume, v. Thus Wo.y, is given in terms of the current pressure, po.j, and past flows by... 

Meanwhile, the backward characteristic of pipe section (2) will also apply, so that we may apply equation... 

Meanwhile the backward characteristic with / = 0 will apply, i.e. equation (19.42), repeated below ... 

Conversely, the backward characteristic will apply for the lines that normally carry fluid away from the Junction. Hence using the superscript (S, s), s = 1, 2,..., S to specify such an efferent pipeline, we may use equation (19.31), setting the distance index i = 0 to imply the beginning of each line ... 

J. Behrens, A. Iske, and M. Kaser (2002) Adaptive meshfree method of backward characteristics for nonlinear transport equations. Meshfree Methods for Partial Differential Equations, M. Griebel and M.A. Schweitzer (eds.). Springer, Berlin, 21-36. 

EIOs), backward wave oscillators (BWOs) or magnetrons are available. Their spectral characteristics may be favourable however, they typically require highly stabilized high-voltage power supplies. Still higher frequencies may be obtained using far-infrared gas lasers pumped for example by a CO- laser [49]. 

Figure 9.42 shows the typical characteristic curve of a centrifugal fan, where the blades are backward curved. The figure also shows the theoretical characteristic curve when the slip factor is 1 and when it is smaller than 1. Characteristic curves for a real fan are closer to the isentropic one at the design point. At this point the efficiency is maximum. 

Non-overloading fan A fan with backward-curved blade.s, which has power characteristics that tend to flatten with increasing flow rates, so that as the maximum volume flow rate is approached the power consumed may become constant or even decrease. The... 

The burst pressure maximum cannot exceed the MAWP of the vessel. Depending on the situation, it may be necessary to work backward to the operating pressure maximum to see if this is usable. Table 7-9 summarizes t) pical rupture disk characteristics noting that the maximum normal operating pressure of the system is shown as a function of the rupture disk bursting pressure, P. ... 

The three types of centrifugal fan blades—radial, backward, and forward—give three characteristic performances. Table 12-12B gives a quick comparison. 

Figure 12-131A. Characteristic curves for backward curved blade. (Used by permission The Howden Fan Company.)...

This type of blade is well suited to streamline flow conditions and is used extensively on ventilating, air conditioning, and clean and dirty process gas streams. The backward blade does not catch dirt easily. The outstanding and important characteristic is the nonoverloading horsepower. This is the only commonly used blade style with this feature. It is important in process control and eliminates the need for oversized motors or other drivers. Speed of operation is high, which allows direct or belt connection to the driver. Certain streamlined blade designs provide the same basic characteristics with more efficient and quieter operation. The usual... 

Nonoverloading characteristics. A backward-curve blade will allow close motoring without fear of overloading in the event of process upsets. 

For a given speed of rotation, there is a linear relation between the head developed and the rate of flow. If the tips of the blades of the impeller are inclined backwards. ft is less than tt/2, tan ft is positive, and therefore the head decreases as the throughput increases. If ft is greater than jt/2 (i.e. the tips of the blades are inclined forwards), the head increases as the delivery increases. The angle of the blade tips therefore profoundly affects the performance and characteristics of the pump. For radial blades the head should be independent of the throughput. 

Filtering and smoothing are related and are in fact complementary. Filtering is more complicated because it involves a forward and a backward Fourier transform. However, in the frequency domain the noise and signal frequencies are distinguished, allowing the design of a filter that is tailor-made for these frequency characteristics. 

The third characteristic refers to the fact that a catalyst does not influence the value of the equilibrium constant because it lowers the activation energy of the forward and backward reactions by the same amount and therefore changes the rates of the forward and backward reactions by the same amount. A catalyst only accelerates the attaining of equilibrium it does not exert any influence whatsoever on the quantitative yield of the products. 

Comments at the end of Example 4.1 also apply here. The result should be correct, and we should find that both the roots of the characteristic polynomial p and the eigenvalues of the matrix a are -0.2 0.98j. We can also check by going backward ... 

Case C is illustrated in Scheme 4.2, where Iq is the diffusional second-order rate constant for the formation of the pair (M. .. Q) from separated M and Q, k i is the first-order backward rate constant for this step, kR is the first-order rate constant for the reaction of (M. .. Q) to form products 3, and km (= 1/%) is the rate constant for intrinsic de-excitation of M. If the interaction between M and Q is weak, the fluorescence characteristics of the pair M. .. Q are the same as those of M (Scheme 4.2). 

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