Unknown flow rate

In this case, the flow rate is to be determined when a given fluid is transported in a given pipe with a known net driving force (e.g., pump head, pressure head, and/or hydrostatic head). The same total variables are involved, and hence the dimensionless variables are the same and are related in the same way as for the unknown driving force problems. The main difference is that now the unknown (Q) appears in two of the dimensionless variables (/ and 7VRe), which requires a different solution strategy. 

The strategy is to redefine the relevant dimensionless variables by combining the original groups in such a way that the unknown variable appears in one group. For example, / and /VRe can be combined to cancel the unknown (Q) as follows  

if we work with the three dimensionless variables fNRe, NRe, and s/D, the unknown (Q) appears in only NRe, which then becomes the unknown (dimensionless) variable. 

A reasonable guess might be based on the assumption that the flow conditions are turbulent, for which the Colebrook equation, Eq. (6-38), applies. 

Using the NRe value from step 4 and the known value of e/D, determine / from the Moody diagram or Churchill equation (if NRc 2000, use / = 16//VRc). 

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The relationship between flow rate, pressure drop, and pipe diameter for water flowing at 60°F in Schedule 40 horizontal pipe is tabulated in Appendix G over a range of pipe velocities that cover the most likely conditions. For this special case, no iteration or other calculation procedures are required for any of the unknown driving force, unknown flow rate, or unknown diameter problems (although interpolation in the table is usually necessary). Note that the friction loss is tabulated in this table as pressure drop (in psi) per 100 ft of pipe, which is equivalent to 100pef/144L in Bernoulli s equation, where p is in lbm/ft3, ef is in ft lbf/lbm, and L is in ft. 

The inclusion of significant fitting friction loss in piping systems requires a somewhat different procedure for the solution of flow problems than that which was used in the absence of fitting losses in Chapter 6. We will consider the same classes of problems as before, i.e. unknown driving force, unknown flow rate, and unknown diameter for Newtonian, power law, and Bingham plastics. The governing equation, as before, is the Bernoulli equation, written in the form... 

These network equations can be solved for the unknown driving force (across each branch) or the unknown flow rate (in each branch of the net-... 

We will illustrate the procedure for solving the three types of pipe flow problems for high-speed gas flows unknown driving force, unknown flow rate, and unknown diameter. 

The procedure for an unknown diameter involves a trial-and-error procedure similar to the one for the unknown flow rate. 

Three classes of problems involving orifices (or other obstruction meters) that the engineer might encounter are similar to the types of problems encountered in pipe flows. These are the unknown pressure drop, unknown flow rate, and unknown orifice diameter problems. Each... 

In the case of an unknown flow rate, the pressure drop across a given orifice is measured for a fluid with known properties, and the flow rate is to be determined. 

Alternatively, it may be necessary to determine the maximum capacity (e.g., flow rate, Q) at which particles of a given size, d, will (or will not) settle out. This can also be obtained directly from the Dallavalle equation in the form of Eq. (11-13), by solving for the unknown flow rate ... 

The scope of coverage includes internal flows of Newtonian and non-Newtonian incompressible fluids, adiabatic and isothermal compressible flows (up to sonic or choking conditions), two-phase (gas-liquid, solid-liquid, and gas-solid) flows, external flows (e.g., drag), and flow in porous media. Applications include dimensional analysis and scale-up, piping systems with fittings for Newtonian and non-Newtonian fluids (for unknown driving force, unknown flow rate, unknown diameter, or most economical diameter), compressible pipe flows up to choked flow, flow measurement and control, pumps, compressors, fluid-particle separation methods (e.g.,... 

An optimal solution of such an optimization model will provide information on all unknown flow rates and temperatures of the streams in the superstructure (i.e., it will automatically determine the network configuration), as well as the optimal areas of the exchangers for a minimum investment cost network. 

The hyperstructure for the illustrative example is shown in Figure 8.21 The variables also shown in Figure 8.21 are the unknown flow rates and temperatures of the streams. Note that the notation is based on capital letters for the hot streams (i.e., F, T) and lower case letters for the cold streams (i.e., /, t), while the superscripts denote the particular hot and cold streams. 

The superstructure for the illustrative example is shown in Figure 9.2. Note that we have two sequences of distillation columns (i.e., column 1 followed by column 2, and column 3 followed by column 4) since the feed stream has three components. In Figure 9.2, the potential process-process matches and the process-utility matches are indicated. The variables shown in Figure 9.2 are the unknown flow rates for the column sequences (i.e., F, F2,F3, F4) and existence variables for the columns (i.e., yi,y2, y3, Vi) and the process-process matches (i.e., yci,R2, yc2,Ri,yc3,R4, yc, R3)-The temperatures of the condensers and reboilers of each column, as well as the existence variables of the process-utility matches are not shown in Figure 9.2. 

Remark 2 Note that at the inlet of each unit (i.e., CSTR, PFR, CSTR approximating PFR) there is a mixer while at the outlet of each unit there is a splitter. There is also a splitter of the feed and a final mixer. As a result, in addition to the unknown flow rates of each stream of the superstructure we have also as unknowns the compositions of outlets of each unit. Finally, the volumes of each reactor unit are treated as unknown variables. 

In a geologic repository, waste form leaching will be affected by unknown flow rates both under normal repository operation and accidental conditions. An understanding of these effects is then necessary in order to predict the geochemical behavior of disposed radioactive waste over the full range of possible scenarios. 

With 11 stages and 5 components the equilibrium-stage model has 143 equations to be solved for 143 variables (the unknown flow rates, temperatures, and mole fractions). Convergence of the computer algorithm was obtained in just four iterations. Computed product flows are shown in Fig. 13-37. 

Before a rotameter can be used to measure an unknown flow rate, a calibration curve of flow rate versus rotameter reading must be prepared. A calibration technique for liquids is illustrated below. A flow rate is set by adjusting the pump speed the rotameter reading is recorded, and the effluent from the rotameter is collected in a graduated cylinder for a limed interval. The procedure is carried out twice for each of several pump settings. 

Calculate the unknown flow rates and compositions of streams 1,2, and 3. 

In the ethane dehydrogenation process, the two unknown flow rates will be determined from balances on atomic carbon and atomic hydrogen. 

Gco.i = nco 0.780 mol CO generated Gc02-2 = Sne o = 6.24 mol CO2 generated Water and oxygen balances complete the calculation of the unknown flow rates. HiO Balance output - generation H20 GH2O.1 + GH202... 

The CNDS subroutine would proceed in a similar manner tor the condenser calculations—first solving material and energy balances to determine unknown flow rates and the heat dut>. and then possibly performing design calculations. 

In Classes 3 and 4, the unknown flow rate, L or V, is calculated from the slope L/V as determined from the graphical construction, and the specified flow rate V or L. 

Two streams of steam are brought into thermal contact as shown in Figure 6- b. The cooler stream enters at 150 and exits at 300 °C and has a flow rate of 1 kg/s. The hot stream enters at 500 °C and exits at 400 °C. All streams are at constant pressure of 1 bar. Determine the unknown flow rate, the amount of heat exchanged between the streams, the rate of entropy generation, and the lost work. 

Solution We will use the NIST WebBook to calculate the properties of nitrogen. The pressures of all streams are known streams 1, 2, 6, 7, 8, and 9 are at 1 bar streams 3,4, and 5 are at 180 bar. Streams 6,7, and 8 is the boiling temperature of nitrogen at 1 bar (-195.91 °C). We notice now that the state of streams 1,7,8, and 9 is fully defined and their properties may be collected from the Webbook (see summaiy table below). In addition to the usual properties (P, T, H, S, and phase) we must also include in the table of streams the flow rates, since these are not the same among all streams. We set the basis to be 1 kg of liquid nitrogen in stream 7, and the unknown flow rate of vapor stream 8 to be x. By straightforward material balance, the flow rate of stream 1 is 1 kg/s, the flow rate of streams 8 and 9 are x, and all other flow rates are x + 1. The known properties of the streams are summarized below ... 

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