Through a horizontal pipeline of variable cross-section

Through a horizontal pipeline of variable cross-section, from a tank A with an excess pressure P0 on the free surface, the liquid enters the open tank B. The height of the liquid level is constant on and off, steady-state fluid movement, diameters and lengths are known. On the pipeline of larger diameter there is a valve. Steel pipes. Determine the flow rate of fluid Q, build a piezometric and pressure line.
Given: glycerin; P0 • = 21kPa = 21000 Pa; ON = 5.8m; HB = ON -3.1 = 2.7m; d1 = 50mm = 0.05m; d2 = d1 30 = 80mm = 0.08m; l1 = 11m; l2 = 12.5m; t = 200C; pipe material - steel; gate valve in the middle of l2
 
Figure 1 - Scheme of the pipeline
Decision
1. We choose the comparison line 0-0, passing through the axis of the pipeline (see figure 1).
2. We compose the Bernoulli equation for sections 1-1 (free surface of the liquid in tank A) and 2-2 (free surface of the liquid in tank B), which are shown in Fig. 1:
                                         (1)
where z1, z2 is the geometric pressure in section 1-1 and 2-2, respectively, m; p1, p2 — fluid pressure in sections 1-1 and 2-2, respectively, Pa; v1, v2 — average flow velocity in sections 1-1 and 2-2, respectively, m / s; α1, α2 - Coriolis coefficient in sections 1-1 and 2-2, respectively; ρ is the density of the liquid, kg / m3; ΣhW1-2 - total pressure loss between sections 1-1 and 2-2.
The geometric height in the sections z1 = ON, z2 = HB.
Pressure in sections: p0 = rat p0 = 98100 21000 = 119100 Pa;
                                         p2 = rat = 98100 Pa.
Density of glycerol at 200С ρ = 1260 kg / m3.
We take the Coriolis coefficients for the cross sections α1 = α2 = 1.
We assume that the flow velocity in sections 1-1 and 2-2 is equal to zero v1 = v2 = 0, because the level in the tanks is constant.
Taking into account the known values, we express from ur (1) the total pressure loss between the sections
                                                 (2)
We substitute the known quantities in (2) and obtain
                                                 (3)
3. The total pressure loss in the pipeline under consideration consists of losses at local resistances and losses along the friction length:
                                                                (4)
4. Local losses
                                     (5)
where hin - local pressure loss at the inlet to the pipe from the pressure tank; hvr - local pressure loss on sudden expansion; hadv - local pressure loss at the valve; hvh - local pressure loss at the outlet of the pipe to tank B.
Local pressure loss according to the Weisbach formula:
                                           (6)
                                           (7)
                                           (8)
                                           (9)
where = 0.5– the coefficient of local resistance of entry into the pipe from the pressure tank; - coefficient of local resistance to sudden expansion of the pipeline at the border of sections with different diameters; = 5.25 - coefficient of local resistance of the valve; = 1,0– coefficient of local resistance of entry into the pipe from the pressure tank; v1, v2 - water flow rate in the pipeline section with a diameter d1, d2.
The coefficient of local resistance to sudden expansion of the pipeline
                                               (10)
where ωу, ωш are the areas of the narrow and wide sections of the pipe, respectively.
 
Substituting expressions (6-9) into (5), we obtain
                                     (eleven)
5. Losses along the length
                                            (12)
where l1, l2 are the lengths of the plots; d1, d2 are the diameters of the sections; -coefficients of hydraulic friction in sections 1 and 2, respectively.
6. Expression (4) can now be written as
                          (thirteen)
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