47. Pipeline with pump

The foregoing discourse dealt essentially with isolated sections of plain and compound pipelines, not with complete systems of liquid supply (with the exception of the simplest gravity flow system). It was mentioned, furthermore, that in aeronautical engineering forced flow by means of pumps is the main method of liquid supply. Let us investigate the operation of a pipeline with a pump and the principles for calculating such pipelines.

A pipeline with pump may be open, if the liquid is pumped from one place to another, or closed (looped) if a constant volume of liquid is recycled.

We shall first consider an open pipeline (Fig. 122) in which the liquid is driven by pump from, say, a reservoir where tbfe pressure is p₀ to the combustion chamber of an engine where the pressure is p₃ or to another reservoir.

The elevation of the pump shaft above the lower level #, is the geodetic suction head, the pipe along which the liquid flows to the pump being the suction, or intake, pipe or line. The elevation of the end section of the pipeline, or the upper level of the liquid (#₂), is the geodetic delivered head and the pipe through which the liquid travels from the pump is called the pressure, or discharge, or outlet pipe or line.

Bernoulli's equation for the motion of the liquid in the suction line, i. e., between sections 0-0 and 1-1, yields

This equation shows that the pump develops suction (lifts the li­quid to the height H_t)y imparts kinetic energy to the liquid and over­comes hydraulic resistances by expending the pressure p_o.As this pressure is usually very limited, it should be expended in such a way as to continuously maintain some pressure reserve p_x at the pump intake, which is essential for the pump to work without cavitation. It stands to reason that suction lines should therefore be calculated with special care.

Equation (11.11) is the basic formula for this.

The following solutions may be required to calculate a suction pipeline.

1. Given dimensions and rate of discharge. To determine absolute pressure at pump intake.

The solution of the problem is a verification estimate for the suc­tion line. The absolute pressure p_t as found from Eq. (11.11) is compared with the minimum permissible pressure for the case under consideration.

2. Given minimum permissible absolute pressure p_imin at pump intake. To determine one of the following permissible limiting quan­ tities:

Determination of the last quantity is of special importance in aircraft hydraulic systems, in which p₀ = p_aim + Δp. Here Δp is the pressure above atmospheric pressure produced by supercharging or the pressure of an inert gas. The minimum permissible atmospher­ic pressure p_{atni min} is determined from pomin, then the maximum permissible altitude of flight for an aircraft equipped with such a system is determined from a standard atmosphere table.

An increase in the pressure p₀ increases the pressure in the whole of the siiction line, which makes for greater altitude range of the system. On the other hand, high pressure in the supply tank means stronger walls and, consequently, greater weight of the tank. There­fore greater altitude range of aircraft hydraulic systems is usually attained by installing an additional pump at the beginning of the suction line to produce a higher pressure in the latter, thereby pre­venting cavitation at the intake of the main pump.

Bernoulli's equation for a liquid in the pressure pipe between sec­tions 2-2 and 3-3 yields

If the pressure pipe discharges into a reservoir there is no velocity head in the right-hand side of the equation, but the loss of head due to enlargement must be taken into account.

The left-hand side of Eq. (11.12) represents the specific energy of the liquid at the pump outlet.

The specific energyat the intake can be found from Eq. (11.11).

Let us find the increase in the specific energy of the liquid in the pump, i. e., determine the energy obtained by every kilogram of the Hquid in passing through the pump This energy is imparted by the pump, and it is called the pumping head, denoted H_pamp. To determine H_pamp, subtract the last equation from Eq. (11.12):

or

where Δz = total geodetic elevation of liquid (see Fig. 122);

kO^m = total hydraulic losses in suction and pressure pipings

If we add to the actual change in elevation Δz the pressure head difference we consider an equivalent added lift

and Eq. (11.13) can be rewritten

Compare the expression (11.13) with the required head formula in (11.6). It is evident that

This equality can be extended to all cases of steady operation of a pump in a pipeline and formulated as a rule: In the case of steady flow in a pipeline the head developed by the pump is equal to the required head. This is the only condition of steady pump perform­ance. Usually it is maintained automatically.

Equation (11.14) is the basis for a method of calculating pipelines with pumps which consists in plotting two curves to the same scale on one diagram—the characteristic curve of the pipeline and the characteristic curve of the pump - and finding their point of intersection (Fig. 123).

Pump characteristics will be discussed in greater detail later on in a special chapter. Still, running ahead somewhat, the following definition must be given: the characteristic of a pump is defined as the dependence of the pumping head on the delivery (rate of dis­charge) at a constant speed of rotation.

The intersection of the pipeline characteristic and the pump char­acteristic locates the point of equality between the required head and the pumping head, i. e., Eq. (11.14). This is known as the oper­ating point. The pumping regime is always maintained, as far as possible, so as to correspond to that point.

To obtain another operating point the opening of the control valve(i. e., the pipeline characteristic), or the pump's rpm must be changed. This will be discussed in detail later on.

It should be remembered, however, that this analytical method of determining the operating point is applicable only when the rota­tion speed of the pump drive does not depend on the power input, i. e., the load on the pump shaft. This is possible when a pump is driven by an alternating-current electric motor or an aircraft engine whose power is many times greater than the power of the pump.

If a pump is driven by a special internal-combustion engine or tur­bine, the power of which depends

on the pump shaft load, the cal­culation must necessarily be different. The curves of required and available power must be plotted against the number of revolutions per minute and their point of intersection will then give the operating speed and power.

For a looping pipeline (Fig.. 124) the geodetic elevation of the liquid is zero (Δz = 0), and, con­ sequently, at v_i = v₂,

i. e., the same equation is valid for the required head and the pump­ing head.

A looping pipeline must always be provided with a so-called expan­sion or compensation tank, usually tapped at the pump intake where pressure is lowest. Without such a tank the absolute pressure inside the pipeline would be unsteady and would also vary with- tempera­ture and due to leakage.

With an expansion tank tapped to the pipeline as shown in Fig. 124 the pressure at the pump intake becomes steady and equal to

When Pi is known, the pressure at any cross-section of the loop can be calculated. If the pressure p₀ in the tank changes, the pressure at any point in the system will change by the same value, i. e., Pas­cal's law of the transmission of pressure in a still liquid (see Sec. 6) is valid. The tank can also be tapped to the loop as shown in Fig. 149.

Example 1. Determine the required head at the outlet of a booster pump in an aircraft fuel system to supply T-l fuel at a rate of G = 1,200 kg/hr from the service tank to the engine fuel pump if the length of the duralumin piping is / = 5 m, diameter d = 15 mm, required pressure at fuel-pump ftitake p% =* = 1.3 kg/cm⁸, kinematic viscosity of kerosene у =» 0.045 cm²/sec, specific weight y_k— 820 kg/m*. Local disturbances in piping are shown in Fig. 125. Neglect elevation of liquid in tank.

Solution, (i) Velocity of flow in pipeline:

(ii) Reynolds number]

(iii) From Konakov's formula, the friction factor k» 0.0328. (iv) From Table 2 in Chapter IX, the loss coefficients of the filter, stop cock, flow meter sensor and elbows are

(v) Pressure drop in pipeline from booster pump to fuel pump:

(vi) Hence the required pressure at the booster pump outlet is

Example 2. Referring to Fig. 126, fuel T-l is delivered from two under-wing tanks to the service tank by higher pressure in the former: Δр = p_uw — p_st =0.2 kg/cm².

Determine the diameter d of the piping if the two under-wing tanks are to be emptied simultaneously and the total rate of discharge of the fuel is G =1500 kg/hr. The volume of each under-wing tank W = 450 lit. The pipingis of duralumin, I = 7m, Kinematic viscosity of kerosene v = 0.045 cm²/sec, specific weight γ_k = 830 kg/m^s. Neglect height of liquid columns in under-wing tanks.

Solution, (i) Time of engine operation until under-wing tanks are emptied:

(ii) Rate of discharge from each tank.

(iii) Assign several pipe diameters: d = 12; 14; 16; and 18 mm.

(iv) Table 2 gives loss coefficients at pipe entrance ζ_entrancei nonreturn valve ζ_valve elbow ζ_elbow tee ζ_tee pipe outlet ζ_itoutie. After determining Re compute the friction factor λt according to Konakov for each value of d.

(v) Determine the required head H_req for each value of d from the formula

and plot curve H_req = f(d) (Fig. 127).

(vi) Dividing Δp by y_ki determine available head and from diagram find required diameter, which is d = 15.7 mm.

Hence, the commercial pipe diameter to ensure the required flow rate for simultaneous delivery of fuel from under-wing tanks is d = 16 mm.

If the pipelines were of different length, their diameters would also have to be different for the given set of conditions and the curves H_req=f(d) would have to be constructed for each pipeline.

Example 3. When an airplane is refuelled under pressure all the tanks must be filled simultaneously. A block diagram of the refuelling system is shown in Fig. 128.

Let all the tanks lie in a horizontal plane at an elevation h_t over the refuel­ler pump. The elevation of the fuel main А В above the pump is h_A. The charac­teristic of the refuelling pump, the length l_si and diameter d_sl of the filling sleeve, the length of all pipelines and the volumes of the tanks are given.

Neglecting the height of the liquid columns in the tanks and the gauge pres­sure in them, solve the following commonly occurring engineering problems:

(I) Determine the refuelling time T if the diameters of the piping are given.

(II) Determine the required pipe diameters d_m, d_v cfo, d_t to ensure the simul­ taneous filling of all the tanks in time T.

(I) The solution is graphoanalytical.

(i) Plot characteristic curve of filling sleeve from pump to the filling connec­tion A.

(ii) Plot characteristic curve for fuel main from A to B.

(iii) Sum both curves according to the rule of compound pipes in series (Fig. 129).

(iv) Plot characteristics for piping from the cross at В to respective tanks.

(v) Sum characteristics in (iv) according to the rule for compound pipes in parallel.

(vi) Summation of the net characteristic of the four parallel pipes and the pipe characteristic from the pump to В gives the characteristic curve of the whole composite pipe system.

(vii) The intersection of the pipeline character­istic with the refuelling pump characteristic gives the pumping head H_pump and the rate of discharge of the pump Q_pamp.

(viii) The rate of discharge into each tank is determined as shown in Fig. 129.

(ix) The refuelling time T (when all tanks are filled simultaneously) is

(II) The solution is graphoanalytical. (i) Determine the rate of discharge Q_pump of the refuelling pump necessary to fill all tanks in time T by dividing the total tank volume by T.

(ii) Locate the operating point on the pump char­acteristic according toQ_pump, i. e., the pumping head H_pamp. The pipe diameters must be so selected as to ensure a rate of discharge equal to Q_pamp when the head is H_pJ{mp,

(II) Determine the rate of discharge through each pipe by dividing the volume of each tank by the time T.

(iv) Determine the loss of head H_5t in the filling sleeve, taking into account the minor losses in the nonreturn valve of the filling connection.

(v) Plot the loss of head as a function of the diameter of the fuel main be­tween A and B: Ha-b = f(d). For this assign several diameter values and for each d determine: the Reynolds number Re, the friction factor l_t, and the loss of head Ял-в, including the difference between the elevation heads of the fuel main and the refueller pump h_A and losses in the sleeve H_sl.

(vi) Plot the head losses as a function of diameter for each pipe to the respe­ctive tanks.

The construction is the same as that of the curve, but the ori­gin of the coordinate axes is taken at point Н_рищр, the positive direction of the diameter axis being to the left and of the head axis, downwards (Fig. 130). Add to thie head losses the head h_tank — h_A.

This rather unusual directing of the coordinate axes and the curves for the respective pipings simplifies the determination of the required diameters.

Assign a diameter d_m of the fuel main and, from the diagram, determine the diameters d_u d% , d_t as shown by arrows in Fie. 130. The diagram can be used to determine several diameters from which tne most economical can be chosen.

Point В on curve shows the loss of head H_B between the pump and the cross. The remaining head is used to overcome the resist­ances to the flow through the pipes to the tanks and the elevation h_iank — h_A.

This method of problem solution can be employed when the pipes to the respective tanks branch at different points; in this case three curves have to be plotted instead of a single curve .