Fluid flow in a duct of varying cross-section.

Conclusions

In conclusion the results showed that as the water travelled through the pipe the static head was uniform until it reached the neck of the pipe. When the water flowed through the neck the static head increased due to the decrease in pressure.

The results also showed that as the water flowed through the neck of the pipe the velocity increased because of the reduction in area, which reduced the pressure.

 

Appendix

Here is the raw data obtained from the experiment for flow rates of 46, 34 and 24 litres / minute.

Table 1

Table 2

Table 3

Specimen calculations;

The velocity (v) was calculated from the continuity equation Q = v A , where A = the cross sectional area of the pipe.

i.e Table 1 tap 4:

v = Q / A

                                      = 46 / πr2

                                   = 46 / π (7.75)2

                                   = 0.0180 mms-1

                                        

When plotting  (hn – h1 ) versus x graph, (hn – h1) was calculated by subtracting the first value for height from the rest of the values for height.  

i.e. Table 1, tap 4:

(hn – h1 ) = (h4 – h1 )

                                           =  46.7 – 140.6

                                          = -93.9 mm

The calculations for the second graph involved multiplying (hn – h1 ) by 2g (19.6) and dividing by v12  

i.e Table 1 tap 4:

2g(hn – h1 ) = 19.6(46.7 – 140.6)

                                v12                                   0.001214

                                         =  -90507.9 mm

The calculation for the third graph was similar to the second graph. The formula is basically the same except  (Vn  / V1)2  -1        is added to the equation.

                                               

i.e. Table 1 tap 4:

2g(hn – h1 )+(Vn  / V1)2 –1 = 19.6(46.7 – 140.6)+(0.074705/0.034842)2 -1

       v12                                                                0.001214

                                                                 

                                  = -90507.9 + 3.5972

                                  = -90504.3 mm

Error estimation

Error in flow rate measurements = + 1 litre / min

Mean value of flow rate = 35 litre / min

Therefore the percentage error in the flow rate = δQ   x 100                                                                                 Q

                                                                

                                                            = 1   x 100

       35

    = 2.9%

Error in pressure readings = + 1 mm

Value of h at the neck (with flow rate of 46 l/min) = 46.7cm or 467 mm

Therefore the percentage error = δh x 100

                                            h

                                         =  1 x100

                                          467

                                         =  0.2 %

Figures 

Figure 1 = (hn – h1 ) versus x which, represents the change in static head.

Figure 2 = 2g(hn – h1 )   versus x, which represents the change in                                v12         normalised static head.         

Figure  3 =   2g(hn – h1 ) + (Vn  / V1)2 –1 versus x, which represents the                                                                                                                                                                                                                                                                          .                                        v12                                                     variation of normalised total            .                                                                head along the duct.