Paraphrasing Lab

ME 3506

Impact Force of a Jet

 

 

 

 

ABSTRACT

 

Whenever a jet of fluid strikes a surface, which can be of different shapes a force is generated based on the law of conservation of momentum. This property of fluid jet finds application in a number of hydraulic machineries. One of the major applications is in the designing of hydraulic turbines. It is of prime importance to know about the impact force generated by the jet over different shapes of the surfaces. An analysis to determine what shape of vane would be best suited to draw maximum efficiency in terms of output from the turbine would be necessary. In this experiment we have tried to study the impact force of jet of water over a flat and a cup vane, and compare it with theoretical results from momentum equation. Results of the experiment were found to be accurately equal to the theoretical values. The variation in actual and experimental values was due to assumptions of vane surface to be frictionless and neglecting the effects of gravity.

 

 

Table of Contents ABSTRACT 2 1. INTRODUCTION 1 2. THEORY 1 3. EXPERIMENTAL APPARATUS 3 4. PROCEDURE 4 5. RESULTS 5 6. CONCLUSION 8 DATA REDUCTION AND COMPARISON WITH THEORY 9 APPENDIX 11 REFERENCES 12

1. INTRODUCTION

 

The jet emerging from the outlet of a nozzle would have the liquid flowing under extreme pressure. If some surface, which could be fixed or moving is placed in the path of this jet a force is exerted by on this surface. This force is due to the impact of jet on the surface and can be determined by Newton’s second law of motion (Impulse-momentum equation).

This property of jets is utilized in generation of electricity in impulse turbines, mainly pelton turbine. The jet of water is produced with the help of a nozzle and is applied tangentially to the cups (or buckets) attached over the circumference of the pelton wheel. When the jet strikes the bucket a force is generated which in turn induces a moment to the wheel. This moment of wheel increases the mechanical energy of the wheel, which is then converted into electrical energy.

Another application of force due to jet is water jet cutter. With the help of water jet cutters metal surfaces can be cut with very smooth final surfaces. This method of metal cutting is highly efficient than other metal cutting techniques because it does not produce any heat and dust.

Thus in order to understand the output of a pelton wheel and for metal cutting process, we need to have an idea about the effect of jets on various surfaces. We also need to understand how the deflection in direction of jet produces the force and how is this force related to the momentum of the fluid flowing through the jet. In this experiment we have tried to study the jet force on a flat surface and a hemispherical surface and compare it with theoretical values.

 

2. THEORY

 

The theoretical relationship for impact of jet on a stationary surface can be established by considering integral forms of continuity and momentum equations. For one dimensional steady incompressible fluid flow condition, the mass conservation equation is given by continuity equation as shown in equation 1.

 

 

 

 

(1)

 

 

Where Q is the volumetric flow rate,

and are velocities at section 1 and 2 respectively, and

and are cross-sectional area of pipe at section 1 and 2.

Figure 1 shows the fluid flow parameters and the geometric parameters for the derivation of theoretical expression of force on stationary surfaces due to jet of water. , is the velocity of jet by which it strikes the surface and deflects away in radially outward direction with velocity of at an angle of . The dashed lines represent the control volume considered for the derivation. It is assumed that the friction between water and the impact surface is negligible and the jet velocity does not vary. Under this assumption and from equation 1 we can write

 

 

(2)

 

Also in horizontal direction, since the flow and the surface is symmetrical, the horizontal reaction will be zero.

Figure 1: Geometric and fluid parameters for impact of jet experiment

 

Applying the integral form of impulse momentum equation we get:

 

  (3)

 

Where is the reaction force in vertical direction. Now, using equation 2 this relationship can be reduced to:

  (4)

Here:

Thus Equation 4 can be written as:

  (5)

Equation 5 represents the theoretical expression for computing the impact force on a stationary surface due to a vertical jet of liquid.

3. EXPERIMENTAL APPARATUS

The schematic diagram for the experimental arrangement is shown in figure 2 and the apparatus diagram is shown in Figure 3. The water is supplied from the pump to a vertical pipe, which ends in a nozzle. This produces a jet of water which strikes the vane, in form of a flat plate or a hemispherical cup shape surface.

Figure 2: Schematic for the experiment arrangement.

 

This arrangement of the nozzle and the vane is contained in a transparent cylinder. This cylinder ends in an outlet, which flows directly to the collection tank. The vane is attached to a lever carrying jockey weight and is restrained by an arrangement of light spring. The jockey weight can be adjusted at a desired length over the lever to bring back the vane to its original balanced position.

 

Picture 479 

 

Figure 3: Jet-Impact Measuring Bench.

 

4. PROCEDURE

 

Step 1: In the first step the lever was set to a balanced position by placing the jockey weight at zero and adjusting the nuts over spring arrangement.

Step 2: Now, jockey weight, diameter of the nozzle, vane height above the nozzle tip and the distance of vane Centre to lever pivot were noted down.

Step 3: Now water was allowed through the nozzle by opening the flow control valve. As the jet stroked the surface the vane moved upwards. Now additional weights were placed bring back the lever to its original balanced position.

Step 4: At this position the flow rate was noted down from the collection tank by measuring taken to collect a fixed weight of water collected in tank, using a stopwatch.

Step 5: The procedure was repeated to get another set of readings at same flow rate. Step 6: Steps 1 to 5 were repeated to get readings for four more flow rates.

Step 7: Steps 1 to 6 were repeated for hemispherical vane.

 

5. RESULTS

 

This section of the report explains the results of the experiment and its comparison with the expected theoretical results. The experiment was performed over the apparatus mentioned in section 3 of the report. Appendix shows the data recorded as per step 2 of section 4.

Based on the steps mentioned in section 4, Table 1 and Table 2 shows the experimental force values for flat surface and Cup shaped surface respectively.

 

Table 1: Experimental observations for flat plate
    First reading Second reading average
Flow rate 1 Force(N) 1.81 1.77 1.79 N
  t1(s) 22.86 22.72 23.2475 s
  t2(s) 22.72 24.69  
         
Flow rate 2 Force(N) 1.51 1.45 1.48
  t1(s) 24.89 24.4 24.9825 s
  t2(s) 24.63 26.01  
         
Flow rate 3 Force(N) 1.22 1.15 1.185
  t1(s) 28.79 28.25 28.52 s
  t2(s)      
         
Flow rate 4 Force(N) 0.93 0.83 0.88
  t1(s) 31.47 34.59 32.5275 s
  t2(s) 31.25 32.8  
         
Flow rate 5 Force(N) 0.61 0.58 0.595
  t1(s) 38.79 39.11 38.675 s
  t2(s) 38.39 38.41  

 

 

Table 2: Experimental observations for cup
    First reading Second reading average
Flow rate 1 Force(N) 3.37 3.27 3.32
  t1(s) 23.61 22.53 22.695
  t2(s) 22.32 22.32  
         
Flow rate 2 Force(N) 2.69 2.65 2.67
  t1(s) 25.06 25.08 25.4575
  t2(s) 25.14 26.55  
         
Flow rate 3 Force(N) 2.19 2.14 2.165
  t1(s) 28.09 28.06 28.175
  t2(s) 28.05 28.5  
         
Flow rate 4 Force(N) 1.7 1.66 1.68
  t1(s) 30.89 32.21 32.0575
  t2(s) 33.51 31.62  
         
Flow rate 5 Force(N) 1.2 1.16 1.18
  t1(s) 39.03 39.14 39.3475
  t2(s) 39.89 39.33  

 

 

The time values in column 3 and 4 of Table 1 and 2, shows the time taken to collect 30 lbs of water in the collection tank. The average force values in column 5 shows the mean value of force to bring the lever to original balanced position. And the average time values in column 5 shows the mean time taken to collect 30 lbs of water for each case of flow rates. It can be observed that for almost similar flow rates the impact force on cup shaped surface is greater as compared to flat surface.

From equation 4 the theoretical impact force on flat plate and cup are shown in Table 3 and Table 4 respectively.

 

Table 3: Theoretical force calculations for flat plate
  Average time flowrate(kg/s) v0(m/s) v1(m/s) Impact force (Theoretical) F @ 15 in.
Flow rate 1 23.248 0.585 8.231 8.189 4.769 1.908
Flow rate 2 24.983 0.545 7.659 7.614 4.123 1.649
Flow rate 3 28.520 0.477 6.709 6.658 3.153 1.261
Flow rate 4 32.528 0.418 5.883 5.824 2.412 0.965
Flow rate 5 38.675 0.352 4.948 4.878 1.692 0.677

 

 

 

Table 4: Theoretical force calculations for flat plate
  Average time flowrate(kg/s) v0(m/s) v1(m/s) Impact force (Theoretical) F @ 15 in.
Flow rate 1 22.6950 0.600 8.431 8.391 10.013 4.005
Flow rate 2 25.4575 0.535 7.516 7.471 7.938 3.175
Flow rate 3 28.1750 0.483 6.791 6.741 6.463 2.585
Flow rate 4 32.0575 0.424 5.969 5.911 4.970 1.988
Flow rate 5 39.3475 0.346 4.863 4.792 3.266 1.306

 

 

The theoretical values were found to be very close to the actual experimental values. This observation was further examined by computing the percentage error (shown in Table 5). The error in two values was more for cup surface as compared to flat surface.

 

Table 5: Comparision between actual and theoretical forces
    Actual Theoretical Error Percentage error
Flat plate Flow rate 1 1.790 1.908 0.118 6.575
  Flow rate 2 1.480 1.649 0.169 11.440
  Flow rate 3 1.185 1.261 0.076 6.414
  Flow rate 4 0.880 0.965 0.085 9.649
  Flow rate 5 0.595 0.677 0.082 13.754
           
Cup surface Flow rate 1 3.320 4.005 0.685 20.643
  Flow rate 2 2.670 3.175 0.505 18.922
  Flow rate 3 2.165 2.585 0.420 19.404
  Flow rate 4 1.680 1.988 0.308 18.332
  Flow rate 5 1.180 1.306 0.126 10.719

 

 

In order to interpret the association between the actual and theoretical force values a linear regression analysis was performed between the two sets of data for both flat and cup surfaces. The result has been shown in Figure 4 and Figure 5 for flat plate and cup surface respectively. The R-squared value for the straight line fitted to the dataset shows a value of 0.99631 for flat plate case and a value of 0.99944 for cup surface case. The R-square values are very close to 1, representing a linear relationship between actual and theoretical force values.

Figure 4: Linear regression between actual and theoretical impact force on flat plate

 

Figure 4: Linear regression between actual and theoretical impact force on cup

 

6. CONCLUSION

This experiment was conducted to examine the impact of jet over stationary solid surfaces of different shapes. This experiment is based on principle of momentum. From the results observed from the experiment and comparing the same with the theoretical computed values it was observed that the experimental data is approximately same as that of theoretical data. It can also be verified from the results in table 3 and table 4 that the net force acting on the plate is proportional to the momentum of the fluid flowing through the nozzle. The percentage error computed in table 5 shows that the error is more for the cup shaped surface. The error being very less the experimental results can be accepted.

The discrepancy in experimental and theoretical impact force values can be attributed to errors and the assumptions involved during the experiment. The contact surface between water and target planes was assumed to be frictionless, which in real posses some amount of friction. Further, the collision of the water molecules with the vanes is not fully elastic and would cause some loss of energy resulting in decrease of velocity.

The actual impact force was observed to be less than the experimental values for all the cases. This can be due the fact that while the water from jet reaches the vane; a part of kinetic energy would convert into potential energy.

7. DATA REDUCTION AND COMPARISON WITH THEORY

The intermediate calculations Based on the raw data are shown in this section of the report. The sample calculations are shown for impact force on flat vane under the flow rate 1 conditions.

 

Theoretical force calculations for flat vane:

Diameter of jet at nozzle = 0.375 in =0.375 x 2.54 = 0.9525 cm

 

Height of vane (h) = 1.375 in = 3.4925 cm

Weight of water collected = 30 lbs

 

 

Average Flow rate calculations for flow rate 1 in case of flat plate:

 

 

 

 

 

Theoretical impact force on the flat plate was computed from equation 5 as following:

 

 

Force on the plate in imperial units:

 

Mean actual impact force:

 

 

 

 

APPENDIX

 

Necessary constants

Distance from jet center to the fulcrum point 6.000 inches
Distance from jet center to force gauge 9.000 inches
Jet diameter 0.375 inches
Distance from nozzle to vane 1.375 inches

 

 

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