# Fluid Mechanics Lab Report; Study the Pipe Flow in both Laminar and Turbulent Flow Regime

### Objective of the Lab

We need to experimentally study the pipe flow in both laminar and turbulent flow regime.

Objective is to arrive at pipe roughness numbers and compare them with literature.

### Formulation

Head loss “ hL”  is directly proportional to the Pipe frictional constant “f” , Length of the pipe ( L) in m, Square of flow velocity.

Head loss “ hL”  is inversely  proportional to the Pipe diameter (D) in m , acceleration due to gravity (g) in m/s*s

Taking Log of above equation, we get

Log(hL)= log(f) + Log(L)- log(D)+ 2*Log(V) – log(2*g)

So plot of Log(hL) vs Log(V) must have linear behaviour and must have slope of 2.

Same phenomenon gets checked out we plot hL vs f*V2 , and verify if we get linear trend in laminar and turbulent flow regimes.

The friction factor is function of

, where e is the average wall roughness height

The Data of test setup

Pipe diameter is 3mm and pipe length 524 mm.

Specific gravity of water is 1000 kg/m3, and Hg 13594 kg/m3

### Procedure for data analysis

From the collected data an analysis program is made in MATLAB and the same is used for data analysis.

The programs codes are attached in appendix.

The Moody chart defines the variation of pipe friction coefficient for pipes of different roughness coefficients.

Moody’s diagram relates friction factor with Reynolds number and relative pipe rough ness (

Figure 1 : Gives the Moody Diagram taken from Wikipedia

### Results

The water flow results plot is shown in figure 2

‘Figure 2 : Water flow head loss coefficient multiplied by V*V Vs Reynolds number

Laminar flow, turbulent flow and transition zones are clearly seen.

We see laminar flow where plot has lower slope from Re number 1000 to 1700

From Reynolds number 2500 and beyond we can see higher slope, and now flow is turbulent.

Between Re number 1700 to 2500 we see slowly rising slope signifying transition zone.

Velocity in m/s for 9 test runs are

0.2947  0.345    0.404    0.491    0.606    0.707    0.7736  0.8842  1.0268

f*[V*V] is 0.00562 0.007          0.00876 0.01097 0.0137 0.0171 0.0214  0.0322  0.0483

The f varies from

 0.064656 0.058966 0.053627 0.045458 0.037279 0.034254 0.035787 0.041164 0.045785

For Re Number range

 884.195 1035.15 1212.61 1473.66 1818.91 2122.07 2321.01 2652.59 3080.42

Figure 3 shows the plots of experimental points for water flow experiments.

When we place the experimental points on Moody Diagram we see that Friction Factor Vs Re number follows Laminar flow path up to Re Number 2000.

 Re f 64/Re RE*f 1000*V*0.003/0.001 884.1949 0.064656 0.072382 57.16806 1035.155 0.058966 0.061826 61.03882 1212.61 0.053627 0.052779 65.02867 1473.658 0.045458 0.043429 66.98954 1818.915 0.037279 0.035186 67.80768 2122.068 0.034254 0.030159 72.68919 2321.012 0.035787 0.027574 83.06248 2652.585 0.041164 0.024127 109.191 3080.421 0.045785 0.020776 141.0384

Mean of First 5 points is = 63.6, so Laminar flow region formulation of 64/Re gets validated.

Figure 3 shows the plots of experimental points for water flow experiments.

At Re Number of 3080 point crosses transition zone, and relative pipe rough ness is predicted as 0.002.

Since pie diameter is 3 mm, the surface roughness is 0.006mm or 6 microns.

As only one data point is present in turbulent zone, the pipe surface rough ness may be get predicted with accuracy.

So further experiments are performed using Hg

For flow of Hg, 6 experimental data points are generated.

RE Number varies from 3000 to 8000, and data falls in turbulent zone.

‘Figure4: Hg flow head loss coefficient multiplied by V*V Vs Reynolds number

Results from MATLAB code

Volume_flow_Hg_cum_p_s =   1.0e-04 *[    0.0758    0.0946    0.1154    0.1471    0.1818    0.2333]

V_mps =[    1.0718    1.3382    1.6324    2.0805    2.5722    3.3010]

Re_nos =   1.0e+04 *[ 2.7310    3.4100    4.1595    5.3013    6.5543    8.4113]

f_Vmps_Vmps = [  0.0039    0.0058    0.0087    0.0130    0.0195    0.0293]

Actual RE Numbers are

27310   34100   41595   53013   65543   84113

V in m/s is

1.071    1.338    1.632    2.08      2.572    3.30

f*V*V is

0.00386 0.0058 0.00869 0.0130 0.0195  0.0293

Giving f as

0.0034 0.00324 0.00326 0.0030 0.00295 0.0269

The points are plotted on the Moody chart as shown in figure

Figure 5 shows the plots of experimental points for Hg flow experiments.

The flow is in turbulent regime.

Relative pipe roughness in in range 0.003 and 0.004. With 0.004, the roughness is 3mm*0.004=0.012 mm or 12 microns. From literature (ref1) it is

 Surface Material Absolute Roughness Coefficient – ε in mm Drawn Brass, Drawn Copper 0.0015 PVC, Plastic Pipes 0.0015 Fiberglass 0.005 Stainless steel 0.015

So surface roughness of 12 micron is reasonable.

Plot of log(hL) vs log( V) is shown in figure 6

Figure 6: Log(velocity) vs Log(hL) plot

We can see that behaviour is liner as expected.

The slope is = (5.6-3.5)/ (1.2-0.1) = 2.1 / 1.1, Close to 2 which is theoretical value.

Figure 7 demonstrates low RE region covered by water flow experiments, which falls in Laminar flow region, and Re number moves to turbulent flow region with Hg flow experiments.

Figure 7 : Friction factor for both water flow and Hg flow experiments

### Conclusions

Nine sets of experiments were performed using water, which covered laminar flow region and transitory flow region. First 5 test points validated friction factor of 64/Re formulation.

During transition region this factor keep rising from 64 to 141. The last point falls beyond transition zone.

For data of flow of Hg, re no varies from 1.0e+04 *[ 2.7310    3.4100    4.1595    5.3013    6.5543    8.4113] and from Moody chart, we could predict surface roughness as 12 microns, which looks reasonable.

### References

1: https://www.enggcyclopedia.com/2011/09/absolute-roughness/

### Appendix A

MATLAB codes

Exp 1 with water flow

Dia_m=0.003;Length_m= 0.524;viscocity= 0.001;

Area_sq_m= (3.14159/4)*(0.003*0.003);

H2O_Head_loss=[50 62.5 78 97.65 122 152.58 190.7 286.5 429.75];

Volume_flow_H2O_cum_p_s=[75/36 100/41 100/35 125/36 150/35 150/30 175/32 200/32 225/31]*0.000001

V_mps=Volume_flow_H2O_cum_p_s/Area_sq_m;

Re_nos=1000*V_mps*Dia_m/viscocity;

plot(Re_nos, f_by_Vmps_Vmps)

Results

>> Lab_code_H2O

Volume_flow_H2O_cum_p_s =

1.0e-05 *

0.2083    0.2439    0.2857    0.3472    0.4286    0.5000    0.5469    0.6250    0.7258

V_mps =

0.2947    0.3451    0.4042    0.4912    0.6063    0.7074    0.7737    0.8842    1.0268

Re_nos =    1.0e+03 *[    0.8842    1.0352    1.2126    1.4737    1.8189    2.1221    2.3210    2.6526    3.0804]

f_by_Vmps_Vmps =[ 0.0056    0.0070    0.0088    0.0110    0.0137    0.0171    0.0214    0.0322    0.0483]

Exp 2 with Hg

Dia_m=0.003;Length_m= 0.524;viscocity= 0.0016;

Area_sq_m= (3.14159/4)*(0.003*0.003);

Hg_Head_loss=[34.38 51.57 77.35 116 174 261];next=6;

Volume_flow_Hg_cum_p_s=[250/33 350/37 450/39 500/34 600/33 700/30]*0.000001

V_mps=Volume_flow_Hg_cum_p_s/Area_sq_m

Re_nos=13590*V_mps*Dia_m/viscocity

%plot(Re_nos, f_Vmps_Vmps)

for i = 1:next

friction_coef(i)= f_Vmps_Vmps(i)/(V_mps(i)*V_mps(i));

end

friction_coef

% plot(Re_nos, friction_coef)

Results

>> Lab_code_Hg

Hg_Head_loss_m =    0.0344    0.0516    0.0774    0.1160    0.1740    0.2610

Volume_flow_Hg_cum_p_s =   1.0e-04 *

0.0758    0.0946    0.1154    0.1471    0.1818    0.2333

V_mps =    1.0718    1.3382    1.6324    2.0805    2.5722    3.3010

Re_nos =   1.0e+04 *

2.7310    3.4100    4.1595    5.3013    6.5543    8.4113

f_Vmps_Vmps =    0.0039    0.0058    0.0087    0.0130    0.0195    0.0293

friction_coef =    0.0034    0.0032    0.0033    0.0030    0.0030    0.0027

>>

Combined code

Dia_m=0.003;Length_m= 0.524;viscocity_Hg= 0.0016;

Area_sq_m= (3.14159/4)*(0.003*0.003);

Hg_Head_loss=[34.38 51.57 77.35 116 174 261];nexp_Hg=6;

Volume_flow_Hg_cum_p_s=[250/33 350/37 450/39 500/34 600/33 700/30]*0.000001

V_mps=Volume_flow_Hg_cum_p_s/Area_sq_m

Re_nos=13590*V_mps*Dia_m/viscocity_Hg

for i = 1:nexp_Hg

friction_coef_Hg(i)= f_Vmps_Vmps(i)/(V_mps(i)*V_mps(i));

end

friction_coef_Hg

figure

subplot(2,2,1);

subplot(2,2,2);

plot(Re_nos, friction_coef_Hg)

hold on

Dia_m=0.003;Length_m= 0.524;viscocity_H2O= 0.001;nexp_H2O=9;

Area_sq_m= (3.14159/4)*(0.003*0.003);

H2O_Head_loss=[50 62.5 78 97.65 122 152.58 190.7 286.5 429.75];

Volume_flow_H2O_cum_p_s=[75/36 100/41 100/35 125/36 150/35 150/30 175/32 200/32 225/31]*0.000001

V_mps_H2O=Volume_flow_H2O_cum_p_s/Area_sq_m

Re_nos_H2O=1000*V_mps_H2O*Dia_m/viscocity_H2O

for i = 1:nexp_H2O

friction_coef_H2O(i)= f_by_Vmps_Vmps(i)/(V_mps_H2O(i)*V_mps_H2O(i));

end

friction_coef_H2O

subplot(2,2,1);

hold on

subplot(2,2,2);

plot(Re_nos_H2O, friction_coef_H2O)

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