Question

Special Assignment #3 (Due date March 12, 2020, Thursday) Objectives: (1) To calculate and plot the velocity profile vz(r) in annulus with Eq. (6.155) using Excel spreadsheet. (2) To numerically integrate the velocity profile from rito ro to obtain volume flowrate Q. You can use any integration tool such as trapezoidal rule in Excel, Mathlab, etc. (3) To compare the volume flowrate obtained from the numerical integration with the value that can be calculated from Eq. (6.156) Given: inner radius = 5 cm; outer radius = 10 cm; liquid viscosity = 0.05 Pa.s dp/dz = -10,000 (Pa/m) Trapezoidal Rules Two students can work together and Submit one report. 1 = ax flx.) + z fino) t.. follows + finn) ax=brada Thas fwax

Answer 1

the below figure shows the calculation of velocity profile in column Vz and it also shows the formula applied in the excel cell to evaluate the velocity as per equation 6.155.

the velocity profile is also depicted in the below figure.

Note: the units for the radius are taken in "m" as to maintain the SI units throughout the calculations.

D23 X V C fx D =($B$2/(4*$B$1))*(C23^2-$B$4^2+(($B$3^2-$B$4^2)*LN(C23/$B$4)/LN($B$4/$B$3))) E E F G H I J K L M A 1 mu 2 dp/dz - B 0.05 -10000 0.05 0.1 The Equation 6.155 has been used to evaluate the velocity profile in the Vz column below Velocity profile (Vz) Flow velocity (m/s) 0.05 0.052 0.054 0.056 < 0.058 0.06 0.062 0.064 0.068 0.07 9 0.066 0.072 0.074 0.076 0.078 0.08 0.082 0.084 80.086 0.088 6 600 € 0.092 0.094 9600 0.05 0.052 11.018823 0.054 20.836742 0.056 29.512025 0.058 37.096802 0.061 43.637902 0.062 49.177545 0.064 53.753929 0.066 57.401724 0.068 60.152494 0.07 62.03506 0.072 63.075804 0.074 63.298941 0.076 62.726746 0.078 61.379761 0.08 59.276964 0.082 56.435931 0.084 52.872962 0.086 48.603212 0.088 43.640786 0.0937.99884 0.092 31.689662 0.094 24.724748 0.096 17.114867 0.098 8.8701204 0.1 0.1 0.098 Variation of radius from inner to outer point (m)

the below figure represents the calcuation as per the objective 2 and 3 asked in the question.

At first the flow rate has been evaluated as per the Trapezoidal rule and its formula is also shown.

Then the flow rate is evaluated as per the equation 6.156 and its formula is highlighted in the formula bar of excel as per the equation 6.156.

The error is evaluated to comapre the results from both the methods.

H50 =-3.14*(B36/(8*B35))*(B3844-B3714-(((B38^2-B3742)^2/LN(B38/B37))) 34 35 mu 36 dp/dz 37 ri 38 ro 39 0.05 -10000 0.05 0.1 25 h = (ro-ri)/n 0.002 S.No 40 41 Using the trapezoidal rule Flow rate (Q) = (h/2)*(f(x0)+F(x25) + 2*(f(x1)+f(x2)+.....+f(x24))) 44 Q 0.98740 m3/s 48 0.05 0.052 0.054 0.056 0.058 0.06 0.062 0.064 0.066 0.068 0.07 0.072 0.074 0.076 0.078 Using the equation 6.156 the flow rate (Q) is Q 0.98897 m3/s del E = 0.98897 -0.98740 = 0.00157 Vz f(x)=Vz*2*pi*r 0 11.018823 3.598306884 20.836742 7.066155996 29.512025 10.37878881 37.09680213.51213916 43.637902 16.44276154 49.177545 19.14776901 53.753929 21.60477908 57.401724 23.7918664 60.15249425.68752117 62.03506 27.27061246 63.075804 | 28.52035571 63.298941 29.41628384 62.726746 | 29.93822151 61.379761 30.06626208 59.27696429.78074692 56.435931 29.0622468 52.872962 27.89154511 48.603212 26.24962265 43.64078624.11764388 37.99884 21.47694434 31.68966218.30901932 24.724748 14.59551337 17.11486710.31821078 8.8701204 5.459026885 0 0 there is an error of 0.00157 m3/s in the calculation of flow rate from the trapezoidal rule and equation 6.156 56 0.08 0.082 0.084 0.086 0.088 0.09 0.092 0.094 0.096 0.098 0.1 67