Ejercicios de Ecuaciones Diferenciales, Resueltos en Matlab

 

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Chapter 2 Excel  Practice  Practice Problems Use Excel  and  and VBA VBA to  to solve the following following problems. Document your solut solutions ions using the Expert Problem Solving  steps  steps outlined in Table 1.2.

1.  1.  Often, the best way to learn a new software application is simply simply to try it. Click through the various Excel  features  features shown on the ribbon tabs at the top of an  Excel  worksheet.  worksheet. Make a list of the top 10 that you find helpful for engineering engineering problems solving. solving. Include a short description description of each feature and a simple example. 2.  2.  Perform the following calculations and report the correct number of significant figures in the results. 12.2  569.08  36  0.00768   678.98  23.4567  9.5 2490  363

 

 12300      1.98  567 

15.6  1012 exp  

log  23  36  2     Answers: 545, 0.0013, 2.72x10 2.72x108 , 2

3.  3.  Use the following Chen-Othmer Equation to estimate the diffusivity of gas species CO 2 (A) in air (B) at 2.0 atm pressure and 303 K temperature (use DIPPR or another reference for the molecular properties): 0.1498T 1.81  D AB



0.1405

 P  TCATCB 

1  M A

  



VC0A.4

1 M B



V C0B.4



2

 

where DAB = diffusion coefficient, cm2/s T = temperature, K P = pressure, atm MA, MB = molecular weight of A and B, g/mol TCA, TCB = critical temperatures of A and B, K VCA, VCB = critical molar volumes of A and B, cm 3/mol  Answer: 0.84 cm2 /s

4.  4.  Rosin, Rammler, and Intelmann derived an equation for determining the minimum particle diameter that can be captured by a gas cyclone: d min



9   B  NV   s

  

 

where dmin = minimum particle diameter, m

 = gas viscosity, kg/m s B = width of rectangular cyclone inlet duct, m  N = number of turns made by the gas stream in the cyclone V = average cyclone inlet velocity, m/s

 

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s = density of solid particles, kg/m3   = density of gas, kg/m3  Calculate the width of the duct to capture a particle with diameter 10-5 m for an inlet gas velocity of 15 m/s, and the number of turns is 8, s = 1500 kg/m3,  = 3.0x102 kg/m3,  = 1.3x10-5 kg/m s.  Answer: 0.4 m

5. 5.   The heat transfer coefficient for heat transfer by a fluid to helical coils in a vessel mixed with a paddle agitator is given by the following Equation: 0.62

0.33

 L2 N      c p          0.87           k     k       w 

hd i

0.14

 

where h = heat transfer coefficient outside of coil, W/m2 K di = inside vessel diameter, m k  =  = thermal conductivity of fluid, W/m K

L = diameter of agitator, m  N = speed of agitator, rps

 = density of fluid, kg/m3   = fluid viscosity, kg/m s w = water viscosity, kg/m s c p = specific heat of the fluid at constant pressure, J/kg K Calculate the heat transfer coefficient if the vessel is filled with benzene at 100F. The internal diameter of the vessel is 5.32 ft, the diameter of the paddle agitator is 3.64 ft, and turns at 80 rpm (See DIPPR or another reference for properties of benzene and water at 100 F).  Answer: 1300 W/m2 K

6. 6.   The height of a liquid in a column column of mercury is 2.493 ft. Assume the density of mercury is 8 848.7 48.7 3 2 lb/ft  and atmospheric pressure is 2116 lb f /ft /ft . Use unit conversions to calculate the gauge pressure pressure in 2 2 lbf /ft /ft  and the absolute pressure in lbf /ft /ft , atm, mm Hg, and Pa.  Answers: 2.116x103 lb f  /ft 2 , 1 atm, 759.854 mmHg, 1.013x10 1.013x105 Pa 7. 7.   Pipes are manufactured in standard sizes or schedules (for wall thickness). Create a look-up table using the worksheet function VLOOKUP.

 Nominal Pipe Size/in

Inside Diameter/in

Capacity (U.S. gpm) at 1 ft/s velocity

1

1.049

2.690

2

2.067

10.45

3

3.068

23.00

4

4.026

39.60

5

5.047

62.30

6

6.065

90.00

 

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Use the lookup table to determine the proper pipe size for the following flow rates at 1 ft/s: 10, 25, 90 gpm  Answers: 2 in, 4 in, 6 in

8.  8.  The price of a used car is $8,000. $8,000. The dealer finance rate is 8.5% APR for four years with 10% d down own  payment. The bank bank finance finance rate is 7.8% for three three years with 20% down payment. Calculate the monthly monthly  payment for each loan ( Hint   Hint : Formulas >Function Library >Financial>PMT).  Answers: Dealer = $197.19/month, $197.19/month, Bank = $249.95/month $249.95/month

9.  9.  Use a t-test to determine if the mean value of the following two sets of measurements is the same within 95% confidence (null hypothesis). ( Hint : Formulas/Function Library/More Functions/Statistical).  Pooled standard deviation

  x  x     x 2

 s p



i

i

j

 xj

2

n1  n2  1

   s12  n1  1  s22  n2  1    n1  n2  2

Experimental t-statistic t exp



 x1  x2

n1n2

 s p

n1  n2

 

Alternatively, get a complete set of descriptive statistics from the Data Tab >Data Analysis >Descriptive  Statistics (Click the following commands: File >Add‐ins >Manage: Excel  Add‐ins>Go>Analysis Tool  Pack>Go).

A

B

179.738

179.864

179.707

179.611

179.731

179.537

179.722

179.903

179.745

179.543

179.731

179.661

179.749 179.705

179.544 179.653

 

179.512

 

179.683  Answer: t-stat = 1.59, t-crit t-crit = 2.12 confirms null hypothesis hypothesis

10.  Create a color contour plot of the “butterfly” 10.  “butterfly” function. Try a = 0.1, 1, and 10.

 f  x, y  

 x2  y 2   sin  x a y   x 2  y 2

a

 

11.   Generate data and plot the following function between 0 ≤  xx ≤ 6. Create two d 11. different ifferent scatter plots with smooth lines using  x intervals of 0.5 and 0.2.

 x   x 2 e x sin  30 x    12.  Find a root to the following function using Goal Seek. 12. 

 

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 x 3

 1  5 cos x    Answer: x = 1.314

13.  Vapor pressure is commonly correlated with temperature using Antoine’s equation: 13.  B   Pv      A    T C   C    mmHg 

log 

(a) (a)   Plot the vapor pressure for acetone (A = 7.02447, B = 1161.0, C = 224) over the temperature range 25 to 75oC. (b)  (b)  Will acetone boil at 50 oC and ambient pressure? pressure? Explain. (c) (c)   Find the normal boiling point for acetone using the Antoine equation.  Answers: (a) Plot the function; function; consider using using Data Table (b) No (c) 56.2 C

14.  Create an Excel  worksheet 14.   worksheet for calculating calculating your grade point averag averagee (GPA). Use Data Validation tools and MATCH, INDEX worksheet reference functions to assign numerical scores to letter grades. 15.   A projectile is launched at an angle of 30 degrees from the horizontal with an initial velocity of 15. 1.8107 cm/hr. The object strikes strikes the top of of a vertical wall. The base of the the wall is a horizontal horizontal distance of 325 ft away. Calculate the height height of the wall in yards. Use the following following equations of motion. Use Excel  to  to solve this problem and include units in your calculations. Horiz Ho rizon ontal tal Travel: Travel:

cos( )  x  Vt cos(

Vertical Travel:

 y  Vt sin    0.5 gt 2  

where t is time and V is velocity. x and y are the horizontal and vertical positions of the projective. The parameter g is gravity acceleration.  Answer: 34.5 yrd

16.   Use the equations of motion from problem 15 to plot  y versus x for the projectile. 16. 17.   Use the following equation and tabulated data for the vapor pressure of methanol in your calculations:  17.

 

 P  exp  A 

 C ln T  DT  E     T  

 B

where P and T have units of Pa and K, respectively, and where A=82.718, B = -6904.5, C = -8.8622, D = 7.4664x10-6, E = 2.

T/(K)

P x 10-3/(Pa)

283.1

7.213

293.1

12.75

303.1

21.57

313.1

35.09

323.1

55.13

333.1

83.97

343.1

124.3

353.1

179.5

 

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Use the vapor pressure equation to generate a range of values for P over the temperature range 280 K ≤ T   ≤ 360 K, with 5 K increments. Plot your results along w with ith the data in an yx scatter plot following the guidelines for graphing in in Section 2.2.6. Plot pressure on the vertical vertical axis and temperature on the hori hori-zontal axis. Use markers for the data and smooth smooth lines for the values from the equat equation. ion. 18.   Refer to the Equation 18. Equation in Problem 17. Use GOAL SEEK to find the temperature when P equals 40 kPa. Use the worksheet function CONVERT to get the temperature in degrees F.  Answers: 315.81 K, 108.79 108.79 F 2

19.  Plot 19.  the Stokes-Einstein diffusivity units m /s versus temperature worksheet over the rangeequation 300  T for  400 K. Use (D) the with following followi ng tabulated parameters. parame ters. in an Excel  

 D 

kT 

3 d 

 

Bolt Bo ltzm zman ann’ n’ss ccon onst stan antt

k  1.38x10-23 J/K

Particle diameter

d

2.00x10-9 m

The viscosity (Pa·s) is a function of temperature according to the following equation.

 

 

   exp  52.84 

3704  5.866 ln T    T  

20.   A vertical cylinder is filled with 5 g of ammonia gas (NH3) and capped by a movable piston. 20. piston. At a temperature of 25 oC and pressure of 1 atm, the piston is 2 cm from the bottom of the cylinder. Find (a) (a)   The volume of gas in the container (b)  (b)  The temperature of the gas when the piston is raised to 5 cm from the bottom by heating at constant pressure Answers:  Answers: V = 7.2 L, T = 745 K 21.   The formula for the dimensionless Reynolds number for a fluid flowing in a round pipe is 21. Re = DV/  where D is the pipe diameter, V is the average fluid velocity,  is the fluid density, and  is the fluid viscosity. Find the Reynolds Reynolds number and type of flow for the following following cases. Assume Re < 2100 for laminar flow and turbulent Re > 10000 for turbulent flow. Otherwise, the flow is in transition from laminar to to turbulent flow. Use an Excel  IF  IF function to determine laminar or turbulent flow. flow. Remember to put “quotations” on text. (a) (a)   Water flowing at 2.1 m/s through a schedule-40 steel pipe (D = 102.3 mm) at 20 oC ( = 998 kg/m3,  = 9.82 x 10 -4 Pa-s). (b)  (b)  Water flowing at 2.1 m/s through a schedule-40 steel pipe at 90 oC ( = 958 kg/m3,  = 3.1 x 10 -4 Pa-s). (c) (c)   Honey flowing at 2.1 m/s through a schedule-40 steel pipe at 20 oC ( = 1400 kg/m3,  = 10 Pa-s) (d)  (d)  Air flowing at 6.3 m/s through a round duct (D = 0.61 m) at 100 oC ( = 0.95 kg/m3,  = 2.1 x 10-5 Pa-s)  Answers: (a) Re = 2.2x105 , Turbulent (b) Re = 6.6x105 , Turbulent (c) Re = 30, Laminar (d) Re = 1.7x105 , Turbulent

22.   A well-mixed tank filled with 3000 liters of hot water is initially at 115 oF. The tank is cooled by 22. adding water with an inlet temperature, temperature, Tin = 35 oF, at a rate of 30 L/min. Water is removed from the tank at the same rate. The energy balance around the w water ater in the tank is

 

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dT dt



 V  V 

Tin  T   

(a) (a)   Calculate the tank water temperature, temperature, T, at 5-minute intervals for the first 6 60 0 minutes. Use Ex’s auto-fill feature. cel ’s (b)  (b)  Plot the water temperature as a function of time using the results from (a). (c) (c)   Calculate the time needed to cool the water tank to 100 oF. (d)  (d)  If the tank is not well mixed, explain whether your equation will predict a time that is too long or too short.  Answer: (c) Time = 20.8 min

23.   A common heuristic (“rule of thumb”) for water flowing in a pipe is that the average velocity should 23.  be about 1 m/s. A pipe is needed to deliver 6,000 6,000 m3 of water per day. Calculate the inside inside pipe diameter in units units of meters and inches. Use Excel ’’ss CONVERT worksheet function.  Answers: Di = 0.3 m, 11.7 in

24.   A 4 m diameter cylindrical storage 24. storage bin has a conical bottom sectio section. n. The total height of tthe he bin (including cone) is constant at 10 m. The height of the cone is h = 3 m. m.

4m 10 m

  r

h

(a) (a)   Calculate the volume of the cone, V, and angle of the cone, θ.  Vcone



1 2  r h   3

    r  h  tan

  (b)  (b)  Calculate the volume of material in the bin when the level of material ranges from 0 to 5 m above the bottom of the cone. cone. Use the Excel  if( ) worksheet ) worksheet function to generate the data. (c) (c)   Plot the material volume volume versus level. Label both axes on the graph. graph. 3  Answer: (a) V cone  = 0.588 rad cone = 12.6 m  , θ  =