Name In Class Worksheet #1 ME 303 Spring 2014 1. Find the primary dimensions of a) T, torque b) P, power c) V 2 /2, kinetic pressure; is the density and V is the velocity of the fluid d d 2. What is the primary dimensions of U, where is the derivative with dt dt respect to time, is density, and U is velocity?
Name 3. Calculate the cost in $ to operate a pump for 1 year. The pump power is 20 hp and it operates for 20 hrs/day. Electricity costs $0.10 per kwh. 4. The power produced by a pump is P = mgh, where m is the mass flow rate, g is the gravitational constant, and h is the pump head. Show this equation is dimensionally homogeneous. 5. A 4 m 3 oxygen tank is at 20 C and 700 kpa. The valve is opened, and some oxygen is released until the pressure in the tank is 500 kpa. Assume that the temperature in the tank does not change during this process. What mass of oxygen was released from the tank?
ME303 Worksheet #2, January 24, 2014 Name 1. Two plates are separated by a gap of 2 mm filled with a fluid. The plates are 250 mm long and 100 mm wide. The bottom plate is fixed and the top plate is moving to the right with a constant velocity of 7 m/s in response to a force of 8N. What is the fluid viscosity. V 250 mm Step 1. Draw a figure that shows the dimensions, value of the forces, velocity of the plates etc. Also draw a figure of what is happening in the fluid gap i.e. the velocity profile. Step 2. Write the appropriate equation. Use algebra to get in proper form Step 3. Identify the known quantities with units. Step 4. Plug into equation with units. 1
ME303 Worksheet #2, January 24, 2014 Name 2. A water bug is suspended on the surface of a pond by surface tension. The bug has 6 legs and each leg is in contact with the water over a length of 5 mm. What is the maximum mass in grams of the bug if it is to avoid sinking? 3. The velocity distribution for water near a wall is u=a(y/b)1/6, where a = 10m/s, b = 2 mm, and y is the distance from the wall in mm. Find the shear stress in the water at y = 1mm. Assume the water is 20 degrees C. 2
ME303 Worksheet #2, January 24, 2014 Name 4. A cylinder of weight 15 N and 200 mm long slides downward in a lubricated pipe, as shown in the figure. The cylinder diameter is 100 mm. The pipe diameter is 100.5 mm. The lubricant is SAE 20W oil at 10 C. What is the rate of descent of the cylinder? 3
ME303 Worksheet #3, January 31, 2014 Name 1. Determine the gage pressure at the center of Pipe A in psi and in kpa. L4 L1 L2 L3 First solve the problem in SI system (use meters, Pa etc.) Step 1. Start at the place where you know the pressure. Here one end is open to the atmosphere so start there (labeled L1). Step 2. What is the pressure of the free surface of mercury (i.e. at L1) in contact with the atmosphere? Express as a gage pressure, call it P 1. P 1 = Step 3. To find pressure P 2 at location L2, use the hydrostatic equation P 1-P 2=-γΔz. As the fluid is mercury use its γ value. Δz is the difference in heights. P 2 = Step 4. Pressure at L2 and L3 are same as they lie on the same horizontal plane. So P 3= Step 5. Now that we know pressure P 3 we can find pressurep 4 at location L4. Write the hydrostatic equation P 4-P 3=-γΔz. This time note that the fluid is water. P 4= Step 6. Pressure P 4 is same as pressure at the center of Pipe A (as they lie on the same horizontal plane). 1
Step 7. As pressure P 1 is expressed in gage pressure, the pressure P 4 will be gage and so do the pressure at the center of Pipe A. Step 8. Present your answer with appropriate units (kpa, psi etc.) Step 9. Now follow steps 1-8 to solve in traditional system (psi etc.). 2
2. Determine force P necessary to just start opening the 2m wide rectangular gate. length of gate? Step 1. Calculate the resultant hydrostatic force using F= A. is the pressure at the depth of the centroid given by. = the height of fluid level measured from the centroid A = area of the gate. A = = = F= A = Step 2. Now calculate center of pressure (y cp) using But first we need to calculate and. Use figure below to find the slant distance is measured from the surface of the liquid to the centroid of the panel (or gate). 3
Note: The gate we have is rectangular The moment of inertia is calculated for a rectangular gate about a horizontal axis (out of plane of the paper) passing through the centroid., w- width, l length = Step 3. Draw free body diagram of the gate. Show applied force P, hinge forces, resultant hydrostatic force etc. Step 4. Write the moment balance equation for the gate about the hinge to find force the P M hinge=0 4
3. As shown in the figure, the mouse can use the mechanical advantage provided by the hydraulic machine to lift up the elephant. Assume the mouse has mass of 10 g and the elephant 5000 kg. Determine the value of D 2 so that the mouse can support the elephant. Given D 1=20 cm. 5
ME303 Worksheet #4, February 7, 2014 Name 1. For the cylindrical gate shown below, what will the magnitude of the reaction at A when l is 2m? Neglect the weight of the gate. Width is 1 m. C Step 1. Start by applying drawing a free body diagram of the body of fluid ABC (figure above). Show all forces, dimensions etc. Step 2. Now find the horizontal component of force F H acting on the side AC using F H = A. is the pressure at the depth of the centroid (of side AC) given by. = the height of fluid level measured from the centroid of AC A = area of the projected gate on vertical plane AC. A = = = F H = A = 1
Step 3. Now find the vertical component of force F V acting on the side BC (figure above) using F V= A. is the pressure at the depth of the centroid (of side BC) given by. = the height of fluid level measured from the centroid of BC. A = area of the projected gate on horizontal plane BC. A = = = F V = A = Step 3. To find the weight of water in volume ABC, we first need to know the volume of water in ABC. Area of section ABC = V ABC = Area of section ABC * Width = Weight W=γV ABC Step 4. Sum all the vertical forces on the body of fluid ABC F v, resultant = 2
Step 5. Now, find the line of action (horizontal force F H) y cp acting on the plane AC using the equation But first we need to calculate and. Use figure below to find the slant distance is measured from the surface of the liquid to the centroid of the panel (here it is centroid of AC). W, w- width of AC, l length of AC = y cp = Step 6. Now, find the line of action (vertical force F V) x cp acting on the plane BC. For this sum the moments about point C. Moment by force F V is M V= Moment by force F V, resultant is M V,R= F V,Resultant* x cp 3
Moment by weight W is M W= W*x w Take x w =0.32 m Upon summing the moments (M W, M V, M V,Resultant) we can find the value of x cp, x cp = Step 7. Draw a similar sketch (shown below) to the fluid body ABC. Step 8. Write the moment balance equation for the gate about the hinge to find reaction force at A R A Remember we have force at A, resultant force on the gate by the fluid, and hinge force. Note: assume there is no reaction force at A along the horizontal plane. M hinge=0 4
2. A cylindrical block of wood 1 m in diameter and 1 m long has a specific weight of 5000 N/m 3. Will it float in water with the ends horizontal? Take specific weight of water 9810 N/m 3 Water h=? Step 1. Draw a free body diagram of the wooden block (show all forces, center of gravity, center of buoyancy etc.) Step 2. Calculate weight of block and buoyancy force (in terms of diameter, h, fluid properties etc.) Calculate the weight of the block W= specific weight of body volume W= Calculate the buoyancy force F B= specific weight of water submerged volume of body [Hint: express submerged volume in terms of unknown variable h] F B = Step 3. Now write the equilibrium equation for the wooden block and solve for unknown variable h F net=0 The value of h= 5
Step 4. The geometric center of the wooden block is its center of gravity G. Find its value? G C Step 5. The center of buoyancy C is geometric center of the submerged section of the wooden block. Find its value? water Step 6. Now that you have the locations of points C and G, find the distance between them CG CG = Step 7. Find the metacentric height GM using the expression where, V = submerged volume of cylindrical block, I oo= area moment of inertia about waterline (in the figure along axis xx) Submerged volume V= Area moment of inertia I oo= Metacentric head GM = Step 8. Based on the sign of metacentric head what do you think? The block will (a) float stable with its ends horizontal (b) not float stable with its ends horizontal 6
EXTRA CREDIT PROBLEM 3. The velocity of water flow in the nozzle shown is given by the following expression: V = 2t 2 /(1-0.5x/L) 2 where, V=velocity in m/s, t= time in s, x=distance along the nozzle, and L= length of nozzle=4 m. When x=0.5l and t=3 s, what is the local acceleration along the centerline? What is the convective acceleration? Assume quasi-one-dimensional flow prevails. Express your acceleration in m/s 2 Step 1. Find local acceleration a l using substitute the values of t, x, L, and V.. First perform partial differentiation and later 7
Step 2. Find convective acceleration a c using substitute the values of t, x, L, and V.. First perform partial differentiation and later 8
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ME303 Worksheet #6, March 7, 2014 Name 1. The water (density ρ=1000 kg/m 3 ) in this jet has a speed V 1 =60 m/s to the right and is deflected by a cone that is moving to the left with a speed of 5 m/s. The diameter of the jet is 10 cm. Determine the external horizontal force F needed to move the cone. Assume negligible friction between the water and the cone (or vane). Neglect gravity. Step 1. Start by drawing the cone and fluid. Show the control volume surrounding the cone and fluid. In your drawing, show all known parameters (i.e. fluid jet velocities, cone speed, flow direction, dimensions etc). Step 2. We now need to choose a reference frame. Let us choose a reference frame that is fixed to the moving cone. So draw X-Y axis on the cone or vane (in step 1). This reference frame makes the analysis simpler. Find the inlet jet velocity V in entering the control volume with respect to our reference frame. Remember our reference is moving, so find the velocity of the jet with respect to the moving cone or reference frame. V in = Tip 1. Assume that v 1 =v 2 =v 3. This assumption can be justified with the Bernoulli equation. In particular, assume inviscid flow and neglect elevation changes, and the Bernoulli equation can be used to prove that the velocity of the fluid jet is constant. 1
Step 3. Using Tip 1. Find outlet jet velocity V out exiting the control volume with respect to our reference frame. V out = Step 4. Calculate the inlet mass flow rate Step 5. In the force diagram (below figure), draw the control volume and label all the external forces acting on the control volume. Remember, you to include the external horizontal force F in the force diagram. = Force diagram Momentum diagram Step 6. For the momentum diagram (above figure), you need to know the magnitudes of momentum flow in and momentum flow out of the control volume. In our problem, we just need to calculate X- direction momentum flow in and momentum flow out. X-direction momentum flow in = = X-direction momentum flow out= = 2
Step 7. Now in the momentum diagram (previous page), show the momentum flow in and momentum flow out of control volume. Remember, momentum flow is a vector --- it has directionality. Step 8. Let s apply linear momentum equation by using the force diagram and momentum diagram (see figures on page 2). Usually we need to write momentum equation for X, Y, and Z directions. However, in this problem X-direction momentum equation will suffice to find the unknown external force F. X-direction momentum equation, Assume steady flow and write the x-direction momentum equation. The external horizontal force F is 3
2. Assume that the gage pressure P is the same at sections 1 and 2 in the horizontal bend shown in the figure (below). The fluid flowing in the bend has density ρ, discharge Q, and velocity V. The cross-sectional area of the pipe is A. Then the magnitude of the force F (neglecting gravity) required at the flanges to hold the bend in place will be a) PA b) PA + ρqv c) 2PA + ρqv d) 2PA + 2 ρqv Step 1. Start by drawing the pipe bend and show the control volume surrounding the pipe bend. Show all known parameters (i.e. fluid jet velocities, dimensions, flow direction, section #, etc.). Step 2. We now need to choose a reference frame. Let us choose a reference frame that is fixed to the ground. So draw X-Y axis (in step 1). Step 3. Calculate the inlet mass flow rate. Step 4. Calculate the pressure force at sections 1 and 2, F 1 or F 2 = Pressure*Cross-sectional area F 1 = F 2 = 4
Step 5. In the force diagram (next page), draw the control volume and label all the external forces acting on the control volume. Remember, you should include the external horizontal force F, Pressure forces F 1, F 2 in the force diagram. = Force diagram Momentum diagram Step 6. For the momentum diagram (above figure), you need to know the magnitudes of momentum flow in and momentum flow out of the control volume. So, let us calculate the momentum flow in and momentum flow out. X-direction momentum flow in = = X-direction momentum flow out= = Step 7. Now in the momentum diagram (above figure), show the momentum flow in and momentum flow out of control volume. Remember, momentum flow is a vector --- it has directionality. Step 8. Let s apply linear momentum equation by using the force diagram and momentum diagram (see above figures). Usually we need to write momentum equation for X, Y, and Z directions. However, in this problem X-direction momentum equation will suffice to find the unknown external force F. X-direction momentum equation, Assume steady flow and write the x-direction momentum equation. 5
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ME303 Worksheet #7, March 28, 2014 Name USE TABLE 8.3 (on the last page of this worksheet) to look at definitions of Re,We, M etc 1. Oil with a kinematic viscosity of 4 10-6 m 2 /s flows through a smooth pipe 12 cm in diameter at 2.3 m/s. What velocity should water have at 20 C in a smooth pipe 5 cm in diameter to be dynamically similar? Step 1. Start by looking at π-groups (Re, M, Fr, We, C F, C f, C p, S t ) and pick the one that is appropriate. Begin by eliminating each π-group that has no relevance with the problem. For example, ask yourselves, do we have free surface effects? If yes, then use We #; or else try other π-groups. Please note: For most fluid problems consider using Re # first. So what other π-groups do you think will be relevant for flow through a pipe? HINT: We have, no compressibility effects, no free-surface effects, no gravitation effects, no oscillation flows in the pipe. Appropriate π-group is: Step 2. Now equate the π-group for model and prototype that satisfies dynamic similitude. Substitute the given values and find the model velocity. π model = π prototype Model velocity is 1
2. A spherical balloon that is to be used in air at 60 F and atmospheric pressure is tested by towing a 1/10 scale model in a lake. The model is 2 ft in diameter, and a drag of 10 lbf is measured when the model is being towed in deep water at 5 ft/s. What drag (in pounds force) can be expected for the prototype in air under dynamically similar conditions? Assume that the water temperature is 60 F. Step 1. Start by looking at π-groups (Re, M, Fr, We, C F, C f, C p, S t ) and pick the one that is appropriate. Begin by eliminating each π-group that has no relevance with the problem. For example, ask yourselves, do we have free surface effects? If yes, then use We #; or else try other π-groups. Please note: For most fluid problems consider using Re # first. So what other π-groups do you think will be relevant for the spherical balloon testing? HINT: We have, no compressibility effects, no free-surface effects, no gravitation effects, no oscillation flows in the pipe. Appropriate π-groups are: Step 2. Now equate the Re# for model and prototype that satisfies dynamic similitude, from where you find velocity ratio V p /V m. Re model = Re prototype The velocity ratio V p /V m = 2
Step 3. Now to find the drag force on the prototype, look for a π-group that has force term in it. The appropriate π-group is: Step 4. Now equate the π-group (step 3) for model and prototype that satisfies dynamic similitude, from where you find force on prototype F p. π model = π prototype The value of force on prototype F p is 3. A 60 cm valve is designed for control of flow in a petroleum pipeline. A 1/3 scale model of the full size valve is to be tested with water in the laboratory. If the prototype flow rate is to be 0.5 m 3 /s, what flow rate should be established in the laboratory test for dynamic similitude to be established? Also the pressure coefficient C p in the model is found to be 1.07, what will be the corresponding C p in the full-scale valve? The relevant fluid properties for the petroleum are S=0.82 and μ=3 10-3 N-s/m 2. The viscosity of water is 1 10-3 N-s/m 2. Step 1. Start by looking at π-groups (Re, M, Fr, We, C F, C f, C p, S t ) and pick the one that is appropriate. Begin by eliminating each π-group that has no relevance with the problem. For example, ask yourselves, do we have free surface effects? If yes, then use We #; or else try other π-groups. Please note: For most fluid problems consider using Re # first. So what other π-groups do you think will be relevant for the petroleum pipe? HINT: We have, no compressibility effects, no free-surface effects, no gravitation effects, no oscillation flows in the petroleum pipe. Appropriate π-group is: 3
Step 2. Now equate the π-group for model and full-scale that satisfies dynamic similitude, from where you find velocity ratio V m /V p. π model = π full-scale The velocity ratio V p /V m = Step 3. Now that you know the velocity ratio V p /V m,, you can find the volume flow rate ratio Q p /Q m. Use volume flow rate Q=Velocity Cross-sectional area The volume flow rate ratio Q p /Q m = Step 4. 4
Using the statement in the box above, what will be the C p for full-scale? C p for full-scale is 5
ME303 Worksheet #8, April 18, 2014 Name 1. As shown, a round tube (microchannel) of diameter 0.5 mm and length 750 mm is connected to plenum. A fan produces a negative gage pressure of -1.5 inch H 2 O in the plenum and draws air (20 C) into the microchannel. What is the mean velocity of air in the microchannel? Assume that the only head loss is in the tube. Step 1. Start by drawing a control volume (below figure) enclosing two points where you know the information (Point 1) and where you want to find the information (Point 2). HINT: Point 1 could be near the mouth of the tube (entrance) where the air is at atmospheric pressure with negligible velocity. Point 2 could be at the exit of the tube where the pressure is known to be -1.5 inch H 2 0 while the velocity is unknown. Step 2. Now, plan on applying the energy balance equation to the control volume. For this, choose a reference line (or datum). 1 (Energy balance equation)
Information at Point 1 Gage Pressure, P 1 = Conversion 1 inch water = 249.2 Pa In SI units, Gage Pressure, P 2 = Velocity, V 1 = Elevation, z 1 = Ask yourself is there any pump between points 1 and 2? Pump head, h p = Information at Point 2 Gage Pressure, P 2 = In SI units, Gage Pressure, P 2 = Velocity, V 2 = Elevation, z 2 = Ask yourself is there any turbine between points 1 and 2? Turbine head, h t = The problem states that the head loss h L is only due to the tube. So the head loss h L =h f Step 3. Let us assume the flow in the tube is laminar, so take α = 2. Now substitute the known values and simplify the energy equation. NOTE: Upon simplifying the energy equation, it will end being up a quadratic equation in V 2! Solve the quadratic equation to find the mean velocity of air in the microchannel, V 2. 2
Mean velocity of air in the microchannel V 2 = Step 4. Since we have assumed the flow is laminar in step 2, this assumption has to be verified. So, calculate the Reynolds number in the microchannel. 3
Re = So therefore the assumption that the flow is laminar is VALID / INVALID (circle the correct option) If Re <2000, the flow is laminar. If Re >3000, the flow is turbulent. Step 5. If the flow is VALID, then we are finished with the problem. If NOT, we have to go back the Step 2 and solve without making assumptions. 2. The sketch shows a test of an electrostatic air filter. The pressure drop for the filter is 3 inches of water when the airspeed is 10 m/s. What is the minor loss coefficient (K) for the filter? Assume air properties at 20 C. Step 1. Start by drawing a control volume (figure on next page) enclosing two points where you know the information (Point 1) and where you want to find the information (Point 2). HINT: Point 1 could be near the upstream of air flow where information is known while Point 2 could be where the information is unknown. For instance, Point 1 could be at A and Point 2 could be at B. Ensure the control volume you draw will surround the electrostatic filter. 4
Step 2. Now, plan on applying the energy balance equation to the control volume. For this, choose a reference line (or datum). (Energy balance equation) Note: You may not know the absolute/ gage pressures at A and B but all we need is the difference of pressure at these two points which is given in the problem. Information at Point 1 Velocity, V 1 = Elevation, z 1 = Ask yourself is there any pump between points 1 and 2? Pump head, h p = Information at Point 2 By applying continuity equation between Points 1 and 2 determine the Velocity V 2. A 1 V 1 =A 2 V 2 (where, A is cross-sectional area) Velocity, V 2 = 5
Elevation, z 2 = Ask yourself is there any turbine between Points 1 and 2? Turbine head, h t = Now, calculate the pressure difference between Points 1 and 2, ΔP = P 1 - P 2 = In SI units, Pressure, ΔP = Step 3. Now substitute the known values and simplify the energy equation. Take α 1 = α 2. NOTE: Use SI units. Upon simplifying the value of h L = Step 4. Now to determine the value of the minor loss coefficient (K), use the definition of head loss. We know the value of h L from Step 3. So substitute the values of V, g, and h L to obtain the value of K. 6
The minor loss coefficient (K) is 7
ME303 Worksheet #9, April 25, 2014 Name 1. How much power is required to move a spherical-shaped submarine of diameter 1.5 m through seawater at a speed of 10 knots? Assume the submarine is fully submerged. Assume all power is being used to overcome drag. Step 1. Start by calculating the Reynolds number. Re=VD/ν Unit Conversion: 1 knot= 0.514 m/s Step 2. Now, use a correlation to estimate the drag coefficient C D of the submarine. As the submarine is spherical-shaped, use correlation for a sphere. One relevant correlation is given by Clift and Gauvin as shown below. Drag coefficient, C D = Step 3. Now, to estimate drag force we need to first find the reference area (A). In this situation, we take the projected area as reference area (A). What is the projected area for a sphere? Projected area, A = 1
Step 4. Using the drag force expression find the force needed to overcome the drag. Drag force, F D =, where V o is the free-stream velocity. Drag force, F D = Step 5. Using the power equation, Power= Force Velocity, estimate the power needed to overcome drag. Ensure your final answer is in kw. Power = 2. Determine the lift of a baseball when pitched at a speed of 38 m/s and with a spin rate of 35 rps. Also determine how much the ball deflects from its original path in time t=0.5 s. Take the circumference and mass of ball to be 0.23 m and 0.15 kg respectively. Assume the axis of rotation is vertical. NOTE: This plot valid for rotating sphere. Step 1. Start by determining the rotational parameter defined by rω/v o. Ensure the values of r, ω (radian/s), and V o are in SI units. Angular velocity, ω (in radian/s) = 2
Rotational parameter= rω/v o = Step 2. For the calculated rotational parameter, determine the value of lift coefficient C L using the chart (previous page). Studies have shown that the C L for baseball is three times of C L obtained from the chart! Coefficient of lift for the baseball, C L = Step 3. Now calculate the lift force F L given by, where V o is the free-stream velocity, and A is the projected area of the ball. Lift force, F L = Step 4. Now find the acceleration of the ball due to lift force. [HINT: Use Newton s second law of motion.] Acceleration, a = Step 5. To estimate deflection, use kinematic equation for an accelerated body, X= ut +0.5at 2 [NOTE: Here we have zero initial velocity in the direction of lift, hence u=0] Deflection, X= 3