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Thermodynamics for Engineers 1st edition Kenneth A. Kroos, Merle C. Potter - Solutions

Using the geometry of Fig. 10.6a, prove that the most efficient circular open channel (maximum hydraulic radius for a given flow area) is a semicircle.
Determine the most efficient value of θ for the vee-shaped channel of Fig. P10.41
Suppose that the side angles of the trapezoidal channel in Prob. 10.39 are reduced to 15° to avoid earth slides. If the bottom flat width is 8 ft,(a) Determine the normal depth and(b) Compare the resulting wetted perimeter with the solution P = 24.1 ft from Prob. 10.39. (Do not reveal this
What are the most efficient dimensions for a riveted-steel rectangular channel to carry 4.8 m3/s of water at a slope of 1:900?
What are the most efficient dimensions for a half-hexagon cast-iron channel to carry 15000 gal/min of water at a slope of 0.16°?
What are the most efficient dimensions for an asphalt trapezoidal channel to carry 3 m3/s of water at a slope of 0.0008?
It is suggested that a channel which reduces erosion has a parabolic shape, as in Fig. P10.46. Formulas for area and perimeter of the parabolic cross-section are as follows [Ref. 7 of Chap. 10]:
Replot Fig. 10.8b in the form of q versus y for constant E. Does the maximum q occur at the critical depth?
A wide, clean-earth river has a flow rate q = 150 ft3/(s•ft). What is the critical depth? If the actual depth is 12 ft, what is the Froude number of the river? Compute the critical slope by (a) Manning’s formula and (b) The Moody chart.
Find the critical depth of the brick channel in Prob. 10.34 for both the 4 €“ and 8-ft widths. Are the normal flows sub- or supercritical?
A pencil point piercing the surface of a rectangular channel flow creates a 25° half-angle wedge like wave, as in Fig. P10.50. If the channel surface is painted steel and the depth is 35 cm, determine(a) The Froude no.;(b) The critical depth; and(c) The critical slope.
An asphalt circular channel, of diameter 75 cm, is flowing half-full at an average velocity of 3.4 m/s. Estimate (a) The volume flow rate; (b) The Froude number; and (c) The critical slope.
Water flows full in an asphalt half hexagon channel of bottom width W. The flow rate is 12 m3/s. Estimate W if the Froude number is exactly 0.6.
For the river flow of Prob. 10.48, find the depth y2 which has the same specific energy as the given depth y1 = 12 ft. These are called conjugate depths. What is Fr2?
A clay tile V-shaped channel has an included angle of 70° and carries 8.5 m3/s. Compute(a) The critical depth,(b) The critical velocity, and(c) The critical slope for uniform flow.
A trapezoidal channel resembles Fig. 10.7 with b = 1 m and θ = 50°. The water depth is 2 m and Q = 32 m3/s. If you stick your fingernail in the surface, as in Fig P10.50, what half-angle wave might appear?
A riveted-steel triangular duct flows partly full as in Fig. P10.56. If the critical depth is 50 cm, compute (a) the critical flow rate; and (b) the critical slope.
For the triangular duct of Fig P10.56, if the critical flow rate is 1.0 m3/s, compute (a) The critical depth; and (b) The critical slope.
A circular corrugated-metal water channel is half-full and in uniform flow when laid on a slope of 0.0118. The average shear stress on the channel walls is 29 Pa. Estimate (a) The channel diameter; (b) The Froude number; and (c) The volume flow rate.
Uniform water flow in a wide brick channel of slope 0.02° moves over a 10-cm bump as in Fig P10.59, a slight depression in the water surface results If the minimum depth over the bump is 50 cm, compute(a) The velocity over the bump;
Modify Prob. 10.59 as follows. Again assuming uniform sub critical approach conditions (V1, y1), find (a) The flow rate and (b) Y2 for which the Froude number Fr2 at the crest of the bump is exactly 0.7.
Modify Prob. 10.59 as follows: Again assuming uniform sub-critical approach flow V1, find (a) The flow rate q; and (b) The height y2 for which the Froude number Fr2 at the crest of the bump is exactly 1.0 (critical).
Consider the flow in a wide channel over a bump, as in Fig. P10.62. One can estimate the water-depth change or transition with frictionless flow. Use continuity and the Bernoulli equation to show that is the drawdown of the water surface realistic in Fig. P10.62? Explain under what conditions the
In Fig P10.62, let Vo = 1 m/s and yo = 1 m. If the maximum bump height is 15 cm, estimate (a) The Froude number over the top of the bump; and (b) The maximum depression in the water surface.
In Fig P10.62, let Vo = 1 m/s and yo = 1 m. If the flow over the top of the bump is exactly critical (Fr = 1), determine the bump height hmax.
Program and solve the differential equation of “frictionless flow over a bump,” for entrance conditions V0 = 1 m/s and y0 = 1 m. Let the bump have the convenient shape h = 0.5hmax [1 − cos (2π x/L)], which simulates Fig. P10.62. Let L = 3 m, and generate a numerical solution for y(x)
In Fig P10.62 let Vo = 6 m/s and yo = 1 m. If the maximum bump height is 35 cm, estimate (a) The Froude number over the top of the bump; and (b) The maximum increase in the water-surface level.
In Fig P10.62 let Vo = 5 m/s and yo = 1 m. If the flow over the top of the bump is exactly critical (Fr = 1), determine the bump height hmax.
Modify Prob. 10.65 to have a supercritical approach condition V0 = 6 m/s and y0 = 1 m. If you have time for only one case, use hmax = 35 cm (Prob. 10.66), for which the maximum Froude number is 1.47. If more time is available, it is instructive to examine a complete family of surface profiles for 1
Given is the flow of a channel of large width b under a sluice gate, as in Fig. P10.69. Assuming frictionless steady flow with negligible upstream kinetic energy, derive a formula for the dimensionless flow ratio 2 3 2 Q /(y1b g) as a function of the ratio y2/y1.Show by differentiation that the
In Fig P10.69 let V1 = 0.75 m/s and V2 = 4.0 m/s. Estimate (a) The flow rate per unit width; (b) Y2; and (c) Fr2.
In Fig P10.69 let y1 = 95 cm and y2 = 50 cm. Estimate the flow rate per unit width if the upstream kinetic energy is (a) neglected; and (b) included.
Water approaches the wide sluice gate in the figure, at V1 = 0.2 m/s and y1 = 1 m. Accounting for upstream kinetic energy, estimate, at outlet section 2,(a) Depth;(b) Velocity; and(c) Froude number.
In Fig P10.69 suppose that y1 = 1.4 m and the gate is raised so that its gap H is 15 cm. Estimate the resulting flow rate per unit width and the downstream depth.
With respect to Fig P10.69, show that, for frictionless flow, the upstream velocity may be related to the water levels by
A tank of water 1 m deep, 3 m long and 4 m wide into the paper has a closed sluice gate on the right side, as in Fig. P10.75. At t = 0 the gate is opened to a gap of 10 cm. Assuming quasi-steady sluice-gate theory, estimate the time required for the water level to drop to 50 cm. Assume free outflow.
In Prob. 10.75 estimate what gap height H would cause the tank level to drop from 1 m to 30 cm in exactly 40 s. Assume free outflow.
Equation 10.41 for the discharge coefficient is for free (nearly frictionless) outflow. If the outlet is drowned, as in Fig. 10.10c, there is dissipation and Cd drops sharply. Fig. P10.77 at right shows data from Ref. 3 on drowned vertical sluice gates. Use this chart to repeat Prob. 10.73, and
Repeat Prob. 10.75, to find the time to drain the tank from 1.0 m to 50 cm, if the gate is drowned downstream at y2 = 40 cm. Again assume gap H = 10 cm.
Show that the Froude number downstream of a hydraulic jump will be given by 1/2 2 1/2 3/2 Fr2=8 Fr1/[(1+8Fr1 ) −1] . Does the formula remain correct if we reverse subscripts 1 and 2? Why?
Water, flowing in a channel at 30-cm depth, undergoes a hydraulic jump of dissipation 71%. Estimate (a) The downstream depth; and (b) The volume flow.
Water flows in a wide channel at q = 25 ft3/s•ft and y1 = 1 ft and undergoes a hydraulic jump. Compute y2, V2, Fr2, hf, the percentage dissipation, and the horsepower dissipated per unit width. What is the critical depth?
Downstream of a wide hydraulic jump the flow is 4 ft deep and has a Froude number of 0.5. Estimate (a) Y1; (b) V1; (c) Fr1; (d) The percent dissipation; and (e) yc.
A wide channel flow undergoes a hydraulic jump from 40 cm to 140 cm. Estimate (a) V1; (b) V2; (c) The critical depth; and (d) The percent dissipation.
Consider the flow under the sluice gate of Fig. P10.84. If y1 = 10 ft and all losses are neglected except the dissipation in the jump, calculate y2 and y3 and the percentage of dissipation, and sketch the flow to scale with the EGL included. The channel is horizontal and wide.
In Prob. 10.72 the exit velocity from the sluice gate is 4.33 m/s. If there is a hydraulic jump at “3” just downstream of section 2, determine the downstream (a) Velocity; (b) Depth; (c) Froude number; and (d) Percent dissipation.
A bore is a hydraulic jump which propagates upstream into a still or slower moving fluid, as in Fig. 10.4a. Suppose that the still water is 2 m deep and the water behind the bore is 3 m deep. Estimate (a) The propagation speed of the bore and (b) The induced water velocity.
A tidal bore may occur when the ocean tide enters an estuary against an oncoming river discharge, such as on the Severn River in England. Suppose that the tidal bore is 10 ft deep and propagates at 13 mi/h upstream into a river which is 7 ft deep. Estimate the river current in kn. Fig. P10.87
For the situation in Fig P10.84, suppose that at section 3 the depth is 2 m and the Froude number is 0.25. Estimate (a) The flow rate per meter of width; (b) Yc; (c) Y1; (d) The percent dissipation in the jump; and (e) The gap height H of the gate.
Water 30 cm deep is in uniform flow down a 1° unfinished-concrete slope when a hydraulic jump occurs, as in Fig. P10.89. If the channel is very wide, estimate the water depth y2 downstream of the jump.
Modify Prob. 10.89 as follows. Suppose that y2 = 1.5 m and y1 = 30 cm but the channel slope is not equal to 1 degree. Determine the proper slope for this condition.
No doubt you used the horizontal-jump formula (10.43) to solve Probs. 10.89 and 10.90, which is reasonable since the slope is so small. However, Chow [ref. 3, p. 425] points out those hydraulic jumps are higher on sloped channels, due to €œthe weight of the fluid in the jump.€
At the bottom of an 80-ft-wide spillway is a horizontal hydraulic jump with water depths 1 ft upstream and 10 ft downstream. Estimate (a) The flow rate; and (b) The horsepower dissipated.
Water in a horizontal channel accelerates smoothly over a bump and then undergoes a hydraulic jump, as in Fig. P10.93. If y1 = 1 m and y3 = 40 cm, estimate (a) V1; (b) V3; (c) Y4; and (d) The bump height h.
For the flow pattern of Fig P10.93, consider the following different, and barely sufficient, data. The upstream velocity V1 = 1.5 m/s, and the bump height h is 27 cm. Find(a) Y1;(b) Y2;(c) Y3; and(d) Y4.
A 10-cm-high bump in a wide horizontal channel creates a hydraulic jump just upstream and the flow pattern in Fig. P10.95. Neglect losses except in the jump, if y3 = 30 cm, estimate(a) V4;(b) Y4;(c) V1; and(d) y1.
Show that the Froude numbers on either side of a hydraulic jump are related by the simple formula Fr2 = Fr1 (y1/y2)3/2.
A brickwork rectangular channel 4 m wide is flowing at 8.0 m3/s on a slope of 0.1°, is this mild, critical, or steep slope? What type of gradually-varied-solution curve are we on if the local water depth is (a) 1 m; (b) 1.5 m; (c) 2 m?
A gravelly-earth wide channel is flowing at 10.0 m3/s per meter on a slope of 0.75°. Is this a mild, critical, or steep slope? What type of gradually-varied-solution curve are we on if the local water depth is (a) 1 m; (b) 2 m; (c) 3 m?
A clay tile V-shaped channel, of included angle 60°, is flowing at 1.98 m3/s on a slope of 0.33°. Is this a mild, critical, or steep slope? What type of gradually-varied solution curve are we on if the local water depth is (a) 1 m; (b) 2 m; (c) 3 m?
If bottom friction is included in the sluice-gate flow of the depths (y1, y2, y3) will vary with x. Sketch the type and shape of gradually-varied solution curve in each region (1,2,3) and show the regions of rapidly-varying flow.
Consider the gradual change from the profile beginning at point a in Fig. P10.101 on a mild slope So1 to a mild but steeper slope So2 downstream, Sketch and label the gradually-varied solution curve(s) y(x) expected.
The wide channel flow in Fig P10.102 changes from a steep slope to one even steeper. Beginning at points a and b, sketch and label the water surface profiles which are expected for gradually varied flow.
A circular painted-steel channel, of radius 50 cm, is running half-full at 1.2 m3/s on a slope of 5 m/km. Determine (a) Whether the slope is mild or steep; and (b) What type of gradually-varied solution applies at this point? (c) Use the approximate method of Eq. (10.52), and a single depth
The rectangular channel flow in Fig P10.104 expands to a cross-section 50% wider. Beginning at points a and b, sketch and label the water-surface profiles which are expected for gradually-varied flow.
In Prob. 10.84 the frictionless solution is y2 = 0.82 ft, which we denote as x = 0 just downstream of the gate. If the channel is horizontal with n = 0.018 and there is no hydraulic jump, compute from gradually-varied theory the downstream distance where y = 2.0 ft.
A rectangular channel with n = 0.018 and a constant slope of 0.0025 increases its width linearly from b to 2b over a distance L, as in Fig. P10.106(a) Determine the variation y(x) along the channel if b = 4 m, L = 250 m, y(0) = 1.05 m, and Q = 7 m3/s.(b) Then, if your computer program is working
A clean-earth wide-channel flow is flowing up an adverse slope with so = −0.002. If the flow rate is q = 4.5 m3/s•m, use gradually-varied theory to compute the distance for the depth to drop from 3.0 to 2.0 meters.
Illustrate Prob. 10.104 with a numerical example. Let the channel be rectangular with a width b1 = 10 m for 0 < x < 100 m, expanding to b2 = 15 m for 100 < x < 250 m. The flow rate is 27 m3/s, and n = 0.012. Compute the water depth at x = 250 m for initial depth y(0) equal to (a) 75 cm and (b)
Figure P10.109 illustrates a free overfall or dropdown flow pattern, where a channel flow accelerates down a slope and falls freely over an abrupt edge. As shown, the flow reaches critical just before the overfall. Between yc and the edge the flow is rapidly varied and does not satisfy gradually
We assumed frictionless flow in solving the bump case, Prob. 10.65, for which V2 = 1.21 m/s and y2 = 0.826 m over the crest when hmax = 15 cm, V1 = 1 m/s, and y1 = 1 m. However, if the bump is long and rough, friction may be important. Repeat Prob. 10.65 for the same bump shape, h = 0.5hmax [1 –
Solve Prob. 10.105 (a horizontal variation along an H-3 curve) by the approximate method of Eq. (10.52), beginning at (x, y) = (0, 0.82 ft) and using a depth increment Δy = 0.2 ft. (The final increment should be Δy = 0.18 ft to bring us exactly to y = 2.0 ft.)
The clean-earth channel in Fig P10.112 is 6 m wide and slopes at 0.3°. Water flows at 30 m3/s in the channel and enters a reservoir so that the channel depth is 3 m just before the entry. Assuming gradually-varied flow, how far is L to a point upstream where y = 2m? What type of curve is the water
Figure P10.113 shows a channel contraction section often called a venturi flume [from Ref. 23 of Chap. 10], because measurements of y1 and y2 can be used to meter the flow rate. Show that if losses are neglected and the flow is one-dimensional and sub-critical, the flow rate is given by Apply this
Investigate the possibility of choking in the venturi flume of Fig. P10.113. Let b1 = 4 ft, b2 = 3 ft, and y1 = 2 ft. Compute the values of y2 and V1 for a flow rate of(a) 30 ft3/s and(b) 35 ft3/s. Explain your vexation.
Gradually varied theory, Eq. (10.49), neglects the effect of width changes, db/dx, assuming that they are small. But they are not small for a short, sharp contraction such as the venturi flume in Fig. P10.113. Show that, for a rectangular section with b = b(x), Eq. (10.49) should be modified as
Investigate the possibility of frictional effects in the venturi flume of Prob. 10.113, part(a) For which the frictionless solution is Q = 9.88 m3/s. Let the contraction be 3 m long and the measurements of y1 and y2 be at positions 3 m upstream and 3 m downstream of the contraction, respectively.
A full-width weir in a horizontal channel is 5 m wide and 80 cm high. The upstream depth is 1.5 m. Estimate the flow rate for (a) A sharp-crested weir; and (b) A round-nosed broad-crested weir.
Using a Bernoulli-type analysis similar to Fig. 10.16a, show that the theoretical discharge of the V-shaped weir in Fig P10.118 is given by Q= 0.7542g1/2 tanα H5/2
Data by A. T. Lenz for water at 20°C (reported in Ref. 19) show a significant increase of discharge coefficient of V-notch weirs (Fig. P10.118) at low heads, for α = 20°, some measured values are as follows:
The rectangular channel in Fig P10.120 contains a V-notch weir as shown. The intent is to meter flow rates between 2.0 and 6.0 m3/s with an upstream hook gage set to measure water depths between 2.0 and 2.75 m. What are the most appropriate values for the notch height Y and the notch half-angle
Water flow in a rectangular channel is to be metered by a thin-plate weir with side contractions, as in Table 10.2b, with L = 6 ft and Y = 1 ft. It is desired to measure flow rates between 1500 and 3000 gal/min with only a 6-in change in upstream water depth. What is the most appropriate length for
In 1952 E. S Crump developed the triangular weir shape shown in Fig. P10.122 [Ref. 19, chap. 4], the front slope is 1:2 to avoid sediment deposition, and the rear slope is 1:5 to maintain a stable tail water flow. The beauty of the design is that it has a unique discharge correlation up to near
The Crump weir in Prob. 10.122 is for modular flow, that is, when the flow rate is independent of downstream tail water. When the weir becomes drowned, the flow rate decreases by the following factor:
Water flows at 600 ft3/s in a rectangular channel 22 ft wide with n ≈ 0.024 and a slope of 0.1°. A dam increases the depth to 15 ft, as in Fig. P10.124. Using gradually varied theory, estimate the distance L upstream at which the water depth will be 10 ft. What type of solution curve are we
The Tupperware dam on the Blackstone River is 12 ft high, 100 ft wide, and sharp-edged. It creates a backwater similar to Fig. P10.124. Assume that the river is a weedy-earth rectangular channel 100 ft wide with a flow rate of 800 ft3/s. Estimate the water-depth 2 mi upstream of the dam if So =
Suppose that the rectangular channel of Fig. P10.120 is made of riveted steel with a flow of 8 m3/s. If α = 30° and Y = 50 cm, estimate, from gradually-varied theory, the water depth 100 meters upstream.
A horizontal gravelly earth channel 2 m wide contains a full-width Crump weir (Fig. P10.122) 1 m high, if the weir is not drowned, estimate, from gradually varied theory, the flow rate for which the water depth 100 m upstream will be 2 m.
A rectangular channel 4 m wide is blocked by a broad crested weir 2 m high, as in Fig. P10.128. The channel is horizontal for 200 m upstream and then slopes at 0.7° as shown. The flow rate is 12 m3/s, and n = 0.03. Compute the water depth y at 300 m upstream from gradually varied theory
Describe the geometry and operation of a human peristaltic PDP which is cherished by every romantic person on earth. How do the two ventricles differ?
What would be the technical classification of the following turbo machines?
A PDP can deliver almost any fluid, but there is always a limiting very-high viscosity for which performance will deteriorate. Can you explain the probable reason?
An interesting turbo machine is the torque converter [58], which combines both a pump and a turbine to change torque between two shafts. Do some research on this concept and describe it, with a report, sketches and performance data, to the class.
What type of pump is shown in Fig P11.5? How does it operate?
Fig P11.6 shows two points a half period apart in the operation of a pump. What type of pump is this? How does it work? Sketch your best guess of flow rate versus time for a few cycles.
A piston PDP has a 5-in diameter and a 2-in stroke and operates at 750 rpm with 92% volumetric efficiency. (a) What is the delivery, in gal/min? (b) If the pump delivers SAE 10W oil at 20°C against a head of 50 ft, what horsepower is required when the overall efficiency is 84%?
A centrifugal pump delivers 550 gal/min of water at 20°C when the brake horsepower is 22 and the efficiency is 71%. (a) Estimate the head rise in ft and the pressure rise in psi. (b) Also estimate the head rise and horsepower if instead the delivery is 550 gal/min of gasoline at 20°C.
Figure P11.9 shows the measured performance of the Vickers Inc. Model PVQ40 piston pump when delivering SAE 10W oil at 180°F (ρ ≈ 910 kg/m3). Make some general observations about these data vis-à-vis Fig. 11.2 and your intuition about PDP behavior.
Suppose that the pump of Fig. P11.9 is run at 1100 r/min against a pressure rise of 210 bar. (a) Using the measured displacement, estimate the theoretical delivery in gal/min. From the chart, estimate (b) The actual delivery; and (c) The overall efficiency.
A pump delivers 1500 L/min of water at 20°C against a pressure rise of 270 kPa. Kinetic and potential energy changes are negligible. If the driving motor supplies 9 kW, what is the overall efficiency?

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