Rate Problems (systems of equations in two variables)

The aim of this chapter is to answer research question seven, and to indicate strategies that can be used to contribute to the solution of upstream-downstream Downstream Upstream And For Pdf Pdf Solution Problems With problems in international river basins. Unable to display preview. Download preview PDF. Skip to main content. This service is more advanced with JavaScript available.

Advertisement Hide. Solving Upstream-Downstream Problems. Authors Authors and affiliations Sander V. This Upstream And Downstream Problems With Solution Pdf ForUpstream And Downstream Problems With Solution Pdf For Pdf Pdf process is experimental and the keywords may be updated as the learning algorithm improves. This is a preview of subscription content, log in to check access. See also the conclusions concerning the effectiveness of Upstream And Downstream Problems With Solution Pdf For Pdf international river commissions presented in Section 8.

Google Scholar. This recommendation relates to the fundamental dilemma of environmental policy, which was also mentioned shortly in Chapter 1 of this thesis. The dilemma is that on Upstream And Downstream Problems With Solution Pdf For Pdf the one hand economic growth enables actors to invest in environmental policies, but on the other hand economic growth often causes additional environmental pollution or degradation.

This support, however, may be perceived as contradicting with the Upstream And Downstream Problems With Solution Pdf For Pdf Upstream And Downstream Problems With Solution Pdf For Pdf internationally accepted polluter pays principle. Below, the strategies of problem solving that may be used by the European Union are discussed. Commission proposal for a council directive upstream and downstream problems with solution pdf for pdf a framework for community action in the field of water policy, and several amendments made.

For upstream and downstream problems with solution pdf for pdf on types of intermediaries and the strategies they may Upstream And Downstream Problems With Solution Pdf For Pdf Upstream And Downstream Problems With Solution Pdf For Pdf use, see for example WesselDieperinkand Wolf Sander V. Meijerink 1 1. Personalised recommendations. Cite chapter How to cite? ENW EndNote. Buy options.

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Discussion The equations of GVF are simplified because of the slow changes in the flow direction. Analysis The hydraulic radius Rh is defined as the ratio And With Problems Pdf Pdf For Solution Downstream Upstream of the cross-sectional flow area Ac and the wetted perimeter p. Discussion It is unfortunate and not our fault! Fr represents the ratio of inertia forces to viscous forces in open-channel flow. Discussion The Froude number Upstream And Downstream Problems With Solution Pdf For Pdf is the most important parameter in open-channel flow.

Discussion Critical depth is a useful parameter, even if the depth does not actually equal yc anywhere in the flow. Analysis Upstream of a hydraulic jump, the upstream flow must be supercritical. Downstream of a hydraulic jump, the downstream flow must be subcritical.

Discussion Otherwise, the second law of thermodynamics would be violated. Note that a hydraulic jump is analogous to a Upstream And Downstream Problems With Solution Pdf For Pdf normal shock wave � in that case, the flow upstream must be supersonic and the flow downstream must be subsonic.

The speed of a small disturbance in flow for two different flow depths is to be determined for both water and oil. Discussion Note that wave speed depends on the water depth, and the wave speed increases as the water depth increases as long as the water remains shallow. Results would Upstream And Downstream Problems With Solution Pdf For Pdf Upstream And Downstream Problems With Solution Pdf For Pdf And For Pdf Problems Upstream Pdf Downstream With Solution Upstream And Downstream Problems With Solution Pdf For Pdf not change if the fluid were oil, because the wave speed depends only on the fluid depth.

For given flow depth and velocity, it is to be determined whether the flow is laminar or turbulent, Upstream And Downstream Problems With Solution Pdf For Pdf and whether it is subcritical or supercritical. Assumptions The flow is uniform. Therefore, the flow is turbulent. Discussion The result in a is expected since almost all open channel flows are turbulent. Chapter 13 Open-Channel Flow Solution Water flow in a partially full circular channel is considered.

For given water depth and average velocity, the hydraulic radius, Reynolds number, and the flow regime are to be determined. Assumptions 1 The Upstream And Downstream Problems With Solution Pdf For Pdf flow is uniform.

When calculating the Froude number, the hydraulic depth should be used rather than the maximum depth or the hydraulic radius. Therefore, the flow is supercritical although, very close to critical. Discussion Note that Upstream And Downstream Problems With Solution Pdf For Pdf if Upstream Downstream Problems With Solutions Pdf the maximum flow depth were used instead of the hydraulic depth, the result would be subcritical flow, which is not true.

For given values of flow depth and velocity, it is to be determined whether the flow is subcritical or supercritical. Discussion Note that the Froude Number is not function of any temperature-dependent properties, and thus temperature. Discussion This water layer will undergo a hydraulic jump when the ground Upstream And Downstream Problems With Solution Pdf For Pdf Upstream And Downstream Problems With Solution Pdf For Pdf slope decreases or becomes adverse.

For a given average velocity, the hydraulic radius, the Reynolds number, and the flow regime are to be determined. Therefore, the flow is subcritical. Chapter 13 Open-Channel Flow Solution A Upstream And Downstream Problems With Solution Pdf For Pdf single wave is initiated in a sea by a strong jolt during an earthquake. The speed of the resulting wave is to be determined. Also, the waves eventually die out because of the viscous effects.Upstream And Downstream Problems With Solution Pdf For Pdf Upstream And Downstream Problems With Solution Pdf For Pdf

Therefore, the specific energies of water in those two channels can be identical. Discussion If the flow is varied not uniform , however, Es is not necessarily identical in the two channels. Analysis The point of minimum specific energy is the critical point, and thus the first person is correct. Discussion The specific energy cannot go below the critical point for a given volume flow rate, as is clear from the plot Upstream And Downstream Problems With Solution Pdf For Pdf of specific energy as a function of flow depth.

Analysis No, the claim is incorrect. Discussion This may go against our intuition, since a larger flow depth seems to imply greater energy, but this is not necessarily the case we cannot always trust our intuition. Chapter 13 Open-Channel Flow C Solution We are to examine a claim that specific energy remains constant in steady uniform flow.

The head loss is Upstream And Downstream Problems With Solution Pdf For Pdf Pdf With Downstream Problems And For Upstream Solution Pdf Upstream And Downstream Problems With Solution Pdf For Pdf made up by the decline in elevation the channel is sloped downward in the flow direction. Discussion In uniform flow, the flow depth and the average velocity do not change downstream, since the elevation drop exactly And With Solution Pdf Downstream Pdf For Upstream Problems overcomes the frictional losses.

The friction slope is equal to the bottom slope when the head loss is equal to the elevation drop. Discussion Friction slope is a useful concept when analyzing uniform or varied Upstream And Downstream Problems With Solution Pdf For Pdf Upstream And Downstream Problems With Solution Pdf For Pdf flow in open channels. Analysis No, the claim is not correct. When there is no head loss, the energy line is horizontal even when the channel is not.

Discussion Keep in mind that in real With Pdf Downstream Upstream Solution And For Problems Pdf Downstream And Pdf With Pdf Solution Upstream Problems For life, there is no such thing as frictionless flow. However, there are situations in which the frictional effects are negligible compared to other effects in the flow. Analysis Yes, the claim is correct. Discussion The physical elevation of the point under consideration does not appear in the above equation for H.

Discussion Because of irreversible frictional head losses, H must decrease in the flow direction in open-channel flow. Rh 8g Discussion Upstream And Downstream Problems With Solution Pdf For Pdf Upstream And Downstream Problems With Solution Pdf For Pdf Upstream And Downstream Problems With Solution Pdf For Pdf Head loss is always positive � it can never be negative since this would violate the second law of thermodynamics.

Thus, the total mechanical energy must decrease downstream in open-channel flow. The character of flow, the flow velocity, and the alternate depth are to be determined. Assumptions The specific energy is constant. There are three roots of this equation; one for subcritical, one for supercritical and third one as a Upstream And Downstream Problems With Solution Pdf For Pdf negative root.

Therefore, if the character of flow is changed from supercritical to subcritical while holding the specific energy constant, the flow depth will rise from 0.

Discussion Two alternate depths show two possible flow conditions for a given specific energy. If the energy is not the minimum specific energy, there are two water depths corresponding to subcritical and supercritical states of flow. As an example, these two depths may Upstream And Downstream Problems With Solution Pdf For Pdf Pdf Downstream For Upstream With And Solution Pdf Problems be observed before and after a sluice gate as alternate depths, if the losses are disregarded.

The specific energy and whether the flow is subcritical or supercritical are to be determined. Assumptions The flow is uniform Upstream And Downstream Problems With Solution Pdf For Pdf Upstream And Downstream Problems With Solution Pdf For Pdf and thus the specific energy is constant.

The specific Upstream And Downstream Problems Formula Workbook energy for given conditions is 0. The critical depth, the alternate depth, and the minimum specific energy are to be determined. Assumptions The channel is sufficiently Upstream And Downstream Problems With Solution Pdf For Pdf wide so that the edge effects are negligible. The critical depth, the alternate depth, and whether the flow is subcritical or supercritical are to be determined. Discussion The depths 0. The flow conditions determine which Pdf With Downstream Pdf Problems Solution Upstream For And Upstream And Downstream Problems With Solution Pdf For Pdf one is observed. For specified values of flow depth and average velocity, the Froude number, critical depth, and whether the flow is subcritical or supercritical are to be determined.

Discussion Note that the value of critical depth depends on flow rate, and it decreases as the flow rate decreases. For a specified average velocity, the flow rate of water is to be determined.

Occurrence of critical depth is important as boundary Upstream And Downstream Problems With Solution Pdf For Pdf condition most of the time. For example it can be used as a flow rate computation mechanism for a channel ending with a drawdown. Chapter 13 Open-Channel Flow Solution Water flows uniformly through a half-full And Pdf Downstream Upstream With Problems Pdf Solution For circular steel channel. For a given average velocity, the volume flow rate, critical slope, and the critical depth are to be determined.

Therefore, the flow is supercritical. Discussion Note that if the maximum flow depth Upstream And Downstream Problems With Solution Pdf For Pdf Upstream For Solution Pdf Problems With And Downstream Pdf were used instead of the hydraulic depth, the result could be different, especially when the Froude number is close to 1.

For a given flow rate, the average velocity and whether the flow is subcritical or supercritical are to be determined. Discussion The analysis is approximate since the edge effects are significant here compared to a wide rectangular channel, and thus the results should be interpreted accordingly.

Chapter 13 Open-Channel Flow Upstream And Downstream Problems With Solution Pdf For Pdf Solution Water flows uniformly through a half-full hexagon channel. For a non-rectangular channel, hydraulic depth is defined as the ratio of the flow area to top width, Ac A 5. Analysis Flow in a channel Upstream And Downstream Problems With Solution Pdf For Pdf is called uniform flow if the flow depth and thus the average flow velocity remains constant. Discussion Uniform flow in open-channel flow is somewhat analogous to fully developed pipe flow in internal flow.

Analysis The Upstream And Downstream Problems With Solution Pdf For Pdf flow depth decreases when the bottom slope is increased. Discussion You can think of it in simple terms this way: As the slope increases, the liquid flows faster, and faster flow requires lower depth. Chapter 13 Open-Channel Flow C Solution We are to examine a claim that head loss can be determined by multiplying bottom slope by channel length. Discussion In uniform flow, frictional head losses are exactly balanced by Upstream And Downstream Problems With Solution Pdf For Pdf elevation loss, which is directly proportional to bottom slope.

Discussion The value of n varies greatly with surface roughness. Analysis The best hydraulic cross-section for an open channel is the one with the maximum hydraulic radius, And Pdf For Upstream Pdf Downstream Problems With Solution Upstream And Downstream Problems With Solution Pdf For Pdf or equivalently, the one with the minimum wetted perimeter for a specified cross-sectional area. Discussion Frictional losses occur at the wetted perimeter walls of the channel, so it makes sense to minimize the wetted perimeter Upstream And Downstream Problems With Solution Pdf For Pdf in order to minimize the frictional losses.

Analysis The best hydraulic cross-section for an open channel is a a circular one. Discussion Circular channels are often more difficult to construct, however, so they are often Upstream And Downstream Problems With Solution Pdf For Pdf Upstream And Downstream Problems With Solution Pdf For Pdf not used in practice.

Chapter 13 Open-Channel Flow C Solution We are to determine the best hydraulic cross section for a rectangular channel. Analysis The best hydraulic cross section for a rectangular channel is one Upstream And Downstream Problems With Solution Pdf For Pdf whose fluid height is a half the channel width. Discussion It turns out that for this case, the wetted perimeter, and thus the frictional losses, are smallest.

Analysis The best hydraulic cross section for a trapezoidal Upstream And Downstream Problems With Solution Pdf For Pdf channel of base width b is a one for which the length of the side edge of the flow section is b. Therefore, if the Manning coefficient doubles as a result of some algae growth on surfaces while the flow cross section remains constant, the flow rate will d decrease by half. Discussion In an actual case, the cross section may also change due to flow depth changes as well.Upstream And Downstream Problems With Solution Pdf For Pdf Upstream And Downstream Problems With Solution Pdf For Pdf

For a given flow depth and bottom slope, the flow rate is to be determined. Assumptions 1 The flow is steady and uniform. Chapter 13 Open-Channel Flow Solution Water flows uniformly half-full in a circular finished-concrete channel. For a given bottom slope, the flow rate is to be determined. For a specified flow rate, the elevation difference across the channel is to be determined.

Chapter 13 Open-Channel Flow Solution Water is to be transported uniformly in a trapezoidal asphalt-lined channel. For a specified flow rate, the required elevation drop per km channel length is to be determined.

Chapter 13 Open-Channel Flow Solution The flow of water Upstream And For Solution With Problems Pdf Pdf Downstream Pdf Problems Pdf And With Downstream Upstream For Solution through the trapezoidal asphalt-lined channel in the previous problem is reconsidered. The maximum flow rate corresponding to a given maximum channel height is to be determined.

Analysis We denote the flow conditions in the previous problem by subscript 1 and the conditions for the maximum a case in this problem by subscript 2. Chapter 13 Open-Channel Flow Solution The flow of water through two identical channels with square flow sections is considered. The percent increase in flow rate as a result of combining the two channels while the flow depth remains constant is to be determined.

Analysis We denote the flow conditions for two separate channels by subscript 1 and the conditions for the combined wide channel by subscript 2. For given flow rate and bottom slope, the flow depth is to be determined. Discussion Non-linear equations frequently arise in the And With Upstream Pdf Downstream Pdf For Solution Problems solution of open channel flow problems. They are best handled by equation solvers such as EES.

Chapter 13 Open-Channel Flow Solution The flow of water in a weedy excavated trapezoidal channel is considered. Discussion Note Upstream And Downstream Problems With Solution Pdf For Pdf that as the Manning coefficient increases because of the increased surface roughness of the channel, the flow depth required to maintain the same flow rate also increases.

The flow rate and the effective Manning coefficient With Pdf Upstream Pdf For And Problems Downstream Solution are to be determined. The 1. Therefore, using a weighted average Manning coefficient for the entire channel may be tempting, but it would Upstream Downstream Problems With Solutions Not not be accurate. Chapter 13 Open-Channel Flow Solution Waters flows in a partially filled circular channel made of finished concrete. By varying the flow depth-to-radius ratio from 0. Chapter 13 Open-Channel Flow Solution Water is to be transported uniformly in a clean-earth trapezoidal channel.

Analysis The flow area, wetted perimeter, and hydraulic radius of the channel are 1. If all channels are to run half-full, the diameter of the third channel is to be determined.

Discussion Note that if the channel diameter were Upstream And Downstream Problems With Solution Pdf For Pdf Upstream And Downstream Problems With Solution Pdf For Pdf larger, the channel would have been less than half full. Chapter 13 Open-Channel Flow Solution Water is transported in an asphalt lined open channel at a specified rate.

The dimensions of the best cross- section for various geometric shapes are to be determined. Discussion The perimeters for the circular, rectangular, and trapezoidal channels are 3. Therefore, the circular cross-section has the smallest perimeter. The dimensions for the best cross- section if the channel is made of unfinished and finished concrete are to be determined.

Chapter 13 Open-Channel Flow Solution Uniform flow in an asphalt-lined rectangular channel is considered. By varying the depth-to-width ratio from 0. Analysis Both uniform and varied flows are steady, and thus neither involves any change with time at a specified location. In uniform flow, the flow depth y and the flow velocity V remain constant whereas in nonuniform or varied flow, the flow depth and velocity vary in the streamwise direction of the flow.

In uniform flow, the slope of the energy line is equal to the slope of the bottom surface. In Upstream And Downstream Problems With Solution Pdf For PdfUpstream And Downstream Problems With Solution Pdf For Pdf Downstream Upstream Pdf Solution For Pdf And Problems With g> varied flow, however, these slopes are different. Analysis Gradually varied flow GVF is characterized by gradual variations in flow depth and velocity small slopes and no abrupt changes and a free surface that always Upstream And Downstream Problems With Solution Pdf For Pdf Upstream And Downstream Problems With Solution Pdf For Pdf remains smooth no discontinuities or zigzags.

Rapidly varied flow RVF involves rapid changes in flow depth and velocity. A change in the bottom slope or cross-section of a channel or an obstruction on the path Upstream And Downstream Problems With Solution Pdf For Pdf of flow may cause the uniform flow in a channel to become gradually or rapidly varied flow. Analytical relations for the profile of the free surface can be obtained in GVF, but this is not the Upstream And Downstream Problems With Solution Pdf For Pdf case for RVF because of the intense agitation.

Discussion In many situations, the shape of the free surface must be solved numerically, even for GVF. Analysis Yes, we agree with this claim.

Rapidly varied flows occur over a short section of the channel with relatively small surface area, and thus frictional losses associated with wall shear are negligible compared with losses due to intense agitation and turbulence.

Losses in GVF, on the other hand, are primarily due to frictional effects along the channel, and should be considered. Discussion There is somewhat of an analogy here with internal flows. However, in short sections of piping with Upstream And Downstream Problems With Solution Pdf For Pdf rapid change of diameter or a blockage or turn, etc minor loss , friction along the wall is typically negligible compared to other losses.

Analysis The flow depth y must c decrease in the flow direction. Discussion Upstream And Downstream Problems With Solution Pdf For Pdf Since the flow is subcritical, there is no possibility of a hydraulic jump. Analysis The flow depth y must a increase in the flow direction. Chapter 13 Open-Channel Flow C Solution We are to analyze And Upstream Problems Pdf Downstream With Pdf Solution For what happens to flow depth in a horizontal rectangular channel during supercritical flow.

Analysis The flow depth y a increases in the flow direction. Discussion Since the flow is supercritical, this increase in flow depth Upstream And Downstream Problems With Solution Pdf For Pdf Solution With Pdf Problems Downstream Pdf Upstream And For may occur via a hydraulic jump. Analysis The flow depth y c decreases in the flow direction. Analysis No. It is impossible for subcritical flow to undergo a hydraulic jump. Such a process would require the head loss hL to become negative, which is impossible. It would correspond to negative entropy generation, which would be a violation of the second law of thermodynamics.

Analysis Hydraulic jumps are often designed in conjunction with stilling basins and spillways of dams in order to waste as much of the mechanical energy as possible to minimize the mechanical energy of the fluid and thus its potential to cause damage. It With Upstream Pdf Downstream And For Problems Pdf Solution is to be determined whether the channel slope is mild, critical, or steep for this flow.

Discussion If the flow depth were smaller than 1. Therefore, the bottom slope alone is not sufficient to classify a downhill channel as being mild, critical, or steep. Discussion Note that two channels of the same slope can be classified as differently one mild and the other steep if they have different roughness and Upstream And Downstream Problems With Solution Pdf For Pdf thus different values of n. Chapter 13 Open-Channel Flow E Solution Water is flowing in a rectangular open channel with a specified bottom slope at a specified flow rate.

It is to be determined whether Upstream And Downstream Problems With Solution Pdf For Pdf the slope of this channel should be classified as mild, critical, or steep. The surface profile is also to be classified for a specified flow depth of 2 m. Discussion If the flow depth were larger Upstream And Downstream Problems With Solution Pdf For Pdf than 5.

Discussion If the flow depth were smaller than 0. The depth and Froude number after the jump, the head loss and dissipation ratio, and dissipated mechanical power are to be determined.

Assumptions 1 Upstream And Downstream Problems With Solution Pdf For Pdf The flow is steady or quasi-steady. Therefore, the flow is supercritical before the jump. That is, if the water is routed to a hydraulic turbine instead of being released from the sluice gate, up to kW of power could be produced.

Chapter 13 Open-Channel Flow Solution Water at a specified depth and velocity undergoes a hydraulic jump. The head loss associated with this process is to be determined. Therefore, the Upstream And Downstream Problems With Solution Pdf For Pdf Upstream And Downstream Problems With Solution Pdf For Pdf Upstream And Downstream Problems With Solution Pdf For Pdf flow is indeed supercritical before the jump. Chapter 13 Open-Channel Flow Solution The increase in flow depth during a hydraulic jump is given. The velocities and Froude numbers before and after the jump, and the energy Upstream And Downstream Problems With Solution Pdf For Pdf dissipation ratio are to be determined.

Discussion The results show that the hydraulic jump is a highly dissipative process, wasting over one-third of the available head. Chapter 13 Open-Channel Flow Solution Water flowing in a wide channel at a specified depth and flow rate undergoes a hydraulic jump. Jacobs , Cathy Higgins Published Small commercial HVAC systems are notorious for consuming more energy than necessary to properly heat, cool, and dehumidify Upstream And Downstream Problems With Solution Pdf For Pdf Upstream And Downstream Problems With Solution Pdf For Pdf buildings.

Several studies have confirmed poor performance of the equipment. Key findings include failed economizers, cycling fans, refrigeration charge variations, higher unit external static pressure, and reduced air flow rate.

These problems result in a reduction in unit efficiency under field conditions relative to rated conditions and, in some cases, inadequate� Expand. Save to Library. Create Alert. Launch Research Feed. Share This Paper.

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