KATHMANDU UNIVERSITYSCHOOL OF ENGINEERINGDEPARTMENT OF CIVIL AND GEOMATICS ENGINEERING

SELF STUDY REPORTPipes in series and parallelComplex Distribution SystemsPipe pump Characteristics and n-curveChannel ConveyanceCompound Channel SectionsGeometric Properties of Common Prismatic ChannelsSubcritical, Critical and Supercritical FlowCritical DepthGeneral Equation of Gradually Varied FlowSubmitted To:Prof. Dr. Ramesh Kumar MaskeyDepartment of Civil and Geomatics EngineeringSubmitted By:Anish PathakRoll 42CIEG III/I

PIPES IN SERIESSeries pipe systems are those with a single flow path; all of the fluid must travel the same route. Because of that, we can use conservation of mass and conservation of energy to develop the generalized form of the Bernoulli Equation

Single pipe size - If the entire pipe run has the same diameter, then the velocity is uniform, so the same Reynolds number, the same relative roughness, and hence the same friction factor applies everywhere. Note that a customary rule of thumb is to ignore minor losses when the ratio L/D > 1,000. The resulting equation for pump head is:

Variable pipe size - When pipes of different sizes are used, the velocity will vary with pipe size since all pipes have the same flow rate. For two pipe sizes 1 and 2 V1 A1 = V2 A2 Or V2 = V1 * (D1/D2)2 This velocity must be used in all calculations for that portion of the pipe with diameter D2. System curves - Since the head loss depends on the flow rate, then there exists a functional dependence between the head loss and the flow rate. This is called the system curve for any pipe system. The relationship is quadratic (for high Reynolds number flows where f is relatively constant) since hL = f(V2). When one is using the system curve to size a pump, it is often more convenient to calculate the pump power hp required as a function of flow rate, since this term will also include any elevation head and velocity head requirements of the pump.

PIPES IN PARALLELConsider a pipe that branches into two parallel pipes and then rejoins, as shown in Fig. 1. A problem involving this configuration might be to determine the division of flow in each pipe, given the total flow rate. No matter which pipe is involved, the pressure difference between the two junction points is the same. Also, the elevation difference between the two junction points is the same. Because hL = (p1/ + z1) - (p2/ + z2), it follows that hL between the two junction points is the same in both of the pipes of the parallel pipe system. Thus,

Then

If f1 and f2 are known, the division of flow can be easily determined. However, some trial-and-error analysis may be required if f1 and f2 are in the range where they are functions of the Reynolds number.

Figure 1 Flow in parallel pipesType I Problems: Calculate Flow rate - If the pressure drop P is known, then the flow rate through any given parallel path can be calculated as if it were a series path: 1. Calculate Vi for each path as a series, Category II problem. 2. Calculate the total flow rate as the sum of the parts. Type II Problems: Calculate Pressure Drop - If the total flow rate is specified, but the head loss for the system is unknown, as is the actual distribution of the flow through the parallel paths, then an iterative solution is required: 1. Assume flow rate through one pipe, and solve for the head loss though that path. 2. Use the calculated head loss to calculate the flow rate through all other paths. 3. Determine total flow for this assume configuration. Use this value when compared to the known total flow rate, to linearly correct the original assumed flow rate. 4. Repeat steps 1-3 until the known total flow rate is predicted. SERIES-PARALLEL NETWORKS (Complex Distribution Systems)Combining the previous parallel system analysis with the detailed analysis of series systems allows complex systems with a multiplicity of flow paths. A systematic approach allows solution of these types of problems: 1. Identify pipe loops 2. Identify nodes where two or pipes are joined 3. Apply the conservation laws 4. Conservation of mass at a node (what comes in equals what goes out) 5. Conservation of energy (sum of pressure drop around a loop must be zero) These techniques must be applied iteratively, which requires that the solution being generated be directed toward the true solution, or convergence. A method by which this can be done is known as the Hardy-Cross method. Hardy-Cross Method - This method assures convergence to the solution. It utilizes the Hazen Williams expression for the head loss (replace velocity with flow rate)

Since the duct area is related to the diameter (circular pipes) and the friction factor is related to the Reynolds number (flow rate and diameter), then the head loss depends only on the diameter and flow rate for a given pipe length, or hf = f(L, D, V) For water flowing through circular pipes, a good approximation is represented by

Here, C is a measure of the relative pipe roughness and is called the Hazen Williams coefficient. It is tabulated below:

The coefficient k1 for the units of V, the resulting value for hf is always in feet of water. Values for k1 are tabulated below:

The Hazen Williams expression for the head loss is very convenient for any application involving water flow through circular pipes.Hardy Cross Method - The Hardy Cross method solves for flow in each leg or loop of series-parallel pipe systems by iterating in two successive steps: 1. Flow rates are assumed so that mass conservation is satisfied. 2. Zero total pressure drops around any loop are used to correct for incorrect assumptions of flow rate. Several commercial software packages are available based on the Hardy Cross method, or a spreadsheet can be developed to perform the iterative calculations.

PUMP CHARACTERISTICS AND EFFICIENCY CURVEThe hydraulic power delivered to the fluid by the real pumps is less than the input mechanical power due to the volumetric, friction, and hydraulic losses. The actual pump flow rate, Q, is less than the theoretical flow, Qt, mainly due to:Internal leakagePump cavitation and aerationFluid compressibilityPartial filling of the pump due to fluid inertiaThe first source of power losses is the internal leakage. When operating under the correct design conditions, the flow losses are mainly due to internal leakage, QL. The leakage flow through the narrow clearances is practically laminar and changes linearly with the pressure difference. The resistance to internal leakage, RL, is proportional to oil viscosity, , and inversely proportional to the cube of the mean clearance, c.QL = P/RLQ = Qt - QLWhere, RL = K /c3The effect of leakage is expressed by the volumetric efficiency, v, defined as follows:

The friction is the second source of power losses. A part of the driving torque is consumed to overcome the friction forces. This part is the friction torque, TF. It depends on the pump speed, delivery pressure, and oil viscosity. Therefore, to build the required pressure, a higher torque should be applied. The friction losses in the pump are evaluated by the mechanical efficiency, m, defined as follows:

Where T Actual pump driving torque, NmTF Friction torque, NmT TF Torque converted to pressure, NmwPump speed, rad/s

The third source of power losses in the pump is the pressure losses in the pumps inner passages. The pressure, built inside the pumping chamber, PC, is greater than the pump exit pressure, P. These losses are caused mainly by the local losses. These pressure losses are evaluated by the hydraulic efficiency, h. Where PC = Pressure inside the pumping chamber, PaP = Pump exit pressure, PaAn expression for the total pump efficiency, T, is deduced as follows

In the steady-state operation, the real displacement pump is described by the following relations:

If the pump input pressure, Pi, is too small compared with the delivery pressure, P, then it may be neglected, and the pressure difference, P, equals the pump exit pressure, P. If so, then

Following figure shows the typical characteristics of an axial piston pump.

Fig. Typical flow and efficiency characteristics of an axial piston pump. CHANNEL CONVEYANCEChannel conceyance, K, is a measure of the carrying capacity of a channel. The K is really an agglomeration of several terms in the Chezy or Mannings equation:

So,

Use of conveyance may be made when calculating discharge and stage in compound channels and also calculating the energy and momentum coefficients in this situation.We can use Mannings formula for discharge to calculate steady uniform flow. Two calculations are usually performed to solve uniform flow problems.1. Discharge from a given depth2. Depth for a given dischargeIn steady uniform flow the flow depth is known as normal depth.As we know, uniform flow can only occur in channels of constant cross-section (prismatic channels) so natural channel can be excluded. However, we will need to use Mannings equation for gradually varied flow in natural channels so application to natural/irregular channels will often be required.COMPOUND CHANNELSWhen the channel shape changes with flow depth such sections are referred to as compound channel sections. Such sections are typical of natural stream sections during flood.

Fig. Compound Channel SectionDuring floods water spills over the flood plain.We need to know discharge at various depths or vice-versa so that we can design channels or determine channel safety for various flood magnitudes.In more realistic situations channel roughness, n, may be different for flood plain and the main channel. In that case: Determine velocity for each sub section Then sum up the discharges for the sections

GEOMETRIC PROPERTIES OF COMMON PRISMATIC CHANNELSSECTIONAREAWETTED PERIMETERHYDRAULIC RADIUS

SUBCRITICAL, CRITICAL AND SUPERCRITICAL FLOWSpecific energy, Es, is defined as the energy of the flow with reference to the channel bed as the datum:

For a rectangular channel of width b, Q/A = q/y. Using this relation we can obtain a relationship for specific energy in terms of depth, y, keeping discharge constant.

The specific energy change with depth can be plotted as shown below. It is also possible to plot a graph with the specific energy fixed and see how Q changes with depth. These two forms are plotted side by side below.

Figure of variation of specific energy and discharge with depth.For a fixed discharge:1. The specific energy is minimum at depth yc.this depth is known as critical depth.2. For all other values of E there are two possible depths. These are called alternate depths. For,subcritical flow y > ycsupercritical flow y < yc

For a fixed specific energy:1. The discharge is a maximum at critical depth, yc2. For all other discharges there are two possible depths of flow for a particular Esi.e. There is a sub-critical depth and a super-critical depth with the same EsFroude NumberThe Froude number is defined for channel as: Its physical significance is the ratio of inertial forces to gravitational forces squared. It can also be interpreted as the ratio of water velocity to wave velocity.The value of Froude number determines the regime of flow sub, super or critical, and the direction in which the disturbances travelFr < 1sub-criticalwater velocity > wave velocityupstream levels affected by downstream controlsFr = 1criticalFr > 1super-criticalwater velocity < wave velocityupstream levels not affected by downstream controlsCRITICAL DEPTHAn equation for critical depth can be obtained by setting the differential of Es to zero.

For a rectangular channel Q = qb, B = b and A = by and taking apha = 1 the equation becomes

As Vcyc = q Substituting this in to the specific energy equation GENERAL EQUATION OF GRADUALLY VARIED FLOWConsider the length of channel dx illustrated in Fig a. All the terms which enter the steady-flow energy equation are shown, and the balance between x and x + dx is

Fig a. Energy balance between two sections in a gradually varied open-channel flow.

Where So is the slope of the channel bottom (positive as shown in Fig. a) and S is the slope of the EGL (which drops due to wall friction losses). To eliminate the velocity derivative, differentiate the continuity relation But dA = bo dy, where bo is the channel width at the surface. From these relations and above equations we obtain

Finally, V2 bo/(gA) is the square of the Froude number of the local channel flow. The final desired form of the gradually varied flow equation isThis equation changes sign according as the Froude number is subcritical or supercritical and is analogous to the one-dimensional gas-dynamic area-change formula.The numerator changes sign according as So is greater or less than S, which is the slope equivalent to uniform flow at the same discharge Q.

where C is the Chzy coefficient. The behavior of equation for gradually varied flow thus depends upon the relative magnitude of the local bottom slope So(x), compared with (1) uniform flow, y = yn, and (2) critical flow, y = yc.