The Bernoulli Equation

Pitot Tubes. The fundamental design of a pitot tube is shown in Eigure 9a. The opening into the flow stream measures the total or stagnation pressure of the stream whereas a wall tap senses static pressure. The velocity at the tip opening, lA can be obtained by the Bernoulli equation ... 

Head-Area Meters. The Bernoulli principle, the basis of closed-pipe differential-pressure flow measurement, can also be appHed to open-channel Hquid flows. When an obstmction is placed in an open channel, the flowing Hquid backs up and, by means of the Bernoulli equation, the flow rate can be shown to be proportional to the head, the exact relationship being a function of the obstmction shape. 

The energy state of soil water can be defined with respect to the Bernoulli equation, neglecting thermal and osmotic energy as... 

The Bernoulli equation can be written for incompressible, inviscid flow along a streamhne, where no shaft work is done. 

Unlike the momentum equation (Eq. [6-11]), the Bernoulli equation is not easily generahzed to multiple inlets or outlets. 

For homogeneous flow in a pipe of diameter D, the differential form of the Bernoulli equation (6-15) rearranges to... 

This example demonstrates the dimensioning of a duct with a frictional incompressible fluid flow. Now the Bernoulli equation can be written as... 

The diameter rfj is solved analogously. The Bernoulli equation at the interval 1-2 is... 

Another procedure for design of an air curtain is proposed by Tamm based on the Bernoulli equation. Recently Partyka proposed another procedure based on the model of Schlichting previously described. 

For fluid flow in the (r, 6) plane, it is reasonable to assume that the fluid is inviscid, as the Reynolds number of the fluid flow usually exceeds O(IO ). Thus Eq. (13.1), with /i, = 0, may be integrated along the streamlines to give the Bernoulli equation as follows ... 

Equation (3.14.2.17) shows the form of the Bernoulli equation that is a first-order differential equation. By substituting (3.14.2.18)... 

The conditions at two different positions along a pipeline (at points 1 and 2) are related by the Bernoulli equation (see Problem 11). For flow in a pipe,... 

Note that if each term of Eq. (5-35) is divided by g, then all terms will have the dimension of length. The result is called the head form of the Bernoulli equation, and each term then represents the equivalent amount of... 

The Bernoulli equation should therefore include this kinetic energy correction factor, i.e.,... 

The pressure P2 is determined by Bernoulli s equation. If the diffuser is horizontal, there is no work done between the inlet and outlet, and the friction loss is small (which is a good assumption for a well designed diffuser), the Bernoulli equation gives... 

Comparing this with the Bernoulli equation [Eq. (5-33)] shows that they are identical, provided... 

We see that there are several ways of interpreting the term ef. From the Bernoulli equation, it represents the lost (i.e., dissipated) energy... 

Looking at the Bernoulli equation, we see that the friction loss (ef) can be made dimensionless by dividing it by the kinetic energy per unit mass of fluid. The result is the dimensionless loss coefficient, K ... 

For plug flow, the Bernoulli equation for this system is... 

This can be solved for (P2 — Pi), which, when inserted into the Bernoulli equation, allows us to solve for ef. 

We note first that the Bernoulli equation can be written... 

We will use the Bernoulli equation in the form of Eq. (6-67) for analyzing pipe flows, and we will use the total volumetric flow rate (Q) as the flow variable instead of the velocity, because this is the usual measure of capacity in a pipeline. For Newtonian fluids, the problem thus reduces to a relation between the three dimensionless variables ... 

For this problem, we want to know the net driving force (DF) that is required to move a given fluid (/a, p) at a specified rate (Q) through a specified pipe (D, L, e). The Bernoulli equation in the form DF = ef applies. 

The energy cost is determined from the pumping energy requirement, which is in turn determined from the Bernoulli equation ... 

The inclusion of significant fitting friction loss in piping systems requires a somewhat different procedure for the solution of flow problems than that which was used in the absence of fitting losses in Chapter 6. We will consider the same classes of problems as before, i.e. unknown driving force, unknown flow rate, and unknown diameter for Newtonian, power law, and Bingham plastics. The governing equation, as before, is the Bernoulli equation, written in the form... 

The appropriate expressions that apply are the Bernoulli equation [Eq. (7-45)], the power law Reynolds number [Eq. (7-40)], the pipe friction factor as a function of 7VRejpl and n [Eq. (6-44)], and the 3-K equation for fitting losses [Eq. (7-38)] with the Reynolds number replaced by A pi- The procedure is... 

It is assumed that the system contains only one size (diameter) of pipe. The Bernoulli equation can be rearranged to give D ... 

A special condition called slack flow can occur when the gravitational driving force exceeds the full pipe friction loss, such as when a liquid is being pumped up and down over hilly terrain. Consider the situation shown in Fig. 7-5, in which the pump upstream provides the driving force to move the liquid up the hill at a flow rate of Q. Since gravity works against the flow on the uphill side and aids the flow on the downhill side, the job of the pump is to get the fluid to the top of the hill. The minimum pressure is at point 2 at the top of the hill, and the flow rate (Q) is determined by the balance between the pump head (Hp = — w/g) and the frictional and gravitational resistance to flow on the uphill side (i.e., the Bernoulli equation applied from point 1 to point 2) ... 

Piping systems often involve interconnected segments in various combinations of series and/or parallel arrangements. The principles required to analyze such systems are the same as those have used for other systems, e.g., the conservation of mass (continuity) and energy (Bernoulli) equations. For each pipe junction or node in the network, continuity tells us that the sum of all the flow rates into the node must equal the sum of all the flow rates out of the node. Also, the total driving force (pressure drop plus gravity head loss, plus pump head) between any two nodes is related to the flow rate and friction loss by the Bernoulli equation applied between the two nodes. 

The suction pressure Ps is determined by applying the Bernoulli equation to the suction line upstream of the pump. For example, if the pressure at the entrance to the upstream suction line is P1 the maximum distance above this point that the pump can be located without cavitating (i.e., the maximum suction lift) is determined by Bernoulli s equation from Px to Ps ... 

Because the fluid velocity and properties change from point to point along the pipe, in order to analyze the flow we apply the differential form of the Bernoulli equation to a differential length of pipe (dL) ... 

Eq. (9-17) reduces identically to the Bernoulli equation for an incompressible fluid in a straight, uniform pipe, which can be written in the form... 

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