Flow through a Pipe

We observe in our model system a difference of 2.9 cm water, which we convert to SI units  

How do we use this experimental result from our model to predict the pressure drop in the real system Use the Euler number  

Although SI units are convenient for calculation, I have no sense for a pressure of 3.2 X 10 Pa. So I convert to more convenient metric units and English units  

as promised, one experiment performed on a model has yielded the pressure drop for the actual system. However, we did not obtain a functional relationship. If any of the parameters of the real system change, we must perform another experiment with the model to predict the new pressure drop. 

In the previous section we derived three II groups to describe fluid flow in pipes  

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FIGURE 4.26 Hear flow through a pipe wall Example I... 

When a fluid is flowing through a pipe, resistance to flow is caused by friction. The pipe bore selected for each section must be such that under any operating conditions, the initial head, either static head of oil in the supply tank or the pump delivery pressure, will be adequate to ensure the required flow rate. Additionally, any change of flow rate and consequent variation in loss of head must not adversely affect the operation of the associated oil-burning equipment. 

The operation of a typical fluid power system is illustrated in Figure 40.1. Oil from a tank or reservoir flows through a pipe into a pump. An electric motor, air motor. 

NOTE 1 mil = 1 thousandth of one inch. It can be seen that only 6 ppm carbon dioxide in condensate flowing through a pipe of 1/4 inch (250 mil) wall thickness will eat away 50% of the wall thickness in only 5 years, at which point the structural integrity of the pipe is essentially lost. 

For a fluid flowing through a pipe the momentum per unit cross-sectional area is given by pu2. This quantity, which is proportional to the inertia force per unit area, is the force required to counterbalance the momentum flux. 

A Newtonian liquid of viscosity 0.1 N s/m2 is flowing through a pipe of 25 mm diameter and 20 m in lenglh, and the pressure drop is 105 N/m2. As a result of a process change a small quantity of polymer is added to the liquid and this causes the liquid to exhibit non-Newtonian characteristics its rheology is described adequately by the power-law model and the flow index is 0.33. The apparent viscosity of the modified fluid is equal to ihc viscosity of the original liquid at a shear rate of 1000 s L... 

Sulphuric acid of density 1300 kg/m3 is flowing through a pipe of 50 mm, internal diameter. A thin-lipped orifice, 10 mm in diameter is fitted in the pipe and the differential pressure shown on a mercury manometer is 0.1 m. Assuming that the leads to the manometer are filled with the acid, calculate (a) the mass flow rate of acid and (b) the approximate drop in pressure caused by the orifice in kN/m2. The coefficient of discharge of the orifice may be taken as 0.61, the density of mercury as 13.550 kg/m3 and the density of the water as OHIO kg/m ... 

Convection. Heat transfer by convection arises from the mixing of elements of fluid. If this mixing occurs as a result of density differences as, for example, when a pool of liquid is heated from below, the process is known as natural convection. If the mixing results from eddy movement in the fluid, for example when a fluid flows through a pipe heated on the outside, it is called forced convection. It is important to note that convection requires mixing of fluid elements, and is not governed by temperature difference alone as is the case in conduction and radiation. 

The discrepancy between the coefficients in equations 11.45 and 11,46 is attributable to the fact that the effect of the curvature of the pipe wall has not been taken into account in applying the equation for flow over a plane surface to flow through a pipe. In addition, it takes no account of the existence of the laminar sub-layer at the walls. 

Calculate the thickness of the laminar sub-layer when benzene flows through a pipe 50 mm in diameter at 2 1/s. What is the velocity of the benzene at the edge of the laminar sub-layer Assume that fully developed flow exists within the pipe and that for benzene, p — 870 kg/m3 and p = 0.7 mN s/m2. 

A liquid is pumped in streamline flow through a pipe of diameter d. At what distance from the centre of the pipe will the fluid be flowing at the average velocity ... 

When a gas flows through a pipe the gas density is a function of the pressure and so is determined by the pressure drop. Equation 5.3 and Figure 5.7 can be used to estimate the pressure drop, but it may be necessary to divide the pipeline into short sections and sum the results. 

The value of the Reynolds number which approximately separates laminar from turbulent flow depends, as previously mentioned, on the particular configuration of the system. Thus the critical value is around 50 for a film of liquid or gas flowing down a flat plate, around 500 for flow around a sphere, and around 2500 for flow through a pipe. The characteristic length in the definition of the Reynolds number is, for example, the diameter of the sphere or of the pipe in two of these examples. 

The distribution of the gas velocity across the profile of a moving column of gas changes in the transition from laminar to turbulent flow. In the case of flow through a pipe of radius ro, the laminar flow variation is given by... 

Figure 4-6 Liquid flowing through a pipe. The frictional flow losses between the fluid and the pipe wall result in a pressure drop across the pipe length. Kinetic energy changes are frequently negligible.

Diagrammatic representation of heads in a liquid flowing through a pipe... 

Thus the kinetic energy per unit mass of a Newtonian fluid in steady laminar flow through a pipe of circular cross section is u2. In terms of head this is u2/g. Therefore for laminar flow, a = i in equation 1.14. 

Consider a fully developed turbulent flow through a pipe of circular cross section. A turbulent boundary layer will exist with a thin viscous sublayer immediately adjacent to the wall, beyond which is the buffer or generation layer and finally the fully turbulent outer part of the boundary layer. 

Using the viscometric data given in Table 3.3 calculate the average velocity for the material flowing through a pipe of diameter 37 mm when the pressure gradient is 1.1 kPa/m. 

Figure 1.5 diagrapimatically represents the heads in a liquid flowing through a pipe. Redraw this diagram with a pump placed between points 1 and 2. 

Suppose a process stream is flowing through a pipe in essentially plug flow and that it takes D minutes for any individual element of fluid to flow from the entrance to the exit of the pipe. Then the pipe represents a deadtime element. 

In a situation such as that described in Fig. 4, there is no difficulty in establishing the value for AT. However, when fluid is flowing through a pipe surrounded by a medium at another temperature, the fluid s temperature changes as it travels down the pipe and, consequently, the temperature difference varies with position along the pipe. In such cases, it can be shown that the correct value of AT to use in eqn. (74) is the logarithmic mean temperature difference, ATy which is given by... 

We can therefore replace dt by dz/u in all of the preceding differential equations for the mass balance in the batch reactor and use these equations to describe reactions during flow through a pipe. This reactor is called the plug-flow tubular reactor, which is the most important continuous reactor encountered in the chemical industry. 

The quantity of gas flowing through a piping element during a unit of time -in accordance with the two concepts for gas quantity desaibed above - can be indicated in either of two ways, these being ... 

Instead of molecules flowing, consider water flowing through a pipe with a number of partially opened valves. The flow of the water in the pipe will be determined by the valve that offers the largest resistance to the flow. Actually, we could come up with a fairly accurate estimate of the flow by calculating the resistance to the flow in this valve, neglecting all the others. 

Liquid Viscosity — The value (in centipoise) is a measure of the ability of a liquid to flow through a pipe or a hole higher values indicate that the liquid flows less readily under a fixed pressure head. For example, heavy oils have higher viscosities (i.e., are more viscous) than gasoline. Liquid viscosities decrease rapidly with an increase in temperature. A basic law of fluid mechanics states that the force per unit area needed to shear a fluid is proportional to the velocity gradient. The constant of proportionality is the viscosity. 

If only rotational viscometric data are available, the design of pipe lines is somewhat more complex. Such data are usually expressed in the form of a relationship between shear stress and shear rate. The shear stress on a cylindrical element of fluid of radius r flowing through a pipe in laminar motion is equal to rAP/2L. If the corresponding shear rate — du/dr) can be expressed analytically, i.e., if the functional relationship... 

Fig. 9. Example of a DANTE-type velocity image of water flowing through a pipe. H 2-D image of a 2.0mm-thick longitudinal slice at (a) zero flow rate, and (b) a flow rate of 486cm min . Flow is from left to right.

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