Manometer equation

The formula that relates the pressure difference P — Pi to the difference in manometer fluid levels is based on the principle that the fluid pressure must be the same at any two points at the same height in a continuous fluid. In particular, the pressure at the height of the lower surface of a manometer fluid is the same in both arms of the manometer. (See Figure 3.4-5.) Writing and equating expressions for the pressures at points (a) and (b) in Rgure 3.4-5 yields the general manometer equation... 

In a differential manometer, fluids 1 and 2 are the same, and consequently pi = p2 = p-The general manometer equation then reduces to... 

There are a number of relatively simple experiments with soap films that illustrate beautifully some of the implications of the Young-Laplace equation. Two of these have already been mentioned. Neglecting gravitational effects, a film stretched across a frame as in Fig. II-1 will be planar because the pressure is the same as both sides of the film. The experiment depicted in Fig. II-2 illustrates the relation between the pressure inside a spherical soap bubble and its radius of curvature by attaching a manometer, AP could be measured directly. 

Example 3 Venturi Flowmeter An incompressible fluid flows through the venturi flowmeter in Fig. 6-7. An equation is needed to relate the flow rate Q to the pressure drop measured by the manometer. This problem can he solved using the mechanical energy balance. In a well-made venturi, viscous losses are neghgihle, the pressure drop is entirely the result of acceleration into the throat, and the flow rate predicted neglecting losses is quite accurate. The inlet area is A and the throat area is a. 

The continuity equation gives V2 = V AJa, and Vj = Q/A. The pressure drop measured by the manometer is pi —p2= (p — p)gA . Substituting these relations into the energy balance and rearranging, the desired expression for the flow rate is found. 

In ventilation applications, where the density of the manometer fluid is much higher than the density of air, the pressure difference Ap can be expressed using the equation... 

The three-fluid manometer illustrated in Fig. 4-P11 is used to measure a very small pressure difference (P1 — P2). The cross-sectional area of each of the reservoirs is A, and that of the manometer legs is a. The three fluids have densities pa, ph, and pc, and the difference in elevation of the interfaces in the reservoir is x. Derive the equation that relates the manometer reading h to the pressure difference P1 — P2. How would the relation be simplified if A al... 

The pitot tube is a device for measuring v(r), the local velocity at a given position in the conduit, as illustrated in Fig. 10-1. The measured velocity is then used in Eq. (10-2) to determine the flow rate. It consists of a differential pressure measuring device (e.g., a manometer, transducer, or DP cell) that measures the pressure difference between two tubes. One tube is attached to a hollow probe that can be positioned at any radial location in the conduit, and the other is attached to the wall of the conduit in the same axial plane as the end of the probe. The local velocity of the streamline that impinges on the end of the probe is v(r). The fluid element that impacts the open end of the probe must come to rest at that point, because there is no flow through the probe or the DP cell this is known as the stagnation point. The Bernoulli equation can be applied to the fluid streamline that impacts the probe tip ... 

From Section 7.5.4, Volume 3, for a manometer in which fractional damping is four times that predicted by Poiseuille s equation ... 

The flow of fluids is most commonly measured using head flowmeters. The operation of these flowmeters is based on the Bernoulli equation. A constriction in the flow path is used to increase the flow velocity. This is accompanied by a decrease in pressure head and since the resultant pressure drop is a function of the flow rate of fluid, the latter can be evaluated. The flowmeters for closed conduits can be used for both gases and liquids. The flowmeters for open conduits can only be used for liquids. Head flowmeters include orifice and venturi meters, flow nozzles, Pitot tubes and weirs. They consist of a primary element which causes the pressure or head loss and a secondary element which measures it. The primary element does not contain any moving parts. The most common secondary elements for closed conduit flowmeters are U-tube manometers and differential pressure transducers. 

It is essential to appreciate that the pressure difference measured by. a manometer automatically eliminates the static head difference. This is shown in Figure 8.1(b). The static head pg(zi — z2) in the pipe is exactly balanced by the extra static head above the right hand limb of the manometer. Consequently, if Ah is calculated from Azm using equation 8.4, no further correction for the static head should be made. 

Equation 8.17 gives the point velocity v in terms of the difference in level between the two arms of the manometer Azmi the density of the flowing fluid p, the density of the immiscible manometer liquid pm and the gravitational acceleration g. 

Points on the desorption isotherm are obtained by first adsorbing at some relative pressure close to unity. Then, stopcock 4 is closed and the calibrated volumes and manifold are evacuated. Stopcock 4 is opened which permits desorption to occur. The equilibrium pressure is obtained from the manometer and the volumes desorbed are calculated using equation (14.8) except that P is now the pressure after desorption. 

When highly accurate Pq measurements are necessary, a platinum resistance thermometer can be placed in the bath adjacent to the sample cell. Equation (13.1) can then be used to calculate the nitrogen vapor pressure. Alternatively and perhaps preferably, the vapor pressure can be measured directly by condensing nitrogen in a cell contained in the coolant and connected directly to a manometer or sensitive pressure gauge. 

Each orifice constructed in the above manner was calibrated with a rotameter and manometer. The set-up used for orifice calibration is shown in Figure 3. The uncorrected flow readings were obtained from the rotameter calibration curve. The corrected flows were then calculated using the following equation ... 

Teorell studied the system (6.1-1)—(6.1.5) graphically, by the isocline method, and also numerically. He recovered most of the features observed experimentally. This study was further elaborated by several investigators. Thus, Kobatake and Fujita [5], [6] criticized the original model for invoking the ad hoc equation (6.1.5). These authors assumed instead instantaneous relaxation of the resistance to its stationary value while preserving the overall order of the relevant ordinary differential equation (ODE) system by including consideration of the mechanical inertia of the liquid column in the manometer tube. 

If the temperature of the sample (Ts in K) and of the vapor space in the manometer (Tm in K) are different, then correct water activity using the following equation (Rizvi, 1995). 

Consider a step change in pressure differential applied to a manometer (Fig. 7.21). From equation 7.52 the displacement of the column of liquid is given by ... 

From equation 7.81, the transformed form of the response of the level of the column of liquid in the manometer tube to a unit step change in pressure differential is ... 

Number of independent equations Number of degrees of freedom Number of independent variables Number of zeros of function Pressure upstream of nozzle in flapper/nozzle system Pressures applied to limbs of manometer tube or pressures downstream and upstream of orifice plate Distillation column pressure Pressure in feedback bellows of pneumatic controller Frictional drag per unit cross-sectional area of manometer tube... 

The clean, dry, calibrated bulb is attached to a vacuum system, such as site A on the line illustrated in Fig. 5.2, and evacuated to 10-3 torr or lower. Gas may be admitted to the calibrated bulb by isolating the working manifold from vacuum and opening the stopcocks and valves leading to one of the upper gas storage bulbs, C, while the pressure is monitored by manometer D. When the desired pressure is reached, the valve on the storage bulb is turned off, pressure and temperature measurements are made, and then the stopcock on the calibrated bulb is turned off. At this stage we know the pressure, temperature, and volume of the gas in the calibrated bulb, which permits the calculation of the number of moles via the ideal gas equation. 

We can now determine the total moles of gas in a sample by condensing this gas into trap E, measuring the pressure on manometer D, measuring the room temperature, and using the volume-versus-pressure plot to determine the volume of the gas so that the ideal gas equation can be applied. 

The subscript "true denotes the corrected pressure values while MappM refers to the observed values after they have been corrected for contributions from the background flux. The function a(T) is a small correction that is obtained by intercomparing each cell manometer with a well-calibrated manometer that is always evacuated and maintained at a fixed temperature. Differentiating equation 2 gives... 

Notice that one end of the U-tube is connected to the smaller diameter section of the Venturi tube, whereas the other end is connected to one of the larger diameter sections. If the fluid is flowing from left to right, according to the Bernoulli equation, v2 > vl and P2 < Pr The fact that the pressure is lower in the narrow part of the tube is the primary scientific basis for the operation of an aspirator. The fluid levels in the manometer will reflect the pressure difference between Pt and Pr A measure of the height difference Ah and a knowledge of the density of the fluid in the manometer will yield the pressure difference between Pj and P2 using ... 

For the maximum flow of 4 1/s, G = 4 kg/s. The largest practicable height of a water manometer will be taken as 1 m and equation 6.21 is then used to calculate the orifice area A0. If the coefficient of discharge CD is taken as 0.6, then ... 

Suppose we have an evacuated vessel (Figure 32.3(a)) into which we introduce a liquid which is to form a volatile component A (Figure 32.3(b)) as part of a liquid mixture. It will be observed that the pure A exerts a vapour pressure, P, when confined alone in the containing vessel. This can also be referred to as the saturated vapour pressure , Psat in the sense that it equates to the saturation of the vapour space with only the pure liquid. The vapour pressure itself can be recorded by means of a manometer attached to the vapour space. This vapour pressure emerges because solvent molecules are volatile and leave the surface of the liquid, so creating a pressure in the vapour (or gas) above the liquid (which is the solvent). 

This equation is a powerful tool for the description of the adsorption data in microporous material. In Figure 6.11, the Dubinin plot of the adsorption isotherm in the range 0.001 < P/P0 < 0.03, describing the adsorption of NH3 at 300 K in the natural clinoptilolite sample HC is shown (see Table 4.1) [25], The adsorption data reported in Figure 6.11 were determined volumetrically in a home-made Pyrex glass vacuum system, consisting of a sample holder, a dead volume, a dose volume, a U-tube manometer, and a thermostat [25,31], It is evident that, in the present case, the experimental data is accurately fitted by Equation 6.20. 

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