University Taster

Chemical Engineering – University Taster

  1. 1. Prerequisites
  2. 2. Chemistry & Thermodynamics
  3. 3. Fluid Dynamics
  4. 4. Separation Processes
  5. 5. Biochemical Engineering
  6. 6. Sustainable Chemical Engineering
  7. 7. Next Steps
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The equations and theories of fluid dynamics become invaluable when looking for meaningful insights into flows within vessels and pipes. This lesson will introduce one method of measuring flow within a pipe, the venturi flowmeter, as well as introducing an equation to measure pressure drop within a pipeline.

Flow Measurements

The Venturi effect describes the reduction in fluid pressure as a result of a fluid passing through a reduction in pipe diameter, as seen in Lesson 3.3 Laws of Conservation. This effect can be used with a venturimeter that uses the pressure drop over such a constriction which can be used to determine the fluid velocity. This is based on Bernoulli’s principle from Lesson 3.3 Laws of Conservation.

Figure 1 shows a venturimeter, consisting of a U-Tube manometer that measures the pressure difference in the two sections of pipe. According to Bernoulli’s principle, energy must be conserved in this process, with the law of mass conservation demanding the fluid velocity must increase with a simultaneous pressure drop.

Figure 1. Constricted pipe section on an incline with a venturimeter fitted to measure the differential pressure between sections. \(P\) (Pa) refers to the pressure at a given point, \(Z\) (m) is the distance of a point from the datum line, and \(\Delta Z_m\) (m) refers to the height above the datum in which the mercury has risen.

According to Pascal’s Law, pressure along the datum line (the dashed line in Figure 1) is constant, giving the first equation shown in Equation 1. Similarly, Bernoulli’s equation can be arranged to also give the pressure drop \((P_1 - P_2)\), as shown in Equation 1, thus equating the two to give the third line in Equation 1. Finally, this can be rearranged to give the velocity difference as the subject.

\(P_1 - P_2 = (Z_2 - \Delta Z_m - Z_1) \rho_wg + \Delta Z_m \rho_m g\)

\(P_1 - P_2 = \rho (\frac{u_2^2 - u_1^2}{2}) + \rho g (Z_2 - Z_1)\)

\((Z_2 - \Delta Z_m - Z_1) \rho_wg + \Delta Z_m \rho_m g = \rho (\frac{u_2^2 - u_1^2}{2}) + \rho g (Z_2 - Z_1)\)

\(u_2^2 - u_1^2 = 2 \Delta Z_m g (\frac{\rho_m}{\rho} - 1)\)

Equation 1. Relating the pressure drop according to Pascal’s Law and Bernoulli’s equation, giving us the velocity difference between two points as a function of the fluid density, as well as the height of fluid within the manometer. \(P\) is the pressure (atm), \(Z\) is the height of a point relative to the datum (m), \(\Delta Z_m\) is the height of fluid in the manometer from the datum (m), \(\rho\) is the fluid density (kg/m3), \(u\) is the fluid velocity (m/s) and \(g\) is the acceleration due to gravity (m/s2).

The volumetric throughput of the equation can be found using the law of conservation of mass, where the volumetric throughput is assumed to remain constant, according to the equation in Equation 2.

\(u_1A_1 = u_2A_2 = Q\)

Equation 2. Mass continuity equation, where \(u\) is the fluid velocity (m/s), \(A\) is the cross-sectional flow area (m2), and \(Q\) is the volumetric flow rate (m3/s).

Thus, the equation in Equation 3 can be used to calculate the volumetric throughput within the venturi flowmeter.

\(Q = A_2 (\frac{2 \Delta Z_m g (\frac{\rho_m}{\rho} - 1)}{1 - (\frac{A_2}{A_1})^2})^{1/2}\)

Equation 3. Equation for the volumetric flow rate, \(Q\) (m3/s) for a venturi flowmeter, \(A\) is the cross-sectional area (m2), \(\Delta Z_m\) is the height of fluid in the manometer from the datum (m), \(\rho\) is the fluid density (kg/m3) and \(g\) is the acceleration due to gravity (m/s2).

The mass flow rate and fluid velocity can then be determined using the equations in Equation 4.

\(\dot{m} = \rho A_2 (\frac{2 \Delta Z_m g (\frac{\rho_m}{\rho} - 1)}{1 - (\frac{A_2}{A_1})^2})^{1/2}\)

\(u = (\frac{2 \Delta Z_m g (\frac{\rho_m}{\rho} - 1)}{1 - (\frac{A_2}{A_1})^2})^{1/2}\)

Equation 4. Equations for mass flow rate (kg/s) and velocity (m/s) in a venturi flowmeter, where \(A\) is the cross-sectional area (m2), \(\Delta Z_m\) is the height of fluid in the manometer from the datum (m), \(\rho\) is the fluid density (kg/m3) and \(g\) is the acceleration due to gravity (m/s2).

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