Question

Water at \(15^{\circ} \mathrm{C}\) is flowing through a \(200-\mathrm{m}\)-long standard 1-in Schedule 40 cast iron pipe with a mass flow rate of \(0.5 \mathrm{~kg} / \mathrm{s}\). If accuracy is an important issue, use the appropriate equation to determine \((a)\) the pressure loss and \((b)\) the pumping power required to overcome the pressure loss. Assume flow is fully developed. Is this a good assumption?

Short answer

In this question, we were asked to find the pressure loss and pumping power required for water flowing through a cast iron pipe and to determine if the flow is fully developed. a) The pressure loss for water flowing through the pipe is calculated to be 4154.8 Pa. b) The pumping power required to overcome the pressure loss is found to be 2077.4 W. c) The assumption of fully developed flow is considered valid since the entrance length (27.9 m) is much smaller than the pipe length (200 m).

Step-by-step solution

We have the following given data: 1. Water at \(15^{\circ} \mathrm{C}\). 2. Pipe length, \(L = 200~\mathrm{m}\). 3. Pipe type: 1-in Schedule 40 cast iron pipe. 4. Mass flow rate, \(m = 0.5~\mathrm{kg/s}\). #Step 2: Calculate Reynolds Number and Determine the Flow Regime#

First, we need to know if the flow is laminar or turbulent. We can determine this by calculating the Reynolds number, \(Re\), using the formula: \(Re = \frac{D V \rho}{\mu}\) Where: - \(D\) is the pipe diameter, - \(V\) is the average fluid velocity, - \(\rho\) is the fluid density, and - \(\mu\) is the fluid dynamic viscosity. Note that we'll need to convert the 1-in Schedule 40 pipe size to meters, and also look up the values for \(\rho\) and \(\mu\) for water at \(15^{\circ} \mathrm{C}\). We'll also find the pipe diameter from standard 1-in Schedule 40 pipe dimensions and calculate average fluid velocity from mass flow rate. #Step 3: Calculate Pipe Diameter, Cross-sectional Area, and Average Fluid Velocity#

A 1-in Schedule 40 pipe has an inner diameter (ID) of approximately 1.049 in. First, let’s convert the pipe diameter to meters: \(D = 1.049~\mathrm{in} \times 0.0254~\mathrm{\frac{m}{in}} = 0.0266~\mathrm{m}\) Now calculate the cross-sectional area of the pipe, \(A\): \(A = \pi \frac{D^2}{4} =\pi \frac{(0.0266)^2}{4} = 5.568\times10^{-4}~\mathrm{m^2}\) And the average fluid velocity, \(V\): \(V = \frac{m}{\rho A} = \frac{0.5~\mathrm{kg/s}}{1000~\mathrm{\frac{kg}{m^3}} \cdot 5.568\times10^{-4}~\mathrm{m^2}} = 0.897~\mathrm{m/s}\) #Step 4: Calculate Reynolds Number#

Now, we'll look up the values of \(\rho\) and \(\mu\) for water at \(15^{\circ} \mathrm{C}\) from standard property tables: \(\rho = 999~\mathrm{\frac{kg}{m^{3}}}\) and \(\mu = 1.14\times10^{-3}~\mathrm{N\cdot s/m^2}\) Plug these values into the Reynolds number formula and calculate: \(Re = \frac{D V \rho}{\mu} = \frac{0.0266~\mathrm{m} \cdot 0.897~\mathrm{m/s} \cdot 999~\mathrm{\frac{kg}{m^{3}}}}{1.14\times10^{-3}~\mathrm{N\cdot s/m^2}} = 20890\) Since \(Re < 20000\), the flow is considered laminar. #Step 5: Calculate Friction Factor For Laminar Flow#

For laminar flow, the friction factor is given by: \(f = \frac{16}{Re} = \frac{16}{20890} = 0.000766\) #Step 6: Calculate Pressure Loss using Darcy-Weisbach Equation#

Now that we have the friction factor, we can use the Darcy-Weisbach equation to find the pressure loss: \(\Delta P = f \frac{L}{D} \frac{1}{2} \rho V^2 = 0.000766 \cdot \frac{200}{0.0266} \cdot \frac{1}{2} \cdot 999 \cdot (0.897)^2 = 4154.8~\mathrm{Pa}\) a) The pressure loss is: \(\Delta P = 4154.8~\mathrm{Pa}\). #Step 7: Calculate Pumping Power Required to Overcome Pressure Loss#

The pumping power can be found using the following formula: \(P_t = \dot{m}\Delta P=0.5 \cdot 4154.8=2077.4 \mathrm{W}\) b) The pumping power required is: \(P_t=2077.4 \mathrm{W}\). #Step 8: Check if Fully Developed Flow is a Good Assumption#

The flow is considered fully developed if the entrance length, \(L_e\), is much smaller than the pipe length, \(L\). The entrance length is given by the formula, \(L_e = 0.05 Re D\). Calculate the entrance length: \(L_e = 0.05 \cdot 20890 \cdot 0.0266 = 27.9~\mathrm{m}\) Since \(L \gg L_e\), the assumption of fully developed flow is a good assumption.

Key concepts

Darcy-Weisbach Equation

Understanding the pressure loss in fluid systems is critical for engineers when designing pipelines and pumping machinery. The Darcy-Weisbach equation is a well-established formula that engineers use to estimate this loss. The equation is:
\[\begin{equation}\Delta P = f \frac{L}{D} \frac{1}{2} \rho V^2\end{equation}\]
where \( \Delta P \) is the pressure drop, \( f \) is the friction factor, \( L \) is the length of the pipe, \( D \) is the hydraulic diameter of the pipe, \( \rho \) (rho) is the density of the fluid, and \( V \) is the flow velocity. The friction factor depends on the flow regime (laminar or turbulent) and must be computed accordingly. For laminar flow, it can be calculated using the equation \( f = \frac{16}{Re} \), where \( Re \) refers to the Reynolds number.

Reynolds Number

When it comes to fluid dynamics, the Reynolds number is a dimensionless quantity that engineers use to predict flow patterns in different fluid flow situations. It's expressed as:
\[\begin{equation}Re = \frac{D V \rho}{\mu}\end{equation}\]
where \( D \) is the pipe's inner diameter, \( V \) is the mean velocity of the fluid, \( \rho \) is the fluid's density, and \( \mu \) is the dynamic viscosity. The value of the Reynolds number indicates whether the flow is laminar (smooth and orderly) or turbulent (chaotic). Flows with a Reynolds number less than 2,000 are typically laminar, those above 4,000 are turbulent, and in between these values, the flow can be transitional.

Laminar Flow

In fluid mechanics, laminar flow is characterized by smooth, parallel layers of fluid that move without disruption between them. It occurs at lower velocities and is governed by viscous forces rather than inertial forces. For flow within pipes, a Reynolds number under 2,000 usually suggests laminar flow. In such a flow, the resistance to motion and the pressure loss can be accurately predicted, which is helpful for designing systems with predictable performance. In applications where accurate metering of fluids is necessary, maintaining laminar flow can be beneficial. For our problem, since the Reynolds number is calculated to be less than 20,000, the flow is considered to be laminar and therefore the friction factor can be determined precisely, providing an accurate estimate of pressure loss.

Pumping Power

The pumping power is the amount of energy required by a pump to move a specific volume of fluid through a pipeline against any resistive forces like friction, which can result in pressure loss. The pumping power can be calculated using the formula:
\[\begin{equation}P_t = \dot{m}\Delta P\end{equation}\]
where \( \dot{m} \) is the mass flow rate of the fluid and \( \Delta P \) is the pressure drop over the length of the system. It's important to calculate the pumping power accurately, as it directly translates into the operational cost of the system. Designing a system with minimal pressure loss and hence lesser pumping power requirement can significantly reduce the energy consumption and thus the operational costs.