Question

A 10 -m-long and 10 -mm-inner-diameter pipe made of commercial steel is used to heat a liquid in an industrial process. The liquid enters the pipe with \(T_{i}=25^{\circ} \mathrm{C}, V=0.8 \mathrm{~m} / \mathrm{s}\). A uniform heat flux is maintained by an electric resistance heater wrapped around the outer surface of the pipe, so that the fluid exits at \(75^{\circ} \mathrm{C}\). Assuming fully developed flow and taking the average fluid properties to be \(\rho=1000 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=\) \(4000 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \mu=2 \times 10^{-3} \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s}, k=0.48 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), and \(\operatorname{Pr}=10\), determine: (a) The required surface heat flux \(\dot{q}_{s}\), produced by the heater (b) The surface temperature at the exit, \(T_{s}\) (c) The pressure loss through the pipe and the minimum power required to overcome the resistance to flow.

Short answer

Question: Determine the required surface heat flux, the surface temperature at the exit, the pressure loss through the pipe, and the minimum power required to overcome the resistance to flow in a 10-meter long and 10-mm-inner diameter pipe with an inlet temperature of 25°C and an outlet temperature of 75°C. Answer: 1. Calculate the mass flow rate and Reynolds number: Mass flow rate, \(\dot{m}= \rho AV\) Reynolds number, \(\text{Re}= \frac{\rho V d}{\mu}\) 2. Calculate the Nusselt number and heat transfer coefficient: Nusselt number, \(\text{Nu}= 0.023\; \text{Re}^{0.8}\; \operatorname{Pr}^{n}\) Heat transfer coefficient, \(h= \frac{\text{Nu}\cdot k}{d}\) 3. Calculate the required surface heat flux: Required surface heat flux, \(\dot{q}_{s}= \frac{\dot{m}c_{p}(T_{o}-T_{i})}{L}\) 4. Calculate the surface temperature at the exit: Surface temperature at the exit, \(T_{s}= T_{o}+ \frac{\dot{q}_{s}}{h A_{s}}\) 5. Determine the pressure loss through the pipe and the minimum power required to overcome the resistance to flow: Pressure loss, \(\Delta P= f \frac{L \rho V^{2}}{2 d}\) Friction factor, \(f= 0.079\; \text{Re}^{-0.25}\) Minimum power required, \(P_{\text{min}}= \frac{\Delta P \dot{V}}{\rho}\)

Step-by-step solution

Calculate the mass flow rate and Reynolds number

First, let's find the mass flow rate of the liquid (\(\dot{m}\)). The mass flow rate can be calculated using the formula: $$\dot{m}=\rho A V$$ where \(\rho=1000 \mathrm{~kg} / \mathrm{m}^{3}\) is the density, \(A\) is the cross-sectional area, and \(V=0.8 \mathrm{~m} / \mathrm{s}\) is the velocity. The cross-sectional area can be found using the formula: $$A=\pi r^{2}$$ where \(r\) is the pipe radius, which can be found from the inner diameter of the pipe (10mm). Now find the Reynolds number (\(\text{Re}\)): $$\text{Re}=\frac{\rho V d}{\mu}$$ where \(d\) = diameter and \(\mu=2 \times 10^{-3} \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s}\) is the dynamic viscosity.

Calculate the Nusselt number and heat transfer coefficient

For fully developed flow in a circular pipe, we can use Dittus-Boelter correlation to find the Nusselt number: $$\text{Nu}=0.023\; \text{Re}^{0.8}\; \operatorname{Pr}^{n}$$ where \(\operatorname{Pr}=10\) is the Prandtl number, and \(n\) = 0.3 for heating case. The heat transfer coefficient, \(h\), can be calculated from the Nusselt number: $$h=\frac{\text{Nu}\cdot k}{d}$$ where \(k=0.48 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) is the thermal conductivity.

Calculate the required surface heat flux

According to the conservation of energy principle, the energy added to the fluid by the heater is equal to the energy increase of the fluid, which can be represented as: $$\dot{q}_{s}=h A_{s}(T_{s} - T_{o})$$ and using the mass flow rate, we have: $$\dot{m}c_{p}(T_{o}-T_{i})=\dot{q}_{s}L$$ Solving for \(\dot{q}_{s}\), we get: $$\dot{q}_{s}=\frac{\dot{m}c_{p}(T_{o}-T_{i})}{L}$$ where \(c_{p}=4000 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\) is the specific heat capacity, \(T_{i}=25^{\circ} \mathrm{C}\), and \(T_{o}=75^{\circ} \mathrm{C}\) are the inlet and outlet temperatures, and \(L=10\;\mathrm{m}\) is the pipe length.

Calculate the surface temperature at the exit

From the heat transfer equation in step 3, find the surface temperature at the exit (\(T_{s}\)) by rearranging the equation: $$T_{s}=T_{o}+\frac{\dot{q}_{s}}{h A_{s}}$$

Determine the pressure loss through the pipe and the minimum power required to overcome the resistance to flow

To calculate the pressure loss, we can use the Darcy-Weisbach equation for fully developed flow in a pipe: $$\Delta P=f \frac{L \rho V^{2}}{2 d}$$ We can estimate the friction factor \(f\) using the Blasius correlation for turbulent flow: $$f=0.079\; \text{Re}^{-0.25}$$ From the pressure loss, the minimum power required to overcome the resistance to flow can be calculated as: $$P_{\text{min}}=\frac{\Delta P \dot{V}}{\rho}$$ where \(\dot{V}=AV\) is the volume flow rate. Calculate the values using the found formulas and parameters.

Key concepts

Mass Flow Rate

The mass flow rate in a pipe system is a crucial measure in many engineering applications, determining how much mass is transported through the pipe per unit of time. It is represented by \(\dot{m}\) and is calculated using the formula:\[ \dot{m} = \rho A V \]where:

• \(\rho\) is the fluid density
• \(A\) is the cross-sectional area of the pipe
• \(V\) is the fluid velocity

To find the cross-sectional area \(A\), you use the diameter of the pipe (in this case, provided in millimeters). Conversion to meters, as seen in this example, is crucial:\[ A = \pi r^2 = \pi \left(\frac{d}{2}\right)^2 \]This calculation provides insight into how the mass of fluid flows through the pipe, influencing the heating and pressure conditions within the system.

Nusselt Number

The Nusselt number plays a significant role in understanding heat transfer within pipe systems. It is a dimensionless number that characterizes how effectively heat is transferred in fluid flow, comparing convective to conductive heat transfer.For calculations in a fully developed flow, we use the Dittus-Boelter equation:\[ \text{Nu} = 0.023 \text{Re}^{0.8} \text{Pr}^{n} \]Here, the values are:

• \(\text{Re}\) for Reynolds number, indicative of flow regime
• \(\text{Pr}\), the Prandtl number, is a material property
• \(n\), a factor (0.3 for heating case)

The Nusselt number helps calculate the convective heat transfer coefficient \(h\) by:\[ h = \frac{\text{Nu} \cdot k}{d} \]where \(k\) is the thermal conductivity of the fluid, and \(d\) is the pipe diameter. This coefficient \(h\) tells us how efficiently the heat energy from the heater is transferred to the liquid, allowing us to further understand and calculate the energy dynamics in the system.

Pressure Loss

Pressure loss refers to the reduction in pressure as fluid flows through the pipe due to frictional forces. It's essential to determine this loss since it directly affects the pumping requirements of the system.Using the Darcy-Weisbach equation, we can calculate pressure loss \(\Delta P\) as follows:\[ \Delta P = f \frac{L \rho V^2}{2 d} \]Key components of this formula:

• \(f\), the friction factor, accounts for the effects of pipe roughness and fluid turbulence.
• \(L\), the pipe length, influences the total resistance encountered by the fluid.
• \(\rho\), \(V\), and \(d\) are previously defined parameters.

Estimating \(f\) typically involves the Blasius correlation for turbulent flow:\[ f = 0.079 \text{Re}^{-0.25} \]Understanding pressure loss helps design piping systems to ensure efficient operation while minimizing energy expenditure.

Darcy-Weisbach Equation

The Darcy-Weisbach equation is fundamental in fluid dynamics for evaluating pressure drops in pipe systems. It illustrates how various factors like fluid velocity, pipe diameter, and flow length influence pressure changes.The equation is:\[ \Delta P = f \frac{L \rho V^2}{2 d} \],connecting:

• Pressure drop \(\Delta P\)
• Friction factor \(f\)
• Parameters \(L\), \(\rho\), \(V\), and \(d\)

The friction factor \(f\), central to this equation, requires proper estimation based on the flow regime, often guided by empirical correlations like Blasius for turbulent conditions.The Darcy-Weisbach equation is indispensable for understanding how energy is dissipated due to friction, enabling engineers to design efficient systems that balance pressure requirements against other operational constraints, such as energy input and desired fluid velocity.