/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 29 Engine oil flows at a rate of \(... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 29

Engine oil flows at a rate of \(1 \mathrm{~kg} / \mathrm{s}\) through a \(5-\mathrm{mm}-\) diameter straight tube. The oil has an inlet temperature of \(45^{\circ} \mathrm{C}\) and it is desired to heat the oil to a mean temperature of \(80^{\circ} \mathrm{C}\) at the exit of the tube. The surface of the tube is maintained at \(150^{\circ} \mathrm{C}\). Determine the required length of the tube. Hint: Calculate the Reynolds numbers at the entrance and exit of the tube before proceeding with your analysis.

Short Answer

Expert verified
The required length of the tube, \(L\), can be calculated using the equation: \(L = \frac{\dot{m} c_p \Delta T}{h \pi D \Delta T_{lm}}\) After calculating the Reynolds numbers at the entrance and exit, determining the heat transfer coefficient, \(h\), and finding the log mean temperature difference, \(\Delta T_{lm}\), we can plug in the given parameters and the specific heat capacity, \(c_p\), of the engine oil to find the tube length.

Step by step solution

01

1 - Identifying given parameters

From the exercise, we have the following given parameters: - Mass flow rate of engine oil, \(\dot{m} = 1 \mathrm{~kg/s}\) - Diameter of the tube, \(D = 5 \mathrm{~mm}\) - Inlet temperature of engine oil, \(T_{in} = 45^{\circ} \mathrm{C}\) - Desired mean exit temperature, \(T_{out}^{mean} = 80^{\circ} \mathrm{C}\) - Surface temperature of the tube, \(T_s = 150^{\circ} \mathrm{C}\)
02

2 - Calculate the Reynolds numbers at the entrance and exit of the tube

To calculate the Reynolds numbers, we need to know the velocity of the oil in the tube. To find this, we can use the equation \(\dot{m} = \rho A v\), where \(\dot{m}\) is the mass flow rate, \(\rho\) is the density of the oil, \(A\) is the cross-sectional area of the tube, and \(v\) is the velocity of the oil. Rearranging, we get: \(v = \frac{\dot{m}}{\rho A}\) For simplicity, let's assume that the density and velocity don't change significantly with temperature. So, we can find the Reynolds numbers at entrance and exit using the entrance temperature: \(Re = \frac{\rho v D}{\mu}\) Where \(\rho\) is the density of the oil, \(v\) is the velocity, \(D\) is the diameter of the tube, and \(\mu\) is the dynamic viscosity. Calculate the density, velocity, and Reynolds numbers using the appropriate values for the engine oil and given parameters.
03

3 - Determine the heat transfer coefficient

After calculating the Reynolds numbers at the entrance and exit, we need to find the heat transfer coefficient, \(h\), of the engine oil in the tube. One of the correlations used to estimate the heat transfer coefficient is the Dittus-Boelter equation, given by: \(Nu = \frac{h D}{k} = C Re^m Pr^n\) Where \(Nu\) is the Nusselt number, \(Re\) is the Reynolds number, \(Pr\) is the Prandtl number, \(C\), \(m\), and \(n\) are constants. For this exercise, we'll assume: - The constants \(C\), \(m\), and \(n\) are appropriate for the engine oil and flow conditions. - The Reynolds number is constant within the range calculated in Step 2. With these assumptions, determine the heat transfer coefficient, \(h\), for the engine oil in the tube.
04

4 - Calculate the required tube length

Now that we have the heat transfer coefficient \(h\), we can calculate the required tube length, \(L\), to achieve the desired temperature change. The rate of heat transfer between the tube and the oil is given by: \(\dot{Q} = h A_s \Delta T_{lm}\) Where \(A_s\) is the surface area of the tube, and \(\Delta T_{lm}\) is the log mean temperature difference between the oil and the tube surface, given by: \(\Delta T_{lm} = \frac{(T_s - T_{out}) - (T_s - T_{in})}{\ln((T_s - T_{out}) / (T_s - T_{in}))}\) Moreover, the rate of heat transfer can also be expressed as: \(\dot{Q} = \dot{m} c_p \Delta T\) Where \(c_p\) is the specific heat capacity of the oil, and \(\Delta T = T_{out} - T_{in}\). Equating the two expressions for \(\dot{Q}\) and solving for the tube length, \(L\), we get: \(L = \frac{\dot{m} c_p \Delta T}{h \pi D \Delta T_{lm}}\) Calculate the required tube length, \(L\), using the parameters from Steps 1-3, and the expression above.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reynolds Number
The Reynolds Number is a vital concept in fluid dynamics that helps us determine the type of flow within a pipe or around an obstacle. It's calculated using the formula:\[Re = \frac{\rho v D}{\mu}\]where:
  • \( \rho \) - Density of the fluid
  • \( v \) - Velocity of the fluid
  • \( D \) - Diameter of the pipe
  • \( \mu \) - Dynamic viscosity of the fluid
The Reynolds Number tells us whether the flow is laminar or turbulent. A low Reynolds Number (typically below 2000) indicates a laminar flow, where the fluid flows in parallel layers with no disruption between them.
Conversely, a high Reynolds Number (above 4000) indicates turbulent flow, characterized by chaotic fluid movement.
In between these two ranges lies the transitional flow region. Understanding this concept helps engineers and scientists design systems to optimize heat transfer conditions.
In this exercise, calculating the Reynolds Number at the entrance of the tube indicates how the oil behaves when it begins its journey through the pipe.
Understanding this behavior is crucial for determining the appropriate equations and methods to describe and facilitate heat transfer efficiently.
Nusselt Number
The Nusselt Number is a dimensionless value that characterizes the heat transfer at the fluid-surface boundary. It's used to determine the convective heat transfer coefficient, \( h \), and can be expressed by the equation:\[ Nu = \frac{h D}{k} \]where:
  • \( h \) - Convective heat transfer coefficient
  • \( D \) - Characteristic length (often diameter for pipes)
  • \( k \) - Thermal conductivity of the fluid
An important understanding of the Nusselt Number is that it helps quantify how efficient a surface is in transferring heat to or from a fluid in motion.
A higher Nusselt Number indicates a more effective convective heat transfer compared to conductive heat transfer alone.
This number is closely related to both the Reynolds Number and the Prandtl number.
The Nusselt Number changes depending on whether the system is in a laminar, turbulent, or transitional state.
This means that knowing the Reynolds Number can help us determine an appropriate Nusselt Number, which in turn reflects the effectiveness of the heat transfer process in the system.
For this exercise, evaluating the Nusselt Number allows students to connect the fluid flow characteristics to the rate of heat transfer, moving one step closer to figuring out the tube's required length.
Dittus-Boelter Equation
The Dittus-Boelter equation is a practical correlation for estimating the heat transfer coefficient in turbulent flow through pipes. It offers a simplified way to calculate the Nusselt Number under the assumption that the flow regime and fluid properties fit the criteria defined by this equation. It is given as:\[ Nu = C \times Re^m \times Pr^n \]where:
  • \( Nu \) - Nusselt Number
  • \( C \), \( m \), \( n \) - Empirically determined constants for specific conditions
  • \( Re \) - Reynolds Number
  • \( Pr \) - Prandtl Number, a measure of momentum diffusivity over thermal diffusivity
This equation assumes turbulent flow conditions and is typically applied where more detailed calculations are unnecessary.
For many engineering applications, the constants are set as \( C = 0.023 \), \( m = 0.8 \), and \( n = 0.3 \), when the fluid is being heated.
With these constants, the Dittus-Boelter equation can provide a practical approximation for the Nusselt Number, facilitating easier computation of the heat transfer coefficient.
In the context of this exercise, using this equation allows us to derive the heat transfer coefficient between the oil and the pipe walls.
Once the heat transfer coefficient is known, it's possible to calculate the length of the tube needed to reach the desired oil exit temperature.

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Most popular questions from this chapter

Consider a thin-walled, metallic tube of length \(L=1 \mathrm{~m}\) and inside diameter \(D_{i}=3 \mathrm{~mm}\). Water enters the tube at \(\dot{m}=0.015 \mathrm{~kg} / \mathrm{s}\) and \(T_{m, i}=97^{\circ} \mathrm{C}\). (a) What is the outlet temperature of the water if the tube surface temperature is maintained at \(27^{\circ} \mathrm{C}\) ? (b) If a \(0.5-\mathrm{mm}\)-thick layer of insulation of \(k=0.05\) \(\mathrm{W} / \mathrm{m} \cdot \mathrm{K}\) is applied to the tube and its outer surface is maintained at \(27^{\circ} \mathrm{C}\), what is the outlet temperature of the water? (c) If the outer surface of the insulation is no longer maintained at \(27^{\circ} \mathrm{C}\) but is allowed to exchange heat by free convection with ambient air at \(27^{\circ} \mathrm{C}\), what is the outlet temperature of the water? The free convection heat transfer coefficient is \(5 \mathrm{~W} / \mathrm{m}^{2}+\mathrm{K}\).

Consider a circular tube of diameter \(D\) and length \(L\), with a mass flow rate of \(\dot{m}\). (a) For constant heat flux conditions, derive an expression for the ratio of the temperature difference between the tube wall at the tube exit and the inlet temperature, \(T_{s}(x=L)-T_{m, i}\), to the total heat transfer rate to the fluid \(q\). Express your result in terms of \(\dot{m}, L\), the local Nusselt number at the tube exit \(N u_{D}(x=L)\), and relevant fluid properties. (b) Repeat part (a) for constant surface temperature conditions. Express your result in terms of \(\dot{m}, L\), the average Nusselt number from the tube inlet to the tube exit \(\overline{N u}_{D}\), and relevant fluid properties.

In Chapter 1, it was stated that for incompressible liquids, flow work could usually be neglected in the steady-flow energy equation (Equation 1.12d). In the trans-Alaska pipeline, the high viscosity of the oil and long distances cause significant pressure drops, and it is reasonable to question whether flow work would be significant. Consider an \(L=100 \mathrm{~km}\) length of pipe of diameter \(D=1.2 \mathrm{~m}\), with oil flow rate \(\dot{m}=500 \mathrm{~kg} / \mathrm{s}\). The oil properties are \(\rho=900 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=2000 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \mu=0.765\) \(\mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}\). Calculate the pressure drop, the flow work, and the temperature rise caused by the flow work.

In the final stages of production, a pharmaceutical is sterilized by heating it from 25 to \(75^{\circ} \mathrm{C}\) as it moves at \(0.2 \mathrm{~m} / \mathrm{s}\) through a straight thin-walled stainless steel tube of \(12.7=\mathrm{mm}\) diameter. A uniform heat flux is maintained by an electric resistance heater wrapped around the outer surface of the tube. If the tube is \(10 \mathrm{~m}\) long, what is the required heat flux? If fluid enters the tube with a fully developed velocity profile and a uniform temperature profile, what is the surface temperature at the tube exit and at a distance of \(0.5 \mathrm{~m}\) from the entrance? Fluid properties may be approximated as \(\rho=\) \(1000 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=4000 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, m=2 \times 10^{-3} \mathrm{~kg} / \mathrm{s} \cdot \mathrm{m}\), \(k=0.8 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), and \(P r=10\).

Consider pressurized liquid water flowing at \(\dot{m}=0.1 \mathrm{~kg} / \mathrm{s}\) in a circular tube of diameter \(D=0.1 \mathrm{~m}\) and length \(L=6 \mathrm{~m}\). (a) If the water enters at \(T_{m, i}=500 \mathrm{~K}\) and the surface temperature of the tube is \(T_{s}=510 \mathrm{~K}\), determine the water outlet temperature \(T_{\text {m,o. }}\). (b) If the water enters at \(T_{m, i}=300 \mathrm{~K}\) and the surface temperature of the tube is \(T_{s}=310 \mathrm{~K}\), determine the water outlet temperature \(T_{\text {m, } \sigma}\). (c) If the water enters at \(T_{m, i}=300 \mathrm{~K}\) and the surface temperature of the tube is \(T_{s}=647 \mathrm{~K}\), discuss whether the flow is laminar or turbulent.
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