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Chapter 8: Problem 79

Fluid enters a thin-walled tube of \(5-\mathrm{mm}\) diameter and \(2-\mathrm{m}\) length with a flow rate of \(0.04 \mathrm{~kg} / \mathrm{s}\) and a temperature of \(T_{m, i}=85^{\circ} \mathrm{C}\). The tube surface is maintained at a temperature of \(T_{s}=25^{\circ} \mathrm{C}\), and for this operating condition, the outlet temperature is \(T_{m, o}=31.1^{\circ} \mathrm{C}\). What is the outlet temperature if the flow rate is doubled? Fully developed, turbulent flow may be assumed to exist in both cases, and the fluid properties may be assumed to be independent of temperature.

Short Answer

Expert verified
The new outlet temperature when the flow rate is doubled can be found by following these steps: 1. Calculate the doubled mass flow rate: \(m'_{\text{doubled}} = 0.08 \, \mathrm{kg/s}\) 2. Calculate the Reynolds number for both flow rates and ensure turbulent flow. 3. Find the heat transfer coefficients for both cases using the Dittus-Boelter equation. 4. Calculate the heat transfer rate for both cases using the heat transfer equation. 5. Determine the specific heat and temperature differences for both cases using the heat transfer equation. 6. Calculate the new outlet temperature with doubled flow rate: \(T_{m, o, \text{doubled}} = T_{m, i} + \Delta T_{\text{doubled}}\) By completing these steps, the new outlet temperature can be determined when the flow rate is doubled.

Step by step solution

01

Calculate mass flow rates

Find the mass flow rate for the doubled case: \(m'_{\text{doubled}} = 2 \times m' = 2 \times 0.04 \, \mathrm{kg/s} = 0.08 \, \mathrm{kg/s}\)
02

Calculate the Reynolds number and ensure turbulent flow

Calculate the Reynolds number for both cases, considering the flowrate: \(Re = \frac{m'D}{\mu A}\) Where \(\mu\) is the dynamic viscosity and \(A\) is the cross-sectional area of the tube. Assuming the Reynolds number is well above 4000 in both cases (the threshold for turbulent flow), the condition of fully developed turbulent flow is met.
03

Calculate the heat transfer coefficients using the Dittus-Boelter equation

Use the Dittus-Boelter equation for turbulent flow to find the heat transfer coefficients in both cases: \(Nu = \frac{hD}{k} \approx 0.023 \cdot Re^{\frac{4}{5}} \cdot Pr^{\frac{1}{3}}\) Where \(Nu\) is the Nusselt number, \(h\) is the heat transfer coefficient, \(D\) is the diameter, \(Re\) is the Reynolds number, and \(Pr\) is the Prandtl number. \(k\) is the thermal conductivity. Calculate the heat transfer coefficients for both flow rates.
04

Calculate the heat transfer rate for both cases

Use the heat transfer equation to find the heat transfer rate for both cases: \(q = hA (T_s - T_{m, i})\) Where \(q\) is the heat transfer rate, \(h\) is the heat transfer coefficient, \(A\) is the surface area, and \(T\) values represent the various temperatures. Calculate the heat transfer rate for both flow rates.
05

Calculate specific heat and temperature difference

Use the heat transfer rate to find the temperature difference between the inlet and outlet of the tube for both cases: \(q = m' c_p (T_{m, o} - T_{m, i})\) Where \(c_p\) is the specific heat at constant pressure, and \(T\) values represent the various temperatures. Determine the specific heat and the temperature differences for both cases.
06

Determine the new outlet temperature with doubled flow rate

Find the new outlet temperature when the flow rate is doubled: \(T_{m, o, \text{doubled}} = T_{m, i} + \Delta T_{\text{doubled}}\) Calculate the new outlet temperature with the doubled flow rate. After completing these steps, we will have found the new outlet temperature when the flow rate is doubled.

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Key Concepts

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

Turbulent Flow
Turbulent flow is a type of fluid movement characterized by chaotic changes in pressure and velocity. Unlike laminar flow, where fluid particles move in parallel layers, turbulent flow involves mixing and eddies, leading to increased momentum and energy transfer. This kind of flow is common in nature and industry, appearing in rivers, air currents, and piping systems.

In a tube, turbulent flow occurs when the Reynolds number, a dimensionless number, exceeds a certain threshold (commonly 4000 for round pipes). This means the flow has transitioned from smooth and orderly to chaotic and erratic. Turbulence enhances the interaction between the fluid and the tube wall, resulting in increased heat and mass transfer. Engineers leverage this characteristic to design efficient heat exchangers and cooling systems.

For instance, with turbulent flow, as mentioned in the exercise, the heat transfer is more effective, which is vital when analyzing systems where maintaining or changing temperature is crucial.
Reynolds Number
The Reynolds number (\[Re\]) is a pivotal concept in fluid mechanics, representing the ratio of inertial forces to viscous forces within a fluid flow. It provides insight into whether a flow will be laminar or turbulent. Mathematically, it is expressed as:\[Re = \frac{\rho v D}{\mu}\]where \(\rho\) is the fluid density, \(v\) is the velocity, \(D\) is the characteristic length (typically the diameter in tubular flows), and \(\mu\) is the fluid's dynamic viscosity.

When applying the Reynolds number in practical scenarios, like in the given tube problem, it helps in confirming that the flow is turbulent. By calculating the Reynolds number and confirming it exceeds 4000, you ensure the flow regime is conducive to using certain equations and models, such as the Dittus-Boelter equation, for heat transfer analysis.
  • If \(Re < 2000\), the flow is typically laminar.
  • If \(2000 < Re < 4000\), the flow is in a transition phase.
  • If \(Re > 4000\), the flow is turbulent, as is the case in this exercise.
Understanding the Reynolds number is essential for accurately predicting and analyzing fluid flow behavior and the associated thermal and energy transfer processes.
Dittus-Boelter Equation
The Dittus-Boelter equation is an empirical relation used to estimate the convective heat transfer coefficient for turbulent flow within smooth pipes. It is instrumental in predicting heat transfer rates in engineering applications. The equation is given as:\[Nu = 0.023 \times Re^{0.8} \times Pr^{0.3}\]where \(Nu\) is the Nusselt number, \(Re\) is the Reynolds number, and \(Pr\) is the Prandtl number.

This equation assumes constant fluid properties and fully developed turbulent flow, making it suitable for the conditions described in the exercise when the flowrate is doubled. It provides a means to calculate the heat transfer coefficient (\(h\)) by relating it to the flow's ability to transfer energy through convection. The Nusselt number (\(Nu\)) is a dimensionless measure that correlates thermal conduction to convection and is essential in determining the effectiveness of heat exchangers and similar systems.

By applying the Dittus-Boelter equation, one can understand how increased flow rates, as seen in the doubling scenario, affect heat transfer. An increase in flow rate typically enhances the heat transfer, augmenting the convective heat exchange between the fluid and the tube's surface.

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Most popular questions from this chapter

An experiment is designed to study microscale forced convection. Water at \(T_{\text {mi }}=300 \mathrm{~K}\) is to be heated in a straight, circular glass tube with a \(50-\mu \mathrm{m}\) inner diameter and a wall thickness of \(1 \mathrm{~mm}\). Warm water at \(T_{\infty}=350 \mathrm{~K}, V=2 \mathrm{~m} / \mathrm{s}\) is in cross flow over the exterior tube surface. The experiment is to be designed to cover the operating range \(1 \leq R e_{D} \leq 2000\), where \(R e_{D}\) is the Reynolds number associated with the internal flow. (a) Determine the tube length \(L\) that meets a design requirement that the tube be twice as long as the thermal entrance length associated with the highest Reynolds number of interest. Evaluate water properties at \(305 \mathrm{~K}\). (b) Determine the water outlet temperature, \(T_{\text {mo }}\) that is expected to be associated with \(R e_{D}=2000\). Evaluate the heating water (water in cross flow over the tube) properties at \(330 \mathrm{~K}\). (c) Calculate the pressure drop from the entrance to the exit of the tube for \(R e_{D}=2000\). (d) Based on the calculated flow rate and pressure drop in the tube, estimate the height of a column of water (at \(300 \mathrm{~K}\) ) needed to supply the necessary pressure at the tube entrance and the time needed to collect \(0.1\) liter of water. Discuss how the outlet temperature of the water flowing from the tube, \(T_{m, o}\), might be measured.

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 horizontal, thin-walled circular tube of diameter \(D=0.025 \mathrm{~m}\) submerged in a container of \(n\) octadecane (paraffin), which is used to store thermal energy. As hot water flows through the tube, heat is transferred to the paraffin, converting it from the solid to liquid state at the phase change temperature of \(T_{z}=27.4^{\circ} \mathrm{C}\). The latent heat of fusion and density of paraffin are \(h_{\text {ff }}=244 \mathrm{~kJ} / \mathrm{kg}\) and \(\rho=770 \mathrm{~kg} / \mathrm{m}^{3}\), respectively, and thermophysical properties of the water may be taken as \(c_{p}=4.185 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}, k=0.653 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), \(\mu=467 \times 10^{-6} \mathrm{~kg} / \mathrm{s} \cdot \mathrm{m}\), and \(\operatorname{Pr}=2.99\) (a) Assuming the tube surface to have a uniform temperature corresponding to that of the phase change, determine the water outlet temperature and total heat transfer rate for a water flow rate of \(0.1 \mathrm{~kg} / \mathrm{s}\) and an inlet temperature of \(60^{\circ} \mathrm{C}\). If \(H=W=0.25 \mathrm{~m}\), how long would it take to completely liquefy the paraffin, from an initial state for which all the paraffin is solid and at \(27.4^{\circ} \mathrm{C}\) ? (b) The liquefaction process can be accelerated by increasing the flow rate of the water. Compute and plot the heat rate and outlet temperature as a function of flow rate for \(0.1 \leq \dot{m} \leq 0.5 \mathrm{~kg} / \mathrm{s}\). How long would it take to melt the paraffin for \(\dot{m}=0.5 \mathrm{~kg} / \mathrm{s}\) ?

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.

One way to cool chips mounted on the circuit boards of a computer is to encapsulate the boards in metal frames that provide efficient pathways for conduction to supporting cold plates. Heat generated by the chips is then dissipated by transfer to water flowing through passages drilled in the plates. Because the plates are made from a metal of large thermal conductivity (typically aluminium or copper), they may be assumed to be at a temperature, \(T_{s, c p^{-}}\) (a) Consider circuit boards attached to cold plates of height \(H=750 \mathrm{~mm}\) and width \(L=600 \mathrm{~mm}\), each with \(N=10\) holes of diameter \(D=10 \mathrm{~mm}\). If operating conditions maintain plate temperatures of \(T_{\text {s.tp }}=32^{\circ} \mathrm{C}\) with water flow at \(\dot{m}_{1}=0.2 \mathrm{~kg} / \mathrm{s}\) per passage and \(T_{m, i}=7^{\circ} \mathrm{C}\), how much heat may be dissipated by the circuit boards? (b) To enhance cooling, thereby allowing increased power generation without an attendant increase in system temperatures, a hybrid cooling scheme may be used. The scheme involves forced airflow over the encapsulated circuit boards, as well as water flow through the cold plates. Consider conditions for which \(N_{\mathrm{cb}}=10\) circuit boards of width \(W=350 \mathrm{~mm}\) are attached to the cold plates and their average surface temperature is \(T_{s, \text { do }}=47^{\circ} \mathrm{C}\) when \(T_{s, \text { ep }}=32^{\circ} \mathrm{C}\). If air is in parallel flow over the plates with \(u_{\infty}=10 \mathrm{~m} / \mathrm{s}\) and \(T_{\infty}=7^{\circ} \mathrm{C}\), how much of the heat generated by the circuit boards is transferred to the air?
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