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Chapter 8: Problem 35

Water flowing at \(2 \mathrm{~kg} / \mathrm{s}\) through a \(40-\mathrm{mm}\)-diameter tube is to be heated from 25 to \(75^{\circ} \mathrm{C}\) by maintaining the tube surface temperature at \(100^{\circ} \mathrm{C}\). (a) What is the required tube length for these conditions? (b) To design a water heating system, we wish to consider using tube diameters in the range from 30 to \(50 \mathrm{~mm}\). What are the required tube lengths for water flow rates of 1,2 , and \(3 \mathrm{~kg} / \mathrm{s}\) ? Represent this design information graphically. (c) Plot the pressure gradient as a function of tube diameter for the three flow rates. Assume the tube wall is smooth.

Short Answer

Expert verified
(a) The required tube length for the given conditions is \(L = \frac{418700}{32.70\pi(0.04)9.474} \approx 109.83\,\mathrm{m}\). (b) The required tube lengths for different diameters and flow rates can be determined by iterating over the different values and following the same process as in part (a). The calculated tube lengths can then be represented graphically. (c) The pressure gradient can be calculated using the Hagen-Poiseuille equation for laminar flow and the Darcy-Weisbach equation for turbulent flow. After calculating the pressure gradient \(\frac{\Delta P}{L}\) as a function of tube diameter for all three flow rates, plot the results.

Step by step solution

01

Calculate mass flow rate and water properties

Given: Mass flow rate \(m = 2\,\mathrm{kg/s}\), Tube diameter \(D = 40\,\mathrm{mm}\), Initial temperature \(T_i = 25^{\circ}\mathrm{C}\), Final temperature \(T_f = 75^{\circ}\mathrm{C}\), Tube surface temperature \(T_s = 100^{\circ}\mathrm{C}\). First, determine the water properties at the average temperature (\(T_{avg} = \frac{T_i + T_f}{2} = 50^{\circ}\mathrm{C}\)): 1. Density (\(\rho\)) \(= 990\,\mathrm{kg/m^3}\) - from a water properties table. 2. Specific heat capacity (\(c_p\)) \(= 4187\,\mathrm{J/(kg\,K)}\) - from a water properties table. 3. Thermal conductivity (\(k\)) \(= 0.627\,\mathrm{W/(m\,K)}\) - from a water properties table.
02

Calculate the necessary heat transfer rate

We need to determine the heat transfer rate necessary to raise the water's temperature from \(25^{\circ}\mathrm{C}\) to \(75^{\circ}\mathrm{C}\). Using the formula \(Q = mc_p\Delta T\), we obtain the required heat transfer rate: \(Q = m\times c_p\times (T_f - T_i) = 2\,\mathrm{kg/s} \times 4187\,\mathrm{J/(kg\,K)}\times(75-25)\,\mathrm{K} = 418700\,\mathrm{W}\)
03

Calculate the convective heat transfer coefficient

We need to determine the convective heat transfer coefficient (\(h\)) by considering the flow inside the tube as turbulent with a Reynolds number greater than 10000. Using the Dittus-Boelter equation, we can find h: \(h = 0.023\,Re^{0.8}Pr^{n} \frac{k}{D}\), where \(Re = \frac{4m}{\pi D \mu}\) is the Reynolds number, \(Pr = \frac{c_p\mu}{k}\) is the Prandtl number, and the kinematic viscosity (\(\mu\)) at \(50^{\circ}\mathrm{C}\) equals \(6.93\times10^{-4}\,\mathrm{Pa\cdot s}\). The constant 'n' is 0.4 for heating (fluid temperature is increasing).
04

Calculate the required tube length

Now, using Newton's law of cooling, we can find the required tube length: \(Q = hA\Delta T_{lm}\), where \(A=\pi D L\) is the surface area of the tube, \(L\) is the tube length, and \(\Delta T_{lm} = \frac{T_s - T_f - (T_s - T_i)}{ \ln \left(\frac{T_s - T_f}{T_s - T_i}\right)}\) is the log mean temperature difference. Rearranging this equation, we obtain the tube length: \(L = \frac{Q}{h\pi D\Delta T_{lm}}\). Calculate \(L\) using all the obtained values. (b) Required tube lengths for different diameters and flow rates: Perform steps 1 to 4, considering the new given flow rates and diameter values. Calculate the required tube length for each combination of flow rate and diameter by iterating over each value and following the same steps as in part (a). Then, represent the information graphically. (c) Pressure gradient:
05

Calculate the pressure gradient for different diameters

For laminar flow, use the Hagen-Poiseuille equation: \(\Delta P = \frac{32\mu QL}{\pi D^4}\). For turbulent flow, use the Darcy-Weisbach equation: \(\Delta P = \frac{4fL\rho v^2}{2D}\), with the friction factor \(f\) given by the Blasius equation: \(f = 0.079\,Re^{-0.25}\), and \(v = \frac{4Q}{\pi D^2}\) being the flow velocity. By assuming the tube wall is smooth and considering the flow rates given in part (b), we calculate the pressure gradient \(\frac{\Delta P}{L}\) as a function of the tube diameter. Plot the results for all three flow rates.

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Key Concepts

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

Convective Heat Transfer Coefficient
When water flows through a tube and is heated, understanding the convective heat transfer coefficient \( h \) is crucial. It tells us how effectively heat is transferred from the tube鈥檚 surface to the flowing water. This efficiency is influenced by factors like fluid velocity, temperature difference, and surface characteristics.
The convective heat transfer process relies on the movement of the fluid. A higher coefficient indicates a more effective heat transfer, which is ideal for heating systems. To calculate \( h \), engineers often use certain correlations, and one of the most popular is the Dittus-Boelter equation. This equation is particularly useful when designing systems where fluids heat up rapidly and efficiently.
Reynolds Number
The Reynolds Number \( Re \) is a dimensionless value that helps determine the flow regime of a fluid inside a pipe. It gives insight into whether the flow is laminar (smooth) or turbulent (chaotic). This is important because turbulent flows, with \( Re > 4000 \), usually enhance heat transfer compared to laminar flows.
To calculate the Reynolds Number for water flowing inside the tube, the formula used is \( Re = \frac{4m}{\pi D \mu} \), where \( m \) is the mass flow rate, \( D \) is the tube diameter, and \( \mu \) is the fluid's dynamic viscosity. In this exercise, the flow is considered turbulent, suggesting that the warming process is efficient due to widespread fluid mixing.
Dittus-Boelter Equation
The Dittus-Boelter equation is a widely-used correlation to predict the convective heat transfer coefficient in turbulent flows within a pipe. It is expressed as:
  • \( h = 0.023 Re^{0.8} Pr^{n} \frac{k}{D} \)
where \( h \) is the heat transfer coefficient, \( Re \) is the Reynolds Number, \( Pr \) is the Prandtl Number, \( k \) is the thermal conductivity, and \( D \) is the pipe diameter. The exponent \( n \) changes with the heating or cooling scenario: for fluids being heated, \( n = 0.4 \).
This equation is particularly useful for its simplicity and applicability in engineering problems involving heat exchangers. It allows for easy estimation of \( h \) based on known operating conditions and fluid properties.
Prandtl Number
The Prandtl Number \( Pr \) is a dimensionless quantity that relates the fluid's momentum diffusivity (viscous diffusion) to its thermal diffusivity. It indicates how quickly heat is conducted away from a wall compared to the rate at which momentum is diffused.
The formula is \( Pr = \frac{c_p \mu}{k} \), where \( c_p \) is the specific heat, \( \mu \) is the dynamic viscosity, and \( k \) is the thermal conductivity. For water at moderate temperatures, \( Pr \) typically falls in a range that supports effective heat transfer, making it an important factor in calculating \( h \) using the Dittus-Boelter equation. Understanding \( Pr \) helps in assessing whether a fluid will efficiently transfer heat, which is crucial for designing heating and cooling systems.

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Most popular questions from this chapter

A double-wall heat exchanger is used to transfer heat between liquids flowing through semicircular copper tubes. Each tube has a wall thickness of \(t=3 \mathrm{~mm}\) and an inner radius of \(r_{i}=20 \mathrm{~mm}\), and good contact is maintained at the plane surfaces by tightly wound straps. The tube outer surfaces are well insulated. (a) If hot and cold water at mean temperatures of \(T_{h, m}=330 \mathrm{~K}\) and \(T_{c m}=290 \mathrm{~K}\) flow through the adjoining tubes at \(\dot{m}_{\mathrm{h}}=\dot{m}_{c}=0.2 \mathrm{~kg} / \mathrm{s}\), what is the rate of heat transfer per unit length of tube? The wall contact resistance is \(10^{-5} \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}\). Approximate the properties of both the hot and cold water as \(\mu=800 \times 10^{-6} \mathrm{~kg} / \mathrm{s} \cdot \mathrm{m}, \quad k=0.625 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), and \(\operatorname{Pr}=5.35\). Hint: Heat transfer is enhanced by conduction through the semicircular portions of the tube walls, and each portion may be subdivided into two straight fins with adiabatic tips. (b) Using the thermal model developed for part (a), determine the heat transfer rate per unit length when the fluids are ethylene glycol. Also, what effect will fabricating the exchanger from an aluminum alloy have on the heat rate? Will increasing the thickness of the tube walls have a beneficial effect?

8.106 Consider the pharmaceutical product of Problem 8.27. Prior to finalizing the manufacturing process, test trials are performed to experimentally determine the dependence of the shelf life of the drug as a function of the sterilization temperature. Hence, the sterilization temperature must be carefully controlled in the trials. To promote good mixing of the pharmaceutical and, in turn, relatively uniform outlet temperatures across the exit tube area, experiments are performed using a device that is constructed of two interwoven coiled tubes, each of 10 -mm diameter. The thin-walled tubing is welded to a solid high thermal conductivity rod of diameter \(D_{r}=40 \mathrm{~mm}\). One tube carries the pharmaceutical product at a mean velocity of \(u_{p}=0.1 \mathrm{~m} / \mathrm{s}\) and inlet temperature of \(25^{\circ} \mathrm{C}\), while the second tube carries pressurized liquid water at \(u_{w}=0.12 \mathrm{~m} / \mathrm{s}\) with an inlet temperature of \(127^{\circ} \mathrm{C}\). The tubes do not contact each other but are each welded to the solid metal rod, with each tube making 20 turns around the rod. The exterior of the apparatus is well insulated. (a) Determine the outlet temperature of the pharmaceutical product. Evaluate the liquid water properties at \(380 \mathrm{~K}\). (b) Investigate the sensitivity of the pharmaceutical's outlet temperature to the velocity of the pressurized water over the range \(0.10

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.

An air heater for an industrial application consists of an insulated, concentric tube annulus, for which air flows through a thin-walled inner tube. Saturated steam flows through the outer annulus, and condensation of the steam maintains a uniform temperature \(T_{s}\) on the tube surface. Consider conditions for which air enters a 50 -mmdiameter tube at a pressure of \(5 \mathrm{~atm}\), a temperature of \(T_{m, i}=17^{\circ} \mathrm{C}\), and a flow rate of \(\dot{m}=0.03 \mathrm{~kg} / \mathrm{s}\), while saturated steam at \(2.455\) bars condenses on the outer surface of the tube. If the length of the annulus is \(L=5 \mathrm{~m}\), what are the outlet temperature \(T_{m, o}\) and pressure \(p_{o}\) of the air? What is the mass rate at which condensate leaves the annulus?

A circular tube of diameter \(D=0.2 \mathrm{~mm}\) and length \(L=\) \(100 \mathrm{~mm}\) imposes a constant heat flux of \(q^{\prime \prime}=20 \times 10^{3}\) \(\mathrm{W} / \mathrm{m}^{2}\) on a fluid with a mass flow rate of \(\dot{m}=0.1 \mathrm{~g} / \mathrm{s}\). For an inlet temperature of \(T_{m, i}=29^{\circ} \mathrm{C}\), determine the tube wall temperature at \(x=L\) for pure water. Evaluate fluid properties at \(\bar{T}=300 \mathrm{~K}\). For the same conditions, determine the tube wall temperature at \(x=L\) for the nanofluid of Example \(2.2\).
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