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Chapter 7: Problem 107

Consider the packed bed of aluminum spheres described in Problem \(5.12\) under conditions for which the bed is charged by hot air with an inlet velocity of \(V=1 \mathrm{~m} / \mathrm{s}\) and temperature of \(T_{g, i}=300^{\circ} \mathrm{C}\), but for which the convection coefficient is not prescribed. If the porosity of the bed is \(\varepsilon=0.40\) and the initial temperature of the spheres is \(T_{i}=25^{\circ} \mathrm{C}\), how long does it take a sphere near the inlet of the bed to accumulate \(90 \%\) of its maximum possible energy?

Short Answer

Expert verified
The short version of the answer is: To find the time it takes for a sphere near the inlet to accumulate \(90\%\) of its maximum possible energy, follow these steps: 1. Calculate the heat transfer between the air and the aluminum spheres using the equation \(Q = hA\Delta T\). 2. Determine the convection heat transfer coefficient, \(h\), using the Nusselt number (\(Nu\)), with the correlation \(Nu \approx 2\) for packed beds. 3. Calculate the sphere's maximum possible energy (\(E_{max}\)) using the equation \(E_{max} = mC\Delta T_{max}\). 4. Calculate the time duration required to accumulate \(90\%\) of the maximum possible energy by applying the first law of thermodynamics and integrating the equation with respect to time: \(t = \frac{\int_0^{0.9E_{max}} dE}{\int_0^\infty hA\Delta T dt}\). Substitute the expressions for \(Q\), \(E_{max}\), and \(h\) from the previous steps, calculate the time duration, and you have your answer.

Step by step solution

01

Calculate the heat transfer between air and aluminum spheres

\ To calculate the heat transfer, we will use the following equation: \[Q=hA\Delta T\] where: - \(Q\) is the heat transfer - \(h\) is the convection heat transfer coefficient (not provided, we'll determine it in Step 2) - \(A\) is the surface area of the aluminum sphere - \(\Delta T\) is the temperature difference between the air and the sphere
02

Determine the convection heat transfer coefficient

\ Since the convection heat transfer coefficient is not provided, we can determine it using the Nusselt number (\(Nu\)). The Nusselt number is the ratio of convection heat transfer to conduction heat transfer, and it is calculated as: \[Nu=\frac{hD}{k}\] where: - \(D\) is the diameter of the aluminum sphere - \(k\) is the thermal conductivity of the fluid (air) For packed beds, we can use the following correlation to get an approximate value of the Nusselt number: \[Nu\approx2\] Thus, we can rearrange the equation to solve for \(h\): \[h=\frac{Nu\cdot k}{D}\]
03

Calculate the sphere's maximum possible energy

\ The maximum possible energy (\(E_{max}\)) of the sphere is reached when its temperature equals the temperature of the hot air at the inlet (\(T_{g, i}\)). Therefore, the maximum possible energy can be calculated using the following equation: \[E_{max}=mC\Delta T_{max}\] where: - \(m\) is the mass of the aluminum sphere - \(C\) is the specific heat capacity of aluminum - \(\Delta T_{max}=T_{g, i}-T_{i}\)
04

Calculate the time duration required to accumulate 90% of the maximum possible energy

\ We need to determine the time duration (\(t\)) required for a sphere to accumulate \(90\%\) of its maximum possible energy. We can do this by applying the first law of thermodynamics, which states that the rate of heat transfer equals the rate of energy accumulation in the sphere: \[\frac{dQ}{dt}=\frac{dE}{dt}\] Rearranging and integrating the equation with respect to time limits, we get: \[t=\frac{\int_0^{0.9E_{max}} dE}{\int_0^\infty hA\Delta T dt}\] Now, substitute the expressions for \(Q\), \(E_{max}\), and \(h\) from the previous steps. Calculate the time duration, and you have your answer.

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Key Concepts

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

Convection Heat Transfer Coefficient
When discussing heat transfer, the convection heat transfer coefficient, often abbreviated as \(h\), plays a crucial role. It is a measure of how effectively heat is transmitted from a fluid to a surface or vice versa. The higher the value of \(h\), the more efficient the heat transfer process. In our packed bed scenario, this coefficient helps determine how quickly aluminum spheres can absorb heat from the air, allowing them to reach equilibrium with the incoming hot air.

The calculation of \(h\) depends on various factors, including the type of fluid, the flow conditions, and the surface characteristics. Since it wasn't initially provided in the problem, we determined it using the Nusselt number, which connects the convective and conductive heat transfers. By rearranging the Nusselt number equation, \(Nu = \frac{hD}{k}\), we can solve for \(h\) once the Nusselt number is known. This enables engineers to estimate \(h\) for non-standard situations like our aluminum-packed bed.
Nusselt Number
The Nusselt number (\(Nu\)) is a dimensionless parameter that provides a bridge between convective and conductive heat transfer in fluid systems. Think of it as an indicator of how effectively heat is being transferred in relation to pure conduction. A higher Nusselt number means better convective heat transfer relative to conduction.

Calculating \(Nu\) often involves correlations derived from empirical data based on the geometry and flow characteristics of the system. These factors might include things like shape, size, and flow type. In many engineering applications, appropriate formulas or empirical relations are used to estimate \(Nu\). In our packed bed case study, a simplified assumption of \(Nu \approx 2\) was used to estimate the convection heat transfer coefficient. Utilizing \(Nu\) ensures effective calculations even in complex systems where direct measurement isn't feasible.
Packed Bed
A packed bed is an arrangement of solid objects, like spheres, that are packed together to enhance processes such as filtration, chemical reactions, or heat exchange. In this problem, aluminum spheres are used in a packed bed to maximize the surface area for heat exchange with hot air.

The design of packed beds takes into account parameters such as void fraction or porosity, which is the fraction of the volume not occupied by spheres. Here, a porosity \(\varepsilon = 0.40\) indicates that 40% of the bed鈥檚 volume is taken up by empty space, allowing for effective air circulation. Packed beds are useful in industry due to their ability to promote bulk reactions and enhance overall heat transfer by increasing the contact area between the fluid and the packed material.

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Most popular questions from this chapter

Consider a liquid metal \((\operatorname{Pr} \leqslant 1)\), with free stream conditions \(u_{\infty}\) and \(T_{\infty}\), in parallel flow over an isothermal flat plate at \(T_{s}\). Assuming that \(u=u_{\infty}\) throughout the thermal boundary layer, write the corresponding form of the boundary layer energy equation. Applying appropriate initial \((x=0)\) and boundary conditions, solve this equation for the boundary layer temperature field, \(T(x, y)\). Use the result to obtain an expression for the local Nusselt number \(\mathrm{Nu}_{x}\). Hint: This problem is analogous to one-dimensional heat transfer in a semiinfinite medium with a sudden change in surface temperature.

A circular pipe of 25 -mm outside diameter is placed in an airstream at \(25^{\circ} \mathrm{C}\) and 1 -atm pressure. The air moves in cross flow over the pipe at \(15 \mathrm{~m} / \mathrm{s}\), while the outer surface of the pipe is maintained at \(100^{\circ} \mathrm{C}\). What is the drag force exerted on the pipe per unit length? What is the rate of heat transfer from the pipe per unit length?

Evaporation of liquid fuel droplets is often studied in the laboratory by using a porous sphere technique in which the fuel is supplied at a rate just sufficient to maintain a completely wetted surface on the sphere. Consider the use of kerosene at \(300 \mathrm{~K}\) with a porous sphere of 1 -mm diameter. At this temperature the kerosene has a saturated vapor density of \(0.015 \mathrm{~kg} / \mathrm{m}^{3}\) and a latent heat of vaporization of \(300 \mathrm{~kJ} / \mathrm{kg}\). The mass diffusivity for the vapor-air mixture is \(10^{-5} \mathrm{~m}^{2} / \mathrm{s}\). If dry, atmospheric air at \(V=15 \mathrm{~m} / \mathrm{s}\) and \(T_{\infty}=300 \mathrm{~K}\) flows over the sphere, what is the minimum mass rate at which kerosene must be supplied to maintain a wetted surface? For this condition, by how much must \(T_{\infty}\) actually exceed \(T_{s}\) to maintain the wetted surface at \(300 \mathrm{~K}\) ?

Air at atmospheric pressure and a temperature of \(25^{\circ} \mathrm{C}\) is in parallel flow at a velocity of \(5 \mathrm{~m} / \mathrm{s}\) over a 1 -m-long flat plate that is heated with a uniform heat flux of \(1250 \mathrm{~W} / \mathrm{m}^{2}\). Assume the flow is fully turbulent over the length of the plate. (a) Calculate the plate surface temperature, \(T_{s}(L)\), and the local convection coefficient, \(h_{x}(L)\), at the trailing edge, \(x=L\). (b) Calculate the average temperature of the plate surface, \(\bar{T}_{s}\). (c) Plot the variation of the surface temperature, \(T_{s}(x)\), and the convection coefficient, \(h_{x}(x)\), with distance on the same graph. Explain the key features of these distributions. Working in groups of two, our students design and perform experiments on forced convection phenomena using the general arrangement shown schematically. The air box consists of two muffin fans, a plenum chamber, and flow straighteners discharging a nearly uniform airstream over the flat test-plate. The objectives of one experiment were to measure the heat transfer coefficient and to compare the results with standard convection correlations. The velocity of the airstream was measured using a thermistorbased anemometer, and thermocouples were used to determine the temperatures of the airstream and the test-plate. With the airstream from the box fully stabilized at \(T_{\infty}=20^{\circ} \mathrm{C}\), an aluminum plate was preheated in a convection oven and quickly mounted in the testplate holder. The subsequent temperature history of the plate was determined from thermocouple measurements, and histories obtained for airstream velocities of 3 and \(9 \mathrm{~m} / \mathrm{s}\) were fitted by the following polynomial: The temperature \(T\) and time \(t\) have units of \({ }^{\circ} \mathrm{C}\) and \(\mathrm{s}\), respectively, and values of the coefficients appropriate for the time interval of the experiments are tabulated as follows: \begin{tabular}{lcc} \hline Velocity \((\mathrm{m} / \mathrm{s})\) & 3 & 9 \\ \hline Elapsed Time (s) & 300 & 160 \\ \(a\left({ }^{\circ} \mathrm{C}\right)\) & \(56.87\) & \(57.00\) \\ \(b\left({ }^{\circ} \mathrm{C} / \mathrm{s}\right)\) & \(-0.1472\) & \(-0.2641\) \\\ \(c\left({ }^{\circ} \mathrm{C} / \mathrm{s}^{2}\right)\) & \(3 \times 10^{-4}\) & \(9 \times 10^{-4}\) \\ \(d\left({ }^{\circ} \mathrm{C} / \mathrm{s}^{3}\right)\) & \(-4 \times 10^{-7}\) & \(-2 \times 10^{-6}\) \\ \(e\left({ }^{\circ} \mathrm{C} / \mathrm{s}^{4}\right)\) & \(2 \times 10^{-10}\) & \(1 \times 10^{-9}\) \\ \hline \end{tabular} The plate is square, \(133 \mathrm{~mm}\) to a side, with a thickness of \(3.2 \mathrm{~mm}\), and is made from a highly polished aluminum alloy \(\left(\rho=2770 \mathrm{~kg} / \mathrm{m}^{3}, \quad c=875 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right.\), \(k=177 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\). (a) Determine the heat transfer coefficients for the two cases, assuming the plate behaves as a spacewise isothermal object. (b) Evaluate the coefficients \(C\) and \(m\) for a correlation of the form $$ \overline{N u_{L}}=C \operatorname{Re}^{m} \operatorname{Pr}^{1 / 3} $$ Compare this result with a standard flat-plate correlation. Comment on the goodness of the comparison and explain any differences.

Highly reflective aluminum coatings may be formed on the surface of a substrate by impacting the surface with molten drops of aluminum. The droplets are discharged from an injector, proceed through an inert gas (helium), and must still be in a molten state at the time of impact. \(V=3 \mathrm{~m} / \mathrm{s}\), and \(T_{i}=1100 \mathrm{~K}\), respectively, traverse a stagnant layer of atmospheric helium that is at a temperature of \(T_{\infty}=300 \mathrm{~K}\). What is the maximum allowable thickness of the helium layer needed to ensure that the temperature of droplets impacting the substrate is greater than or equal to the melting point of aluminum \(\left(T_{f} \geq T_{\text {mp }}=933 \mathrm{~K}\right)\) ? Properties of the molten aluminum may be approximated as \(\rho=2500 \mathrm{~kg} / \mathrm{m}^{3}, c=\) \(1200 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), and \(k=200 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\).
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