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Chapter 7: Problem 28

In the production of sheet metals or plastics, it is customary to cool the material before it leaves the production process for storage or shipment to the customer. Typically, the process is continuous, with a sheet of thickness \(\delta\) and width \(W\) cooled as it transits the distance \(L\) between two rollers at a velocity \(V\). In this problem, we consider cooling of an aluminum alloy (2024-T6) by an airstream moving at a velocity \(u_{\infty}\) in counter flow over the top surface of the sheet. A turbulence promoter is used to provide turbulent boundary layer development over the entire surface. (a) By applying conservation of energy to a differential control surface of length \(d x\), which either moves with the sheet or is stationary and through which the sheet passes, derive a differential equation that governs the temperature distribution along the sheet. Because of the low emissivity of the aluminum, radiation effects may be neglected. Express your result in terms of the velocity, thickness, and properties of the sheet \(\left(V, \delta, \rho, c_{p}\right)\), the local convection coefficient \(h_{x}\) associated with the counter flow, and the air temperature. For a known temperature of the sheet \(\left(T_{i}\right)\) at the onset of cooling and a negligible effect of the sheet velocity on boundary layer development, solve the equation to obtain an expression for the outlet temperature \(T_{a}\). (b) For \(\delta=2 \mathrm{~mm}, V=0.10 \mathrm{~m} / \mathrm{s}, L=5 \mathrm{~m}, W=1 \mathrm{~m}\), \(u_{\infty}=20 \mathrm{~m} / \mathrm{s}, T_{\infty}=20^{\circ} \mathrm{C}\), and \(T_{i}=300^{\circ} \mathrm{C}\), what is the outlet temperature \(T_{a}\) ?

Short Answer

Expert verified
The outlet temperature \(T_a\) for the aluminum sheet is approximately \(71.61^{\circ} C\).

Step by step solution

01

Derivation of the Differential Equation

Using the conservation of energy principle for the differential control surface of length \(d x\), we can write the energy balance equation: \(V \delta \rho c_{p} \left( T(x)-T_{i} \right) = -h_x (T(x)-T_\infty)d xW\) Now, let's find a differential equation for the temperature distribution: \(\frac{d (T(x)-T_{i})}{d x} = \frac{-h_x (T(x)-T_\infty)}{V \delta \rho c_{p}}\) Since the sheet velocity has negligible effect on boundary layer development, we can assume that \(h_{x}\) is a constant, and rewrite the equation as: \(\frac{d (T(x)-T_{i})}{d x} = -k(T(x)-T_\infty)\), where \(k = \frac{h_x}{V \delta \rho c_{p}}\)
02

Solving the Differential Equation

Now, let's solve this first-order linear ordinary differential equation: \((T(x)-T_{i})'= -k(T(x)-T_\infty)\) We can rewrite the equation in the form: \(\frac{d(T(x)-T_{i})}{(T(x)-T_\infty)} = -kdx\) Now, integrate both sides: \[\int \frac{d(T(x)-T_{i})}{(T(x)-T_\infty)}= -\int kdx\] \[\ln{(T(x) - T_\infty)} = -kx + C\] To eliminate the logarithm, exponentiate both sides of the equation: \(T(x) - T_\infty = e^{-kx} e^{C}\) \(T(x) = T_\infty + e^{-kx} e^{C}\) Now, let's find the constant \(C\). We know that when \(x=0\), the temperature is \(T_i\). So, we have: \(T_i = T_\infty + e^{-k(0)} e^{C} \Rightarrow e^C = T_i - T_\infty\) Thus, the temperature distribution is given by: \(T(x) = T_\infty + (T_i - T_\infty)e^{-kx}\) The outlet temperature when the sheet exits the cooling zone is \(T_a = T(L)\), so we have: \(T_a = T_\infty + (T_i - T_\infty)e^{-kL}\) Now, we will substitute the given values and find \(T_{a}\).
03

Substituting the given parameters

Using the given problem parameters: \(L = 5 \ m\), \(V = 0.10 \ \frac{m}{s}\), \(\delta = 2 \ mm\), \(W = 1 \ m\), \(u_{\infty} = 20 \ \frac{m}{s}\), \(T_{i} = 300^{\circ} C\) and \(T_{\infty} = 20^{\circ} C\), we can find the outlet temperature \(T_{a}\). First, we need to find the convection coefficient \(h_x\) for the aluminum sheet. The properties of the aluminum alloy are: \(\rho = 2700 \ \frac{kg}{m^3}\), \(c_p = 903 \ \frac{J}{kg \cdot K}\), and the thermal conductivity \(k_{al} = 167 \ \frac{W}{m \cdot K}\). Assuming that the convection coefficient is roughly equal to the heat transfer coefficient, based on mixing-length theory, we can approximate it as: \(h_x = \frac{k_{al}}{(Pr \delta / 2)}\), where \(Pr\) is the Prandtl number. For air, we can assume the Prandtl number to be approximately \(Pr = 0.72\). Then, we can find \(h_x\): \(h_x = \frac{167}{(0.72 \cdot 0.002 / 2)} \approx 23194 \ \frac{W}{m^2 \cdot K}\). Now, we can find the value of \(k\): \(k = \frac{23194}{(0.10 \cdot 0.002 \cdot 2700 \cdot 903)} \approx 0.354 \ \frac{1}{m}\). Finally, we can find the outlet temperature \(T_a\): \(T_a = 20 + (300 - 20)e^{-(0.354)(5)} \approx 20 + 280e^{-1.77} \approx 20 + 51.61 \approx 71.61^{\circ} \text{C}\). Hence, the outlet temperature \(T_{a}\) for the aluminum sheet is about \(71.61^{\circ} \text{C}\).

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Key Concepts

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

Conservation of Energy
The principle of Conservation of Energy is a fundamental concept in physics stating that energy cannot be created or destroyed, only transformed from one form to another. In the context of this exercise, conservation of energy is applied to the cooling process of an aluminum sheet as it moves through a control surface. The energy balance considers the rate at which energy enters, leaves, and accumulates within the sheet as it exchanges heat with the surrounding air. This kind of analysis is crucial for determining how much the sheet's temperature changes over its transit between the rollers. The initial mathematical expression for the energy balance is:
  • The term \(V \delta \rho c_{p} (T(x) - T_i)\) represents the heat stored in the sheet per unit length.
  • The term \(-h_x (T(x) - T_\infty)d xW\) accounts for the convective heat loss to the air across an infinitesimal section of the sheet.
By equating these expressions, we preserve the conservation of energy by ensuring that the heat lost through convection equals the heat reduced from the sheet.
Differential Equation
A Differential Equation is an equation involving derivatives of a function, which in this case, describes how the temperature of the aluminum sheet changes as it moves. The derived differential equation for temperature distribution along the sheet is expressed as:\[ \frac{d(T(x) - T_i)}{d x} = -k(T(x) - T_\infty) \]By solving this type of first-order linear ordinary differential equation, we gain insight into how the temperature of the moving sheet declines due to cooling. The solution involves integrating both sides of the equation to find a relationship for the temperature at any position \(x\) on the sheet. The process of solving this differential equation involves finding a specific function that describes the behavior of \(T(x)\) depending on initial boundary conditions provided (like the initial sheet temperature \(T_i\) at \(x=0\)). These conditions help solve for any constants of integration and provide a complete solution for temperature distribution.
Convection Coefficient
The Convection Coefficient \(h_x\) is a measure of the convective heat transfer between the surface of the sheet and the moving air above it. It represents how efficiently heat is transferred away from the sheet. The value of \(h_x\) influences how much heat is lost and consequently affects the final temperature of the sheet. To determine \(h_x\), we estimate it using properties like the Prandtl number \(Pr\) for air and the turbulent flow conditions. The calculation is given by:\[h_x = \frac{k_{al}}{(Pr \delta / 2)}\]Where \(k_{al}\) is the thermal conductivity of the aluminum. With a known \(Pr=0.72\) and sheet thickness \(\delta\), this semi-empirical formula gives an approximation of \(h_x\), which helps in predicting the cooling rate and outlet temperature of the sheet over its length.
Temperature Distribution
Understanding the Temperature Distribution along the length of the sheet is essential for ensuring the final product meets desired temperature specifications. The temperature distribution is expressed through the equation derived from solving the differential equation:\[T(x) = T_\infty + (T_i - T_\infty)e^{-kx}\]This equation illustrates how the sheet's temperature adapts as it moves and cools. Here, \(T(x)\) represents the temperature at position \(x\), \(T_i\) is the initial temperature, and \(T_\infty\) is the ambient air temperature. The exponential term \(e^{-kx}\) indicates how quickly the temperature approaches the air temperature, controlled by the rate coefficient \(k\) which was previously calculated.Finally, the outlet temperature \(T_a\) is simply \(T(L)\), the temperature at the end of the cooling process, which helps inform whether further cooling adjustments are needed for optimal product specifications.

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Most popular questions from this chapter

Steel (AISI 1010) plates of thickness \(\delta=6 \mathrm{~mm}\) and length \(L=1 \mathrm{~m}\) on a side are conveyed from a heat treatment process and are concurrently cooled by atmospheric air of velocity \(u_{\infty}=10 \mathrm{~m} / \mathrm{s}\) and \(T_{x}=20^{\circ} \mathrm{C}\) in parallel flow over the plates. For an initial plate temperature of \(T_{i}=300^{\circ} \mathrm{C}\), what is the rate of heat transfer from the plate? What is the corresponding rate of change of the plate temperature? The velocity of the air is much larger than that of the plate.

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.

Consider steady, parallel flow of atmospheric air over a flat plate. The air has a temperature and free stream velocity of \(300 \mathrm{~K}\) and \(25 \mathrm{~m} / \mathrm{s}\). (a) Evaluate the boundary layer thickness at distances of \(x=1,10\), and \(100 \mathrm{~mm}\) from the leading edge. If a second plate were installed parallel to and at a distance of \(3 \mathrm{~mm}\) from the first plate, what is the distance from the leading edge at which boundary layer merger would occur? (b) Evaluate the surface shear stress and the \(y\)-velocity component at the outer edge of the boundary layer for the single plate at \(x=1,10\), and \(100 \mathrm{~mm}\). (c) Comment on the validity of the boundary layer approximations.

The use of rock pile thermal energy storage systems has been considered for solar energy and industrial process heat applications. A particular system involves a cylindrical container, \(2 \mathrm{~m}\) long by \(1 \mathrm{~m}\) in diameter, in which nearly spherical rocks of \(0.03-\mathrm{m}\) diameter are packed. The bed has a void space of \(0.42\), and the density and specific heat of the rock are \(\rho=2300 \mathrm{~kg} / \mathrm{m}^{3}\) and \(c_{p}=879 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), respectively. Consider conditions for which atmospheric air is supplied to the rock pile at a steady flow rate of \(1 \mathrm{~kg} / \mathrm{s}\) and a temperature of \(90^{\circ} \mathrm{C}\). The air flows in the axial direction through the container. If the rock is at a temperature of \(25^{\circ} \mathrm{C}\), what is the total rate of heat transfer from the air to the rock pile?

A long, cylindrical, electrical heating element of diameter \(D=10 \mathrm{~mm}\), thermal conductivity \(k=240 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), density \(\rho=2700 \mathrm{~kg} / \mathrm{m}^{3}\), and specific heat \(c_{p}=900 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\) is installed in a duct for which air moves in cross flow over the heater at a temperature and velocity of \(27^{\circ} \mathrm{C}\) and \(10 \mathrm{~m} / \mathrm{s}\), respectively. (a) Neglecting radiation, estimate the steady-state surface temperature when, per unit length of the heater, electrical energy is being dissipated at a rate of \(1000 \mathrm{~W} / \mathrm{m}\). (b) If the heater is activated from an initial temperature of \(27^{\circ} \mathrm{C}\), estimate the time required for the surface temperature to come within \(10^{\circ} \mathrm{C}\) of its steady-state value.
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