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Chapter 5: Problem 9

A solid steel sphere (AISI 1010 ), \(300 \mathrm{~mm}\) in diameter, is coated with a dielectric material layer of thickness \(2 \mathrm{~mm}\) and thermal conductivity \(0.04 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). The coated sphere is initially at a uniform temperature of \(500^{\circ} \mathrm{C}\) and is suddenly quenched in a large oil bath for which \(T_{\infty}=100^{\circ} \mathrm{C}\) and \(h=3300 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Estimate the time required for the coated sphere temperature to reach \(140^{\circ} \mathrm{C}\). Hint: Neglect the effect of energy storage in the dielectric material, since its thermal capacitance \((\rho c V)\) is small compared to that of the steel sphere

Short Answer

Expert verified
The time required for the coated sphere temperature to reach \(140^{\circ}\mathrm{C}\) is approximately \(214.41\, \mathrm{s}\).

Step by step solution

01

Identify the properties of the materials

First, we need to gather the properties of AISI 1010 steel and the given dielectric material. The relevant properties are: - Diameter of the steel sphere: \(D = 300 \, \mathrm{mm}\) - Thickness of the dielectric coating: \(\delta = 2\, \mathrm{mm}\) - Thermal conductivity of the dielectric material: \(k = 0.04 \, \mathrm{W/m\cdot K}\) - Bulk oil temperature: \(T_\infty = 100 ^\circ\mathrm{C}\) - Heat transfer coefficient: \(h = 3300 \, \mathrm{W/m^2 \cdot K}\) - Initial temperature of the sphere: \(T_i = 500^\circ\mathrm{C}\) - Desired temperature of the sphere: \(T_f = 140^\circ\mathrm{C}\)
02

Apply heat transfer equations and establish the temperature as a function of time

Next, we apply the lumped capacitance analysis for transient conduction. Since the hint advises us to neglect the effect of energy storage in the dielectric material, we can calculate the temperature as a function of time as: \[ln\left(\frac{T-T_\infty}{T_i - T_\infty}\right) = -\frac{h A t}{\rho c_p V}\] Where \(A\) is the surface area of the coated sphere, \(\rho\) is the density of AISI 1010 steel, \(c_p\) is the specific heat capacity, and \(V\) is the volume of the sphere.
03

Solve for the required time

We want to solve for time (\(t\)) when the temperature (\(T\)) reaches 140掳C. Rearrange the equation for \(t\): \[t = -\frac{ln\left(\frac{T-T_\infty}{T_i - T_\infty}\right)\rho c_p V}{h A}\] Now, plug in the known values and properties of the AISI 1010 steel (\(\rho = 7.85 \times 10^3 \, \mathrm{kg/m^3}\) and \(c_p = 434 \, \mathrm{J/kg\cdot K}\)). Also, calculate the surface area of the coated sphere: \[A = 4\pi\left(\frac{D}{2} + \delta\right)^2 = 4\pi\left(0.151\, \mathrm{m}\right)^2 \approx 0.287 \, \mathrm{m^2}\] And the volume of the steel sphere: \[V = \frac{4}{3}\pi\left(\frac{D}{2}\right)^3 = \frac{4}{3}\pi\left(0.15\, \mathrm{m}\right)^3 \approx 1.41 \times 10^{-2} \, \mathrm{m^3}\] Plug all the values into the equation for \(t\): \[t = -\frac{ln\left(\frac{140-100}{500 - 100}\right)(7.85 \times 10^3 \, \mathrm{kg/m^3})(434 \, \mathrm{J/kg\cdot K})(1.41 \times 10^{-2} \, \mathrm{m^3})}{3300 \, \mathrm{W/m^2 \cdot K}(0.287 \, \mathrm{m^2})}\] Solve for \(t\): \[t \approx 214.41\, \mathrm{s}\] It takes approximately 214.41 seconds for the coated sphere temperature to reach \(140^{\circ}\mathrm{C}\).

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Key Concepts

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

Lumped Capacitance Model
The Lumped Capacitance Model is a simplified approach used in transient heat conduction analysis. This model assumes that the temperature within an object changes uniformly with time. Thus, it allows for the heat conduction problem to be reduced to a function that changes with time only, instead of both time and space. The Lumped Capacitance Model is valid when the Biot number (Bi), defined as the ratio of conductive to convective heat transfer, is much less than 1. The Biot number is given by the formula:\[ \text{Bi} = \frac{hL_c}{k} \] where \(h\) is the heat transfer coefficient, \(L_c\) is the characteristic length of the object, and \(k\) is the thermal conductivity of the material. If the Biot number is small, the object's resistance to heat conduction is negligible compared to its surface resistance to heat transfer. Therefore, the object can be treated as if it has a uniform temperature at any given time. This simplifies the calculation of the time required to cool or heat the object through convection.
Thermal Conductivity
Thermal Conductivity, denoted as \(k\), is a physical property of materials that measures their ability to conduct heat. It is typically expressed in units of \(\mathrm{W/m \, \cdot \, K}\). Materials with high thermal conductivity are good heat conductors and transfer heat quickly, while those with low thermal conductivity are good insulators. In the given exercise, the dielectric material coating has a thermal conductivity of \(0.04 \, \mathrm{W/m \cdot K}\), indicating it is a relatively poor conductor of heat.

This lower thermal conductivity is why energy storage in the dielectric layer can be neglected, as mentioned in the problem's hint. Compared to the high thermal conductivity of carbon steel (AISI 1010), which is around \(51.9 \, \mathrm{W/m \cdot K}\), the dielectric material primarily acts as an insulating barrier. This property impacts the rate at which heat flows through the material and is essential for determining how quickly temperature changes in the system.
  • Understanding thermal conductivity helps in selecting materials for thermal insulation or conduction.
  • Designing systems like heat exchangers requires materials with suitable thermal conductivities to ensure efficient heat transfer.
Heat Transfer Coefficient
The Heat Transfer Coefficient, symbolized as \(h\), is a measure that quantifies the convective heat transfer between a fluid and a solid surface. It reflects how well heat is transferred from the sphere to the surrounding oil bath in the exercise. The heat transfer coefficient depends on factors like fluid properties, flow regime, and surface characteristics.

In the given problem, \(h = 3300 \, \mathrm{W/m^2 \cdot K}\), indicating a high rate of heat transfer due to convection. The convective heat transfer process occurs when the fluid motion increases the contact between the fluid and the surface, enhancing thermal exchange. To apply the lumped capacitance model effectively, a high heat transfer coefficient ensures rapid thermal equilibrium between the solid and the fluid, allowing us to assume a uniform temperature within the sphere.
  • A high heat transfer coefficient is favorable for applications that require fast heating or cooling, such as quenching processes.
  • For comprehensive analysis, factors like the Reynold's number and Nusselt's number, which relate to flow and thermal dynamics, can provide deeper insights.

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Most popular questions from this chapter

Steel is sequentially heated and cooled (annealed) to relieve stresses and to make it less brittle. Consider a 100 -mm-thick plate \(\left(k=45 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho=7800 \mathrm{~kg} / \mathrm{m}^{3}\right.\), \(c_{p}=500 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\) ) that is initially at a uniform temperature of \(300^{\circ} \mathrm{C}\) and is heated (on both sides) in a gas-fired furnace for which \(T_{\infty}=700^{\circ} \mathrm{C}\) and \(h=\) \(500 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). How long will it take for a minimum temperature of \(550^{\circ} \mathrm{C}\) to be reached in the plate?

Standards for firewalls may be based on their thermal response to a prescribed radiant heat flux. Consider a \(0.25\)-m-thick concrete wall \(\left(\rho=2300 \mathrm{~kg} / \mathrm{m}^{3}\right.\), \(c=880 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, k=1.4 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\), which is at an initial temperature of \(T_{i}=25^{\circ} \mathrm{C}\) and irradiated at one surface by lamps that provide a uniform heat flux of \(q_{s}^{\prime \prime}=10^{4} \mathrm{~W} / \mathrm{m}^{2}\). The absorptivity of the surface to the irradiation is \(\alpha_{s}=1.0\). If building code requirements dictate that the temperatures of the irradiated and back surfaces must not exceed \(325^{\circ} \mathrm{C}\) and \(25^{\circ} \mathrm{C}\), respectively, after \(30 \mathrm{~min}\) of heating, will the requirements be met?

During transient operation, the steel nozzle of a rocket engine must not exceed a maximum allowable operating temperature of \(1500 \mathrm{~K}\) when exposed to combustion gases characterized by a temperature of \(2300 \mathrm{~K}\) and a convection coefficient of \(5000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). To extend the duration of engine operation, it is proposed that a ceramic thermal barrier coating \((k=10 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), \(\alpha=6 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\) ) be applied to the interior surface of the nozzle. (a) If the ceramic coating is \(10 \mathrm{~mm}\) thick and at initial temperature of \(300 \mathrm{~K}\), obtain a conservative estimate of the maximum allowable duration of engine operation. The nozzle radius is much larger than the combined wall and coating thickness. (b) Compute and plot the inner and outer surface temperatures of the coating as a function of time for \(0 \leq t \leq 150 \mathrm{~s}\). Repeat the calculations for a coating thickness of \(40 \mathrm{~mm}\).

Consider a thin electrical heater attached to a plate and backed by insulation. Initially, the heater and plate are at the temperature of the ambient air, \(T_{\infty}\). Suddenly, the power to the heater is activated, yielding a constant heat flux \(q_{o}^{\prime \prime}\left(\mathrm{W} / \mathrm{m}^{2}\right)\) at the inner surface of the plate. (a) Sketch and label, on \(T \leftarrow x\) coordinates, the temperature distributions: initial, steady-state, and at two intermediate times. (b) Sketch the heat flux at the outer surface \(q_{x}^{\prime \prime}(L, t)\) as a function of time.

A spherical vessel used as a reactor for producing pharmaceuticals has a 5 -mm-thick stainless steel wall \((k=17 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) and an inner diameter of \(D_{i}=1.0 \mathrm{~m}\). During production, the vessel is filled with reactants for which \(\rho=1100 \mathrm{~kg} / \mathrm{m}^{3}\) and \(c=2400 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), while exothermic reactions release energy at a volumetric rate of \(\dot{q}=10^{4} \mathrm{~W} / \mathrm{m}^{3}\). As first approximations, the reactants may be assumed to be well stirred and the thermal capacitance of the vessel may be neglected. (a) The exterior surface of the vessel is exposed to ambient air \(\left(T_{\infty}=25^{\circ} \mathrm{C}\right)\) for which a convection coefficient of \(h=6 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) may be assumed. If the initial temperature of the reactants is \(25^{\circ} \mathrm{C}\), what is the temperature of the reactants after \(5 \mathrm{~h}\) of process time? What is the corresponding temperature at the outer surface of the vessel? (b) Explore the effect of varying the convection coefficient on transient thermal conditions within the reactor.
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