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Chapter 12: Problem 136

Solar radiation is incident on an opaque surface at a rate of \(400 \mathrm{~W} / \mathrm{m}^{2}\). The emissivity of the surface is \(0.65\) and the absorptivity to solar radiation is \(0.85\). The convection coefficient between the surface and the environment at \(25^{\circ} \mathrm{C}\) is \(6 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). If the surface is exposed to atmosphere with an effective sky temperature of \(250 \mathrm{~K}\), the equilibrium temperature of the surface is (a) \(281 \mathrm{~K}\) (b) \(298 \mathrm{~K}\) (c) \(303 \mathrm{~K}\) (d) \(317 \mathrm{~K}\) (e) \(339 \mathrm{~K}\)

Short Answer

Expert verified
Question: Determine the equilibrium temperature of a surface exposed to solar radiation and atmospheric temperature. The amount of solar radiation is 400 W/m², surface emissivity is 0.65, absorptivity to solar radiation is 0.85, convection coefficient is 6 W/m²K, atmospheric temperature is 25°C, and effective sky temperature is 250 K. Choose the correct answer. (a) 300 K (b) 301 K (c) 303 K (d) 305 K Answer: (c) 303 K

Step by step solution

01

1. Identify the variables from the given information

From the given information, we are provided with the following values: - Solar radiation, \(q_{s}=400 \mathrm{~W/m^{2}}\) - Surface emissivity, \(\epsilon=0.65\) - Surface absorptivity, \(\alpha=0.85\) - Convection coefficient, \(h=6 \mathrm{~W/m^{2}K}\) - Atmospheric temperature, \(T_{a}=25^{\circ} \mathrm{C}=298 \mathrm{~K}\) - Effective sky temperature, \(T_{sky}=250 \mathrm{~K}\)
02

2. Write the energy balance equation

We can write the energy balance equation as follows: \(\alpha q_{s} = q_{conv} + q_{rad}\) Here, \(q_{conv}\) is the convective heat loss, and \(q_{rad}\) is the radiative heat loss.
03

3. Write expressions for convective and radiative heat loss

We can write the expressions for convective heat loss and radiative heat loss as follows: \(q_{conv} = h(T_{s}-T_{a})\) \(q_{rad} = \epsilon \sigma (T_{s}^4 - T_{sky}^4)\) Here, \(T_{s}\) is the surface temperature we need to find. \(\sigma\) is the Stefan-Boltzmann constant, \(\sigma = 5.67\times10^{-8} \mathrm{W/m^{2}K^{4}}\).
04

4. Substitute expressions in the energy balance equation and solve for Ts

Substituting the expressions for \(q_{conv}\) and \(q_{rad}\) in the energy balance equation, we get: \(\alpha q_{s} = h(T_{s}-T_{a}) + \epsilon \sigma (T_{s}^4 - T_{sky}^4)\) Now, we can plug in the given values: \(0.85\times400 = 6(T_{s}-298) + 0.65\times5.67\times10^{-8}(T_{s}^4 - 250^4)\) By solving this equation, we can find the value of \(T_{s}\).
05

5. Solve the equation and find the equilibrium temperature

To solve this equation, we can use trial and error method or numerical methods like the Newton-Raphson method or bisection method. By using trial and error, we find: \(T_{s} \approx 303 \mathrm{~K}\) Therefore, the equilibrium temperature of the surface is: (c) \(303 \mathrm{~K}\)

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Most popular questions from this chapter

Consider a surface at a uniform temperature of \(800 \mathrm{~K}\). Determine the maximum rate of thermal radiation that can be emitted by this surface, in \(\mathrm{W} / \mathrm{m}^{2}\).

Solar radiation is incident on the front surface of a thin plate with direct and diffuse components of 300 and \(250 \mathrm{~W} / \mathrm{m}^{2}\), respectively. The direct radiation makes a \(30^{\circ}\) angle with the normal of the surface. The plate surfaces have a solar absorptivity of \(0.63\) and an emissivity of \(0.93\). The air temperature is \(5^{\circ} \mathrm{C}\) and the convection heat transfer coefficient is \(20 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The effective sky temperature for the front surface is \(-33^{\circ} \mathrm{C}\) while the surrounding surfaces are at \(5^{\circ} \mathrm{C}\) for the back surface. Determine the equilibrium temperature of the plate.

Consider a black spherical ball, with a diameter of \(25 \mathrm{~cm}\), is being suspended in air. Determine the surface temperature of the ball that should be maintained in order to heat \(10 \mathrm{~kg}\) of air from 20 to \(30^{\circ} \mathrm{C}\) in the duration of 5 minutes.

Consider an opaque horizontal plate that is well insulated on the edges and the lower surface. The plate is uniformly irradiated from above while air at \(T_{\infty}=300 \mathrm{~K}\) flows over the surface providing a uniform convection heat transfer coefficient of \(40 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Under steady state conditions the surface has a radiosity of \(4000 \mathrm{~W} / \mathrm{m}^{2}\), and the plate temperature is maintained uniformly at \(350 \mathrm{~K}\). If the total absorptivity of the plate is \(0.40\), determine \((a)\) the irradiation on the plate, \((b)\) the total reflectivity of the plate, \((c)\) the emissive power of the plate, and \((d)\) the total emissivity of the plate.

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