Question

A blackbody plate \(\left(A_{1}=5 \mathrm{~cm}^{2}\right)\) is subjected to a uniform heat flux of \(1000 \mathrm{~W} / \mathrm{m}^{2}\) on the bottom, while the top surface is exposed to an ambient surrounding at \(5^{\circ} \mathrm{C}\). The heat transfer coefficient due to natural convection on the plate surface is \(5 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). A radiometer is placed \(1 \mathrm{~m}\) above the plate normal to the direction of viewing from the plate. Determine the irradiation that the radiometer would detect.

Short answer

Based on the given information and calculations, the radiometer placed 1 meter above the blackbody plate would detect an irradiation of approximately \(0.02193~W/m^2\).

Step-by-step solution

Calculate the total heat transfer to the plate (Q)

First, we need to convert the given heat flux \(q''=1000~W/m^2\) to find the total heat transfer to the plate (in Watts). To do this, we multiply the heat flux by the area of the plate \(A_1\) in meters squared. The area of the plate \(A_1 = 5~cm^2\) must be converted to meters squared. $$A_1 = 5 \times 10^{-4}~m^2$$ Now we can find the total heat transfer (Q) by multiplying the heat flux and area: $$Q = q'' \cdot A_1 = 1000~W/m^2 \cdot 5 \times 10^{-4}~m^2 = 0.5~W$$

Calculate the plate temperature

Next, we can determine the plate's temperature. We know that the heat transfer due to natural convection (Q_conv) is described by the following formula: $$Q_{conv} = h \cdot A_1 \cdot (T_p - T_{\infty})$$ Where \(h = 5~W/m^2 K\) is the heat transfer coefficient, \(T_p\) is the plate's temperature in Kelvin, and \(T_{\infty} = 5^\circ C = 278.15~K\) is the ambient temperature. From step 1, we know Q=0.5W. So we can rearrange the equation for Q_conv and solve for \(T_p\): $$T_p = \frac{Q}{h \cdot A_1} + T_{\infty} = \frac{0.5~W}{5~W/m^2K \cdot 5 \times 10^{-4}~m^2} + 278.15~K = 428.15~K$$

Calculate heat transfer by radiation

Now that we have the plate's temperature, we can calculate the heat transfer by radiation (Q_rad), using the Stefan-Boltzmann Law: $$Q_{rad} = \sigma \cdot A_1 \cdot (T_p^4 - T_{\infty}^4)$$ Where \(\sigma = 5.67 \times 10^{-8}~W/m^2K^4\) is the Stefan-Boltzmann constant. Plugging in the values, we get: $$Q_{rad} = 5.67 \times 10^{-8}~W/m^2K^4 \cdot 5 \times 10^{-4}~m^2 \cdot (428.15^4 - 278.15^4~K^4) = 0.27569~W$$

Calculate the irradiation detected by the radiometer

Finally, we can calculate the irradiation (E) detected by the radiometer at a distance of 1 meter using the following equation: $$E = \frac{Q_{rad}}{A_2}$$ Where \(A_2\) is the area of the sphere with a radius of 1 meter: $$A_2 = 4\pi r^2 = 4\pi (1~m)^2 = 12.566~m^2 $$ Now, we can find the irradiation detected by the radiometer: $$E = \frac{0.27569~W}{12.566~m^2} = 0.02193~W/m^2$$ Thus, the radiometer would detect an irradiation of approximately \(0.02193~W/m^2\).

Key concepts

Blackbody Radiation

Blackbody radiation is a crucial concept in understanding how objects emit energy. A blackbody absorbs all incident radiation, regardless of frequency or angle, and in turn, it radiates energy in a characteristic spectrum determined solely by its temperature. This emitted radiation is called blackbody radiation.

At any temperature above absolute zero, an object radiates energy, and this energy is proportional to the fourth power of the blackbody's absolute temperature. Such objects emit energy across the electromagnetic spectrum. The emitted spectrum can range from visible light to infrared.

Blackbody radiation forms the foundation for the laws governing thermal radiation, such as the Stefan-Boltzmann Law. This law helps calculate how much energy a surface emits, which is vital in engineering, environmental studies, and astrophysics. In many real-world applications, especially in thermal engineering, blackbody laws enable us to predict heat exchanges and design efficient thermal systems.

Convection Heat Transfer

Convection heat transfer involves the transfer of heat between a solid surface and a fluid flowing over it. This mechanism is critical in systems where fluid motion is involved, such as in air conditioning, heating, and cooling of electronic devices.

Convection can be natural or forced. In natural convection, fluid motion is generated by buoyancy forces arising from temperature differences in the fluid. For instance, as the fluid near a hot surface warms, it becomes less dense and rises, forming a natural circulation pattern that promotes heat transfer. This is evident in the given exercise, where the heat transfer coefficient (5 W/m²·K) is a measure of the natural convection efficiency on the plate's surface.

Understanding convection is essential for calculating the temperature around heat-emitting objects and ensuring adequate thermal comfort in living spaces. You use it when setting up heating systems or designing components that require temperature management, such as computer processors.

Stefan-Boltzmann Law

The Stefan-Boltzmann Law plays a pivotal role in understanding heat radiation from surfaces. It relates the total energy radiated per unit surface area of a blackbody to the fourth power of its temperature. The mathematical representation of this law is given by:

\[ Q_{rad} = \sigma \cdot A \cdot T^4 \]

Where \(\sigma = 5.67 \times 10^{-8} W/m²K⁴\) is the Stefan-Boltzmann constant, \(A\) is the surface area, and \(T\) is the temperature in Kelvin. This law provides a precise method for quantifying the radiated heat from a blackbody and applies not just to idealized conditions, but also closely to real-world objects, provided their surfaces behave similarly to a blackbody.

In practice, when designers and engineers calculate heat losses or design thermal radiators, they turn to the Stefan-Boltzmann Law. It's essential in climate modeling, as well as for developing heating and cooling mechanisms that depend on radiative transfer.

Radiometry

Radiometry is the science of measuring electromagnetic radiation, including visible light. In this context, it captures the power of electromagnetic radiation - crucial for understanding and harnessing heat transfer processes in physics and engineering.

Radiometers, as mentioned in the exercise, are instruments used for measuring the radiative energy emitted by surfaces. They detect the intensity of radiation at various distances from the source, helping to quantify how much energy reaches a certain point. It allows engineers and scientists to measure the power emitted from any heat source, such as the blackbody plate in the given exercise.

Radiometry is pivotal in applications ranging from climate science, where it helps in measuring solar radiation, to astronomy for observing celestial sources. The radiometer's calculated irradiation, as in the step-by-step exercise, is essential for ensuring precise energy measurements, crucial for accurate data in scientific studies and energy resource management.