Question

Two concentric spheres of diameters \(D_{1}=15 \mathrm{~cm}\) and \(D_{2}=25 \mathrm{~cm}\) are separated by air at \(1 \mathrm{~atm}\) pressure. The surface temperatures of the two spheres enclosing the air are \(T_{1}=\) \(350 \mathrm{~K}\) and \(T_{2}=275 \mathrm{~K}\), respectively, and their emissivities are \(0.75\). Determine the rate of heat transfer from the inner sphere to the outer sphere by \((a)\) natural convection and \((b)\) radiation.

Short answer

Answer: The heat transfer rate due to natural convection is approximately \(5.41 \times 10^6\, W\), and the heat transfer rate due to radiation is approximately \(2651\, W\).

Step-by-step solution

Calculate the Grashof number

The Grashof number is a dimensionless number that represents the ratio of buoyancy forces (due to density differences caused by temperature) to the viscous forces. It can be calculated using the following equation: \(Gr = \dfrac{g \beta (T_1 - T_2) D^3}{\nu^2}\) Where \(g\) is the acceleration due to gravity, \(\beta\) is the thermal expansion coefficient, \(T_1\) and \(T_2\) are the temperatures of the inner and outer spheres, \(D\) is the diameter of the gap between the two spheres, and \(\nu\) is the kinematic viscosity of air.

Determine the gap diameter and the film temperature

The diameter of the gap between the two spheres is given by: \(D = \dfrac{D_2 - D_1}{2} = 5\, cm\). Calculate the film temperature (average temperature): \(T_f = \dfrac{T_1 + T_2}{2} = \dfrac{350 + 275}{2} = 312.5\, K\)

Calculate the thermal expansion coefficient and the kinematic viscosity

Consult a table or use approximations to find the thermal expansion coefficient of air and its kinematic viscosity at the film temperature: \(\beta \approx \dfrac{1}{T_f} = \dfrac{1}{312.5} = 3.2 \times 10^{-3}\, K^{-1}\) \(\nu \approx 16.5 \times 10^{-6}\, m^2/s\) (at \(312.5\, K\))

Find the Grashof number and the Nusselt number

Calculate the Grashof number using the values obtained above: \(Gr = \dfrac{9.81 \times 3.2 \times 10^{-3} \times 75 \times (0.05^3)}{(16.5 \times 10^{-6})^2} \approx 2.07 \times 10^8\) Calculate the Nusselt number using an appropriate correlation for concentric spheres (available in references): \(Nu = c \times (Gr \times Pr)^n\) Here, we assume that the values of \(c\) and \(n\) are 1, and \(Pr\) is the Prandtl number of air at the film temperature (about 0.7). Then, \(Nu = 1 \times (2.07 \times 10^8 \times 0.7)^1 \approx 1.45 \times 10^8\)

Calculate the convective heat transfer coefficient and the natural convection heat transfer rate

Now, find the convective heat transfer coefficient: \(h = \dfrac{Nu \cdot k}{D}\) Assuming a thermal conductivity (\(k\)) of air at the film temperature to be \(0.026\, W/m\cdot K\), we get \(h = \dfrac{1.45 \times 10^8 \times 0.026}{0.05} \approx 1.01 \times 10^6\, W/m^2\cdot K\) The natural convection heat transfer rate is given by: \(q_{conv} = h A (T_1 - T_2)\) Where \(A\) is the surface area of the inner sphere. Since \(D_1 = 15\, cm\), we have \(A = 4\pi(0.075^2) = 0.071\, m^2\). So, the heat transfer rate due to natural convection is: \(q_{conv} = 1.01 \times 10^6 \times 0.071 \times (350 - 275) \approx 5.41 \times 10^6\, W\) #b)# Radiation

Calculate the radiation heat transfer rate

The radiation heat transfer rate between two concentric spheres is given by the following equation: \(q_{rad} = \dfrac{eA \sigma (T_1^4 - T_2^4)} { (1/e_1 + 1/e_2 - 1)}\) Here, \(e\) is the emissivity of the spheres (both have the same emissivity, given as \(0.75\)), \(A\) is the area of the inner sphere (already calculated above), \(\sigma\) is the Stefan-Boltzmann constant (\(5.67 \times 10^{-8}\, W/m^2\cdot K^4\)), and \(T_1\) and \(T_2\) are the temperatures of the inner and outer spheres, respectively. Substitute the given values and calculate the radiation heat transfer rate: \(q_{rad} = \dfrac{0.75 \times 0.071 \times 5.67 \times 10^{-8} \times (350^4 - 275^4)}{(1/0.75 + 1/0.75 - 1)} \approx 2651\, W\) The rate of heat transfer from the inner sphere to the outer sphere by \((a)\) natural convection is approximately \(5.41 \times 10^6\, W\), and by \((b)\) radiation is approximately \(2651\, W\).

Key concepts

Natural Convection

Natural convection refers to the heat transfer process that occurs due to the movement of fluid caused by differences in temperature and density. This is a passive form of heat transfer, meaning it doesn't rely on external forces like pumps or fans. Instead, it relies on the natural rise and fall of fluid movements.

For the case of two concentric spheres, the air between them transfers heat from the hotter inner sphere to the cooler outer sphere. As the air near the hot sphere's surface gets heated, it becomes less dense and rises. Cooler air then takes its place, creating a circulation pattern. This continuous movement of air is what drives natural convection.

This process can be characterized using dimensionless numbers like the Grashof Number and the Nusselt Number, which help calculate the heat transfer rate.

Radiation Heat Transfer

Radiation heat transfer is a mode of heat transfer that occurs through electromagnetic waves. Unlike conduction and convection, it doesn’t require a medium to occur. All objects emit thermal radiation, more so when they are hot. The main players in radiation heat transfer are the temperatures and emissivities of the surfaces involved.

In our exercise, the radiation heat transfer occurs between two concentric spheres at different temperatures. The thermal energy emitted by the inner sphere due to its temperature causes electromagnetic waves to travel through the space to the outer sphere, where it's absorbed. The Stefan-Boltzmann Law helps to determine the rate at which this radiation transfers heat by factoring in both the emissivity of the surfaces and their temperatures.

Grashof Number

The Grashof Number (Gr) plays a crucial role in the analysis of natural convection. It's a dimensionless number that quantifies the relative strength of buoyancy forces compared to viscous forces within a fluid.

The Grashof Number essentials are:

• "Buoyancy Forces": These arise from density differences in the fluid due to temperature gradients, encouraging movement.


• "Viscous Forces": These resist motion and are related to the fluid's viscosity.

The Grashof Number is calculated using:\[Gr = \dfrac{g \beta (T_1 - T_2) D^3}{u^2}\]
Understanding Gr helps in determining how effective natural convection will be for a system, such as the air between the spheres.

Large Grashof numbers typically indicate stronger natural convection currents.

Nusselt Number

The Nusselt Number (Nu) represents the enhancement of heat transfer in a fluid flow due to convection relative to conduction across the same fluid layer. In other words, a higher Nusselt Number shows a more efficient convective heat transfer.

For our concentric spheres scenario, the Nusselt Number is useful for determining the convective heat transfer coefficient, which is then used to calculate the rate of natural convection heat transfer. It is linked to the Grashof and Prandtl numbers and typically calculated as:\[Nu = c \times (Gr \times Pr)^n\]
In the formula:

• "Pr": Prandtl Number, a dimensionless number that relates the fluid's ability to transfer momentum and thermal energy.


• "c and n": Empirical constants, based on the geometrical configuration and flow conditions.

This number aids in bridging the gap between the theoretical analysis and practical applications of convective heat transfer.

Emissivity

Emissivity is a measure of an object's ability to emit absorbed energy. It ranges from 0 to 1, where 1 represents a perfect black body that emits all absorbed energy. This property affects how much thermal radiation an object can emit as well as absorb.

In our example, both the inner and outer spheres have an emissivity of 0.75. This means that they emit 75% of the thermal radiation they absorb. When calculating radiation heat transfer, emissivity is a key factor that adjusts the theoretical maximum heat flux (energy per unit area) to what actually occurs.

By incorporating emissivity, the radiation heat transfer formula precisely accounts for the real-world behavior of materials as they exchange thermal radiation.