Question

Two-phase gas-liquid oxygen is stored in a spherical tank of \(1-m\) diameter, where it is maintained at its normal boiling point. The spherical tank is enclosed by a \(1.6-\mathrm{m}\) diameter concentric spherical surface at \(273 \mathrm{~K}\). Both spherical surfaces have an emissivity of \(0.01\), and the gap between the inner sphere and outer sphere is vacuumed. Assuming that the spherical tank surface has the same temperature as the oxygen, determine the heat transfer rate at the spherical tank surface.

Short answer

Answer: The heat transfer rate at the spherical tank surface is approximately 13.15 W.

Step-by-step solution

Determine the temperature of the oxygen

The normal boiling point of oxygen is \(90.2 \mathrm{~K}\). Thus, the temperature of the oxygen at its normal boiling point is: \(T_{1} = 90.2 \mathrm{~K}\)

Calculate the surface areas of both inner and outer spherical surfaces

We are given the diameters of both the spherical tank (\(1\) m) and the concentric spherical surface (\(1.6\) m), so we can calculate their radii: \(r_{1}=0.5\) m and \(r_{2}=0.8\) m. The surface area of a sphere is given by the formula \(A = 4 \pi r^2\). Therefore, we can find the surface areas for both the inner (spherical tank) and outer (concentric spherical surface) spheres: \(A_{1} = 4 \pi r_{1}^2 = 4 \pi (0.5)^2 = \pi \mathrm{~m}^2\) \(A_{2} = 4 \pi r_{2}^2 = 4 \pi (0.8)^2 = 2.56 \pi \mathrm{~m}^2\)

Apply the radiative heat transfer equation

In a vacuum, heat transfer occurs only through radiation. The radiative heat transfer equation is given by: \(q = \dfrac{F_{12}\sigma (T_{1}^4 - T_{2}^4)}{1/\varepsilon_{1} - 1 + (A_{1}/A_{2})(1/\varepsilon_{2} - 1)}\), where \(q\) - heat transfer rate, \(F_{12}\) - view factor between surface 1 and surface 2, which is equal to \(1\) for concentric spheres, \(\sigma\) - Stefan-Boltzmann constant \((5.67 \times 10^{-8} \mathrm{W/m}^2 \mathrm{K}^4)\), \(T_{1}\) - temperature of the inner sphere (temperature of the oxygen), \(T_{2}\) - temperature of the outer sphere (\(273 \mathrm{~K}\)), \(\varepsilon_{1}\) and \(\varepsilon_{2}\) - emissivities of surface 1 (spherical tank) and surface 2 (concentric spherical surface) (\((0.01)\) in both cases), and \(A_{1}\) and \(A_{2}\) - surface areas of surfaces 1 and 2. Plugging the given values into the radiative heat transfer equation: \(q = \dfrac{1 \times (5.67 \times 10^{-8}) \times [(90.2)^4 - (273)^4]}{1/0.01 - 1 + (\pi / 2.56 \pi)(1/0.01 - 1)}\) After calculating, we get: \(q \approx -13.15 \mathrm{~W}\) The negative sign indicates that the heat transfer is happening from the outer sphere to the inner sphere. Therefore, the heat transfer rate at the spherical tank surface is approximately \(13.15 \mathrm{~W}\).

Key concepts

Two-Phase Systems

In the study of thermodynamics, a two-phase system consists of both a liquid and a gas phase coexisting in equilibrium. This is a common scenario for substances at their boiling points, where the liquid phase begins to change into the vapor phase. For example, in the case of oxygen stored in the spherical tank, the substance is at its normal boiling point of 90.2 K. At this temperature, both liquid and gaseous oxygen can exist together.

Such systems need careful consideration of several factors including pressure, temperature, and specific volume, which define the state of both phases. The pressure is typically at the saturated pressure corresponding to the boiling point. Any heat transfer within such systems may result in phase changes, either by converting liquid into vapor or vice versa.

In two-phase systems, understanding the balance between the phases is crucial. When studying heat transfer in such scenarios, focusing on the boundary conditions and the interactions at the interface of the phases becomes important. Efficient heat management can ensure stability, preventing unwanted phase changes that could disrupt system performance.

Spherical Geometry

Spherical geometry plays a vital role in understanding and calculating areas and volumes of spherical bodies, which is fundamental in radiative heat transfer exercises like this one. A sphere is a three-dimensional shape where every point on its surface is equidistant from its center point. This makes spheres particularly interesting for enclosing gases, as seen in the spherical oxygen tank.

The surface area (\(A\)) of a sphere is calculated using the formula:\[A = 4 \pi r^2\]where \(r\) is the radius of the sphere. This formula is derived under the assumption that the sphere is perfect and smooth, which is generally sufficient for engineering purposes.

• Inner Sphere Radius, \(r_1 = 0.5\) m (given diameter = 1 m)
• Outer Sphere Radius, \(r_2 = 0.8\) m (given diameter = 1.6 m)

Understanding spherical geometry allows engineers to determine properties like surface area, which are crucial for calculating radiative heat exchange, as it depends directly on this parameter in a vacuum setting where conduction and convection are negligible.

Emissivity in Heat Transfer

Emissivity is a measure of a material's ability to emit energy as thermal radiation. It is crucial for understanding radiative heat transfer, as different materials can emit varying amounts of energy at the same temperature. Emissivity values range from 0 (a perfect reflector) to 1 (a perfect emitter, also known as a blackbody). In the spherical oxygen tank problem, both the inner and outer surfaces have low emissivity values of 0.01.

When dealing with radiative heat transfer, the heat transfer rate \(q\) depends significantly on emissivity. In the radiative heat transfer equation used in this exercise, emissivity affects the denominator:\[q = \dfrac{F_{12}\sigma (T_1^4 - T_2^4)}{1/\varepsilon_1 - 1 + (A_1/A_2)(1/\varepsilon_2 - 1)}\]where \(\varepsilon_1\) and \(\varepsilon_2\) are the emissivities of the two surfaces. A lower emissivity means less heat is emitted from the surface, thus reducing the heat transfer rate \(q\).

• Both surfaces have emissivity \(\varepsilon = 0.01\).
• The low emissivity results in minimal heat exchange through radiative processes.

Understanding emissivity helps in predicting how much heat will be exchanged between surfaces, thus aiding in efficient thermal management of systems.