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Latent heat capsules consist of a thin-walled spherical shell within which a solid-liquid, phase-change material \((\mathrm{PCM})\) of melting point \(T_{\mathrm{mp}}\) and latent heat of fusion \(h_{s f}\) is enclosed. As shown schematically, the capsules may be packed in a cylindrical vessel through which there is fluid flow. If the \(\mathrm{PCM}\) is in its solid state and \(T_{\mathrm{mp}}T_{i}\), energy is released from the \(\mathrm{PCM}\) as it freezes and heat is transferred to the fluid. In either situation, all of the capsules within the packed bed would remain at \(T_{\mathrm{mp}}\) through much of the phase change process, in which case the fluid outlet temperature would remain at a fixed value \(T_{o^{*}}\). Consider an application for which air at atmospheric pressure is chilled by passing it through a packed bed \((\varepsilon=0.5)\) of capsules \(\left(D_{c}=50 \mathrm{~mm}\right)\) containing an organic compound with a melting point of \(T_{\text {mp }}=4^{\circ} \mathrm{C}\). The air enters a cylindrical vessel \(\left(L_{v}=D_{v}=0.40 \mathrm{~m}\right)\) at \(T_{i}=25^{\circ} \mathrm{C}\) and \(V=1.0 \mathrm{~m} / \mathrm{s}\). (a) If the PCM in each capsule is in the solid state at \(T_{\text {map }}\) as melting occurs within the capsule, what is the outlet temperature of the air? If the density and latent heat of fusion of the \(\mathrm{PCM}\) are \(\rho=\) \(1200 \mathrm{~kg} / \mathrm{m}^{3}\) and \(h_{s f}=165 \mathrm{~kJ} / \mathrm{kg}\), what is the mass rate \((\mathrm{kg} / \mathrm{s})\) at which the \(\mathrm{PCM}\) is converted from solid to liquid in the vessel? (b) Explore the effect of the inlet air velocity and capsule diameter on the outlet temperature. (c) At what location in the vessel will complete melting of the PCM in a capsule first occur? Once complete melting begins to occur, how will the outlet temperature vary with time and what is its asymptotic value?

Step by step solution

01

Basic heat transfer relationship

Let's write down the basic heat transfer relationship for the energy transferred from the air to the phase change material inside the capsules. Q = m_h * C_p_h * (T_i - T_mp) = m_c * h_sf The left side of the equation represents the thermal energy transferred from the air to the capsules, and the right side represents the energy stored in the PCM as it melts. Here, Q is the heat transferred, m_h is the mass flow rate of air, C_p_h is the specific heat capacity of air, T_i is the air inlet temperature, T_mp is the melting point of the PCM, m_c is the mass flow rate of the PCM, and h_sf is the latent heat of fusion of the PCM.

02

Outlet temperature of the air

Given that the PCM maintains a fairly constant temperature during phase change, the outlet temperature (T_o) of the air remains constant (assumed to be the melting point of PCM). So, T_o = T_mp = 4 掳C

03

Calculate mass flow rate of PCM

Now we rearrange the basic heat transfer relationship to find the mass flow rate of the PCM at which it is converted from solid to liquid: m_c = (m_h * C_p_h * (T_i - T_mp)) / h_sf We are given T_i = 25 掳C, T_mp = 4 掳C, C_p_h = 1005 J/kgK (specific heat capacity of air), h_sf = 165 kJ/kg (latent heat of fusion of PCM), and V = 1.0 m/s (air velocity). First, we need to find the mass flow rate of the air (m_h). We know the volumetric flow rate (V) and can use the relationship m_h = V * 蟻_h, where 蟻_h is the density of air (approximately 1.184 kg/m^3 at 25 掳C and atmospheric pressure). m_h = V * 蟻_h = 1.0 m/s * 1.184 kg/m^3 = 1.184 kg/s Now we can calculate the mass flow rate of the PCM as it melts: m_c = (1.184 kg/s * 1005 J/kgK * (25 掳C - 4 掳C)) / (165 kJ/kg * 1000 J/kJ) m_c 鈮� 0.0129 kg/s The mass flow rate of the PCM as it melts is approximately 0.0129 kg/s.

04

Explore the effect of inlet air velocity and capsule diameter on outlet temperature

The effect of the inlet air velocity and capsule diameter on the outlet temperature can be investigated by performing a sensitivity analysis. Vary the air velocity and the capsule diameter and study their impact on the outlet temperature. It is expected that as you increase the inlet air velocity and/or decrease the capsule diameter, the outlet temperature will increase, as the rate of heat transfer between the air and the PCM will also increase. Conversely, if you decrease the air velocity and/or increase the capsule diameter, the outlet temperature will decrease due to a decrease in heat transfer rate.

05

Determine the location of complete melting and variations in outlet temperature

To determine the location in the vessel where complete melting of the PCM in a capsule first occurs, we need to find the heat transfer rate between the air and the spheres and determine the point where the energy transferred to the PCM is enough to completely melt its entire mass. As for the variation in outlet temperature with time, once complete melting begins to occur in capsules, the outlet temperature will slowly rise towards the air inlet temperature as heat continues to be absorbed by the PCM capsules and transferred from the air. This trend will continue until an asymptotic value is approached, which is the point where the outlet temperature no longer changes significantly. To find the asymptotic value, you would need to model the heat transfer behavior as the capsules become fully melted and reach an equilibrium state with the inlet air. This may involve complex heat transfer models and simulations and is beyond the scope of this exercise.

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Key Concepts

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

Phase-Change Material (PCM)

Phase-Change Materials (PCMs) are substances that store and release thermal energy during the process of melting and freezing, which is known as a phase transition. A classic example of PCM is the melting of ice into water; the ice absorbs heat without a change in temperature until it completely turns to water. PCMs are characterized by their high latent heat of fusion, the energy absorbed or released during the change of state at a constant temperature.

In the context of latent heat capsules, PCMs are enclosed within containers to absorb heat as they melt, effectively cooling a substance or space. They then release this stored heat as they solidify, providing a source of heat. This dual capability makes PCMs highly efficient for temperature control in various applications such as thermal energy storage, electronic cooling, and in this case, air conditioning.

To enhance the understanding of PCM function, let's digest how it works inside a capsule. A PCM within a latent heat capsule undergoes a transformation from solid to liquid as it absorbs heat from its environment. During this process, the temperature of the PCM does not rise despite the heat absorption. Instead, the energy is utilized to alter the material's phase from solid to liquid. This process goes on until the PCM fully melts. Similarly, as the PCM returns from liquid to solid, energy is released, maintaining a constant temperature throughout the transition.

Heat Transfer in PCM

Understanding how heat transfer operates in PCMs is essential for utilizing these materials effectively. Heat transfer in PCMs occurs predominantly through conduction and convection. Conduction is the transfer of heat through a solid or stationary fluid, while convection is the transport of heat by the movement of a fluid.

Within the latent heat capsules, the shell conducts heat to the PCM from the surrounding fluid, like air or water. As the PCM approaches its melting point, it begins to absorb heat but does not change its temperature. Instead, the energy goes into the phase change process. This phase change occurs at a consistent temperature, ensuring that the fluid passing over the PCM capsule, such as air, can be cooled or heated to a stable temperature.

If the outlet air temperature remains constant, it indicates that the PCM is either absorbing or releasing heat at a rate that matches the energy being transferred by the air stream. Once the PCM completes its phase transition, the heat transfer dynamics change, and the PCM will then start to increase in temperature if more heat is applied, or decrease if it is losing heat.

Latent Heat of Fusion

The latent heat of fusion is a crucial concept when discussing phase-change materials, as it quantifies the amount of energy required to change a substance from the solid phase to the liquid phase without changing its temperature.

For PCMs, the latent heat of fusion is particularly significant because it represents the energy storage capacity of the material. The higher the latent heat of fusion, the more energy the PCM can store during the melting process. This property is harnessed in the latent heat capsule to absorb or release large quantities of heat at a regulated temperature, which is the melting point of the PCM.

Utilizing the latent heat of fusion, the equation \( Q = m_c \times h_{sf} \) described in the steps depicts the relationship between the energy absorbed by the PCM (\( Q \)), the mass of the PCM undergoing the phase change (\( m_c \)), and the PCM's latent heat of fusion (\( h_{sf} \)). This equation allows us to calculate the mass flow rate of the PCM as it transitions between solid and liquid states, and is pertinent in designing thermal systems involving PCMs to ensure they meet the required thermal energy storage or dissipation profiles.