91Ó°ÊÓ

Consider a horizontal, thin-walled circular tube of diameler \(D=0.025 \mathrm{~m}\) submerged in a container of \(n\)-octadecane (paraffin), which is used to store thermal energy. As hot water flows through the tube, heat is transferred to the paraffin, converting it from the solid to liq: uid state at the phase change temperature of \(T_{\mathrm{s}}=\) \(27.4^{\circ} \mathrm{C}\). The latent heat of fusion and density of paraffin are \(h_{t j}=244 \mathrm{~kJ} / \mathrm{kg}\) and \(\rho=770 \mathrm{~kg} / \mathrm{m}^{3}\), respectively, and thermophysical properties of the water may be taken as \(c_{\mathrm{r}}=4.185 \mathrm{~kJ} / \mathrm{kg}+\mathrm{K}, k=0.653 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \mu=467 \times\) \(10^{-6} \mathrm{~kg} / \mathrm{s} \cdot \mathrm{m}\), and \(\operatorname{Pr}=2.99\). (a) Assuming the tube surface to have a uniform temperature corresponding to that of the phase change, determine the water outlet temperature and total heat transfer rate for a water flow rate of \(0,1 \mathrm{~kg} / \mathrm{s}\) and an inlet temperature of \(60^{\circ} \mathrm{C}\). If \(H=W=\) \(0.25 \mathrm{~m}\), how long would it take to completely liquefy the paraffin, from an initial state for which all the paraffin is solid and at \(27.4^{\circ} \mathrm{C}\) ? (b) The liquefaction process can be accelerated by increasing the flow rate of the water. Compute and plot the heat rate and outlet temperature as a function of flow rate for \(0.1 \leq m \leq 0.5 \mathrm{~kg} / \mathrm{s}\). How long would it take to melt the paraffin for in \(=0.5 \mathrm{~kg} / \mathrm{s}\) ?

Step by step solution

01

Understand and Visualize the Problem

Before starting the calculations, we need to understand that we have a circular tube submerged in paraffin, with hot water flowing through it. Heat from the water will melt the paraffin. We start with some given properties and the need to find the water outlet temperature and heat transfer rate, then calculate the time for melting all the paraffin.

02

Calculate the Heat Transfer Rate

First, we need to use the equation for total heat transfer rate to the paraffin, which is given by: \[ q = mc_p (T_{in} - T_{out}) \] Where:- \(q\) is the heat transfer rate, - \(m = 0.1 \mathrm{~kg/s}\) is the mass flow rate of water,- \(c_p = 4.185 \mathrm{~kJ/kg \, K}\) is the specific heat of water, and - \(T_{in} = 60^{\circ}C\) is the inlet temperature of the water.

03

Determine Water Outlet Temperature

Given that the tube's surface temperature is equal to the phase change temperature of the paraffin, which is \(T_s = 27.4^{\circ}C\), and we assume all the heat is exchanged between the water and paraffin, around this temperature. Therefore, during this process: \[ mc_p (T_{in} - T_s) = q \] From this equation, solve for the outlet temperature \(T_{out}\) using the heat transfer rate.

04

Calculate Time to Melt Paraffin

To calculate the time it takes to melt all the paraffin, use the total energy required, which is determined by the latent heat of fusion: \[ Q = H \times W \times D \times \rho \times h_{fj} \] Where:- \(H=0.25 \,m\) and \(W=0.25 \,m\) are the dimensions of paraffin,- \(D=0.025\,m\) is the tube diameter,- \(\rho = 770 \mathrm{~kg/m^3}\) is the density,- \(h_{fj} = 244 \mathrm{~kJ/kg}\), the latent heat,After calculating \(Q\), the time \(t\) is equal to \(Q/q\).

05

Plot and Analysis for Variable Flow Rates

For varying the flow rate \(m\) from \(0.1\) to \(0.5\, \mathrm{~kg/s}\), repeat the above calculations to get \( T_{out} \) and \( q \) for each flow rate. Use these to plot \( q \) and \( T_{out} \) as functions of flow rates. Finally, calculate the time for complete melting with \( m=0.5 \mathrm{~kg/s}\).

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Key Concepts

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

Latent Heat of Fusion

The latent heat of fusion is a key factor in processes where a substance changes its state, such as melting. It is the amount of heat required to convert a solid into a liquid without changing its temperature. In this exercise, we focus on melting paraffin using the heat transferred from hot water flowing through a tube. Paraffin has a latent heat of fusion of 244 kJ/kg. This means for every kilogram of paraffin, 244 kJ of energy is needed to transition from solid to liquid at the phase change temperature of 27.4°C.

When calculating the time required to melt all the paraffin, it is essential to consider not just the heat transferred but how much of that heat contributes to the phase change. Only heat that exceeds the paraffin's phase change temperature acts to convert the solid paraffin into liquid.

• The latent heat is a specific property of the material.
• Paraffin's transition occurs at a stable temperature, precisely 27.4°C.
• Calculating energy needs involves considering both temperature and this heat property.

Phase Change

The phase change is the process of a substance transitioning between different states of matter, such as solid to liquid. In the given problem, the phase change occurs as paraffin melts due to heat from the water flowing through the tube. The temperature at which this transition occurs is known as the phase change temperature, which for paraffin is set at 27.4°C.

Understanding phase change is crucial as it dictates how and where heat energy from the water affects the paraffin. During the phase change, the paraffin remains at a constant temperature, absorbing heat to transition its structure without raising in temperature.

• Phase change absorbs energy but keeps temperature constant.
• For paraffin, phase change involves transforming solid to liquid.
• Heat energy directly contributes to breaking intermolecular bonds.

Mass Flow Rate

The mass flow rate is the amount of mass passing through a section of the system per unit of time. In this situation, it's the flow rate of water through the tube submerged in paraffin. The flow rate is a critical parameter because it influences the rate of heat transfer to the paraffin. A higher mass flow rate means more water passes through the tube, generating more heat available per unit time.

In the exercise, varying the flow rate helps us to understand its impact on melting paraffin. By increasing the water flow rate from 0.1 kg/s to 0.5 kg/s, more heat can be supplied, reducing the time needed to melt all the paraffin.

• Directly affects the amount of heat transferred per second.
• Higher flow rates potentially increase the efficiency of heat transfer.
• Adjusting flow rate is a way to control the process speed.

Paraffin Properties

Paraffin properties, such as its density and latent heat, critically influence how it absorbs heat and changes state. The density of paraffin is 770 kg/m³, an important trait that determines the amount of paraffin around the tube, which directly relates to the mass that needs exiting heat to melt.

Density combined with the latent heat of fusion helps calculate how much energy is required to melt the entire mass of paraffin. These properties allow precise calculations essential for efficient thermal energy storage using paraffin.

• Density influences the mass in a given volume.
• Latent heat defines energy required for state change.
• Together, they help compute total heat absorption needs.