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Chapter 5: Problem 97

When a molten metal is cast in a mold that is a poor conductor, the dominant resistance to heat flow is within the mold wall. Consider conditions for which a liquid metal is solidifying in a thick-walled mold of thermal conductivity \(k_{v}\) and thermal diffusivity \(\alpha_{w}\). The density and latent heat of fusion of the metal are designated as \(\rho\) and \(h_{s f}\), respectively, and in both its molten and solid states, the thermal conductivity of the metal is very much larger than that of the mold. Just before the start of solidification \((S=0)\), the mold wall is everywhere at an initial uniform temperature \(T_{i}\) and the molten metal is everywhere at its fusion (melting point) temperature of \(T_{f}\). Following the start of solidification, there is conduction heat transfer into the mold wall and the thickness of the solidified metal \(S\) increases with time \(t\). (a) Sketch the one-dimensional temperature distribution, \(T(x)\), in the mold wall and the metal at \(t=0\) and at two subsequent times during the solidification. Clearly indicate any underlying assumptions. (b) Obtain a relation for the variation of the solid layer thickness \(S\) with time \(t\), expressing your result in terms of appropriate parameters of the system.

Short Answer

Expert verified
In summary, the one-dimensional temperature distribution in the mold wall and metal at different solidification times consists of three main stages: initial point (t=0), early stages of solidification, and later stages of solidification. The main assumptions made include one-dimensional heat transfer and the metal's thermal conductivity being much larger than the mold's. The relation for the solid layer thickness, S, as a function of time, t, is given by: \(S(t) = \frac{-k_{v}(T_f-T_i)t}{\rho h_{sf}}\), which describes how the solidified metal layer's thickness increases during the solidification process.

Step by step solution

01

Sketch 1 - t=0

At the initial point, t=0, we have the molten metal at fusion temperature Tf and the mold at initial uniform temperature Ti. Since there is no solidification, the temperature distribution in the metal remains constant, while the temperature in the mold decreases from Tf to Ti as we move further inside the mold wall. As time progresses and solidification starts, the temperature distribution in both the mold and metal will change. Conduction heat transfer into the mold wall will cause the thickness of the solidified metal to increase.
02

Sketch 2 - Early stages of solidification

At early stages of solidification, the metal near the mold wall starts solidifying, while the temperature in the mold decreases moving inward from the metal-mold interface. The temperature remains constant within the solidified metal area, which is now between the molten metal and the mold wall.
03

Sketch 3 - Later stages of solidification

As solidification continues, the thickness of the solidified metal layer (S) increases, pushing the remaining molten metal further away from the mold wall. The temperature distribution within the mold wall remains similar to the previous stages but moved further inside as a result of the increased solidified metal layer thickness. #Assumptions#: 1. The thermal conductivity of the metal is much larger than that of the mold, allowing us to neglect the metal's thermal resistance. 2. Temperature distribution is one-dimensional. #b)# Relation for the solid layer thickness (S) with time (t) To obtain the relation for S as a function of time, we can apply Fourier's law of heat conduction on the mold wall at the interface with the solidified metal.
04

Apply Fourier's Law

Fourier's Law states that: \(q=-k_{v} \frac{dT}{dx}\), where q is the heat flux, \(k_{v}\) is the thermal conductivity of the mold wall, and \(\frac{dT}{dx}\) is the temperature gradient in the mold wall at the interface with the solidified metal.
05

Write an equation for total heat flux

The total heat flux transferring from the solidified metal layer (S) into the mold wall can also be written as: \(q=\rho h_{sf}\frac{dS}{dt}\), where \(\rho\) is the density of the metal, \(h_{sf}\) is the latent heat of fusion, and \(\frac{dS}{dt}\) is the rate of increase in solid layer thickness.
06

Equate the two heat flux expressions

Equating the two expressions for the heat flux, we have: \(-k_{v} \frac{dT}{dx}=\rho h_{sf}\frac{dS}{dt}\).
07

Rearrange and integrate

Rearrange the expression above and integrate both sides with respect to time: \(\int_{0}^{t} \frac{dS}{dt} dt = -\frac{k_{v}}{\rho h_{sf}} \int_{T_{i}}^{T_{f}} dx\), resulting in \(S(t) = -\frac{k_{v}}{\rho h_{sf}} (T_f-T_i)\int_{0}^{t} dt\). By integrating, we obtain the relation between the solid layer thickness, S, as a function of time, t: \(S(t) = \frac{-k_{v}(T_f-T_i)t}{\rho h_{sf}}\). This equation describes how the thickness of the solidified metal layer increases with time during the solidification process.

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Key Concepts

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

Heat Transfer in Molds
The process of solidification of metals in molds is a classic heat transfer problem involving the complex interaction between a liquid metal and its solidifying front. In this context, the dominant resistance to heat flow typically occurs within the mold wall, especially when the mold is a poor conductor.

When molten metal cools inside a mold, the heat must escape through the mold walls for solidification to occur. The ability of the mold to conduct heat away from the molten metal is a critical factor in how quickly the metal solidifies. The rate at which heat is transferred depends on the thermal conductivity of the mold material, symbolized as \(k_v\), and its thermal diffusivity, represented by \(\alpha_w\).

During solidification, heat flows from the molten metal, which is at its fusion or melting point temperature \(T_f\), into the cooler mold wall at a temperature \(T_i\). This heat flow is governed by the properties of the mold material and the shape and thickness of the mold walls. To ensure even and defect-free casting, it is essential to understand and control this heat transfer process.

In designing molds, one must consider the mold's material properties, including its ability to withstand high temperatures without degrading and its thermal properties to efficiently manage the heat transfer. This allows for predicting the solidification rate of the metal, which is crucial for determining the cycle time of the casting process and the overall quality of the cast metal parts.
Fourier's Law of Heat Conduction
Fourier's law of heat conduction serves as the foundation for understanding how heat moves through materials. This law states that the heat flux, \(q\), through a material is proportional to the negative gradient of the temperature, symbolized as \(-\frac{dT}{dx}\), and is directly related to the material's thermal conductivity, \(k_v\). Mathematically, Fourier's law is represented by the equation \(q = -k_v \frac{dT}{dx}\).

This simple yet powerful principle implies that heat will move from regions of higher temperature to regions of lower temperature, and the rate at which it does so is dependent on the thermal conductivity of the material through which it is passing. In the context of heat transfer in molds, Fourier's law explains how heat is conducted from the hot, solidifying metal into the cooler mold walls.

The relationship described by Fourier's law allows engineers to design and analyze cooling systems for molds, ensuring the solidification process is efficient and that the final metal product has the desired properties. It can also help in making predictions about the solid layer thickness over time during the cooling process and is crucial in designing molds with optimal thickness and materials for controlled cooling.
Latent Heat of Fusion
The latent heat of fusion, \(h_{sf}\), is an inherent property of materials that signifies the amount of heat required to change a unit mass from solid to liquid or vice versa at constant pressure and temperature, without changing its temperature. In the context of metal casting, it represents the heat that must be extracted from the molten metal for solidification to occur, without a change in the metal's temperature.

During solidification, the latent heat of fusion is released by the metal as it transitions from liquid to solid. This heat needs to be effectively conducted away through the mold for the metal to solidify. The amount of energy involved in this process is substantial; thus, understanding and accounting for the latent heat is crucial in designing molds and cooling systems that can handle this energy transfer.

The equation \(q = \rho h_{sf} \frac{dS}{dt}\) shows the relation between the heat flux and the rate of increase in solid layer thickness. It is essential to consider this when calculating the cooling time for a particular metal and mold combination. By managing the latent heat effectively, manufacturers can control the solidification rate, which influences the microstructure and mechanical properties of the final metal product. A detailed comprehension of latent heat of fusion helps metallurgists to ensure that casted materials have the appropriate characteristics for their intended applications.

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Most popular questions from this chapter

A support rod \(\left(k=15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \alpha=4.0 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\right)\) of diameter \(D=15 \mathrm{~mm}\) and length \(L=100 \mathrm{~mm}\) spans a channel whose walls are maintained at a temperature of \(T_{b}=300 \mathrm{~K}\). Suddenly, the rod is exposed to a cross flow of hot gases for which \(T_{\infty}=600 \mathrm{~K}\) and \(h=75 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The channel walls are cooled and remain at \(300 \mathrm{~K}\). (a) Using an appropriate numerical technique, determine the thermal response of the rod to the convective heating. Plot the midspan temperature as a function of elapsed time. Using an appropriate analytical model of the rod, determine the steadystate temperature distribution, and compare the result with that obtained numerically for very long elapsed times. (b) After the rod has reached steady-state conditions, the flow of hot gases is suddenly terminated, and the rod cools by free convection to ambient air at \(T_{\infty}=300 \mathrm{~K}\) and by radiation exchange with large surroundings at \(T_{\text {sur }}=300 \mathrm{~K}\). The free convection coefficient can be expressed as \(h\left(\mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\right)=C\) \(\Delta T^{n}\), where \(C=4.4 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}^{1.188}\) and \(n=0.188\). The emissivity of the rod is \(0.5\). Determine the subsequent thermal response of the rod. Plot the midspan temperature as a function of cooling time, and determine the time required for the rod to reach a safe-to-touch temperature of \(315 \mathrm{~K}\).

A plane wall \(\left(\rho=4000 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=500 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right.\), \(k=10 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) of thickness \(L=20 \mathrm{~mm}\) initially has a linear, steady-state temperature distribution with boundaries maintained at \(T_{1}=0^{\circ} \mathrm{C}\) and \(T_{2}=100^{\circ} \mathrm{C}\). Suddenly, an electric current is passed through the wall, causing uniform energy generation at a rate \(\dot{q}=2 \times 10^{7} \mathrm{~W} / \mathrm{m}^{3}\). The boundary conditions \(T_{1}\) and \(T_{2}\) remain fixed. (a) On \(T-x\) coordinates, sketch temperature distributions for the following cases: (i) initial condition \((t \leq 0)\); (ii) steady-state conditions \((t \rightarrow \infty\) ), assuming that the maximum temperature in the wall exceeds \(T_{2}\); and (iii) for two intermediate times. Label all important features of the distributions. (b) For the system of three nodal points shown schematically (1, \(m, 2)\), define an appropriate control volume for node \(m\) and, identifying all relevant processes, derive the corresponding finitedifference equation using either the explicit or implicit method. (c) With a time increment of \(\Delta t=5 \mathrm{~s}\), use the finitedifference method to obtain values of \(T_{m}\) for the first \(45 \mathrm{~s}\) of elapsed time. Determine the corresponding heat fluxes at the boundaries, that is, \(q_{x}^{\prime \prime}\) \((0,45 \mathrm{~s})\) and \(q_{x}^{\prime \prime}(20 \mathrm{~mm}, 45 \mathrm{~s})\). (d) To determine the effect of mesh size, repeat your analysis using grids of 5 and 11 nodal points ( \(\Delta x=5.0\) and \(2.0 \mathrm{~mm}\), respectively).

Consider the series solution, Equation \(5.42\), for the plane wall with convection. Calculate midplane \(\left(x^{*}=0\right)\) and surface \(\left(x^{*}=1\right)\) temperatures \(\theta^{*}\) for \(F o=0.1\) and 1 , using \(B i=0.1,1\), and 10 . Consider only the first four eigenvalues. Based on these results, discuss the validity of the approximate solutions, Equations \(5.43\) and \(5.44\).

One end of a stainless steel (AISI 316) rod of diameter \(10 \mathrm{~mm}\) and length \(0.16 \mathrm{~m}\) is inserted into a fixture maintained at \(200^{\circ} \mathrm{C}\). The rod, covered with an insulating sleeve, reaches a uniform temperature throughout its length. When the sleeve is removed, the rod is subjected to ambient air at \(25^{\circ} \mathrm{C}\) such that the convection heat transfer coefficient is \(30 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). (a) Using the explicit finite-difference technique with a space increment of \(\Delta x=0.016 \mathrm{~m}\), estimate the time required for the midlength of the rod to reach \(100^{\circ} \mathrm{C}\). (b) With \(\Delta x=0.016 \mathrm{~m}\) and \(\Delta t=10 \mathrm{~s}\), compute \(T(x, t)\) for \(0 \leq t \leq t_{1}\), where \(t_{1}\) is the time required for the midlength of the rod to reach \(50^{\circ} \mathrm{C}\). Plot the temperature distribution for \(t=0,200 \mathrm{~s}, 400 \mathrm{~s}\), and \(t_{1}\).

Two plates of the same material and thickness \(L\) are at different initial temperatures \(T_{i, 1}\) and \(T_{i, 2}\), where \(T_{i, 2}>T_{i, 1}\). Their faces are suddenly brought into contact. The external surfaces of the two plates are insulated. (a) Let a dimensionless temperature be defined as \(T *(F o) \equiv\left(T-T_{i, 1}\right) /\left(T_{i, 2}-T_{i, 1}\right)\). Neglecting the thermal contact resistance at the interface between the plates, what are the steady-state dimensionless temperatures of each of the two plates, \(T_{s s, 1}^{*}\) and \(T_{s s, 2}^{*}\) ? What is the dimensionless interface temperature \(T_{\text {in }}^{*}\) at any time? (b) An effective overall heat transfer coefficient between the two plates can be defined based on the instantaneous, spatially averaged dimensionless plate temperatures, \(U_{\mathrm{eff}}^{*} \equiv q^{*} /\left(\bar{T}_{2}^{*}-\bar{T}_{1}^{*}\right)\). Noting that a dimensionless heat transfer rate to or from either of the two plates may be expressed as \(q^{*}=d\left(Q / Q_{o}\right) / d F o\), determine an expression for \(U_{\text {eif }}^{*}\) for \(F o>0.2\).
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