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Chapter 2: Problem 22

Starting with an energy balance on a rectangular volume element, derive the one-dimensional transient heat conduction equation for a plane wall with constant thermal conductivity and no heat generation.

Short Answer

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Based on the given step-by-step solution, provide a short answer explaining the one-dimensional transient heat conduction equation for a plane wall with constant thermal conductivity and no heat generation. The one-dimensional transient heat conduction equation for a plane wall with constant thermal conductivity and no heat generation is derived from an energy balance considering the net heat rate into a volume element and the rate of change of energy storage within the element. It is given by: ρ * c_p * ∂T / ∂t = k * ∂²T / ∂x² where ρ is the density, c_p is the specific heat at constant pressure, k is the thermal conductivity, T is the temperature, x is the position, and t is the time.

Step by step solution

01

Define a volume element

Consider a rectangular volume element in the plane wall of thickness Δx, height h, and width w in the x-direction. Define T as the temperature of the material at position x and time t.
02

Perform an energy balance

Analyze the energy change within the volume element between its two faces at x and x + Δx due to heat conduction. Incoming heat flux at x is q_x and outgoing heat flux at x + Δx is q_(x+Δx). The net heat rate into the volume element, Δq_net, is the difference between the rates of incoming and outgoing heat fluxes: Δq_net = q_x - q_(x+Δx)
03

Apply Fourier's law of heat conduction

Apply Fourier's law of heat conduction to express the heat flux in terms of temperature gradient and thermal conductivity (k). q = -k * ∂T / ∂x Calculate the incoming and outgoing heat fluxes at x and x + Δx: q_x = -k * ∂T / ∂x |_(x) q_(x+Δx) = -k * ∂T / ∂x |_(x+Δx) Plug the heat fluxes into the net heat rate equation: Δq_net = -k * ∂T / ∂x |_(x) - (-k * ∂T / ∂x |_(x+Δx)) = -k * ( ∂T / ∂x |_(x+Δx) - ∂T / ∂x |_(x))
04

Relate net heat rate to energy storage

The net heat rate into the volume element equals the rate of change of energy storage within the volume element. The energy storage per unit volume in the element is given by ρ*c_p * ∂T / ∂t, where ρ is the density and c_p is the specific heat at constant pressure of the wall material. The rate of change of energy storage in the volume element = (ρ * c_p * ∂T / ∂t) * h * w * Δx
05

Combine energy balance with energy storage

Equate the rate of change of energy storage to the net heat rate into the volume element: (ρ * c_p * ∂T / ∂t) * h * w * Δx = -k * ( ∂T / ∂x |_(x+Δx) - ∂T / ∂x |_(x)) * h * w Divide both sides by h * w * Δx: ρ * c_p * ∂T / ∂t = -k * ( ∂T / ∂x |_(x+Δx) - ∂T / ∂x |_(x)) / Δx
06

Take the limit as Δx approaches 0

Take the limit of the above equation as Δx approaches 0 to obtain the one-dimensional transient heat conduction equation: ρ * c_p * ∂T / ∂t = k * ∂²T / ∂x² This is the one-dimensional transient heat conduction equation for a plane wall with constant thermal conductivity and no heat generation.

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Most popular questions from this chapter

A spherical container of inner radius \(r_{1}=2 \mathrm{~m}\), outer radius \(r_{2}=2.1 \mathrm{~m}\), and thermal conductivity \(k=30 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) is filled with iced water at \(0^{\circ} \mathrm{C}\). The container is gaining heat by convection from the surrounding air at \(T_{\infty}=25^{\circ} \mathrm{C}\) with a heat transfer coefficient of \(h=18 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Assuming the inner surface temperature of the container to be \(0^{\circ} \mathrm{C},(a)\) express the differential equation and the boundary conditions for steady one- dimensional heat conduction through the container, \((b)\) obtain a relation for the variation of temperature in the container by solving the differential equation, and \((c)\) evaluate the rate of heat gain to the iced water.

A spherical vessel is filled with chemicals undergoing an exothermic reaction. The reaction provides a uniform heat flux on the inner surface of the vessel. The inner diameter of the vessel is \(5 \mathrm{~m}\) and its inner surface temperature is at \(120^{\circ} \mathrm{C}\). The wall of the vessel has a variable thermal conductivity given as \(k(T)=k_{0}(1+\beta T)\), where \(k_{0}=1.01 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), \(\beta=0.0018 \mathrm{~K}^{-1}\), and \(T\) is in \(\mathrm{K}\). The vessel is situated in a surrounding with an ambient temperature of \(15^{\circ} \mathrm{C}\), the vessel's outer surface experiences convection heat transfer with a coefficient of \(80 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). To prevent thermal burn on skin tissues, the outer surface temperature of the vessel should be kept below \(50^{\circ} \mathrm{C}\). Determine the minimum wall thickness of the vessel so that the outer surface temperature is \(50^{\circ} \mathrm{C}\) or lower.

Consider a steam pipe of length \(L=30 \mathrm{ft}\), inner radius \(r_{1}=2\) in, outer radius \(r_{2}=2.4\) in, and thermal conductivity \(k=7.2 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}\). Steam is flowing through the pipe at an average temperature of \(300^{\circ} \mathrm{F}\), and the average convection heat transfer coefficient on the inner surface is given to be \(h=12.5 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2}{ }^{\circ} \mathrm{F}\). If the average temperature on the outer surfaces of the pipe is \(T_{2}=175^{\circ} \mathrm{F},(a)\) express the differential equation and the boundary conditions for steady one-dimensional heat conduction through the pipe, \((b)\) obtain a relation for the variation of temperature in the pipe by solving the differential equation, and \((c)\) evaluate the rate of heat loss from the steam through the pipe.

In subsea oil and natural gas production, hydrocarbon fluids may leave the reservoir with a temperature of \(70^{\circ} \mathrm{C}\) and flow in subsea surrounding of \(5^{\circ} \mathrm{C}\). As a result of the temperature difference between the reservoir and the subsea surrounding, the knowledge of heat transfer is critical to prevent gas hydrate and wax deposition blockages. Consider a subsea pipeline with inner diameter of \(0.5 \mathrm{~m}\) and wall thickness of \(8 \mathrm{~mm}\) is used for transporting liquid hydrocarbon at an average temperature of \(70^{\circ} \mathrm{C}\), and the average convection heat transfer coefficient on the inner pipeline surface is estimated to be \(250 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The subsea surrounding has a temperature of \(5^{\circ} \mathrm{C}\) and the average convection heat transfer coefficient on the outer pipeline surface is estimated to be \(150 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). If the pipeline is made of material with thermal conductivity of \(60 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), by using the heat conduction equation (a) obtain the temperature variation in the pipeline wall, \((b)\) determine the inner surface temperature of the pipeline, \((c)\) obtain the mathematical expression for the rate of heat loss from the liquid hydrocarbon in the pipeline, and \((d)\) determine the heat flux through the outer pipeline surface.

Consider a steam pipe of length \(L\), inner radius \(r_{1}\), outer radius \(r_{2}\), and constant thermal conductivity \(k\). Steam flows inside the pipe at an average temperature of \(T_{i}\) with a convection heat transfer coefficient of \(h_{i}\). The outer surface of the pipe is exposed to convection to the surrounding air at a temperature of \(T_{0}\) with a heat transfer coefficient of \(h_{o^{*}}\) Assuming steady one-dimensional heat conduction through the pipe, \((a)\) express the differential equation and the boundary conditions for heat conduction through the pipe material, \((b)\) obtain a relation for the variation of temperature in the pipe material by solving the differential equation, and (c) obtain a relation for the temperature of the outer surface of the pipe.
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