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Chapter 2: Problem 59

A plane wall has constant properties, no internal heat generation, and is initially at a uniform temperature \(T_{i \cdot}\) Suddenly, the surface at \(x=L\) is heated by a fluid at \(T_{\infty}\) having a convection coefficient \(h\). At the same instant, the electrical heater is energized, providing a constant heat flux \(q_{o}^{\prime \prime}\) at \(x=0\). (a) On \(T-x\) coordinates, sketch the temperature distributions for the following conditions: initial condition \((t \leq 0)\), steady-state condition \((t \rightarrow \infty)\), and for two intermediate times. (b) On \(q_{x}^{\prime \prime}-x\) coordinates, sketch the heat flux corresponding to the four temperature distributions of part (a). (c) On \(q_{x}^{n}-t\) coordinates, sketch the heat flux at the locations \(x=0\) and \(x=L\). That is, show qualitatively how \(q_{x}^{\prime \prime}(0, t)\) and \(q_{x}^{\prime \prime}(L, t)\) vary with time. (d) Derive an expression for the steady-state temperature at the heater surface, \(T(0, \infty)\), in terms of \(q_{o}^{\prime \prime}\), \(T_{\infty}, k, h\), and \(L\).

Short Answer

Expert verified
The steady-state temperature at the heater surface, T(0, 鈭�), can be obtained through the following expression: \(T(0, \infty) = T^\infty + \frac{q鈧�''}{h} + \frac{k}{hL} q鈧�''\), where \(T^\infty\) is the fluid temperature, \(q鈧�''\) is the constant heat flux, k is the thermal conductivity, h is the convection coefficient, and L is the wall thickness. This formula considers the temperature and heat flux distributions under various conditions.

Step by step solution

01

Initial condition

 As the entire wall is initially at a uniform temperature, Ti, the temperature distribution on the T-x coordinates will be a horizontal line.
02

Steady-state condition

At steady-state, temperature distribution is linear between the two boundaries. The surface at x=L is heated by the fluid at T鈭�, and electrical heater at x=0 is providing a constant heat flux q鈧�鈥测��. Draw a straight line from the heated surface (x=0) to the cooled surface (x=L) on T-x coordinates.
03

Intermediate times

For the intermediate times, the temperature distribution will be an increasingly linear curve as time passes. Sketch two curves in between the initial and steady-state conditions; these will show the temperature distribution for the intermediate times. (b) Sketching Heat Flux distributions:
04

Heat flux for the initial condition

The heat flux for the initial condition will be zero as there is no heat transfer initially, so this will be a horizontal line at qx'' = 0.
05

Heat flux for the steady-state condition

At steady-state, the heat flux corresponding to the temperature distributions of part (a) will be constant along the x-direction since qx'' is a constant given by Fourier's Law. So, draw a horizontal line at the constant heat flux value.
06

Heat flux for intermediate times

For the intermediate times, the heat flux will gradually reach the steady-state value from an initially zero heat flux. Sketch two curves that show the heat flux distribution for the intermediate times. (c) Sketching Heat Flux at x=0 and x=L with respect to time:
07

Heat flux at x=0

On the qx鈥测��-t coordinates, sketch a horizontal line starting from the value of q鈧�鈥测�� at t=0, representing the heat flux at x=0, which remains constant with time.
08

Heat flux at x=L

At x=L, the heat flux varies with time and converges to a steady-state value as time goes to infinity. Sketch a curve starting from zero heat flux at t=0 that approaches the steady-state value of qx'' as t鈫掆垶. (d) Deriving steady-state temperature at the heater surface, T(0, 鈭�):
09

Applying Fourier's Law along the x-direction

Apply Fourier's Law to compute the heat transfer across the wall: \(-k \frac{dT}{dx}|_{x=0} = q鈧�''\)
10

Applying Convection and Fourier's Law at x=L

Apply convection and Fourier's Law at x = L: \(-k \frac{dT}{dx}|_{x=L} = h(T^\infty - T(L))\)
11

Setting up the steady-state temperature equation

We have two equations and two unknowns (T(0) and T(L)). Solve these equations to find T(0, 鈭�) in terms of the given variables: \(T(0, \infty) = T(L, \infty) - \frac{k}{hL} (-q鈧�'' - h(T^\infty - T(L)))\)
12

Simplify the expression

Simplify the expression to get the final formula for T(0, 鈭�): \(T(0, \infty) = T^\infty + \frac{q鈧�''}{h} + \frac{k}{hL} q鈧�''\) This expression gives the steady-state temperature at the heater surface in terms of given variables q鈧�鈥测��, T鈭�, k, h, and L.

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Key Concepts

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

Convection Coefficient
The convection coefficient, often denoted as \(h\), plays a critical role in problems involving heat transfer between a solid surface and a fluid in motion. It quantifies the rate at which heat is transferred per unit area per unit temperature difference between the surface and the fluid.

For instance, in our textbook exercise, the surface at \(x=L\) is being cooled by a fluid, and the convection coefficient helps determine how efficiently the fluid can remove heat from the surface. In real-life applications, it's a parameter influenced by fluid properties, flow velocity, and the nature of the surface itself.

When dealing with heat transfer problems, understanding how to calculate the convection coefficient鈥攚hether through empirical correlations for different flow conditions or from experimental data鈥攊s essential for accurate thermal analysis and design.
Heat Flux
Heat flux, represented by \(q''\), is a measure of the rate of heat energy transfer through a given surface area. It is usually expressed in watts per square meter \(\text{W/m}^2\). In our heat transfer scenario, we're looking at a constant heat flux \(q_{o}^{''}\) applied to the surface at \(x=0\), simulating an electrical heater.

Understanding heat flux is important for managing the thermal performance of systems and ensuring materials are not exposed to temperatures that may cause damage. During analysis, you would sketch how heat flux changes with position and time to visualize the process and better design the thermal system.
Fourier's Law
Fourier's Law is the cornerstone of heat conduction analysis and states that the heat transfer rate through a material is proportional to the negative gradient of temperature and the area through which heat is flowing. Mathematically, it's given by:\[q'' = -k \frac{dT}{dx}\]

In the exercise, Fourier's Law is used to connect the steady-state heat flux \(q_{o}^{''}\) with the temperature gradient at the heater surface. This law also informs the slope of the temperature distribution in steady-state conditions, as seen in the exercise's sketches.
Steady-State Temperature
Steady-state temperature refers to a condition where the temperature in a system does not change with time. In other words, the temperature distribution becomes time-independent. For the given exercise, we derive the steady-state temperature at the heater surface in part (d), which is the temperature after the system has stabilized and there are no further changes in temperature over time.

Reaching a steady-state condition is critical in many engineering applications for performance analysis and system design, ensuring that operational temperatures do not exceed safe or functional limits for the materials involved.
Thermal Conductivity
Thermal conductivity, denoted by \(k\), is a material property that indicates how well a material conducts heat. It is central to the exercise as it directly affects the rate at which heat is transferred through the wall by conduction.

In the provided solution, \(k\) is used to relate the heat flux \(q_{o}^{''}\) to the temperature gradient at the surface of the heater. Thermal conductivity varies between materials and is critical when selecting materials for their thermal management properties in designing and analyzing heat transfer systems.

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Most popular questions from this chapter

A pan is used to boil water by placing it on a stove, from which heat is transferred at a fixed rate \(q_{\sigma}\). There are two stages to the process. In Stage 1, the water is taken from its initial (room) temperature \(T_{i}\) to the boiling point, as heat is transferred from the pan by natural convection. During this stage, a constant value of the convection coefficient \(h\) may be assumed, while the bulk temperature of the water increases with time, \(T_{\infty}=T_{\infty}(t)\). In Stage 2, the water has come to a boil, and its temperature remains at a fixed value, \(T_{\infty}=T_{b}\), as heating continues. Consider a pan bottom of thickness \(L\) and diameter \(D\), with a coordinate system corresponding to \(x=0\) and \(x=L\) for the surfaces in contact with the stove and water, respectively. (a) Write the form of the heat equation and the boundary/ initial conditions that determine the variation of temperature with position and time, \(T(x, t)\), in the pan bottom during Stage 1. Express your result in terms of the parameters \(q_{o}, D, L, h\), and \(T_{\infty}\), as well as appropriate properties of the pan material. (b) During Stage 2, the surface of the pan in contact with the water is at a fixed temperature, \(T(L, t)=\) \(T_{L}>T_{b}\). Write the form of the heat equation and boundary conditions that determine the temperature distribution \(T(x)\) in the pan bottom. Express your result in terms of the parameters \(q_{o}, D, L\), and \(T_{L}\), as well as appropriate properties of the pan material.

A plane wall of thickness \(L=0.1 \mathrm{~m}\) experiences uniform volumetric heating at a rate \(\dot{q}\). One surface of the wall \((x=0)\) is insulated, and the other surface is exposed to \(\mathrm{a}\) fluid at \(T_{\infty}=20^{\circ} \mathrm{C}\), with convection heat transfer characterized by \(h=1000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Initially, the temperature distribution in the wall is \(T(x, 0)=a+b x^{2}\), where \(a=300^{\circ} \mathrm{C}, b=-1.0 \times 10^{40} \mathrm{C} / \mathrm{m}^{2}\), and \(x\) is in meters. Suddenly, the volumetric heat generation is deactivated ( \(\dot{q}=0\) for \(t \geq 0\) ), while convection heat transfer continues to occur at \(x=L\). The properties of the wall are \(\rho=7000 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=450 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), and \(k=90 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). (a) Determine the magnitude of the volumetric energy generation rate \(\dot{q}\) associated with the initial condition \((t<0)\). (b) On \(T-x\) coordinates, sketch the temperature distribution for the following conditions: initial condition \((t<0)\), steady-state condition \((t \rightarrow \infty)\), and two intermediate conditions. (c) On \(q_{x}^{\prime \prime}-t\) coordinates, sketch the variation with time of the heat flux at the boundary exposed to the convection process, \(q_{x}^{\prime \prime}(L, t)\). Calculate the corresponding value of the heat flux at \(t=0, q_{x}^{\prime \prime}(L, 0)\). (d) Calculate the amount of energy removed from the wall per unit area \(\left(\mathrm{J} / \mathrm{m}^{2}\right)\) by the fluid stream as the wall cools from its initial to steady-state condition.

A chemically reacting mixture is stored in a thin-walled spherical container of radius \(r_{1}=200 \mathrm{~mm}\), and the exothermic reaction generates heat at a uniform, but temperaturedependent volumetric rate of \(\dot{q}=\dot{q}_{o} \exp \left(-A / T_{o}\right)\), where \(\dot{q}_{o}=5000 \mathrm{~W} / \mathrm{m}^{3}, A=75 \mathrm{~K}\), and \(T_{o}\) is the mixture temperature in kelvins. The vessel is enclosed by an insulating material of outer radius \(r_{2}\), thermal conductivity \(k\), and emissivity \(\varepsilon\). The outer surface of the insulation experiences convection heat transfer and net radiation exchange with the adjoining air and large surroundings, respectively. (a) Write the steady-state form of the heat diffusion equation for the insulation. Verify that this equation is satisfied by the temperature distribution $$ T(r)=T_{s, 1}-\left(T_{s, 1}-T_{s, 2}\right)\left[\frac{1-\left(r_{1} / r\right)}{1-\left(r_{1} / r_{2}\right)}\right] $$ Sketch the temperature distribution, \(T(r)\), labeling key features. (b) Applying Fourier's law, show that the rate of heat transfer by conduction through the insulation may be expressed as $$ q_{r}=\frac{4 \pi k\left(T_{s, 1}-T_{s, 2}\right)}{\left(1 / r_{1}\right)-\left(1 / r_{2}\right)} $$ Applying an energy balance to a control surface about the container, obtain an alternative expression for \(q_{r}\), expressing your result in terms of \(\dot{q}\) and \(r_{1}\). (c) Applying an energy balance to a control surface placed around the outer surface of the insulation, obtain an expression from which \(T_{s, 2}\) may be determined as a function of \(\dot{q}, r_{1}, h, T_{\infty}, \varepsilon\), and \(T_{\text {sur }}\) (d) The process engineer wishes to maintain a reactor temperature of \(T_{o}=T\left(r_{1}\right)=95^{\circ} \mathrm{C}\) under conditions for which \(k=0.05 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, r_{2}=208 \mathrm{~mm}, h=5\) \(\mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}, \varepsilon=0.9, T_{\infty}=25^{\circ} \mathrm{C}\), and \(T_{\text {sur }}=35^{\circ} \mathrm{C}\). What is the actual reactor temperature and the outer surface temperature \(T_{s, 2}\) of the insulation? (e) Compute and plot the variation of \(T_{s, 2}\) with \(r_{2}\) for \(201 \leq r_{2} \leq 210 \mathrm{~mm}\). The engineer is concerned about potential burn injuries to personnel who may come into contact with the exposed surface of the insulation. Is increasing the insulation thickness a practical solution to maintaining \(T_{s, 2} \leq 45^{\circ} \mathrm{C}\) ? What other parameter could be varied to reduce \(T_{s, 2}\) ?

The steady-state temperature distribution in a onedimensional wall of thermal conductivity \(k\) and thickness \(L\) is of the form \(T=a x^{3}+b x^{2}+c x+d .\) Derive expressions for the heat generation rate per unit volume in the wall and the heat fluxes at the two wall faces \((x=0, L)\).

A hot water pipe with outside radius \(r_{1}\) has a temperature \(T_{1}\). A thick insulation, applied to reduce the heat loss, has an outer radius \(r_{2}\) and temperature \(T_{2}\). On \(T-r\) coordinates, sketch the temperature distribution in the insulation for one-dimensional, steady-state heat transfer with constant properties. Give a brief explanation, justifying the shape of your curve.
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