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Chapter 5: Problem 4

A plate of thickness \(2 L\), surface area \(A_{s}\), mass \(M\), and specific heat \(c_{p}\), initially at a uniform temperature \(T_{i}\), is suddenly heated on both surfaces by a convection process \(\left(T_{\infty}, h\right)\) for a period of time \(t_{o}\), following which the plate is insulated. Assume that the midplane temperature does not reach \(T_{\infty}\) within this period of time. (a) Assuming \(B i \geqslant 1\) for the heating process, sketch and label, on \(T-x\) coordinates, the following temperature distributions: initial, steady- state \((t \rightarrow \infty), T\left(x, t_{o}\right)\), and at two intermediate times between \(t=t_{o}\) and \(t \rightarrow \infty\). (b) Sketch and label, on \(T+\) coordinates, the midplane and exposed surface temperature distributions. (c) Repeat parts (a) and (b) assuming \(B i \ll 1\) for the plate. (d) Derive an expression for the steady-state temperature \(T(x, \infty)=T_{f}\), leaving your result in terms of plate parameters \(\left(M, c_{p}\right)\), thermal conditions \(\left(T_{i}, T_{\infty}\right.\), \(h)\), the surface temperature \(T(L, t)\), and the heating time \(t_{o}\).

Short Answer

Expert verified
The temperature distribution of the plate for various time instants and different Biot numbers can be analyzed by sketching the distributions on T-x and T-t coordinates. For \(Bi \geq 1\), the temperature distributions will have slopes within the plate and will vary from the initial uniform temperature to a linear steady-state distribution. For \(Bi \ll 1\), the entire plate will have a nearly uniform temperature equal to \(T_\infty\) for all instances. The steady-state temperature, \(T_f\), can be expressed as: \[T_f = \frac{A_s \cdot 2 \cdot h (T_\infty - T(L,t)) \cdot t_o}{M c_p} + T_i\] where \(A_s\) is the surface area, \(M\) is the mass, \(c_p\) is the specific heat, \(h\) is the convection coefficient, \(T_\infty\) is the ambient temperature, \(T(L, t)\) is the surface temperature at a position \(L\), and \(t_o\) is the heating time.

Step by step solution

01

(a) Sketching temperature distributions for Bi 鈮� 1

We have a plate of thickness \(2L\), and the Biot number is given as \(Bi \geq 1\), meaning that conduction in the plate is important compared to convection from the surface. To sketch the temperature distribution for various time instants, we need to analyze the convection process. Initially, the plate's temperature is uniform (Ti); thus, the initial temperature distribution is a straight line at Ti. At steady-state, the temperature distribution will be linear, with the maximum temperature at the exposed surfaces and decreasing towards the midplane. At time \(t = t_o\), the temperature distribution will show an increase in temperature at both surfaces due to exposure to heating processes. At intermediate times between \(t = t_o\) and \(t \rightarrow \infty\), we'll observe a gradual transition from \(t = t_o\) to the steady-state.
02

(b) Sketching midplane and exposed surface temperature distributions for Bi 鈮� 1

Here, we plot the temperature vs. time at the midplane and exposed surfaces of the plate. Initially, both exposed surfaces and the midplane have the same temperature (Ti). When heating starts, the temperature at the exposed surfaces will rise quickly due to convection, while the midplane temperature will increase at a slower rate. At steady-state, the exposed surface will have the maximum temperature, and the midplane temperature will reach a relatively stable value.
03

(c) Sketching temperature distributions for Bi

For \(Bi \ll 1\), the conduction inside the plate is negligible, and the dominant mode of heat transfer is convection on both surfaces. Therefore, the initial, steady-state, intermediate, and \(t = t_o\) temperature distributions will show almost no slope in the plate, and the entire plate will have a nearly uniform temperature equal to \(T_\infty\).
04

(d) Derivation of the steady-state temperature T(x, 鈭�)=Tf

We know that the total energy in the plate at steady-state would be: \[m c_p (T_f - T_i)\] This energy will come from the energy provided through convection on both surfaces. The rate of convection on each surface can be given by the equation: \[q'' = h (T_\infty - T(L,t))\] As both surfaces undergo convection for time \(t_o\), the total energy supplied by convective heat transfer is: \[A_s \cdot 2 \cdot q'' \cdot t_o\] Setting the total energy in the plate equal to the convection energy, we can write: \[\begin{aligned} m c_p (T_f - T_i) &= A_s \cdot 2 \cdot q'' \cdot t_o \\ M c_p (T_f - T_i) &= A_s \cdot 2 \cdot h (T_\infty - T(L,t)) \cdot t_o \end{aligned}\] Now, we solve for the steady-state temperature, Tf: \[T_f = \frac{A_s \cdot 2 \cdot h (T_\infty - T(L,t)) \cdot t_o}{M c_p} + T_i\] This equation represents the steady-state temperature in terms of the plate parameters (M, cp), thermal conditions (Ti, T鈧溾垶, h), the surface temperature \(T(L, t)\), and the heating time \(t_o\).

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Key Concepts

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

Biot Number
The Biot number (Bi) is a dimensionless parameter that compares the rate of heat conduction within a body to the rate of convective heat transfer across the body's boundary to the surrounding fluid. It is defined as:
\[ Bi = \frac{hL_c}{k} \]
where \(h\) is the convective heat transfer coefficient, \(L_c\) is the characteristic length (typically, the volume of the body divided by the surface area through which heat is transferred), and \(k\) is the thermal conductivity of the material.
In the context of the exercise, assuming \(Bi \geq 1\) signifies that conduction resistance within the plate is significant compared to the boundary resistance by convection. Conversely, when \(Bi \ll 1\), it means that convection is the dominant mode of heat transfer, and temperature variation across the thickness of the plate is negligible. This determines the shape of the temperature distribution within the plate over time.
Steady-State Temperature Distribution
When analyzing heat transfer problems, steady-state temperature distribution is a condition where the temperature in a body does not change with time. This occurs after a sufficiently long period of exposure to a constant thermal environment. In steady state, all internal heat generation or absorption is balanced by the heat transfer at the boundaries.
For the plate in the exercise, the steady-state temperature distribution is reached when the temperatures at any given point in the plate no longer change with time. Given that the Biot number is greater than or equal to one, the temperature profile across the plate's thickness is linear. The greatest temperatures are found at the surfaces exposed to convection and gradually decrease towards the center of the plate. The plot of this distribution would graphically show a line with a slope that reflects the rate of temperature decrease from the surface to the midplane.
Midplane Temperature
In the context of heat transfer through a plate, the midplane temperature refers to the temperature at the center of the plate's thickness. This temperature is significant because it represents the slowest heating point in a scenario where heat is applied to the opposite faces of the plate, as in the given exercise.
For the case of \(Bi \geq 1\), the heat conduction within the plate cannot be ignored, leading to a slower increase in the midplane's temperature compared to the exposed surfaces. A plot of this increase would show a time-temperature curve that starts at the initial temperature \(T_i\), gradually increases as heat penetrates the plate, and eventually flattens as it approaches the steady-state condition. The temperature at the center of the plate is crucial for determining when the material has achieved thermal equilibrium.
Thermal Energy Balance
The concept of thermal energy balance is used to establish an equilibrium between the heat entering a system and the heat leaving it. It's an application of the first law of thermodynamics to heat transfer problems.
In the case of the plate undergoing a convection process on both surfaces, we consider the energy balance at steady state. The amount of thermal energy stored in the plate is equal to the total energy transferred from the surroundings through convective heat transfer:
\[M c_p (T_f - T_i) = A_s \cdot 2 \cdot h (T_{\infty} - T(L,t)) \cdot t_o\]
In this equation, \(M\) is the mass of the plate, \(c_p\) is the specific heat capacity, \(T_f\) is the final steady-state temperature, and \(T_i\) is the initial temperature. The right side represents convective heat absorbed over the surface area \(A_s\), with \(h\) being the convective heat transfer coefficient and \(T_{\infty}\) and \(T(L,t)\) representing the bulk fluid temperature and the temperature at the surface of the plate at time \(t\), respectively.

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Most popular questions from this chapter

Small spherical particles of diameter \(D=50 \mu \mathrm{m}\) contain a fluorescent material that, when irradiated with white light, emits at a wavelength corresponding to the material's temperature. Hence the color of the particle varies with its temperature. Because the small particles are neutrally buoyant in liquid water, a researcher wishes to use them to measure instantaneous local water temperatures in a turbulent flow by observing their emitted color. If the particles are characterized by a density, specific heat, and thermal conductivity of \(\rho=999 \mathrm{~kg} / \mathrm{m}^{3}\), \(k=1.2 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), and \(c_{p}=1200 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), respectively, determine the time constant of the particles. Hint: Since the particles travel with the flow, heat transfer between the particle and the fluid occurs by conduction. Assume lumped capacitance behavior.

The objective of this problem is to develop thermal models for estimating the steady-state temperature and the transient temperature history of the electrical transformer shown. The external transformer geometry is approximately cubical, with a length of \(32 \mathrm{~mm}\) to a side. The combined mass of the iron and copper in the transformer is \(0.28 \mathrm{~kg}\), and its weighted-average specific heat is \(400 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\). The transformer dissipates \(4.0 \mathrm{~W}\) and is operating in ambient air at \(T_{\infty}=20^{\circ} \mathrm{C}\), with a convection coefficient of \(10 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). List and justify the assumptions made in your analysis, and discuss limitations of the models. (a) Beginning with a properly defined control volume, develop a model for estimating the steady-state temperature of the transformer, \(T(\infty)\). Evaluate \(T(\infty)\) for the prescribed operating conditions. (b) Develop a model for estimating the thermal response (temperature history) of the transformer if it is initially at a temperature of \(T_{i}=T_{\infty}\) and power is suddenly applied. Determine the time required for the transformer to come within \(5^{\circ} \mathrm{C}\) of its steady-state operating temperature.

A soda lime glass sphere of diameter \(D_{1}=25 \mathrm{~mm}\) is encased in a bakelite spherical shell of thickness \(L=\) \(10 \mathrm{~mm}\). The composite sphere is initially at a uniform temperature, \(T_{i}=40^{\circ} \mathrm{C}\), and is exposed to a fluid at \(T_{\infty}=10^{\circ} \mathrm{C}\) with \(h=30 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Determine the center temperature of the glass at \(t=200 \mathrm{~s}\). Neglect the thermal contact resistance at the interface between the two materials.

A spherical vessel used as a reactor for producing pharmaceuticals has a 5 -mm-thick stainless steel wall \((k=17 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) and an inner diameter of \(D_{i}=1.0 \mathrm{~m}\). During production, the vessel is filled with reactants for which \(\rho=1100 \mathrm{~kg} / \mathrm{m}^{3}\) and \(c=2400 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), while exothermic reactions release energy at a volumetric rate of \(\dot{q}=10^{4} \mathrm{~W} / \mathrm{m}^{3}\). As first approximations, the reactants may be assumed to be well stirred and the thermal capacitance of the vessel may be neglected. (a) The exterior surface of the vessel is exposed to ambient air \(\left(T_{\infty}=25^{\circ} \mathrm{C}\right)\) for which a convection coefficient of \(h=6 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) may be assumed. If the initial temperature of the reactants is \(25^{\circ} \mathrm{C}\), what is the temperature of the reactants after \(5 \mathrm{~h}\) of process time? What is the corresponding temperature at the outer surface of the vessel? (b) Explore the effect of varying the convection coefficient on transient thermal conditions within the reactor. A spherical vessel used as a reactor for producing pharmaceuticals has a 5 -mm-thick stainless steel wall \((k=17 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) and an inner diameter of \(D_{i}=1.0 \mathrm{~m}\). During production, the vessel is filled with reactants for which \(\rho=1100 \mathrm{~kg} / \mathrm{m}^{3}\) and \(c=2400 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), while exothermic reactions release energy at a volumetric rate of \(\dot{q}=10^{4} \mathrm{~W} / \mathrm{m}^{3}\). As first approximations, the reactants may be assumed to be well stirred and the thermal capacitance of the vessel may be neglected. (a) The exterior surface of the vessel is exposed to ambient air \(\left(T_{\infty}=25^{\circ} \mathrm{C}\right)\) for which a convection coefficient of \(h=6 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) may be assumed. If the initial temperature of the reactants is \(25^{\circ} \mathrm{C}\), what is the temperature of the reactants after \(5 \mathrm{~h}\) of process time? What is the corresponding temperature at the outer surface of the vessel? (b) Explore the effect of varying the convection coefficient on transient thermal conditions within the reactor.

Common transmission failures result from the glazing of clutch surfaces by deposition of oil oxidation and decomposition products. Both the oxidation and decomposition processes depend on temperature histories of the surfaces. Because it is difficult to measure these surface temperatures during operation, it is useful to develop models to predict clutch-interface thermal behavior. The relative velocity between mating clutch plates, from the initial engagement to the zero-sliding (lock-up) condition, generates heat that is transferred to the plates. The relative velocity decreases at a constant rate during this period, producing a heat flux that is initially very large and decreases linearly with time, until lock-up occurs. Accordingly, \(q_{f}^{\prime \prime}=q_{o}^{\prime \prime}=\left[1-\left(t / t_{\mathrm{lu}}\right)\right]\), where \(q_{o}^{\prime \prime}=1.6 \times 10^{7} \mathrm{~W} / \mathrm{m}^{2}\) and \(t_{1 \mathrm{u}}=100 \mathrm{~ms}\) is the lock-up time. The plates have an initial uniform temperature of \(T_{i}=40^{\circ} \mathrm{C}\), when the prescribed frictional heat flux is suddenly applied to the surfaces. The reaction plate is fabricated from steel, while the composite plate has a thinner steel center section bonded to low- conductivity friction material layers. The thermophysical properties are \(\rho_{s}=\) \(7800 \mathrm{~kg} / \mathrm{m}^{3}, c_{\mathrm{s}}=500 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), and \(k_{s}=40 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) for the steel and \(\rho_{\mathrm{im}}=1150 \mathrm{~kg} / \mathrm{m}^{3}, c_{\mathrm{fm}}=1650 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), and \(k_{\mathrm{fm}}=4 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) for the friction material. (a) On \(T-t\) coordinates, sketch the temperature history at the midplane of the reaction plate, at the interface between the clutch pair, and at the midplane of the composite plate. Identify key features. (b) Perform an energy balance on the clutch pair over the time interval \(\Delta t=t_{\mathrm{lu}}\) to determine the steadystate temperature resulting from clutch engagement. Assume negligible heat transfer from the plates to the surroundings. (c) Compute and plot the three temperature histories of interest using the finite-element method of FEHT or the finite-difference method of IHT (with \(\Delta x=0.1 \mathrm{~mm}\) and \(\Delta t=1 \mathrm{~ms}\) ). Calculate and plot the frictional heat fluxes to the reaction and composite plates, \(q_{\mathrm{rp}}^{\prime \prime}\) and \(q_{\mathrm{cp}}^{\prime \prime}\), respectively, as a function of time. Comment on features of the temperature and heat flux histories. Validate your model by comparing predictions with the results from part (b). Note: Use of both \(F E H T\) and \(I H T\) requires creation of a look-up data table for prescribing the heat flux as a function of time.
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