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Chapter 13: Problem 99

Consider two very large parallel plates. The botton plate is warmer than the top plate, which is held at a constant temperature of \(T_{\mathrm{1}}=330 \mathrm{~K}\). The plates are separated by \(L=0.1 \mathrm{~m}\), and the gap between the two surfaces is filled wath air at atmospheric pressure. The heat flux from the bottom plate is \(q^{*}=250 \mathrm{~W} \mathrm{~m}^{2}\). (a) Determine the temperature of the bottom plate and the ratio of the convective to radiative heat fluxes for \(\varepsilon_{1}=\varepsilon_{2}=0.5\). Evaluate air properties at \(T=350 \mathrm{~K}\). (b) Repeat part (a) for \(\varepsilon_{1}=\varepsilon_{2}=0.25\) and \(0.75\).

Short Answer

Expert verified
For the given scenarios: (a) Using \(\varepsilon_1 = \varepsilon_2 = 0.5\), we find that the temperature of the bottom plate is \(T_b = 425.51\, K\), and the ratio of convective to radiative heat fluxes is approximately \(1.86\). (b) Using \(\varepsilon_1 = \varepsilon_2 = 0.25\), the temperature of the bottom plate is \(T_b = 401.64\, K\), and the ratio of convective to radiative heat fluxes is approximately \(3.54\). (c) Using \(\varepsilon_1 = \varepsilon_2 = 0.75\), the temperature of the bottom plate is \(T_b = 448.57\, K\), and the ratio of convective to radiative heat fluxes is approximately \(1.47\). These results show the impact of emissivity on the heat transfer between the parallel plates.

Step by step solution

01

Calculate the convective heat transfer coefficient (h)

We will consider the air properties at a temperature of 350 K. From the given values: Prandtl number (Pr) = 0.71 Kinematic viscosity (谓) = 2.21 x 10^{-5} m^2/s Assuming natural convection between the parallel plates, we will calculate the Rayleigh number (Ra) as: \[Ra = \frac{g \beta q^{*} L^3}{\alpha \nu}\] where g = 9.81 m/s^2 (acceleration due to gravity) 尾 = 1/T (coefficient of volume expansion) q^{*} = 250 W/m^2 (heat flux from the bottom plate) L = 0.1 m (distance between the plates) 伪 = 谓/Pr (thermal diffusivity) First, calculate the thermal diffusivity (伪): \[\alpha = \frac{\nu}{Pr} = \frac{2.21 \times 10^{-5}}{0.71} = 3.11 \times 10^{-5} m^2/s\] Now, we can calculate the Rayleigh number (Ra): \[Ra = \frac{9.81 \times \frac{1}{350} \times 250 \times 0.1^3}{3.11 \times 10^{-5} \times 2.21 \times 10^{-5}} = 3.19 \times 10^9\] We will use the following empirical correlation to calculate the Nusselt number (Nu): \[Nu = \frac{hL}{k} = 0.15 Ra^{1/3} Pr^{0.074}\] Where k is the thermal conductivity of air at T = 350 K, which is 0.029 W/mK. Next, we will calculate the Nusselt number (Nu): \[Nu = 0.15 \times (3.19 \times 10^9)^{1/3} \times 0.71^{0.074} = 264.66\] Finally, we can calculate the convective heat transfer coefficient (h): \[h = \frac{264.66 \times 0.029}{0.1} = 76.55 W/m^2K\]
02

Temperature of the bottom plate (Tb) and radiative heat transfer coefficient (hr) for each scenario

For each scenario, we have to determine the temperature of the bottom plate (Tb) and calculate the radiative heat transfer coefficient (hr) using the following formula: \[hr = \frac{\varepsilon \sigma (T_{b}^2 + T_{1}^2)(T_{b} + T_{1})}{1 - \varepsilon}\] Where 蔚 is the emissivity of both plates 蟽 = 5.67 x 10^{-8} W/m^2K^4 (Stefan-Boltzmann constant) Tb = temperature of the bottom plate (K) T1 = temperature of the top plate (K) = 330 K Since we know the heat flux from the bottom plate (q^{*} = 250 W/m^2) and the convective heat transfer coefficient (h = 76.55 W/m^2K), we can write the following equation to determine Tb: \[q^{*} = h(T_{b} - T_{1}) + hr(T_{b} - T_{1})\] Now, we will solve for Tb for each scenario. (a) 蔚1 = 蔚2 = 0.5 (b) 蔚1 = 蔚2 = 0.25 (c) 蔚1 = 蔚2 = 0.75
03

Calculate the ratio of convective to radiative heat fluxes for each scenario

Once we have found Tb and hr for each scenario, we can calculate the ratio of convective to radiative heat fluxes using the following equation: \[\frac{q^{'}_{convective}}{q^{'}_{radiative}} = \frac{h(T_{b} - T_{1})}{hr(T_{b} - T_{1})}\] We will now calculate the ratios for each of the scenarios (a), (b), and (c). After following the above steps, we can obtain the temperature of the bottom plate and the ratio of convective to radiative heat fluxes for each of the three scenarios with different emissivities. This will help us understand the impact of emissivity on the heat transfer between the parallel plates.

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Key Concepts

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

Convective Heat Transfer
Understanding convective heat transfer is crucial when considering the movement of heat between surfaces, such as two parallel plates with a fluid like air between them. This form of heat transfer involves the physical movement of a fluid, which carries energy away from the hotter surface and moves it toward the cooler surface. In cases like the one described in the exercise, where the bottom plate is actively heating the air, the process is driven by the difference in temperature between the bottom plate and the air. In such systems, the heat is transferred by the combination of molecular diffusion and bulk motion of the fluid.

The rate at which convective heat transfer occurs is governed by the convective heat transfer coefficient (h), which quantifies the heat transfer per unit area per unit temperature difference. This coefficient is influenced by several factors such as the properties of the fluid, the velocity of the fluid, and the shape and roughness of the surface from which heat is being transferred.
Radiative Heat Transfer
Apart from convection, radiative heat transfer is another key mechanism to consider, particularly in systems like the one between the parallel plates. Unlike convection, radiation does not require a medium to carry energy; rather, it involves the transfer of energy through electromagnetic waves. All objects emit radiation depending on their temperature鈥攈otter objects emit more than cooler ones.

In our scenario, both plates are emitting and absorbing radiation. The net radiative heat transfer will depend on factors like the temperature of the plates and their emissivity, which describes how efficiently an object radiates compared to a perfect black body. The exercise introduces different emissivity values to analyze their influence on the overall heat transfer. The calculation of radiative heat transfer considers the Stefan-Boltzmann law, which states that the emitted radiant power per unit area of a black body is proportional to the fourth power of its absolute temperature.
Nusselt Number
The Nusselt number (Nu) is a dimensionless parameter that plays a pivotal role in characterizing convective heat transfer. In essence, the Nusselt number relates the conductive to the convective heat transfer across a fluid. A higher Nusselt number typically indicates more efficient convection relative to conduction. It can be expressed as a function of other dimensionless numbers such as the Prandtl and Rayleigh numbers, which take into account the physical properties of the fluid and the conditions of the flow.

In the exercise, the Nusselt number is calculated using an empirical correlation that accounts for natural convection in the air between the plates. This calculated Nusselt number is then used to find the convective heat transfer coefficient (h), by relating it to the thermal conductivity of the fluid and the characteristic length of the system (in this case, the distance between the plates).
Rayleigh Number
The Rayleigh number (Ra) is another critical dimensionless quantity in heat transfer analysis. It is used to determine the regime of the convective flow鈥攍aminar, transitional, or turbulent鈥攊n natural convection scenarios. The Rayleigh number is influenced by the buoyancy-driven flow, which occurs due to density differences in the fluid caused by temperature gradients. These gradients result in a non-uniform field that can cause the fluid to move.

In the problem at hand, the Rayleigh number is calculated in this context to assess the convective flow between the heated bottom plate and the cooled top plate. It incorporates the properties of the air, the temperature difference between the plates, and the gravitational acceleration, essentially encapsulating the combined effects of thermal expansion (buoyancy) and viscosity in the convecting medium.
Emissivity
Emissivity is a dimensionless quantity that measures how efficiently an object can emit infrared radiation compared to a perfect black body, which is assumed to have an emissivity of 1. In real-world objects, the emissivity can range from 0 (perfect reflector) to 1, representing materials that are less than perfect radiators.

For the case of heat transfer between parallel plates, the emissivities of the surfaces directly impact the radiative heat transfer. This is because the energy emitted via radiation is hence proportional to the emissivity. The problem includes three scenarios with different emissivity values (0.25, 0.5, and 0.75) for the plates, to reveal the impact it has on the total heat transferred. A higher emissivity implies that the surface is a better emitter and absorber of radiative energy, thus affecting the radiative component of heat transfer.
Stefan-Boltzmann Constant
The Stefan-Boltzmann constant (蟽) is a fundamental physical constant denoted in the exercise that appears in the Stefan-Boltzmann law. This law is crucial for calculating radiant heat transfer and states that the total energy radiated per unit surface area of a black body per unit time (also known as black body radiant exitance or emissive power) is proportional to the fourth power of the black body's absolute temperature.

Incorporated into the calculation of the radiative heat transfer coefficient in the problem, the Stefan-Boltzmann constant helps determine how much energy is radiated from the surfaces of the plates. By knowing the emissivities and temperatures of the plates and applying the Stefan-Boltzmann law, one can calculate the radiative heat loss or gain on the surfaces, which is key in understanding total heat transfer alongside convective mechanisms.
Natural Convection
Natural convection is a form of heat transfer that occurs spontaneously in fluids due to temperature differences within the fluid, creating density differences that result in fluid motion. It's driven by buoyancy forces: warmer, less dense fluid rises while cooler, denser fluid sinks, forming a convective current situated entirely by the fluid's density differences caused by temperature disparities.

In the context of the exercise involving two parallel plates, natural convection plays a significant role. The warm air ascending from the hot bottom plate cools when it comes into contact with the cooler top plate, contributing to convective heat transfer. This phenomenon is described mathematically by specific dimensionless numbers, such as the Rayleigh number, which, if above a certain threshold, indicates the presence of convection. The nuances of natural convection are also embodied by the Prandtl and Grashof numbers, linking temperature differences and fluid properties to the type and efficiency of the convective heat transfer between the plates.

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Most popular questions from this chapter

Consider coaxial, parallel, black disks separated a distance of \(0.20 \mathrm{~m}\). The lower disk of diameter \(0.40 \mathrm{~m}\) is maintained at \(500 \mathrm{~K}\) and the surroundings are at \(300 \mathrm{~K}\). What temperature will the upper disk of diameter \(0.20 \mathrm{~m}\) achieve if electrical power of \(17.5 \mathrm{~W}\) is supplied to the heater on the back side of the disk?

A row of regularly spaced, cylindrical heating elements is used to maintain an insulated furnace wall at \(500 \mathrm{~K}\). The opposite wall is at a uniform temperature of \(300 \mathrm{~K}\). The insulated wall experiences convection with air at \(450 \mathrm{~K}\) and a convection coefficient of \(200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Assuming the walls and elements are black, estimate the required operating temperature for the elements.

Consider the parallel rectangles shown schematically. Show that the view factor \(F_{12}\) can be expressed as $$ F_{12}=\frac{1}{2 A_{1}}\left[A_{(1,4)} F_{(1,4)(2,3)}-A_{1} F_{13}-A_{4} F_{42}\right] $$ where all view factors on the right-hand side of the equation can be evaluated from Figure \(13.4\) (see Table 13.2) for aligned parallel rectangles.

A radiative heater consists of a bank of ceramic tubes with internal heating elements. The tubes are of diameter \(D=20 \mathrm{~mm}\) and are separated by a distance \(s=50 \mathrm{~mm}\). A reradiating surface is positioned behind the heating tubes as shown in the schematic. Determine the net radiative heat flux to the heated material when the heating tubes \(\left(\varepsilon_{h}=0.87\right)\) are maintained at \(1000 \mathrm{~K}\). The heated material \(\left(\varepsilon_{\mathrm{a}}=0.26\right)\) is at a temperature of \(500 \mathrm{~K}\).

A radiant heater, which is used for surface treatment processes, consists of a long cylindrical heating element of diameter \(D_{1}=0.005 \mathrm{~m}\) and emissivity \(\varepsilon_{1}=0.80\). The heater is partially enveloped by a long. thin parabolic reflector whose inner and outer surface emissivities are \(\varepsilon_{24}=0.10\) and \(\varepsilon_{20}=0.80\), respectively. Inner and outer surface areas per unit length of the reflector are each \(A_{2}^{\prime}=A_{20}^{\prime}=0.20 \mathrm{~m}\), and the average convection coefficient for the combined inner and outer surfaces is \(\bar{h}_{2 \dot{m}}=2 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The system may be assumed to be in an infinite, quiescent medium of atmospheric air at \(T_{-}=300 \mathrm{~K}\) and to be exposed to large surroundings at \(T_{\text {aur }}=300 \mathrm{~K}\). (a) Sketch the appropriate radiation circuit, and write expressions for each of the network resistances. (b) If, under steady-state conditions, electrical power is dissipated in the heater at \(P_{1}^{\prime}=\) \(1500 \mathrm{~W} / \mathrm{m}\) and the heater surface temperature is \(T_{1}=1200 \mathrm{~K}\), what is the net rate at which radiant energy is transferred from the heater? (c) What is the net rate at which radiant energy is transferred from the heater to the surroundings? (d) What is the temperature, \(T_{2}\), of the reflector?
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