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The wall of a drying oven is constructed by sandwiching an insulation material of thermal conductivity \(k=\) \(0.05 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) between thin metal shects. The oven air is at \(T_{s j}=300^{\circ} \mathrm{C}\), and the corresponding convection coefficient is \(h_{4}=30 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The inner wall surface absorbs a radiant flux of \(q_{\text {iat }}^{\prime \prime}=100 \mathrm{~W} / \mathrm{m}^{2}\) from hotter objects within the oven. The room air is at \(T_{m e}=25^{\circ} \mathrm{C}\), and the overall coefficient for convection and radiation from the outer surface is \(h_{e}=10 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). (a) Draw the thermal circuit for the wall and label all temperatures, heat rates, and thermal resistances. (b) What insulation thickness \(L\) is required to maintain the outer wall surface at a sofe-to-touch temperature of \(T_{e}=40^{\circ} \mathrm{C}\) ?

Step by step solution

01

Understand the Problem

We need to determine the thickness of insulation required to ensure the oven's outer surface temperature is safe. The thermal conductivity of the insulation, convection coefficients of inner and outer surfaces, radiant heat flux, and temperature of the surrounding air are given.

02

Model the System with a Thermal Circuit

Draw a thermal circuit to represent heat transfer through the oven wall. The components are: 1. Radiant heat absorbed by the inner surface, 2. Series of thermal resistances due to insulation and surfaces, and 3. Convection heat losses to the outer air.

03

Apply Thermal Resistance Formulas

The thermal resistance for the insulation is given by \( R_{insulation} = \frac{L}{k} \), where \( L \) is the thickness. Convective resistances at the surfaces are \( R_{inner} = \frac{1}{h_{i}} \) and \( R_{outer} = \frac{1}{h_{e}} \). Here, \( h_{i} = 30 \) W/m²⋅K and \( h_{e} = 10 \) W/m²⋅K.

04

Set Up Energy Balance Equation

The net heat transfer, after accounting for the radiant heat absorbed, is given by \[ q_{net} = T_{sj} - T_{e} \times \left( R_{inner} + R_{insulation} + R_{outer} \right) \] Substitute \( T_{sj} = 300^{\circ}C \), \( T_{e} = 40^{\circ}C \), and solve for \( q_{net} \).

05

Solve for Insulation Thickness \( L \)

Rearrange the energy balance equation to solve for \( L \): \[ L = \frac{(T_{sj} - T_{e}) \times (R_{inner} + R_{outer}) - q_{net}}{k} \] Plug in all known values to compute \( L \), ensuring the outer surface temperature is \( 40^{\circ}C \).

06

Final Calculation and Result Verification

Verify the calculated insulation thickness \( L \) ensures the outer wall temperature is \( 40^{\circ}C \) by substituting back into the heat transfer equations, checking for energy conservation and consistency with boundary conditions.

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Key Concepts

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

Thermal Circuit

In the world of heat transfer, a thermal circuit helps us visualize and analyze how heat flows through different materials or components. Imagine it like an electrical circuit, but instead of electricity, we are dealing with heat. In the case of the oven wall, our thermal circuit consists of several key elements that allow us to see where the heat travels and how it interacts with each part of the wall.

• Heat Sources and Sinks: Here, we start with the radiant heat absorbed by the oven's inner surface. This acts like a battery in our circuit, providing thermal energy.
• Thermal Resistances: These are equivalent to resistors in an electrical circuit. Each resistance represents a barrier to heat flow, including the insulation material sandwiched between metal sheets, as well as the convective resistances on both inner and outer surfaces.
• Temperature Nodes: These points show us the temperature at various stages in the heat flow path, like the oven air temperature and the target outer surface temperature.

To effectively analyze the heat transfer, we label each component with its respective temperatures and resistances. This approach allows us to set up equations that quantify the heat flow, helping us determine the necessary insulation properties.

Convection Heat Transfer

Convection is one of the key mechanisms for heat exchange between a surface and a moving fluid, such as the air inside and outside the oven. In our context, convection heat transfer occurs at both the inner and outer surfaces of the oven wall.

• Inner Surface Convection: The convection heat transfer coefficient here is relatively high at 30 W/m²⋅K, indicating efficient heat exchange between the oven air and the wall's inner surface.
• Outer Surface Convection: On the opposite side, with a coefficient of 10 W/m²⋅K, heat transfers away from the outer wall surface into the surrounding room air. This lower coefficient reflects a less intensive heat exchange, partly due to the lower temperature difference and the outside environment being calmer than the oven’s interior.

This interplay of convection on either side of the wall is crucial to understanding how heat moves in relation to the surrounding environments, and influences the insulation's role in maintaining safe-to-touch temperatures.

Insulation Material

The insulation material is the key player in reducing heat transfer through the oven wall. Its purpose is to provide a barrier that minimizes heat loss from the hot interior of the oven to the cooler outside environment.

• Thermal Conductivity: The thermal conductivity of the insulation is given as 0.05 W/mâ‹…K, which indicates how easily heat can pass through it. A lower value means better insulation as it resists heat flow.
• Role in Safety: By controlling how much heat reaches the outer wall, insulation ensures that the surface does not become dangerously hot, keeping it safe to touch.

The thickness of this insulation layer is crucial because it directly influences its ability to maintain a low outer surface temperature. The right thickness, calculated using thermal resistance equations, ensures effective insulation that conserves energy while maintaining safety standards.

Thermal Resistance

Thermal resistance is a measure of a material's ability to resist heat flow. In our oven wall problem, we deal with different types of thermal resistance, each playing a unique role in controlling the overall heat transfer.

• Insulation Thermal Resistance: This is calculated by dividing the insulation thickness by its thermal conductivity, given by the formula: \[ R_{insulation} = \frac{L}{k} \]
• Convective Resistances: These resistances occur at the surfaces exposed to air, calculated as the inverse of the convection coefficients:
  • Inner Surface: \( R_{inner} = \frac{1}{h_{i}} \)
  • Outer Surface: \( R_{outer} = \frac{1}{h_{e}} \)

Understanding these resistances helps establish a thermal balance. The sum of these resistances determines how much heat flows and what temperature differences are needed across them. In our application, balancing these helps ensure the outer wall remains at a safe temperature, guiding us toward the correct insulation thickness.