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A new building to be located in a cold climate is being designed with a basement that has an \(L=200\)-mm-thick wall. Inner and outer basement wall temperatures are \(T_{i}=20^{\circ} \mathrm{C}\) and \(T_{o}=0^{\circ} \mathrm{C}\), respectively. The architect can specify the wall material to be either aerated concrete block with \(k_{\mathrm{ac}}=0.15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), or stone mix concrete. To reduce the conduction heat flux through the stone mix wall to a level equivalent to that of the aerated concrete wall, what thickness of extruded polystyrene sheet must be applied onto the inner surface of the stone mix con-crete wall? Floor dimensions of the basement are \(20 \mathrm{~m} \times 30 \mathrm{~m}\), and the expected rental rate is \(\$ 50 / \mathrm{m}^{2} /\) month. What is the yearly cost, in terms of lost rental income, if the stone mix concrete wall with polystyrene insulation is specified?

Step by step solution

01

Determine heat flux through the aerated concrete wall

The formula for conduction heat flux is given by: \(q = \frac{kA(T_i-T_o)}{L}\), where q is the heat flux, k is the thermal conductivity, A is the wall area, L is the length or thickness, and \(T_i\) and \(T_o\) are the inner and outer temperatures, respectively. First, we calculate the heat flux through the aerated concrete wall: \[ q_{ac} = \frac{k_{ac}(T_i - T_o)}{L} \]

02

Write the equation for heat flux through the stone mix wall and insulation

The combined thickness of the stone mix wall and insulation is represented by L_total. The equations for heat flux through these two parts can be written as follows: \(q_{sm} = \frac{q_{ac}}{A} = \frac{k_{stone}(T_i - T_{insulation})}{L_{stone}}\) \(q_{insulation} = \frac{q_{ac}}{A} = \frac{k_{insulation}(T_{insulation} - T_o)}{L_{insulation}}\) Where \(q_{sm}\) and \(q_{insulation}\) are the heat fluxes through the stone mix wall and extruded polystyrene insulation, respectively, and \(T_{insulation}\) is the temperature at the interface between the stone mix wall and insulation.

03

Solve the system of equations to find the thickness of the insulation

Since both heat fluxes must be equal to the heat flux through the aerated concrete wall, we can solve the system of equations for \(L_{insulation}\) while eliminating the unknowns such as \(T_{insulation}\) and \(q_{ac}\). We have: \[ q_{ac} = k_{stone} \frac{(T_i - T_{insulation})}{L_{stone}} = k_{insulation} \frac{(T_{insulation} - T_o)}{L_{insulation}} \] Divide both sides by \(k_{stone}k_{insulation}\), solve for \(L_{insulation}\) and substitute given values: \[ L_{insulation} = \frac{k_{stone}(T_i - T_o) L_{stone}}{k_{insulation}L + k_{stone}L_{stone}} \]

04

Calculate the annual cost of lost rental income

To find the lost rental income per year due to the stone mix concrete wall, we first need to find the area of the wall that is not usable for renting. We can find this value by multiplying the floor dimensions (20 m x 30 m) by the difference in thickness between the aerated concrete wall and the stone mix wall with insulation. Then, we can multiply this value by the rental rate and calculate the costs for a year: Annual Lost Rental Income = \((L_{stone}+ L_{insulation} - L) \times 20 \times 30 \times 50 \times 12\) Plug in the values of thickness and solve for the annual cost of lost rental income. Whith these steps, the task is complete.

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Key Concepts

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

Thermal Conductivity

When studying heat transfer, thermal conductivity plays a crucial role. It measures how well a material can conduct heat. Thermal conductivity is denoted by the symbol \( k \) and its unit is Watts per meter per Kelvin (W/m·K).
One way to visualize this is by imagining a metal rod partially submerged in boiling water. The end in the water will heat up first, and, depending on the material, the opposite end will eventually warm up too. A high thermal conductivity means heat travels quickly from hot to cold, whereas a low thermal conductivity indicates slower heat transfer.
Different materials have different levels of thermal conductivity. For instance:

• Metals, like copper, typically have high thermal conductivity, which is why they're often used in applications where heat needs to be effectively managed.
• Insulating materials, such as aerated concrete or extruded polystyrene, have low thermal conductivity, making them ideal for maintaining temperatures, like insulating a building against cold weather.

In the original exercise, the thermal conductivity values of aerated concrete and stone mix concrete are or would be instrumental in determining the thickness of additional insulation required to achieve desired heat retention.

Conduction Heat Flux

Conduction heat flux describes the rate at which heat is transferred through a material. It results from the process of heat conduction—the transfer of thermal energy from high temperature areas to low temperature regions within a material.
Mathematically, conduction heat flux, denoted as \( q \), is expressed as:
\[ q = \frac{kA(T_i - T_o)}{L} \] where:

• \( k \) is thermal conductivity
• \( A \) is the surface area through which heat transfer occurs
• \( T_i \) and \( T_o \) are the initial and final temperatures, respectively
• \( L \) is the thickness or distance the heat travels through

Using the formula, we can calculate how much heat is being transferred per unit time. In the context of the exercise, reducing conduction heat flux through the wall would mean adding insulation to lower the rate of heat loss, making it align with the lower heat flux of the aerated concrete wall.

Building Insulation

Building insulation is critical in controlling heat flow in structures, especially in climates with extreme temperatures. It provides a barrier to heat flow, reducing the amount of energy required to maintain a comfortable environment indoors. Effective building insulation reduces heating and cooling costs and improves comfort for occupants.
When selecting insulating materials, factors to consider include:

• Thermal conductivity: A lower value indicates better insulation.
• Thickness: Greater thickness typically provides better insulation.
• Cost: It's essential to balance material costs with performance.

In this exercise, extruded polystyrene is considered for its insulation properties when applied to a stone mix wall. The challenge is adjusting the wall's combined thickness to achieve the same conduction heat flux as with aerated concrete. This involves identifying how thick the polystyrene sheet must be to match the insulation level of the aerated concrete wall, thereby minimizing the rental income loss resulting from additional wall thickness.