Calculate the rate of loss from a room through a glass window of area `1.5 m^(2)` and thickness `2.5 xx 10^(-3)` m, when the temperature of the room is `25^(@) C` and that of the air outside is `10^(@) C`. Assume that the inner glass surface is at the room temperature. [ Thermal conductivity of glass `=1.2 "Wm"^(-1) "K"^(-1)` ]

Answer

To calculate the rate of heat loss from a room through a glass window, we can use the formula derived from Fourier's law of heat conduction:

\[ Q' = \frac{(T_1 - T_2) \cdot A}{R_{th}} \]

Where:
- \( Q' \) = rate of heat loss (W)
- \( T_1 \) = temperature of the room (°C)
- \( T_2 \) = temperature outside (°C)
- \( A \) = area of the window (m²)
- \( R_{th} \) = thermal resistance (K/W)

### Step 1: Identify the given values
- Area of the window, \( A = 1.5 \, m^2 \)
- Thickness of the glass, \( L = 2.5 \times 10^{-3} \, m \)
- Temperature inside the room, \( T_1 = 25 \, °C \)
- Temperature outside, \( T_2 = 10 \, °C \)
- Thermal conductivity of glass, \( k = 1.2 \, Wm^{-1}K^{-1} \)

### Step 2: Calculate the temperature difference
\[ \Delta T = T_1 - T_2 = 25 - 10 = 15 \, °C \]

### Step 3: Calculate the thermal resistance \( R_{th} \)
The thermal resistance \( R_{th} \) can be calculated using the formula:
\[ R_{th} = \frac{L}{k \cdot A} \]
Substituting the values:
\[ R_{th} = \frac{2.5 \times 10^{-3}}{1.2 \cdot 1.5} \]

### Step 4: Calculate \( R_{th} \)
First, calculate the denominator:
\[ 1.2 \cdot 1.5 = 1.8 \]
Now substitute back:
\[ R_{th} = \frac{2.5 \times 10^{-3}}{1.8} \approx 1.3889 \times 10^{-3} \, K/W \]

### Step 5: Calculate the rate of heat loss \( Q' \)
Now, substitute \( \Delta T \) and \( R_{th} \) back into the heat loss formula:
\[ Q' = \frac{(15) \cdot (1.5)}{R_{th}} = \frac{15 \cdot 1.5}{1.3889 \times 10^{-3}} \]

### Step 6: Calculate \( Q' \)
Calculating the numerator:
\[ 15 \cdot 1.5 = 22.5 \]
Now substitute:
\[ Q' = \frac{22.5}{1.3889 \times 10^{-3}} \approx 16256.5 \, W \]

### Step 7: Convert to kilowatts
\[ Q' \approx 16.26 \, kW \]

### Final Answer
The rate of heat loss from the room through the glass window is approximately **16.26 kW**.