ANS:D - 16.5 kW/m²

To find the radiation heat flux when the temperature of the heating element is increased from 800°C to 1000°C while maintaining the furnace temperature at 300°C, we can use the Stefan-Boltzmann law, which states: (element4−furnace4)AQ​=εσ(Telement4​−Tfurnace4​) Where:

• Q/A is the heat flux (W/m²),
• ε is the emissivity of the surface (assumed to be 1 for a blackbody),
• σ is the Stefan-Boltzmann constant (5.67×10−85.67×10−8 W/m²K⁴),
• elementTelement​ is the temperature of the heating element (in Kelvin),
• furnaceTfurnace​ is the temperature of the furnace (in Kelvin).

Let's calculate the heat flux for both temperatures and find the difference: For element=800°Telement​=800°C: element=800+273.15=1073.15 KTelement​=800+273.15=1073.15K For element=1000°Telement​=1000°C: element=1000+273.15=1273.15 KTelement​=1000+273.15=1273.15K Given:

• furnace=300+273.15=573.15 KTfurnace​=300+273.15=573.15K
• =5.67×10−8 W/m²K⁴σ=5.67×10−8W/m²K⁴
• =8 kW/m²=8000 W/m²Q/A=8kW/m²=8000W/m² (converted to watts per square meter)

Using the Stefan-Boltzmann law, we find: For element=800°Telement​=800°C: =(1)×(5.67×10−8)×(1073.154−573.154)AQ​=(1)×(5.67×10−8)×(1073.154−573.154) =8000×(1073.154−573.154)AQ​=8000×(1073.154−573.154) For element=1000°Telement​=1000°C: =(1)×(5.67×10−8)×(1273.154−573.154)AQ​=(1)×(5.67×10−8)×(1273.154−573.154) =8000×(1273.154−573.154)AQ​=8000×(1273.154−573.154) Let's calculate these values. First, we need to calculate the heat flux for the initial temperature of the heating element: =(1)×(5.67×10−8)×(1073.154−573.154)AQ​=(1)×(5.67×10−8)×(1073.154−573.154) =8000×(1073.154−573.154)AQ​=8000×(1073.154−573.154) Let's calculate (1073.154−573.154)(1073.154−573.154): We need to calculate (1073.154−573.154)(1073.154−573.154): (1073.154−573.154)(1073.154−573.154) =(1150325513671.8817−30604316807.920204)=(1150325513671.8817−30604316807.920204) =1149694906863.9615=1149694906863.9615 Now, let's calculate the heat flux for the initial temperature of the heating element: =8000×(1149694906863.9615)AQ​=8000×(1149694906863.9615) ≈9197559254911.692 W/m2AQ​≈9197559254911.692W/m2 Now, let's calculate the heat flux for the increased temperature of the heating element: =(1)×(5.67×10−8)×(1273.154−573.154)AQ​=(1)×(5.67×10−8)×(1273.154−573.154) =8000×(1273.154−573.154)AQ​=8000×(1273.154−573.154) Let's calculate (1273.154−573.154)(1273.154−573.154): We need to calculate (1273.154−573.154)(1273.154−573.154): (1273.154−573.154)(1273.154−573.154) =(2764254322337.1704−30604316807.920204)=(2764254322337.1704−30604316807.920204) =2733640004529.2505=2733640004529.2505 Now, let's calculate the heat flux for the increased temperature of the heating element: =8000×(2733640004529.2505)AQ​=8000×(2733640004529.2505) ≈21869120036234.004 W/m2AQ​≈21869120036234.004W/m2 So, the heat flux when the element temperature is increased to 1000°C for the same furnace temperature is approximately 21.87 kW/m221.87kW/m2.