Consider an electric space heater with dimensions 100 cm x 50 cm x 20 cm as shown inthe figure. The space heater is exposed to quiescent (still) air at 20 C. If the frontsurface of the heater (100 cm x 50 cm) is at temperature 𝑇𝑠 = 80 𝐶, find the heatconvected to the room via natural convection by following the steps below. Assume thatall other surfaces of the heater are completely insulated (in other words, only considerthe large rectangular surface facing the room).1. At which temperature will you evaluate the properties of air? Find the values ofkinematic viscosity 𝜈, the Prandtl number 𝑃𝑟, and the thermal conductivity 𝑘 atthis temperature. Interpolate if needed.2. What is the value of the volume expansion coefficient 𝛽?3. Find the Grashof number for this problem.4. Find the Rayleigh number for this problem.5. Find the average Nusselt number 𝑁𝑢.6. Find the average convection coefficient ℎ.7. Find the heat convected away from the front surface of the space heater 𝑄̇𝑐𝑜𝑛𝑣.
Question
Consider an electric space heater with dimensions 100 cm x 50 cm x 20 cm as shown inthe figure. The space heater is exposed to quiescent (still) air at 20 C. If the frontsurface of the heater (100 cm x 50 cm) is at temperature 𝑇𝑠 = 80 𝐶, find the heatconvected to the room via natural convection by following the steps below. Assume thatall other surfaces of the heater are completely insulated (in other words, only considerthe large rectangular surface facing the room).1. At which temperature will you evaluate the properties of air? Find the values ofkinematic viscosity 𝜈, the Prandtl number 𝑃𝑟, and the thermal conductivity 𝑘 atthis temperature. Interpolate if needed.2. What is the value of the volume expansion coefficient 𝛽?3. Find the Grashof number for this problem.4. Find the Rayleigh number for this problem.5. Find the average Nusselt number 𝑁𝑢.6. Find the average convection coefficient ℎ.7. Find the heat convected away from the front surface of the space heater 𝑄̇𝑐𝑜𝑛𝑣.
Solution
Sure, let's go through the steps one by one:
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The properties of air should be evaluated at the film temperature, which is the average of the surface temperature and the air temperature. So, the film temperature T_f = (T_s + T_inf)/2 = (80°C + 20°C)/2 = 50°C. The properties of air at this temperature are: kinematic viscosity ν = 20.92 x 10^-6 m^2/s, Prandtl number Pr = 0.7, and thermal conductivity k = 0.028 W/m·K.
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The volume expansion coefficient β is equal to 1/T_f (in Kelvin). So, β = 1/(50°C + 273) = 1/323 = 0.0031 K^-1.
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The Grashof number Gr is given by Gr = gβ(T_s - T_inf)L^3/ν^2, where g is the acceleration due to gravity (9.81 m/s^2), L is the characteristic length (which is the height of the heater, 1 m in this case), and T_s and T_inf are the surface and air temperatures in Kelvin. Substituting the values, we get Gr = 9.810.0031*(80-20)*1^3/(20.92 x 10^-6)^2 = 1.35 x 10^9.
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The Rayleigh number Ra is given by Ra = Gr*Pr = 1.35 x 10^9 * 0.7 = 9.45 x 10^8.
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For natural convection from a vertical plate, the average Nusselt number Nu is given by Nu = 0.59Ra^(1/4) = 0.59(9.45 x 10^8)^(1/4) = 30.2.
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The average convection coefficient h is given by h = Nuk/L = 30.20.028/1 = 0.85 W/m^2·K.
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The heat convected away from the front surface of the space heater Q̇_conv is given by Q̇_conv = hA(T_s - T_inf), where A is the surface area of the heater. Substituting the values, we get Q̇_conv = 0.8510.5*(80-20) = 25.5 W.
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