Solving Heat Transfer Problem Involving Black Bodies and Convection Coefficient

[Carlo09]

Ok please can someone help or at least point me in the right direction with this question please?

I have air flowing through a pipe which is heated to 477 degrees via burners on the outside. The pipe itself is heated to 550 degrees and there is a thermocouple inside the air flow. However, the thermocouple receives heat via radiation from the pipe wall and assuming both the wall and thermocouple are black bodies with respect to thermal radiation. the convection heat transfer coefficient is 160 w/m2 k

I need to show that the air temp is actually 477 degrees when the thermocouple shows 508 degrees!?

Ok so i konw black bodies absorb all and/or emit all. Therefore using Eb=Boltzmans constant *T^4 I see that the wall is getting 26031.49 w/m2 of energy and then is emiting all of this also, correct? I know the convection coefficient is 160 as given above but I am not given an area to use so how can I do this?

Thank you so much to whoever can help me.

Edit* we also know that it takes 6.1kw of energy to heat the air from 25 = 477 degrees and the flowrate of the air is 100m^3/Hr

 

[Navin A S]

Assuming the thermocouple is in good contact with the air flow, you can use the convection heat transfer coefficient to calculate the heat transfer from the wall to the air. Since both the wall and thermocouple are black bodies, you can also use the Boltzmann constant to calculate the radiation heat transfer from the wall to the thermocouple. Using the given information, you can calculate the total heat transfer to the air (convection + radiation) as well as the total heat transfer to the thermocouple (radiation only). From there, you can solve for the air temperature and the temperature of the thermocouple.For example, the convection heat transfer rate from the wall to the air can be calculated as:Q_conv = h*A*(T_wall - T_air)Where h is the convection heat transfer coefficient, A is the surface area of the wall, and T_wall and T_air are the wall and air temperatures respectively. The radiation heat transfer rate from the wall to the air can be calculated as:Q_rad = epsilon * sigma * A * (T_wall^4 - T_air^4)where epsilon is the emissivity of the wall, sigma is the Boltzmann constant, and A is the surface area of the wall.Similarly, the radiation heat transfer rate from the wall to the thermocouple can be calculated as:Q_rad = epsilon * sigma * A * (T_wall^4 - T_therm^4)where epsilon is the emissivity of the wall, sigma is the Boltzmann constant, and A is the surface area of the wall.Once you have all of these equations, you can solve for T_air and T_therm using the given information.

 

[oswaler]

I would approach this problem by first understanding the basics of heat transfer and the concept of black bodies. A black body is an idealized object that absorbs all radiation incident upon it and emits all radiation that it absorbs. This means that the pipe wall and the thermocouple will both absorb and emit thermal radiation at a rate determined by their temperatures.

Next, I would consider the heat transfer mechanisms involved in this problem. In this case, we have both convection and radiation heat transfer occurring. Convection is the transfer of heat through the movement of a fluid, in this case the air flowing through the pipe. Radiation is the transfer of heat through electromagnetic waves. Both of these mechanisms contribute to the overall heat transfer from the pipe wall to the thermocouple.

To solve this problem, we need to use the equations and principles of heat transfer to determine the temperature of the air in the pipe. We know that the air is heated to 477 degrees by the burners on the outside of the pipe. We also know that the pipe wall and the thermocouple are both black bodies, so we can use the Stefan-Boltzmann law to calculate the radiative heat transfer between them.

To account for the convection heat transfer, we need to use the convection coefficient provided in the problem. However, as you mentioned, we do not have an area to use for this calculation. In this case, we can assume a representative area, such as the cross-sectional area of the pipe, and use that in our calculations.

Using the principles of heat transfer, we can determine the rate of heat transfer from the pipe wall to the thermocouple and then use this information to calculate the temperature of the air inside the pipe. This will allow us to compare it to the temperature measured by the thermocouple and verify if it is indeed 477 degrees, as expected.

Overall, solving this heat transfer problem involves understanding the basic principles of heat transfer, using appropriate equations and assumptions, and carefully considering the different heat transfer mechanisms involved. With this approach, we can accurately determine the temperature of the air inside the pipe and validate our results.