Question

Consider a surface of area \(A\) at which the convection and radiation heat transfer coefficients are \(h_{\text {conv }}\) and \(h_{\text {radt }}\) respectively. Explain how you would determine \((a)\) the single equivalent heat transfer coefficient, and \((b)\) the equivalent thermal resistance. Assume the medium and the surrounding surfaces are at the same temperature.

Short answer

Question: A surface has convection and radiation heat transfer coefficients, \(h_{\text {conv }}\) and \(h_{\text {radt }}\), respectively. Both mechanisms occur simultaneously, with medium and surrounding surfaces at the same temperature. Determine (a) a single equivalent heat transfer coefficient and (b) the equivalent thermal resistance. Answer: (a) The equivalent heat transfer coefficient can be found using the formula \(h_{eq} = h_{conv} + h_{radt}\). (b) The equivalent thermal resistance can be determined using the formula \(R_{eq} = \frac{1}{h_{eq}\cdot A}\), where A is the surface area.

Step-by-step solution

Understand the problem with given parameters

Since the surface has both convection and radiation heat transfer coefficients, the total heat transfer will be the summation of both processes. In an equivalent circuit, we can treat both heat transfer mechanisms as parallel processes.

Calculate the total heat transfer rate

To find the total heat transfer rate (\(Q_{total}\)), we need to sum the heat transfer rates of convection (\(Q_{conv}\)) and radiation (\(Q_{radt}\)). The heat transfer rate of both processes can be expressed as: \(Q_{conv} = h_{conv} \cdot A \cdot\Delta T\) \(Q_{radt} = h_{radt} \cdot A \cdot\Delta T\) where \(\Delta T\) is the temperature difference between the surface and surrounding medium. Therefore, the total heat transfer rate is: \(Q_{total} = Q_{conv} + Q_{radt} = A \cdot (h_{conv} + h_{radt})\cdot\Delta T\).

Find the equivalent heat transfer coefficient

By observing the total heat transfer rate equation, we can notice that the equivalent heat transfer coefficient (\(h_{eq}\)) is given by the summation of individual coefficients, \(h_{eq} = h_{conv} + h_{radt}\).

Determine the equivalent thermal resistance

The thermal resistance for each heat transfer mechanism can be calculated using the following equations: \(R_{conv} = \frac{1}{h_{conv}\cdot A}\), \(R_{radt} = \frac{1}{h_{radt}\cdot A}\). Since the heat transfer mechanisms occur in parallel, the equivalent thermal resistance (\(R_{eq}\)) can be found by finding the parallel resistance: \(\frac{1}{R_{eq}} = \frac{1}{R_{conv}} + \frac{1}{R_{radt}}\) \(\frac{1}{R_{eq}} = \frac{h_{conv}\cdot A + h_{radt}\cdot A}{A}\) \(R_{eq} = \frac{1}{h_{eq}\cdot A}\) where \(h_{eq}\) is the equivalent heat transfer coefficient obtained in Step 3.