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Chapter 15: Problem 20

Biot number, Bi, is a dimensionless number that describes the relative effect of convection to conduction heat transfer. This number is given as $$ B i=\frac{h L}{k} $$ where \(\mathrm{h}\) is the heat transfer coefficient \(\left[\mathrm{W} /\left(\mathrm{m}^{2}{ }^{\circ} \mathrm{C}\right)\right]\), and \(\mathrm{L}\) is the characteristic length [m]. What is the unit of thermal conductivity, \(k\) ?

Short Answer

Expert verified
Short Answer: The unit of thermal conductivity, k, is W/(m°C) (watts per meter degree Celsius).

Step by step solution

01

Rewrite the Biot number formula to solve for k

The Biot number, Bi, is given by the formula: $$ B i=\frac{h L}{k} $$ To solve for the thermal conductivity, \(k\), we can rearrange this formula by multiplying both sides by \(k\) and dividing both sides by the Biot number, Bi: $$ k = \frac{hL}{Bi} $$
02

Analyze the units of each term in the equation

Let's identify the units of each term involved in the equation: - \(h\): heat transfer coefficient, with unit \(W/(m^2 °C)\) - \(L\): characteristic length, with unit \(m\) - \(Bi\): Biot number, a dimensionless quantity (no unit)
03

Determine the unit of k by analyzing the equation

Now, we can determine the unit of the thermal conductivity \(k\). Using the equation for \(k\) and the units of each term: $$ [\text{k}] = \frac{[\text{W/(m}^{2}\text{°C) * m}]}{[\text{dimensionless}]} $$ Since the Biot number, Bi, is dimensionless, the term in the denominator does not affect the units. Simplifying the expression in the numerator gives us the unit for \(k\): $$ [\text{k}] = \frac{\text{W}}{\text{m} * \text{°C}} $$ The unit of thermal conductivity, \(k\), is W/(m°C).

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Most popular questions from this chapter

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