HELP, please. How can I get to that solution by solving the equation other than by trial and error?

Question

Example 3.38. An electric cable of 12 mm diameter is insulated to increase the current capacity. Due to insulation the current carrying capacity is increased by 15% without increasing cable surface temperature above 70°C. The environmental temperature is 30°C. Assume that the heat transfer coefficient from the bare or insulated cable is 14 W/m2.K. Calculate the conductivity of insulating material.

Considering 2nd and 3rd terms of equation,

we get 1 + ln (11.90k) = 9k or

ln (11.9k) – 9k + 1 = 0

By trial and error Let k = 0.1 W/m.K, then, 1.19 – 0.9 + 1 ≠ 0

Let k = 0.2, then 0.8671 – 1.9422 + 1 = (– 0.075) ≠ 0.031

Let k = 0.2158, satisfy the equation, therefore, Thermal conductivity of insulating material

is 0.2158 W/m.K. ans.

Answer 1

Hi Jessie!

Thanks for your question.

The exact solution of the equation

ln(11.9k) - 9k + 1 = 0

cannot be written in terms of elementary functions like e^() or ln(). Instead, we have to introduce a nonelementary function called the Lambert W function. The Lambert W function W(x) is defined implicitly as the solution of

We^W = x, provided x >= -1/e.

Let's see how the solution of your problem can be written using the Lambert W function. First, I'm going to make a series of algebraic manipulations:

ln(11.9k) - 9k + 1 = 0

ln(11.9k) = 9k - 1

e^ln(11.9k) = e^{9k - 1}

11.9k = e^9ke^-1

ke^-9k = 1/(11.9e)

-9ke^-9k = -9/(11.9e).

Now let's change our variables. Let w = -9k. Then, we have

we^w = -9/(11.9e).

Since -9/(11.9e) >= -1/e, we can conclude that w = W(-9/(11.9e), i.e., the Lambert W function evaluated at -9/(11.9e). Since w = -9k, we have

-9k = W(-9/(11.9e)),

giving us k = -W(-9/(11.9e))/9. We can use a scientific calculator to evaluate the W function, giving us k ≈ 0.0473335073.