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.

Unfortunately, the exact solution of

ln (11.9k) – 9k + 1 = 0

cannot be written in terms of elementary functions like e^() or ln(), but we can express an exact solution if we use a nonelementary function called the Lambert W function. The Lambert W function w = W(x) is defined implicitly as the solution of

we^w = x, provided x >= -1/e.

Let's see how we can write the exact solution of your original equation using the Lambert W function. First, I'll start with 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

11.9ke^-9k = 1/e

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

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

Let's let w = -9k. Then we have

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

Since -9/(11.9e) >= -1/e, we can write w in terms of the Lambert W function:

w = W(-9/(11.9e)).

Using w = -9k, we find

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

This is the exact solution. We can use a good quality scientific calculator to evaluate the Lambert W function, giving us

k = 0.0473335073...

There are other methods that can approximate the solution of this equation without knowing the exact solution derived above. See, for example, Newton's method or fixed point iteration. Both of these numerical methods will work well for this problem.