Question

Maintaining pipe wall temperature. Maintaining a constant pipe wall temperature in some hot-process applications is critical. A technique that utilizes bolt-on trace elements to maintain temperature was presented in the Journal of Heat Transfer (November 2000). Without bolt-on trace elements, the pipe wall temperature of a switch condenser used to produce plastic has a uniform distribution ranging from 260° to 290°F. When several bolt-on trace elements are attached to the piping, the wall temperature is uniform from 278° to 285°F.

a. Ideally, the pipe wall temperature should range between 280° and 284°F. What is the probability that the temperature will fall in this ideal range when no bolt-on trace elements are used? When bolt-on trace elements are attached to the pipe?

b. When the temperature is 268°F or lower, the hot liquid plastic hardens (or plates), causing a buildup in the piping. What is the probability of plastic plating when no bolt-on trace elements are used? When bolt-on trace elements are attachedto the pipe?

Short answer

a. The probability that the temperature will fall is 0.133 when no bolt-on trace elements are used.

The probability that the temperature will fall is 0.571 when bolt-on trace elements are used.

b. The probability of plastic plating is 0.267 when no bolt-on trace elements are used.

The probability of plastic plating is 0 when bolt-on trace elements are used.

Step-by-step solution

Given Information

When without bolt on-trace elements, the pipe wall temperature is uniform from $260°F to 290°F$

Let x be a random variable that follows uniform distribution.

The p.d.f is given below

$\begin{array}{rcl}f\left(x\right)& =& \frac{1}{290-260}; 260⩽x⩽290\\ & =& \frac{1}{30}; 260⩽x⩽290\end{array}$

When several bolt-on trace element are attached to the piping, the wall temperature has a uniform distribution ranging from $278°F to 285°F$.

The p.d.f is given by

$\begin{array}{rcl}f\left(x\right)& =& \frac{1}{285-278}; 278⩽x⩽285\\ & =& \frac{1}{7}; 278⩽x⩽285\end{array}$

(a) Identify the probability for given conditions

When, no bolt-on trace elements are used.

The probability that the temperature will fall in range between

$\begin{array}{rcl}p\left(280<x<284\right)& =& \underset{280}{\overset{284}{\int }}\frac{1}{30}dx\\ & =& \frac{1}{30}{\left.x\right]}_{280}^{284}\\ & =& \frac{1}{30}\left(284-280\right)\\ & =& 0.133\\ & & \end{array}$

Hence, the probability is 0.133.

When, bolt-on trace elements are used

The probability that the temperature will fall in range between $280°Fand 284°F$

$\begin{array}{rcl}p\left(280<x<284\right)& =& \underset{280}{\overset{284}{\int }}\frac{1}{7}dx\\ & =& \frac{1}{7}{\left.x\right]}_{280}^{284}\\ & =& \frac{1}{7}\left(284-280\right)\\ & =& 0.571\end{array}$

Thus, the probability is0.571.

(b) Compute the probability

When, no bolt-on trace elements are used.

The probability of plastic plating is

$\begin{array}{rcl}p\left(x⩽268\right)& =& \underset{260}{\overset{268}{\int }}\frac{1}{30}dx\\ & =& \frac{1}{30}{\left.x\right]}_{260}^{268}\\ & =& \frac{1}{30}\left(268-260\right)\\ & =& 0.267\end{array}$

Therefore, the probability is 0.267.

When, the bolt-on trace elements are used.

The probability of plastic plating is zeroas the value of x is not in between the range when the temperature is lower or$268°F$.