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Chapter 1: Q33E (page 1)

An important concern in the study of heat transfer is to determine the steady-state temperature distribution of a thin plate when the temperature around the boundary is known. Assume the plate shown in the figure represents a cross section of a metal beam, with negligible heat flow in the direction perpendicular to the plate. Let \({T_1},...,{T_4}\) denote the temperatures at the four interior nodes of the mesh in the figure. The temperature at a node is approximately equal to the average of the four nearest nodes鈥攖o the left, above, to the right, and below. For instance,

\({T_1} = \left( {10 + 20 + {T_2} + {T_4}} \right)/4\), or \(4{T_1} - {T_2} - {T_4} = 30\)

33. Write a system of four equations whose solution gives estimates

for the temperatures \({T_1},...,{T_4}\).

Short Answer

Expert verified

The system of equations whose solution gives the estimates for the temperatures \({T_1},...,{T_4}\)is shown below:

\(\begin{array}{c}4{T_1} - {T_2} - {T_4} = 30\\ - {T_1} + 4{T_2} - {T_3} = 60\\ - {T_2} + 4{T_3} - {T_4} = 70\\ - {T_1} - {T_3} + 4{T_4} = 40\end{array}\)

Step by step solution

01

Write the second equation for temperature 2

From the figure, it is observed that for temperature 2, i.e., \({T_2}\) (lies at the interior nodes of the mesh), the nearest temperatures are \({T_1}\), \(20^\circ \), \(40^\circ \), and \({T_3}\).

It is given thatthe temperature at a node is approximately equal to the average of the four nearest nodes. So, temperature \({T_2}\) at a node is approximately equal to the average of the four temperatures at the nearest nodes \({T_1}\), \(20^\circ \), \(40^\circ \), and \({T_3}\) as shown below:

\({T_2} = \left( {{T_1} + 20 + 40 + {T_3}} \right)/4\)

Simplify the above expression.

\(4{T_2} - {T_1} - {T_3} = 60\)

02

Write the second equation for temperature 3

From the figure, it is observed that for temperature 3, i.e., \({T_3}\) (lies at the interior nodes of the mesh), the nearest temperatures are \({T_4}\), \(30^\circ \), \(40^\circ \), and \({T_2}\).

It is given thatthe temperature at a node is approximately equal to the average of the four nearest nodes. So, the temperature \({T_3}\) at a node is approximately equal to the average of the four temperatures at the nearest nodes \({T_4}\), \(30^\circ \), \(40^\circ \), and \({T_2}\) as shown below:

\({T_3} = \left( {{T_4} + 30 + 40 + {T_2}} \right)/4\)

Simplify the above expression.

\(4{T_3} - {T_4} - {T_2} = 70\)

03

Write the second equation for temperature 4

From the figure, it is observed that for temperature 4, i.e., \({T_4}\) (lies at the interior nodes of the mesh), the nearest temperatures are \({T_1}\), \(10^\circ \), \(30^\circ \), and \({T_3}\).

It is given thatthe temperature at a node is approximately equal to the average of the four nearest nodes. So, temperature \({T_4}\) at a node is approximately equal to the average of the four temperatures at the nearest nodes \({T_1}\), \(10^\circ \), \(30^\circ \), and \({T_3}\) as shown below:

\({T_4} = \left( {{T_1} + 10 + 30 + {T_3}} \right)/4\)

Simplify the above expression.

\(4{T_4} - {T_1} - {T_3} = 40\)

04

Rearrange the obtained system of equations

The obtained system of equations is \(4{T_2} - {T_1} - {T_3} = 60\), \(4{T_3} - {T_4} - {T_2} = 70\),and \(4{T_4} - {T_1} - {T_3} = 40\). Also, the given equation is \(4{T_1} - {T_2} - {T_4} = 30\).

Rearrange all the equations as shown below:

\(\begin{array}{c}4{T_1} - {T_2} - {T_4} = 30\\ - {T_1} + 4{T_2} - {T_3} = 60\\ - {T_2} + 4{T_3} - {T_4} = 70\\ - {T_1} - {T_3} + 4{T_4} = 40\end{array}\)

Thus, the system of four equations whose solution gives the estimates for the temperatures \({T_1},...,{T_4}\)is shown below:

\(\begin{array}{c}4{T_1} - {T_2} - {T_4} = 30\\ - {T_1} + 4{T_2} - {T_3} = 60\\ - {T_2} + 4{T_3} - {T_4} = 70\\ - {T_1} - {T_3} + 4{T_4} = 40\end{array}\)

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Most popular questions from this chapter

In Exercises 5-8, write a matrix equation that determines the loop currents. [M] If MATLAB or another matrix program is available, solve the system for the loop currents.

Let \({\bf{u}}\) and \({\bf{v}}\) be vectors in\({\mathbb{R}^{\bf{n}}}\). It can be shown that the set \({\bf{P}}\) of all points in the parallelogram determined by \({\bf{u}}\) and \({\bf{v}}\) has the form \({\bf{av}} + {\bf{bv}}\), for \({\bf{0}} \le {\bf{a}} \le {\bf{1}}\), \({\bf{0}} \le {\bf{b}} \le {\bf{1}}\). Let \({\bf{T}}:{\mathbb{R}^{\bf{n}}} \to {\mathbb{R}^{\bf{n}}}\) be a linear transformation. Explain why the image of a point in \({\bf{P}}\) under the transformation \({\bf{T}}\) lies in the parallelogram determined by \({\bf{T}}\left( {\bf{u}} \right)\) and \({\bf{T}}\left( {\bf{v}} \right)\).

Find the general solutions of the systems whose augmented matrices are given as

12. \(\left[ {\begin{array}{*{20}{c}}1&{ - 7}&0&6&5\\0&0&1&{ - 2}&{ - 3}\\{ - 1}&7&{ - 4}&2&7\end{array}} \right]\).

In Exercises 9 and 10, (a) for what values of h is \({{\bf{v}}_3}\) inSpan \(\left\{ {{{\bf{v}}_1},{{\bf{v}}_2}} \right\}\), and (b) for what values of \(h\) is \(\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3}} \right\}\) linearlydependent? Justify each answer.

10. \({{\bf{v}}_1} = \left[ {\begin{array}{*{20}{c}}1\\{ - 5}\\{ - 3}\end{array}} \right]\), \({{\bf{v}}_2} = \left[ {\begin{array}{*{20}{c}}{ - 2}\\{10}\\6\end{array}} \right]\), \({{\bf{v}}_3} = \left[ {\begin{array}{*{20}{c}}2\\{ - 9}\\h\end{array}} \right]\)

Let \(A = \left[ {\begin{array}{*{20}{c}}1&0&{ - 4}\\0&3&{ - 2}\\{ - 2}&6&3\end{array}} \right]\) and \(b = \left[ {\begin{array}{*{20}{c}}4\\1\\{ - 4}\end{array}} \right]\). Denote the columns of \(A\) by \({{\mathop{\rm a}\nolimits} _1},{a_2},{a_3}\) and let \(W = {\mathop{\rm Span}\nolimits} \left\{ {{a_1},{a_2},{a_3}} \right\}\).

  1. Is \(b\) in \(\left\{ {{a_1},{a_2},{a_3}} \right\}\)? How many vectors are in \(\left\{ {{a_1},{a_2},{a_3}} \right\}\)?
  2. Is \(b\) in \(W\)? How many vectors are in W.
  3. Show that \({a_1}\) is in W.[Hint: Row operations are unnecessary.]
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