Download e-book for kindle: Analytical Methods for Heat Transfer and Fluid Flow Problems by Bernhard Weigand
By Bernhard Weigand
This e-book describes invaluable analytical tools through employing them to real-world difficulties instead of fixing the standard over-simplified lecture room difficulties. The ebook demonstrates the applicability of analytical tools even for advanced difficulties and courses the reader to a extra intuitive knowing of ways and solutions.
Although the answer of Partial Differential Equations by means of numerical equipment is the traditional perform in industries, analytical equipment are nonetheless vital for the serious evaluate of effects derived from complex desktop simulations and the advance of the underlying numerical options. Literature dedicated to analytical equipment, even though, usually makes a speciality of theoretical and mathematical elements and is accordingly dead to so much engineers. Analytical tools for warmth move and Fluid circulation difficulties addresses engineers and engineering students.
The moment variation has been up to date, the chapters on non-linear difficulties and on axial warmth conduction difficulties have been prolonged. And labored out examples have been included.
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Extra info for Analytical Methods for Heat Transfer and Fluid Flow Problems
30 2 Linear Partial Differential Equations Fig. 6 Eigenvalues of the eigenfunction sin(λ) The solution for ΘT is obtained from the Eqs. 80) as À Á HT ¼ C2 exp Àk2~t C3 sinðk~xÞ For simplicity, we combine the constants C2 and C3 and have À Á HT ¼ C sinðk~xÞ exp Àk2~t ð2:83Þ ð2:84Þ Now we try to fulﬁll the initial condition using Eq. 84). Inserting Eq. 84) into Eq. 69) results in À Á ~t ¼ 0 : HT ¼ H0 ð~xÞ À ~x ¼ C sinðk~xÞ exp Àk2 0 ð2:85Þ ) H0 ð~xÞ À ~x ¼ C sinðk~xÞ From the equation above, one can see that Eq.
This is a suitable assumption if the temperature differences in the problem are not to large. Then the velocity distribution for the pipe and channel flow can be calculated analytically from the momentum equations. A detailed description is given in Appendix A. For a laminar flow, very simple expressions are obtained: Laminar Pipe Flow r 2 u ¼2 1À u R Fig. 1 Heat Transfer in Pipe and Channel Flows … 49 Laminar Flow in a Planar Channel y 2 u 3 1À ¼ u 2 h ð3:2Þ where u is the mean velocity of the pipe or channel flow.
This means that the solution will be given by HT ¼ 1 X À Á Cn sinðkn~xÞ exp Àk2n~t ð2:87Þ n¼1 This solution has to fulﬁl the initial condition. Inserting Eq. 87) into Eq. 69) results in H0 ð~xÞ À ~x ¼ 1 X Cn sinðkn~xÞ ð2:88Þ n¼1 which means that we have to represent the function H0 ð~xÞ À ~x by a Fourier series (see Stephenson 1986, Myint-U and Debnath 1987, Zauderer 1989, Sommerfeld 1978). 3 Separation of Variables 31 In order to obtain the unknown coefﬁcients Cn, from Eq. 88), we multiply both sides of the equation by sinðkm~xÞ and integrate the resulting expressions across the region of interest for ~x between zero and one.
Analytical Methods for Heat Transfer and Fluid Flow Problems by Bernhard Weigand