{"id":489,"date":"2022-12-11T23:09:08","date_gmt":"2022-12-11T23:09:08","guid":{"rendered":"https:\/\/books.gw-project.org\/introduction-to-karst-aquifers\/chapter\/exercise-14-solution\/"},"modified":"2023-03-23T18:04:31","modified_gmt":"2023-03-23T18:04:31","slug":"exercise-14-solution","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/introduction-to-karst-aquifers\/chapter\/exercise-14-solution\/","title":{"raw":"Exercise 14 Solution","rendered":"Exercise 14 Solution"},"content":{"raw":"
\r\n

a. The hydraulic radius for image a) is<\/p>\r\n

[latex]\\displaystyle \\frac{\\textrm{Area}}{\\textrm{Circumference}}=\\frac{\\pi r^{2}}{2\\pi r}=\\frac{r}{2}=\\frac{2}{2}=1\\;\\textrm{meter}[\/latex]<\/p>\r\n

for image\u00a0b) the hydraulic radius is<\/p>\r\n

[latex]\\displaystyle \\frac{\\textrm{Area}}{\\textrm{Wetted perimeter}}=\\frac{(3.6)(3.8)+(0.2)(3.8)}{3.6+2\\sqrt{0.2^{2}+3.8^{2}}}[\/latex]<\/p>\r\n

= [latex]\\displaystyle \\frac{13.68+0.76}{3.6+2\\sqrt{14.48}}=\\frac{14.44}{3.6+7.61}=1.29\\;\\textrm{meter}[\/latex]<\/p>\r\n

b. For image a) (0.005 m3<\/sup>\/s)\/(12.57\u00a0m2<\/sup>)\u00a0=\u00a00.000398\u00a0m\/s<\/p>\r\n

and for image\u00a0b) (0.005\u00a0m3<\/sup>\/s)\/(14.44\u00a0m2<\/sup>)\u00a0=\u00a00.000346\u00a0m\/s<\/p>\r\n

c.<\/p>\r\n

[latex]\\displaystyle Re=\\frac{VD\\rho }{\\mu }=\\frac{VD}{\\nu }[\/latex]<\/p>\r\n

For the pipe (a) D is equal to the pipe diameter of 4\u00a0m, but for the open channel (b), we would use the equation for a pipe diameter that has the same hydraulic radius<\/p>\r\n

r<\/em> = 2 (1.29) = 2.58, then D = 2r<\/em>\u00a0=\u00a05.16\u00a0m<\/p>\r\n

for image\u00a0a) Re<\/em>= (0.000398 m\/s)(4 m)\/(10\u22126<\/sup> m2<\/sup>\/s)\u00a0=\u00a01600<\/p>\r\n

and for image\u00a0b) Re<\/em> = (0.000346 m\/s)(5.16 m)\/(10\u22126<\/sup> m2<\/sup>\/s)\u00a0=\u00a01800<\/p>\r\n

d. for image a) Most likely laminar as Re<\/em><\u00a02100<\/p>\r\n

for image\u00a0b) Most likely turbulent as Re<\/em>\u00a0>\u00a0500<\/p>\r\n

Return to Exercise 14<\/a><\/p>\r\n

Return to where text linked to Exercise 14<\/a><\/p>\r\n\r\n<\/div>","rendered":"

\n

a. The hydraulic radius for image a) is<\/p>\n

\"\displaystyle \frac{\textrm{Area}}{\textrm{Circumference}}=\frac{\pi r^{2}}{2\pi r}=\frac{r}{2}=\frac{2}{2}=1\;\textrm{meter}\"<\/p>\n

for image\u00a0b) the hydraulic radius is<\/p>\n

\"\displaystyle \frac{\textrm{Area}}{\textrm{Wetted perimeter}}=\frac{(3.6)(3.8)+(0.2)(3.8)}{3.6+2\sqrt{0.2^{2}+3.8^{2}}}\"<\/p>\n

= \"\displaystyle \frac{13.68+0.76}{3.6+2\sqrt{14.48}}=\frac{14.44}{3.6+7.61}=1.29\;\textrm{meter}\"<\/p>\n

b. For image a) (0.005 m3<\/sup>\/s)\/(12.57\u00a0m2<\/sup>)\u00a0=\u00a00.000398\u00a0m\/s<\/p>\n

and for image\u00a0b) (0.005\u00a0m3<\/sup>\/s)\/(14.44\u00a0m2<\/sup>)\u00a0=\u00a00.000346\u00a0m\/s<\/p>\n

c.<\/p>\n

\"\displaystyle Re=\frac{VD\rho }{\mu }=\frac{VD}{\nu }\"<\/p>\n

For the pipe (a) D is equal to the pipe diameter of 4\u00a0m, but for the open channel (b), we would use the equation for a pipe diameter that has the same hydraulic radius<\/p>\n

r<\/em> = 2 (1.29) = 2.58, then D = 2r<\/em>\u00a0=\u00a05.16\u00a0m<\/p>\n

for image\u00a0a) Re<\/em>= (0.000398 m\/s)(4 m)\/(10\u22126<\/sup> m2<\/sup>\/s)\u00a0=\u00a01600<\/p>\n

and for image\u00a0b) Re<\/em> = (0.000346 m\/s)(5.16 m)\/(10\u22126<\/sup> m2<\/sup>\/s)\u00a0=\u00a01800<\/p>\n

d. for image a) Most likely laminar as Re<\/em><\u00a02100<\/p>\n

for image\u00a0b) Most likely turbulent as Re<\/em>\u00a0>\u00a0500<\/p>\n

Return to Exercise 14<\/a><\/p>\n

Return to where text linked to Exercise 14<\/a><\/p>\n<\/div>\n","protected":false},"author":1,"menu_order":61,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-489","chapter","type-chapter","status-publish","hentry"],"part":580,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/introduction-to-karst-aquifers\/wp-json\/pressbooks\/v2\/chapters\/489","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/books.gw-project.org\/introduction-to-karst-aquifers\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/books.gw-project.org\/introduction-to-karst-aquifers\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/books.gw-project.org\/introduction-to-karst-aquifers\/wp-json\/wp\/v2\/users\/1"}],"version-history":[{"count":11,"href":"https:\/\/books.gw-project.org\/introduction-to-karst-aquifers\/wp-json\/pressbooks\/v2\/chapters\/489\/revisions"}],"predecessor-version":[{"id":934,"href":"https:\/\/books.gw-project.org\/introduction-to-karst-aquifers\/wp-json\/pressbooks\/v2\/chapters\/489\/revisions\/934"}],"part":[{"href":"https:\/\/books.gw-project.org\/introduction-to-karst-aquifers\/wp-json\/pressbooks\/v2\/parts\/580"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/introduction-to-karst-aquifers\/wp-json\/pressbooks\/v2\/chapters\/489\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/introduction-to-karst-aquifers\/wp-json\/wp\/v2\/media?parent=489"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/books.gw-project.org\/introduction-to-karst-aquifers\/wp-json\/pressbooks\/v2\/chapter-type?post=489"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/introduction-to-karst-aquifers\/wp-json\/wp\/v2\/contributor?post=489"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/introduction-to-karst-aquifers\/wp-json\/wp\/v2\/license?post=489"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}