d<\/em> is the pipe diameter.<\/p>\r\nIt is a fundamental equation in fluid mechanics, and I believe it is often tested in university exams and certifications.<\/p>\r\n
This article explains the derivation of the Hagen-Poiseuille equation.<\/p>\r\n\r\n\r\n\r\n\r\n\r\n
Derivation of Hagen-Poiseuille’s equation<\/h2>\r\n
![\"\"](\"https:\/\/chemical-engineering-review.com\/en\/wp-content\/uploads\/sites\/2\/2022\/03\/hagen-poiseuille-equation1.png\")
<\/p>\r\n
Consider the piping flow shown in the figure above.<\/p>\r\n
The equation is formulated in terms of the equilibrium of forces in a small cylindrical region in the pipe.<\/p>\r\n
$$Pressure\\; acting\\; on\\; the\\; inflow\\; surface=p\u30fb\u03c0r^{2}$$<\/p>\r\n
\r\n\r\n<\/p>\r\n
$$Pressure\\; acting\\; on\\; the\\; outflow\\; surface=(p+\\frac{dp}{dx}dx)\u30fb\u03c0r^{2}$$<\/p>\r\n
\r\n\r\n<\/p>\r\n
$$Shear\\; stress\\; acting\\; on\\; the\\; cylindrical\\; side=\u03c4\u30fb2\u03c0rdx$$<\/p>\r\n
Since these three forces are balanced in the developed flow, Eq. (2) holds.<\/p>\r\n
$$p\u30fb\u03c0r^{2}-(p+\\frac{dp}{dx}dx)\u30fb\u03c0r^{2}-\u03c4\u30fb2\u03c0rdx=0\u30fb\u30fb\u30fb(2)$$<\/p>\r\n
Eq. (2) can be rearranged into Eq. (3).<\/p>\r\n
$$\u03c4=-\\frac{dp}{dx}\\frac{r}{2}\u30fb\u30fb\u30fb(3)$$<\/p>\r\n
Here, Eq. (4) follows from Newton’s viscosity law.<\/p>\r\n
$$\u03c4=-\u03bc\\frac{du}{dr}\u30fb\u30fb\u30fb(4)$$<\/p>\r\n
Substituting Eq. (4) into Eq. (3) yields Eq. (5).<\/p>\r\n
$$-\\frac{dp}{dx}\\frac{r}{2}=-\u03bc\\frac{du}{dr}$$<\/p>\r\n
\r\n\r\n<\/p>\r\n
$$\\frac{du}{dr}=\\frac{dp}{dx}\\frac{r}{2\u03bc}\u30fb\u30fb\u30fb(5)$$<\/p>\r\n
Integrate equation (5) from r=R to r. When r=R, u=0 since it is a wall surface, and when r=r, u=u.<\/p>\r\n
$$\\int_{0}^{u}du=\\int_{R}^{r}\\frac{dp}{dx}\\frac{r}{2\u03bc}dr$$<\/p>\r\n
\r\n\r\n<\/p>\r\n
$$u=-\\frac{1}{4\u03bc}\\frac{dp}{dx}(R^{2}-r^{2})$$<\/p>\r\n
\r\n\r\n<\/p>\r\n
$$u=\\frac{R^{2}}{4\u03bc}(-\\frac{dp}{dx})(1-\\frac{r^{2}}{R^{2}})\u30fb\u30fb\u30fb(6)$$<\/p>\r\n
Eq. (6) is the expression for the velocity distribution in a laminar pipe flow.<\/p>\r\n
Incidentally, since the maximum flow velocity u<\/em>max<\/sub> is at the center of the pipe, it is expressed by Eq. (7) when r = 0 in Eq. (6).<\/p>\r\n$$u_{max}=\\frac{R^{2}}{4\u03bc}(-\\frac{dp}{dx})\u30fb\u30fb\u30fb(7)$$<\/p>\r\n
<\/p>\r\n
An equation is then formulated for the volumetric flow rate Q<\/em> of the fluid flowing through the pipe.<\/p>\r\nThe volume flow rate can be calculated by integrating the flow velocity u at any position r in Eq. (6) over the entire pipe cross section.<\/p>\r\n
$$\\begin{align}Q&=\\int_{0}^{R}u\u30fb2\u03c0rdr\\\\&
=\\frac{\u03c0R^{2}}{2\u03bc}(-\\frac{dp}{dx})\\int_{0}^{R}r(1-\\frac{r^{2}}{R^{2}})dr\\\\&
=\\frac{\u03c0R^{2}}{2\u03bc}(-\\frac{dp}{dx})[\\frac{r^{2}}{2}-\\frac{r^{4}}{4R^{2}}]^{R}_{0}\\\\&
=\\frac{\u03c0R^{4}}{8\u03bc}(-\\frac{dp}{dx})\u30fb\u30fb\u30fb(8)\\end{align}$$<\/p>\r\n
Dividing the volumetric flow rate Q<\/em> calculated by Eq. (8) by the cross-sectional area of the pipe, Eq. (9), which represents the cross-sectional average flow velocity, can be calculated.<\/p>\r\n$$\\bar{u}=\\frac{Q}{\u03c0R^{2}}=\\frac{R^{2}}{8\u03bc}(-\\frac{dp}{dx})\u30fb\u30fb\u30fb(9)$$<\/p>\r\n
Incidentally, comparing the maximum flow velocity in Eq. (7) with Eq. (9), Eq. (10) holds.<\/p>\r\n
$$\\bar{u}=\\frac{1}{2}u_{max}\u30fb\u30fb\u30fb(10)$$<\/p>\r\n
Keep in mind that it is well known that the cross-sectional average velocity is 1\/2 of the maximum velocity in a laminar pipe flow.<\/p>\r\n
<\/p>\r\n
Next, for the pressure gradient in Eq. (9), if pressure loss is applied by \u0394p<\/em> in the section of pipe length L, Eq. (11) is obtained.<\/p>\r\n$$-\\frac{dp}{dx}=\\frac{\u0394p}{L}\u30fb\u30fb\u30fb(11)$$<\/p>\r\n
Substituting equation (11) into equation (9), we obtain the following equation.<\/p>\r\n
$$\\bar{u}=\\frac{R^{2}\u0394p}{8\u03bcL}$$<\/p>\r\n
\r\n\r\n<\/p>\r\n
$$\u0394p=\\frac{8\u03bcL\\bar{u}}{R^{2}}$$<\/p>\r\n
Rewrite the radius R<\/em> of the pipe as a diameter d<\/em> (R<\/em>=d<\/em>\/2).<\/p>\r\n$$\u0394p=\\frac{32\u03bcL\\bar{u}}{d^{2}}\u30fb\u30fb\u30fb(1)$$<\/p>\r\n
We were able to derive the Eq. (1) of this article.<\/p>\r\n
In conclusion<\/h2>\r\n
Derived the Hagen-Poiseuille’s equation for use in laminar pipe flow.<\/p>\r\n
Understanding the derivation process makes it easier to remember the equation. Let’s derive it at least once.<\/p>","protected":false},"excerpt":{"rendered":"
The equation relating pressure drop and flow velocity in laminar pipe flow is called the Hagen-Poiseuille equation. This article explains the derivation of the Hagen-Poiseuille equation.<\/p>\n","protected":false},"author":1,"featured_media":236,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[7],"tags":[],"class_list":["post-227","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-fluid-dynamics"],"_links":{"self":[{"href":"https:\/\/chemical-engineering-review.com\/en\/wp-json\/wp\/v2\/posts\/227","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/chemical-engineering-review.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/chemical-engineering-review.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/chemical-engineering-review.com\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/chemical-engineering-review.com\/en\/wp-json\/wp\/v2\/comments?post=227"}],"version-history":[{"count":13,"href":"https:\/\/chemical-engineering-review.com\/en\/wp-json\/wp\/v2\/posts\/227\/revisions"}],"predecessor-version":[{"id":245,"href":"https:\/\/chemical-engineering-review.com\/en\/wp-json\/wp\/v2\/posts\/227\/revisions\/245"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/chemical-engineering-review.com\/en\/wp-json\/wp\/v2\/media\/236"}],"wp:attachment":[{"href":"https:\/\/chemical-engineering-review.com\/en\/wp-json\/wp\/v2\/media?parent=227"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/chemical-engineering-review.com\/en\/wp-json\/wp\/v2\/categories?post=227"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/chemical-engineering-review.com\/en\/wp-json\/wp\/v2\/tags?post=227"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}