(1) <\/span> <\/span> Where <\/a><\/p>\n (2) <\/span> <\/span> Working out the value for pressure:<\/p>\n <\/a><\/p>\n (3) <\/span> <\/span> Due to flow is going out the pipe it is certain that: <\/a><\/p>\n (4) <\/span> <\/span> Taking into account the inequality in characteristic values, it is possible to present the next equation:<\/p>\n <\/a><\/p>\n (5) <\/span> <\/span> Due to flow is going out the pipe it is certain that:<\/p>\n <\/a><\/p>\n (6) <\/span> <\/span> Taking into account the inequality in characteristic values, it is possible to present the next equation:<\/p>\n <\/a><\/p>\n (7) <\/span> <\/span> Then<\/p>\n <\/a><\/p>\n (8) <\/span> <\/span> In conclusion:<\/p>\n <\/span> <\/span> Similarly to outgoing flow, next cases are obtained and showed in the equations below.<\/li>\n <\/span> <\/span> <\/span> <\/span> Due to physical phenomenon depends on flow direction; the employed model will also depend on it. In the next paragraphs are presented the necessary hypothesis for outgoing flow and ingoing flow.<\/p>\n When flow is going out the pipe, it can be considered that pressure loss between the last cross sectional area of the pipe and the exterior are not important. So, it is assumed that pressure will be the same; according to this, the working fluid is getting laminar at constant pressure up till it loses its velocity by friction with surrounding air. <\/a><\/p>\n (9) <\/span> <\/span> Where working out the speed of sound <\/a><\/p>\n (10) <\/span> <\/span> At supersonic situation, <\/a><\/p>\n (11) <\/span> <\/span> In general situation pressure losses are produced and the streamline suffers a contraction that implies a reduction in flow velocity and so a reduction in the ingoing mass flow. These losses are taken into account by means of a velocity loss coefficient. <\/a><\/p>\n (12) <\/span> <\/span> The incident characteristic equation for ingoing flow is:<\/p>\n <\/a><\/p>\n (13) <\/span> <\/span> And the loss velocity coefficient is:<\/p>\n <\/a><\/p>\n (14) <\/span> <\/span> Where The introduction of the auxiliary variable ![\"\begin{equation*}](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-7f9009635fca759519eb8162759f5f7b_l3.png\" "\\"Rendered")
<\/p>\n![\"A_A\"](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-41e8326ea5c7185333f4ac45f8805157_l3.png\" "\\"Rendered")
is the entropy level and ![\"A\"](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-783721c5f28851887dd4199f225c37be_l3.png\" "\\"Rendered")
dimensionless speed of sound, that is related with
\ncharacteristics variables.<\/p>\n![\"\begin{equation*}](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-64cea345cfc209e011dbe406dddb011d_l3.png\" "\\"Rendered")
<\/p>\n![\"\begin{equation*}](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-c672ee618e5ebd9bc554d13ce622ec57_l3.png\" "\\"Rendered")
<\/p>\n![\"\begin{equation*} \left| {U_1 } \right| = \frac{{\lambda _{in,1} - \lambda _{out,1} }}{{\gamma - 1}} \to \left| {U_1 } \right| > 0 \to \lambda _{in,1} > \lambda _{out,1}](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-dc18ef25e598e33fad81cd5f7ebb89aa_l3.png\" "\\"Rendered")
<\/p>\n![\"\begin{equation*}](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-6c1c9eddf7c65899cc1465ffc9abf2db_l3.png\" "\\"Rendered")
<\/p>\n![\"\begin{equation*}](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-9965946e6740e690b0e9fdd9d9a34293_l3.png\" "\\"Rendered")
<\/p>\n![\"\begin{equation*}](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-c3ec78111dcdeadcd4be60328620dcab_l3.png\" "\\"Rendered")
<\/p>\n![\"\begin{equation*}](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-d485f855a9e6c4dc5733f0a2ac31f229_l3.png\" "\\"Rendered")
<\/p>\n\n
![\"\[\frac{\hat\lambda_{in}}{\hat A_A}>\left[\frac{p_a}{p_{ref}}\right]^{\frac{\gamma-1}{2\gamma}}\]\"](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-604b6c5464e33982a739fee2e7518875_l3.png\" "\\"Rendered")
<\/p>\n![\"\[\frac{\hat\lambda_{in}}{\hat A_A}=\left[\frac{p_a}{p_{ref}}\right]^{\frac{\gamma-1}{2\gamma}}\]\"](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-457252e88e0cff975c7e6b96f4ab6b34_l3.png\" "\\"Rendered")
<\/p>\n<\/li>\n![\"\[\frac{\hat\lambda_{in}}{\hat A_A}<\left[\frac{p_a}{p_{ref}}\right]^{\frac{\gamma-1}{2\gamma}}\]\"](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-b19202822a4ab8391a4f2012085cad20_l3.png\" "\\"Rendered")
<\/p>\n<\/li>\n<\/ul>\nPhisical model for open end boundary<\/h1>\n
Flow going out<\/h2>\n
Subsonic flow<\/h3>\n
\nOwing to ![\"\lambda_{in}\"](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-873422093fd0f853fbf42d698a7ef8b7_l3.png\" "\\"Rendered")
and are known in this situation, it is only necessary to determine the ![\"\lambda_{out}\"](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-3a7e7291095b69b15d9e979a08f5e09b_l3.png\" "\\"Rendered")
value. Remembering that pressure at the end of pipe has been supposed the same as the exterior one, we have:<\/p>\n![\"\begin{equation*}](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-9c67d522bec974da592e2b066ac462d5_l3.png\" "\\"Rendered")
<\/p>\n , and replacing it in the definition of characteristic variables, it is obtained finally:<\/p>\n![\"\begin{equation*}](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-8b2e4e298c586d92275e561b497baea5_l3.png\" "\\"Rendered")
<\/p>\nSupersonic Flow<\/h3>\n
![\"A<\left|U\right|\"](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-31a707ca7d6b00c1f24ae4b4a23af578_l3.png\" "\\"Rendered")
,it is simply imposed that flow velocity at the outgoing section is equal to the speed of sound ![\"A=\left|U\right|\"](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-0e9da5baa9473eb20a7d8fce164646ef_l3.png\" "\\"Rendered")
(sonic condition).The condition of sonic limit leads to:<\/p>\n![\"\begin{equation*}](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-6d1f33b5c36dbc73f04fde39dd262189_l3.png\" "\\"Rendered")
<\/p>\nFlow going into<\/h2>\n
Subsonic flow<\/h3>\n
\nIn this case the energy equation is presented between the final section and the exterior:<\/p>\n![\"\begin{equation*}](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-06a3173ae11bdd44a7def05e5398f7ee_l3.png\" "\\"Rendered")
<\/p>\n![\"\begin{equation*}](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-8be0b0bbfd76ad1f97e45077efee8d0b_l3.png\" "\\"Rendered")
<\/p>\n![\"\begin{equation*}](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-b56a323e813e0531a989d73f0694384d_l3.png\" "\\"Rendered")
<\/p>\n![\"\left|U_s\right|\"](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-27b72725a2b4a757175bdbe8b19818b0_l3.png\" "\\"Rendered")
is the highest speed and it corresponds to the isentropic case and ![\"\psi\"](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-090e1cdfe71809de76486a20cd425d1f_l3.png\" "\\"Rendered")
has to be obtained experimentally.<\/p>\n![\"|U_s|\"](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-76d0f55fe1eabd5494c1668b13cdefe0_l3.png\" "\\"Rendered")
requires additional equations that correspond to suppose an isentropic evolution between the exterior and pipe:<\/p>\n
![\"\begin{equation*}](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-e8b655baee8611bbf25cb2d38d8784b7_l3.png\" "\\"Rendered")
<\/p>\n![\"\begin{equation*}](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-4b51b11aa838206d42bf6e2c80cc6f65_l3.png\" "\\"Rendered")
<\/p>\n![\"\begin{equation*}](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-b6a279e82db6854b5736c1adb0fc838d_l3.png\" "\\"Rendered")
<\/p>\n![\"\left|U\right|\"](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-f5924c04d2536fe5fcf495ead13a9696_l3.png\" "\\"Rendered")
is obtained the value of ![\"\left|U\right|=\psi\left|U_s\right|\"](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-4cb6417850f4c44f14f43eab88076322_l3.png\" "\\"Rendered")
<\/p>\n![\"\[A_s=\sqrt{A_a^2-\frac{\gamma-1}{2}\left|U\right|^2}\]\"](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-0b2af5444b129814154fcd5d1a897637_l3.png\" "\\"Rendered")
<\/p>\n![\"A_a\"](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-01d80fbe0b7f5566466707684ed7a045_l3.png\" "\\"Rendered")
are known, ![\"\[A=\sqrt{A_a^2-\frac{\gamma-1}{2}\left|U\right|^2}\]\"](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-bf652f283a300c41de9188a9afee4199_l3.png\" "\\"Rendered")
<\/p>\n![\"\[A_A=A\frac{A_A_a}{A_s}\]\"](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-89692e7cd7b06a99674cf6c088a670c1_l3.png\" "\\"Rendered")
<\/p>\n![\"\[\lambda_{in}=A-\frac{\gamma-1}{2}\left|U\right|\]\"](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-cadfb1d4eaaa544f34c2bf00698a587a_l3.png\" "\\"Rendered")
<\/p>\n![\"\[A=\left|U\right|\]\"](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-5461ff7e8745f57e1c5cfd334a0d940b_l3.png\" "\\"Rendered")
<\/p>\n![\"\[A=\sqrt{\frac{2}{\gamma+1}}A_a\]\"](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-f16e10bf0412d48ae98d436c57183136_l3.png\" "\\"Rendered")
<\/p>\n![\"\[A_A=\frac{\hat A_A \cdot A}{\hat \lambda_{in}-\frac{\gamma-1}{2}\left|U\right|}=\frac{\hat A_A \cdot A}{\hat \lambda_{in}-\frac{\gamma-1}{2}A}\]\"](\"https:\/\/openwam.webs.upv.es\/docs\/wp-content\/ql-cache\/quicklatex.com-87995cfaa901b0fb5bb09decb2a9026f_l3.png\" "\\"Rendered")
<\/p>\n