Lesson 5 : Critical Flow in Open Channel
Topic: Critical flow in Open Channel
Question: In the last lecture we learnt about the specific energy and specific energy diagram, Can anyone remember the definition of specific energy?
Thesis: In today’s lecture we will learn about the critical flow and its criteria.
Main Points: 1. Critical flow 2. Criteria of critical flow
Transition word+ first main point: First, let’s recall the condition of critical flow, the flow is called critical flow when the Froude number is equal to unity.
Explanation: From the definition of specific energy we have come to know that the energy per unit weight of water measured with respect to the channel bottom and the equation can be written as: =y+V^2/2g , if we replace the velocity v from continuity equation, we will have the following equation of specific energy: E=y+Q^2/〖2gA〗^2 ..(1). If we differentiate the equation with respect to the depth of water, dE/dy=1+Q^2/2g(-2)(A^(-3))dA/dy or dE/dy=1-Q^2/(gA^3 )(dA/dy), if we assume an elementary water area near the free surface, then dA=Bdy, so that dA/dy = B, since the hydraulic radius, D = A/B and Froude number, Fr=V/√gD, we can obtain from the above equation dE/dy=1-(V^2 B)/gA=1-V^2/g(A/B) =V^2/gD=1-〖Fr〗^2, now for the minimum specific energy, dE/dy = 0. And hence 1-Fr2 = 0 and Fr =1, which is the criteria for the critical flow. Thus, at the critical state of flow, the specific energy is minimum for a given discharge.
Questions for students about this main point: Can you tell me what will happen if the specific energy remains constant however, the discharge varies?
Transition word + second main point: In order to answer this question, we have to differentiate the above equation with respect to Q considering the E as constant. If you differentiate it will also be the critical flow for maximum discharge.
Explanation: Now let’s find out another property of critical flow. By this time we know very well that the flow is said to be in a critical state when the Froude number is unity. So, from the definition of Froude number, Fr =1, Fr=V/√gD=1=V/√gD= Vc2 = gDc, 〖V_c〗^2/2g= D_c/2, meaning that at the critical state of flow, the velocity head is equal to one half of hydraulic depth. There are some other properties of critical flow such as the specific force will be minimum also.
Question for student about this main point: So, for a rectangular channel, what will be the minimum specific energy? Ec=y_c+〖V_c〗^2/2g= y_c+y_c/2 = 1.5yc
Conclusion: In conclusion, we have learnt the properties of critical flow such as the specific energy and force are minimum but the discharge is maximum at critical state of flow. Moreover, the velocity head is one half of hydraulic depth and for rectangular channel Sp. Energy is 1.5 times the depth.