Convection Heat Transfer

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1 School of Mechanical Engineering Sharif University of Technology Convection Heat Transfer Instructor: M.H.Saidi PhD., Professor Lecture #3 Fall

2 2 Convection Heat Transfer

3 Summary of second session: vthermodynamics laws van introduction to scale analysis This session: Scale analysis- Introduction to boundary layer flow 3

4 RULES OF SCALE ANALYSIS 4

5 RULES OF SCALE ANALYSIS 5

6 RULES OF SCALE ANALYSIS 6

7 RULES OF SCALE ANALYSIS c = a + b O(a) > O(b) O(c) = O(a) 7

8 RULES OF SCALE ANALYSIS c = a + b O(a) = O(b) O(a) O(b) O(c) 8

9 RULES OF SCALE ANALYSIS p = ab O(p) = O(a)O(b) r = a/b O(p) = O(a)/O(b) 9

10 HEATLINES FOR VISUALIZING CONVECTION (x,y) 10

11 HEATLINES FOR VISUALIZING CONVECTION 11

12 12 Convection Heat Transfer

13 LAMINAR BOUNDARY LAYER FLOW As students and researchers, we can learn important lessons from the history of boundary layer theory: 13

14 LAMINAR BOUNDARY LAYER FLOW FUNDAMENTAL PROBLEM IN CONVECTIVE HEAT TRANSFER Velocity and temperature boundary layers near a plate parallel to a uniform flow 14

15 LAMINAR BOUNDARY LAYER FLOW FUNDAMENTAL PROBLEM IN CONVECTIVE HEAT TRANSFER If this flat plate is the plate fin protruding from a heat exchanger surface into the stream that bathes it, we want to know: 1. The net force exerted by the stream on the plate 2. The resistance to the transfer of heat from the plate to the stream 15

16 LAMINAR BOUNDARY LAYER FLOW FUNDAMENTAL PROBLEM IN CONVECTIVE HEAT TRANSFER we are interested in calculating the total force and toal heat transfer as follows: 16

17 LAMINAR BOUNDARY LAYER FLOW FUNDAMENTAL PROBLEM IN CONVECTIVE HEAT TRANSFER The no-slip condition implies that since the 0 < y < 0 Fluid layer is motionless, the heat transfer from the wall to the fluid is first governed by pure conduction. Therefore, in place of eq. (4), we can write the statement for pure conduction through the fluid layer immediately adjacent to the wall. 17

18 LAMINAR BOUNDARY LAYER FLOW FUNDAMENTAL PROBLEM IN CONVECTIVE HEAT TRANSFER Modeling the flow as incompressible and of constant property (Chapter 1), the complete mathematical statement of this problem consists of the following: 18

19 LAMINAR BOUNDARY LAYER FLOW FUNDAMENTAL PROBLEM IN CONVECTIVE HEAT TRANSFER for four unknowns (u,v,p,t), subject to the following boundary conditions: 19

20 LAMINAR BOUNDARY LAYER FLOW CONCEPT OF BOUNDARY LAYER we have the freedom to think that the velocity change from u =0 to u =U and the temperature change from T=T0 T=T occurs in a space situated relatively close to the solid wall. The free stream is characterized by: Let d be the order of magnitude of the distance in which u changes from 0 at the wall to roughly U in the free stream. 20

21 LAMINAR BOUNDARY LAYER FLOW CONCEPT OF BOUNDARY LAYER In the d L region, then, the longitudinal momentum equation (8) accounts for the competition between three types of forces: From the mass continuity equation, we have: 21

22 LAMINAR BOUNDARY LAYER FLOW CONCEPT OF BOUNDARY LAYER we learn that the inertia terms in eq. (14) are both of order 2 U / L ; hence, neither can be neglected at the expense of the other. However, if the boundary layer region d L is slender, such that: The last scale in eq. (14) is the scale most representative of the 2 2 friction force. Neglecting the term U / x at the expense of the 2 2 U / y term in the x momentum equation (8) yields: 22

23 Next session: vlaminar Boundary Layer Flow(Velocity and Temperature fields) 23

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