HEAT TRANSFER Recall: fluid mechanics 们au Heat Transfer Su Yongkang School of Mechanical Engineering
Heat Transfer Su Yongkang School of Mechanical Engineering # 1 HEAT TRANSFER Recall: Fluid Mechanics
What is Convective Heat Transfer? You have already experienced it. Difficulty lies in generalizing our experience; filtering it down to a few laws: learning how to apply these laws to systems we engineers design and use Here is what i want you to do: If a person masters the fundamentals ofhis subject and has learned to think and work independently, he will surely find his way and besides will better be able to adapt himselfto progress and changes than the person whose training principally consists in the acquiring of detailed knowledge. Albert einstein So. please read ahead and come prepared with good questions for the class. Heat Transfer Su Yongkang School of Mechanical Engineering
Heat Transfer Su Yongkang School of Mechanical Engineering # 2 What is Convective Heat Transfer? ─ You have already experienced it. ─ Difficulty lies in generalizing our experience; filtering it down to a few laws; learning how to apply these laws to systems we engineers design and use. ─ Here is what I want you to do: ─ If a person masters the fundamentals of his subject and has learned to think and work independently, he will surely find his way and besides will better be able to adapt himself to progress and changes than the person whose training principally consists in the acquiring of detailed knowledge. ─ – Albert Einstein ─ So, please read ahead and come prepared with good questions for the class
1. Introduction to DIMENSIONAL ANALYSIS Dimensions Each quantitative aspect provides a number and a unit F or example =5m/s The three basic dimensions are l. t and m Alternatively, L, T, and F could be used We can write F÷LT F三MLT The notation is used to indicate dimensional equality Heat Transfer Su Yongkang School of Mechanical Engineering
Heat Transfer Su Yongkang School of Mechanical Engineering # 3 1. Introduction to DIMENSIONAL ANALYSIS 1) Dimensions Each quantitative aspect provides a number and a unit. For example, The three basic dimensions are L, T, and M. Alternatively, L, T, and F could be used. We can write The notation is used to indicate dimensional equality. V = 5m/s 2 1 − − = = F MLT V LT =
2) Dimensional homogeneity Fundamental premise All theoretically derived equations are dimensionally homogeneous----that is, the dimensions of the left side of the equation must be the same as those on the right side and all additive separate terms must have the same dimensions For example, the velocity equation 0 +at In terms of dimensions the equation is LT=lt+lt ----dimensional homogeneous Heat Transfer Su Yongkang School of Mechanical Engineering
Heat Transfer Su Yongkang School of Mechanical Engineering # 4 2) Dimensional homogeneity Fundamental premise: All theoretically derived equations are dimensionally homogeneous----that is, the dimensions of the left side of the equation must be the same as those on the right side, and all additive separate terms must have the same dimensions. For example, the velocity equation, In terms of dimensions the equation is ----dimensional homogeneous −1 −1 −1 LT = LT + LT V =V + at 0
3)Dimensional analysis A problem An incompressible. Newtonian fluid steady flow, through a long, smooth-walled, horizontal circular pipe \P,=f(D, p, u, r) The nature of function is unknown and the experiments are necessary We can recollect these variables into dimensionless products DAp OVD oV2 Variables from 5 to 2 #5 Heat Transfer Su Yongkang School of Mechanical Engineering
Heat Transfer Su Yongkang School of Mechanical Engineering # 5 p f (D V ) l = , ,, 3) Dimensional analysis A problem. An incompressible, Newtonian fluid, steady flow, through a long, smooth-walled, horizontal, circular pipe. The nature of function is unknown and the experiments are necessary. We can recollect these variables into dimensionless products, Variables from 5 to 2. = VD V D pl 2
Here D△ L(F/L (FL T)L7)2 FLTO pVD. (FL T(LT )L F 0r00 The results will be independent of the system of units This type of analysis is called -o---dimensional analysis Heat Transfer Su Yongkang School of Mechanical Engineering
Heat Transfer Su Yongkang School of Mechanical Engineering # 6 Here, The results will be independent of the system of units. This type of analysis is called -----dimensional analysis. 0 0 0 2 4 2 1 0 0 0 4 2 1 2 3 2 ( ) ( )( ) ( )( ) ( / ) F L T FL T VD FL T LT L F L T FL T LT L F L V D pl = = = = − − − − −
4)Buckingham Pi theorem If an equation involving k variables is dimensionally homogeneous it can be reduced to a relationship among k-r independent dimensionless products, where r is the minimum number of reference dimensions required to describe the variables Heat Transfer Su Yongkang School of Mechanical Engineering
Heat Transfer Su Yongkang School of Mechanical Engineering # 7 4) Buckingham Pi theorem If an equation involving k variables is dimensionally homogeneous, it can be reduced to a relationship among k-r independent dimensionless products, where r is the minimum number of reference dimensions required to describe the variables
Here we use the symbol ii to represent a dimensionless product For equation f(u 2:35 We can rearrange to I1=(2,2,→) Usually, the reference dimensions required to describe the variables will be the basic dimensions m.l. and t or f.l. andt In some cases, maybe only two are required, or just one determination of pi terms??? Heat Transfer Su Yongkang School of Mechanical Engineering
Heat Transfer Su Yongkang School of Mechanical Engineering # 8 Here, we use the symbol to represent a dimensionless product. For equation, We can rearrange to, ( ) k u f u ,u ,...,u 1 = 2 3 ( ) = k−r , ,..., 1 2 3 Usually, the reference dimensions required to describe the variables will be the basic dimensions M, L, and T or F, L, and T. In some cases, maybe only two are required, or just one. determination of Pi terms???
2. The Navier-Stokes Equations Combine the differential equations of motion. the stress-deformation relationships and the continuity equation u0=-+/+八 ouou ou +i-+v-+ at Ox ay az a at a—aa y av a2 +-+v—+1 c厂=+/8,+(ax=++a +i-+v-+W 8 Heat Transfer Su Yongkang School of Mechanical Engineering
Heat Transfer Su Yongkang School of Mechanical Engineering # 9 2. The Navier-Stokes Equations Combine the differential equations of motion, the stress-deformation relationships and the continuity equation. + + + + = − + + + + + + + = − + + + + + + + = − + + + 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 z w y w x w g z p z w w y w v x w u t w z v y v x v g y p z v w y v v x v u t v z u y u x u g x p z u w y u v x u u t u z y x
Here, four unknowns(u, V, W, p. We know the conservation of mass equation, +— --four equations Nonlinear, second order, partial differential equations #10 Heat Transfer Su Yongkang School of Mechanical Engineering
Heat Transfer Su Yongkang School of Mechanical Engineering # 10 Here, four unknowns (u, v, w, p.) We know the conservation of mass equation, ----------four equations. Nonlinear, second order, partial differential equations. = 0 + + z w y v x u
