# Introduction to Heat and Mass Transfer. Week 12

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1 Introduction to Heat and Mass Transfer Week 12

2 Next Topic Convective Heat Transfer» Heat and Mass Transfer Analogy» Evaporative Cooling» Types of Flows

3 Heat and Mass Transfer Analogy Equations governing the development of thermal and concentration boundary layers analogous Advection terms balanced by diffusion terms corresponding to heat and mass transport Thermal boundary layer governed by Pr while concentration boundary layer governed by Sc; Re governs both For similar boundary conditions for a particular geometry, we find that the functional form of solutions for thermal and concentration boundary layers same

4 Heat and Mass Transfer Analogy (contd.) Under similar geometric and boundary conditions: Nu f x, Re, Pr f x, Re Pr L L Sh f x Re Sc f x Re Sc,,, L Since the functional form is the same, we can write: Nu Sh Pr Sc Convective heat transfer coefficient and convective mass transfer coefficient are related via: h h m k D Le AB n n n C Le p 1n L n n

5 Heat and Mass Transfer Analogy (contd.) With the negligible pressure gradient in the flow direction and Pr = Sc = 1; velocity, thermal and concentration boundary layer equations exactly similar We must have functional forms of non-dimensional velocity, temperature and concentration exactly the same Under these conditions, friction coefficient, Nusselt number and Sherwood number are related via: ReL C Nu Sh f 2

6 Heat and Mass Transfer Analogy (contd.) Modifying Nusselt and Sherwood numbers: St Nu Re Pr L The relationship between relevant parameters: C f 2 The above relationship valid for the negligible pressure gradient and Pr = Sc = 1; however, using correction factors (such as Colburn j factor) extendable to wider Pr and Sc St St St m m Sh Re Sc L

7 Evaporative Cooling Flow of gas over a liquid surface is important in many engineering applications During evaporation (phase change) the liquid loses internal energy and undergoes cooling In steady state, however, the liquid is heated via sensible energy gain (due to convection and/or radiation) from gas Heat and mass transfer analogy can be applied for understanding the transport processes in this situation

8 Evaporative Cooling (contd.)

9 Example A process involves evaporation of water from a liquid film that forms on a contoured surface. The convection heat transfer correlation is known to be: Nu L = 0.43Re L 0.58 Pr 0.4.» For dry air at 27C flowing with a velocity of 10 m/s, what is the rate of evaporation from 1 m 2 surface area having a characteristic length of 1 m? Approximate the density of saturated vapor as sat = kg/m 3.» What is the steady state temperature of the liquid film?

10 Closure Coverage thus far..» talked about the analogy between heat and mass transfer under geometrically similar conditions Heat and mass transfer analogy and its utility in engineering calculations of friction, energy and mass Nu Sh Pr Sc n n ReL C Nu Sh f 2 C f 2 St St m Evaporative cooling and related applications

11 Closure (contd.) Heat and Mass Transfer Analogy Nu Sh Pr Sc n n h h m k D Le AB n C Le p 1n Heat and mass transfer analogy (special case)(reynolds analogy) ReL C Nu Sh f 2 Modified heat and mass transfer analogy (modified Reynolds analogy) C f 2 St St m

12 Questions What are the limitations of the heat and mass transfer analogy (Reynolds analogy)? What is the modified heat and mass transfer analogy? What is the significance of evaporative cooling in practical engineering applications?

13 Next Topic Types of flow» Based on fluid viscosity» Based on fluid density» Based on fluid velocity» Based on boundary layer development

14 Types of Flows Based on fluid viscosity» Viscous» Inviscid Based on fluid density» Compressible» Incompressible Based on fluid velocity» Laminar» Turbulent Based on boundary layer development» External» Internal

15 Laminar vs. Turbulent Turbulent u Laminar Transition Turbulent Layer Buffer Layer Laminar Sublayer Leading Edge x

16 Laminar vs. Turbulent (contd.) Laminar regions characterized by ordered fluid motion Transition regions extremely difficult to describe completely most cases experiments required Turbulent regions characterized by disordered fluid motion originated from few disturbances Turbulent boundary layers thicker and velocity profiles flatters than laminar boundary layers Turbulent fluctuations result in better mixing of fluid leading to surface friction and convection transport rates higher than laminar regions

17 External Flow Empirical Method Flat Plate

18 Problem Objective For any convective heat transfer problem: For any convective mass transfer problem: How to get the functional form?» Experiments» Theory» Computations Nu f x, Re, Pr L Nu f Re, Pr Sh f x, Re, Sc L Sh f Re, Sc L L

19 Empirical Method Using an appropriate experiment, we can write: Nu L m n CRe Pr L Sh L m n CRe Sc L Note that the values of coefficient C and exponents m and n vary with the surface geometry and type of flow Usually fluid properties based on mean boundary layer temperature or film temperature T film T T 2 Alternatively, we can evaluate all fluid properties at ambient temperature and apply a correction factor s

20 Theory Flat Plate: Laminar Flow We solve boundary layer conservation equations to obtain solutions of velocity, thermal and concentration boundary layer for parallel flow over a flat plate Using an appropriate similarity variable and employing the Blasius method, we first solve velocity boundary layer:» VBL Thickness flat plate laminar 5x Re 1/ 2 x» Friction Coefficient C flat plate 0.664Re, f x laminar x 1/ 2

21 Complementary Materials Forced Convection Cooling

22 Heat Flux in Convection

23 Heat Transfer Coefficient

24 Heat Transfer Coefficient

25 Boundary Layer Equations

26 Boundary Layer Equations

27 Velocity Profiles in Laminar Flat Plate Flow

28 Blasius (Similarity) Solution

29 Boundary Layer Thickness

30 Friction Coefficient

31 Temperature Profile from Energy Equation

32 Local Heat Transfer Coefficient

33 Nusselt Number

34 Nusselt Number

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