Introduction to Heat and Mass Transfer. Week 10

Transcription

1 Introduction to Heat and Mass Transfer Week 10

2 Concentration Boundary Layer No concentration jump condition requires species adjacent to surface to have same concentration as at the surface Owing to concentration gradients depending on fluid motion, the layers away from the wall have lower concentration Through the concentration boundary layer, concentration gradients and diffusive transfer rates are large Outside the boundary layer, concentration gradients and diffusive transfer rates are relatively small Concentration boundary layer grows with distance from the leading edge

3 Concentration Boundary Layer (contd.) C A, y c Leading Edge x c y when y C A = C A,S ; x = 0 C C A, s C A C A, s A, 0.99

4 Concentration Boundary Layer (contd.) Using Fick s law at the surface, we can write: C " A N D A, s AB y y 0 The above mass transports away from the surface via advection process Local convective mass transfer coefficient along the surface is given as: h m C D D C C C C A A " AB AB N y y As, y0 y0 A, s A, A, s A, A, s A,

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6 Example

7 HW # 6 prob. 2 On a summer day the air temperature is 32C and the relative humidity is 40%. Water evaporates from the surface of a lake at the rate of 0.2 kg/hr per m 2 of water surface area. The temperature of water is 32C.» Determine the convective mass transfer coefficient

8 Closure In this lecture, we» talked about the basic mechanisms for convective heat and mass transfer» discussed the importance of boundary layer concepts in dealing with convective transport problems

9 Closure (contd.) Basic heat and mass convection mechanism Local and average convective heat transfer coefficient (h) and convective mass transfer coefficient (h m ) Velocity Boundary Layer (momentum transfer)» shear stress velocity gradient dynamic viscosity Thermal Boundary Layer (energy/heat transfer)» heat flux temperature gradient thermal diffusivity Concentration Boundary Layer (mass transfer)» mass flux concentration gradient binary diffusivity

10 Next Topic Convective Heat Transfer» Convection Transfer Equations» Boundary Layer Approximations

11 Convection Transfer Equations Consider motion of a fluid over a solid surface with velocity, temperature and concentration gradients We will consider steady, two-dimensional flow of a viscous, incompressible Newtonian fluid Equations governing the convective transport of momentum, energy and mass are based on the following principles:» Conservation of Mass» Conservation of Momentum» Conservation of Energy» Conservation of Species

12 Continuity Equation Conservation of mass requires that mass can neither be created nor be destroyed Considering a differential control volume and overall mass conservation for that control volume u x y 0

13 Momentum Equation Application of Newton s second law of motion requires that the net force acting on the fluid must be balanced by the net rate of momentum We consider two types of forces:» surface forces (pressure, viscous/normal effects)» volumetric forces (gravity, electric, magnetic effects) 2 2 u u p u u u X 2 2 x y x x y 2 2 p u Y 2 2 x y y x y

14 Energy Equation Conservation of energy requires that energy can neither be created nor be destroyed x y x y 2 2. T T T T C u k q p 2 2 Viscous dissipation owing to irreversible conversion of mechanical work to thermal energy u u 2 y x x y

15 Species Continuity Equation Conservation of mass applied to a specific species in a mixture of given species Species continuity useful for understanding mass transfer x y x y 2 2. C C C C A A A A u D N AB 2 2 A

16 Boundary Layer Approximations Boundary layer thickness typically very small and gradients through the boundary layer very large The following approximations are applicable:» Velocity and its gradients u» Temperature gradients u u,, y x x y T T y x» Concentration gradients C y A C x A

17 Boundary Layer Equations Continuity: Momentum: Energy: Species: u x y x y 2 u u 1 p u 2 T T T u u x y y 2 C p y u u 0 x y 2 p y 2 C C C A A A DAB 2 x y y 2 0

18 Example

19 HW # 6 prob. 3 Consider Couette flow for which the moving plate is maintained at a uniform temperature while the stationary plate is insulated.» Determine the temperature of the stationary plate in terms of fluid properties and temperature as well as speed of the moving plate» Obtain an expression for the heat flux entering the moving plate

20 HW # 6 prob. 4 Consider a 5-cm diameter shaft rotating at 2500 rpm in a 10-cm long bearing with a clearance of 0.5 mm.» Determine the power required to rotate the shaft if the fluid in the gap is air at 40C at 1 atm» How much power is required to rotate the shaft if the fluid is water at 40C at 1 atm?» Estimate the power requirement to rotate the shaft if the fluid is oil at 40C at 1 atm

21 Questions For what types of fluids and flows is the viscous dissipation term in energy equation likely to be significant? For steady two-dimensional flow, what are the boundary layer approximations?

22 Closure Coverage thus far..» talked about the fundamental convection transfer equations governing transport of mass, momentum and energy in fluids Continuity Equation Momentum Equation Energy Equation Species Continuity Equation» described the conditions and relevant approximations applicable in boundary layers