# Outlines. simple relations of fluid dynamics Boundary layer analysis. Important for basic understanding of convection heat transfer

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1 Forced Convection

2 Outlines To examine the methods of calculating convection heat transfer (particularly, the ways of predicting the value of convection heat transfer coefficient, h) Convection heat transfer requires an energy balance along with an analysis of the fluid dynamics of the problem concern. So that. The discussion will consider; simple relations of fluid dynamics Boundary layer analysis Important for basic understanding of convection heat transfer

3 The region of flow that develops from the leading edge of the plate in which the effects of viscosity boundary layer (The y position where the boundary layer ends and the velocity become 0.99 of the free-stream value) Initially the boundary layer development is laminar but at some critical distance from the leading edge (depending on the flow field & fluid properties), small disturbance in the flow begin to become amplified, and a transition process takes place until the flow become turbulent. occurs when where

4 Flat plate Turbulence flow Tube Laminar flow Turbulence flow For transition between laminar & turbulence d = Tube diameter

5 Re. no. (Tube) Other Form: Define the mass velocity as So that, mass flow rate The Reynolds no. may also written

6 Inviscid Flow Classification of Fluid Flows Although no real fluid is inviscid, in some instance the fluid may be treated as such, and useful to present some of the equation that apply in these circumstances (the flow at a sufficiently large distance from the plate will behave as a nonviscous flow system). or Fundamentally a Dynamic Equation (The Bernoulli equation) Where,

7 To solve convection heat transfer coefficient, h we have to: 1)Identify the type of fluid involve to get the fluid properties 2)State the process

8 1) Type of Fluid- Fluid Properties # Specify the equation state of fluid to calculate pressure drop in compressible flow. An ideal gas Where: e = Internal Energy i = Entalpy Gas Air Where:

9 2) State the Process Example: Reversible Adiabatic Flow through a nozzle The relation involved which is relating the properties at some points in the flow to the Mach no. & stagnation properties Where: For an ideal gas: a: Local velocity of sound For air behaving as an ideal gas:

10 Example 5.1: Water flow in a diffuser Water at 20 C flows at 8 kg/s through the diffuser arrangement shown in Figure, the diameter at section 1 is 3.0 cm and the diameter at section 2 is 7.0 cm. Determine the increase in static pressure between sections 1 and 2. Assume frictionless flow.

11 Solution: The flow cross-sectional areas are: Known the density of water at 20 C is 1000 kg/m 3, so calculate the velocities from the mass-continuity relation: The pressure different obtain from the Bernoulli equation

12 Example 5.2: Isentropic Expansion of Air Air at 300 C and 0.7 Mpa pressure is expanded isentropically from a tank until the velocity is 300 m/s. Determine the static temp., pressure and Mach number of the air at the high-velocity condition. ϒ= 1.4 for air.

13 Laminar Boundary Layer on Flat Plate From the analytical analysis by making a force and momentum balance on the element yield the momentum equation for the laminar boundary layer with constant properties. Can be solved for many boundary conditions. For development in this chapter, we shall satisfied with an approximate analysis that furnishes an easier solution without a loss in physical understanding of process involved.

14 Laminar Boundary Layer on Flat Plate Consider the boundary layer flow system as shown: The free-stream velocity outside the boundary layer is u and the boundary layer thickness is. We wish to make a momentum-and-force balance on the control volume bounded by the plane 1, 2, A- A and the solid wall. The boundary layer thickness, Mass flow rate Where: So that

15 Example 5.3: Mass Flow & Boundary-Layer Thickness Air at 27 C and 1 atm flows over a flat plate at a speed of 2 m/s. Calculate the boundarylayer thickness at distances of 20 cm and 40 cm from the leading edge of the plate. Calculate the mass flow that enters the boundary layer between x=20 cm and x= 40 cm. The viscosity of air at 27 C is 1.85 x 10-5 kg/ m. s. Assume unit depth in the z direction.

16 The Thermal Boundary layer Exist when temperature gradient are present in the flow. If the fluid properties were constant throughout the flow, an appreciable variation between the wall and free stream condition which is film temp. T f define as: Used T f to get the fluid properties from properties fluids table

17 The Thermal Boundary layer Consider the system shown T w : The temp. of the wall T : The temp. of the fluid outside the thermal boundary layer : Thickness of the thermal boundary layer Basic: convection/conduction Case 1 For the plate heated over its entire length h x : Heat transfer coefficient in term of the distance from the leading edge of the plate 0.6 < Pr < 50 Re x Pr > 100 Used the average heat transfer coefficient (by integrating over the length of the plate

18 The Thermal Boundary layer Case 2 For the plate heated starts at or Where:

19 The Thermal Boundary layer Constant Heat Flux (q w = q/a) To find the distribution of the plate surface temp. and The average temp. difference along the plate So that From these equation, can be produce equation

20 To determine heat flow, q and heat flux, q w (laminar Flow) Heat flow, q Determine: 1) Film temp. There is an appreciable variation between wall & free stream condition, so that, it is recommended that the properties be evaluated at film temp. 2) The properties of fluid at T f such as kinematic viscosity, thermal conduction coefficient, heat capacity, Prandle no. 3) Re x at x=x L 4) Choice Nusselt No. equation based on system either For the plate heated over its entire length or for the plate heated starts at

21 Heat flow, q For the plate heated over its entire length 0.6 < Pr < 50 Re x Pr > 100 5) Heat transfer coefficient, h 6) The average heat transfer coefficient, 7) Heat Flow,

22 Heat flow, q For the plate heated starts at or 5) Heat transfer coefficient, h 6) Heat Flow,

23 Heat flux, qw

24 Example 5.4: Isothermal Flat Plate Heated Over Entire Length Air at 27 C and 1 atm flows over a flat plate at a speed of 2 m/s. The viscosity of air at 27 C is 1.85 x 10-5 kg/ m. s. Assume unit depth in the z direction and the plate is heated over its entire length to a temp. of 60 C. Calculate the heat transferred in (a) The first 20 cm of the plate and (b) The first 40 cm of the plate.

25 Example 5.5: Flat Plate with Constant Heat Flux (a) (b) A 1.0 KW heater is constructed of a glass plate with an electrically conducting film that produces a constant heat flux. The plate is 60 cm by 60 cm and placed in an airstream at 27 C, 1 atm with u = 5 m/s. Calculate The average temp. different along the plate and The temperature difference at the trailing edge.

26 Example 5.6: Plate with Unheated Starting Length Air at 1 atm and 300 K flows across a 20-cm-square plate at a free stream velocity of 20 m/s. The last half of the plate is heated to constant temp. of 350 K. Calculate the heat lost by the plate

27 Example 5.7: Oil Flow Over Heated Flat Plate Engine oil at 20 C is forced over a 20-cm-square plate at a velocity of 1.2 m/s. The plate is heated to a uniform temp. of 60 C. Calculate the heat lost by the plate.

28 The relation between fluid friction & heat transfer The shear stress at the wall be expressed in term of a friction coefficient, C f The relation between fluid friction and heat transfer for Laminar flow on a flat plate Where: This is important relation between friction & heat transfer is the drag force (D) which is depends on the average shear stress. The average of shear stress is a friction coeffiecient C fx Drag Force, D = (shear stress) (Area)

29 Example 5.8: Drag Force on a Flat Plate Air at 27 C and 1 atm flows over a flat plate at a speed of 2 m/s. Calculate the boundary-layer thickness at distances of 20 cm and 40 cm from the leading edge of the plate. Calculate the mass flow that enters the boundary layer between x=20 cm and x= 40 cm. The viscosity of air at 27 C is 1.85 x 10-5 kg/ m. s. Assume unit depth in the z direction and the plate is heated over its entire length to a temp. of 60 C. Compute the drag forced exerted on the first 40 cm of the plate using the analogy between fluid friction and heat transfer

30 Turbulent-boundary-layer heat transfer (q) 1) Determine either the flow is turbulent region or not. Check based on Re. no. Turbulent Region 2) Heat transfer (q) from the plate is: Where:

31 Turbulent-boundary-layer thickness ( ) The boundary layer thickness measured when >Re< The boundary layer is fully turbulent from the edge of the plate The boundary layer follows a laminar growth pattern up to Rcrit= 5 x 10 5 and turbulent growth thereafter (Transition point)

32 Example 5.9: Turbulent Heat Transfer from Isothermal Flat Plate Air at 20 C and 1 atm flows over a flat plate at 35 m/s. The plate is 75 cm long and is maintained at 60 C. Assume unit depth in the z direction, calculate a) the heat transfer from the plate. b) the turbulent boundary layer thickness at the end of the plate assuming that it develops (i) from the leading edge of the plate and (ii) from the transition point at Recrit = 5 x 10 5.

33 Heat Transfer in Laminar Tube Flow Consider the tube flow system Aim to calculate the heat transfer under developed flow condition (Laminar Flow) Consider the fluid element derive to get the velocity and temp. distribution

34 Heat Transfer in Laminar Tube Flow From the analysis, the analytical solution give; The velocity at the center of the tube The velocity distribution The temperature distribution

35 Total heat transfer in term of bulk- temperature different The total energy added (energy balance) The heat added, dq can be expressed in term of a bulk temp. different or h; The total heat transfer Note: When the statement is made that a fluid enters a tube used the Bulk Temp. to determine fluid properties

36 The Bulk Temperature In tube flow, convection heat transfer coefficient, h defined by: Why used T b? Where, T w : The wall temp. T b : Bulk temp. For most tube flow heat transfer problem, the topic is the total energy transferred to the fluid. At any x position, the temp. that is indicative of the total energy of the flow is an integrated mass energy average temp. over the entire flow area. The bulk temp. is representative of the total energy of the flow at the particular location

37 The Bulk Temperature From the analysis, the analytical solution give; The bulk temp The wall temp Heat transfer coefficient Heat transfer coefficient in term of the Nusselt No.

38 Heat Transfer in Laminar Tube Flow The relation to used to calculate heat transfer in laminar tube flow (The empirical relation) Fully developed laminar flow in tubes at constant wall temp. # used for long & smooth tube Fully developed laminar flow in tubes at constant wall temp. # used for short & smooth tube # the fluid properties are evaluated at mean bulk temp. of the fluid. Used if Where: Peclet number (Pe)

39 Heat Transfer in Laminar Tube Flow For rough tubes (relation fluid friction and heat transfer), expressed in term of the Stanton Number: Where;

40 Heat Transfer in Laminar Tube Flow To calculated local and average Nusselt No. for laminar entrance regions for the case of a fully developed velocity profile used Graph with inverse Graetz number

41 Local & average Nusselt No. for circular tube thermal entrance regions in fully developed laminar flow

42 Turbulent Flow in a Tube Velocity profile for turbulent flow in a tube To determine heat transfer analytically should know the temp. distribution in the flow # to obtain temp. distribution, the analysis must take into consideration the effect of the turbulent eddies in the transfer of heat and momentum)

43 Turbulent Flow in a Tube From the analysis, the analytical solution give; Where, Relates the heat transfer rate to the friction loss in tube flow Heat transfer coefficient in term of the Nusselt No. or Pr 1.0 Pr 2/3 Relation for turbulent heat transfer in smooth tube # from this analytical solution, shows that h higher than those observed in experiment

44 Turbulent Flow in a Tube Correct relation to used to calculate heat transfer in turbulent tube flow (The empirical relation) is: If wide temp. different are present in the flow also change in the fluid properties between the wall of the tube & the central flow used Note: All the empirical relation here apply to fully developed turbulent flow in tubes

45 Turbulent Flow in a Tube More accurate although more complicated, the expression for fully developed turbulent flow in smooth tube is; or All the properties using in this equation based on T f

46 Turbulent Flow in a Tube For the entrance region (The flow is not developed), used:

47 Example 6.1: Turbulent Heat Transfer in a Tube Air at 2 atm and 200 C is heated as it flows through a tube with a diameter on 1 in (2.54 cm) at a velocity of 10 m/s. a) Calculated the heat transfer per unit length of tube if a constant-heat-flux condition is maintained at the wall and the wall temp. is 20 C above the air temp., all along the length of the tube. b) How much would the bulk temp. increase over a 3m length of the tube?

48 Example 6.2: Heating of Water in Laminar Tube Flow Water at 60 C enters a tube of 1-in (2.54 cm) diameter at a mean flow velocity of 2 cm/s. Calculate the exit water temp. if the tube is 3.0 m long and the wall temp. is constant at 80 C.

49 Example 6.3: Heating of Air in Laminar Tube Flow for Constant Heat Flux Air at 1 atm and 27 C enters a 5.0 mm diameter smooth tube with a velocity of 3.0 m/s. The length of the tube is 10 cm. A constant heat flux is imposed on the tube wall. a) Calculate the heat transfer if the exit bulk temp. is 77 C b) Calculate the exit wall temp. and the value of h at exit

50 Example 6.4: Heating of Air with Isothermal Tube Wall Air at 1 atm and 27 C enters a 5.0 mm diameter smooth tube with a velocity of 3.0 m/s. The length of the tube is 10 cm. A constant wall temp. is imposed on the tube wall. a) Calculate the heat transfer if the exit bulk temp. is 77 C b) Calculate the exit wall temp. and the value of h at exit

51 Example 6.5: Heat Transfer in a Rough Tube A 2.0 cm diameter tube having a relative roughness of is maintained at a constant wall temp. of 90 C. Water enters the tube at 40 C and leaves at 60 C. if the entering velocity is 3 m/s, calculate the length of tube necessary to accomplish the heating.

52 Liquid Metal Heat Transfer # Liquid metal High heat transfer rate because of the higher thermal conductivities of liquid metal. # used in heat exchanger so can compact design the HE. The relation for calculation of h in fully developed turbulent flow of liquid metal in smooth tubes with uniform heat flux at the wall: All properties at the bulk temp. The relation for calculation heat transfer to liquid metal in tubes with constant wall temp.:

53 Liquid Metal Heat Transfer The relation for calculation of h in fully developed turbulent flow of liquid metal in smooth tubes with constant heat flux condition: for:

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