1. Introduction to Convection Heat and Mass Transfer
Heat Transfer
Introduction to Convection Heat and Mass Transfer

2. Convection Fundamental Concepts -Convection Transfer
Consider the flow of a fluid past a flat surface.
The local heat flux is:
Where h is the local heat transfer coefficient (W/m2 K). q” and h vary along the surface. The total heat transfer rate over the entire surface is: L x q”
Ts , As
U∞ , T∞
Average convection heat
transfer coefficient
where

3. Results similar to convection heat transfer may be obtained for convection mass transfer when a fluid with molar concentration C A, ∞ (kmol/m3) or density ρ A,∞ (kg/m3) for species A flows over a surface with uniform concentration C A,s or density ρ A,s for species A as shown below:
The molar flux NA” (kmol/m2 s) and mass flux mA” (kg/ m2 s) are expressed, respectively as: L x
NA”
CA,s , As
U∞ , CA, ∞ dx
Where hm = local convection mass transfer coefficient
ρA = Mc CA = density
MA = molecular weight of species A

4. Where R = universal gas constant = 8.314 kJ / kmol K
Total molar transfer rate NA (kmol/s) and total mass transfer rate mA (kg/s) are also expressed, respectively, as follows:
Where = average mass transfer coefficient, m/s
Assuming saturated state of species A at surface temperature Ts, the density ρ A,∞ may be obtained directly from thermodynamic tables ( for water: table A.6(text))
At the corresponding saturation pressure Psat the molar concentration may be obtained from equation of state for an ideal gas
Where R = universal gas constant
= kJ / kmol K

5. Convection boundary layer
Convection can be forced or natural
It is due to the random motion of molecules (diffusion) and the bulk motion of the fluid particles
Convection could be internal or external
Examples – Airflow over wings, buildings, a bank of heat exchangers (external)
The Velocity boundary Layer
Velocity B.L. → Region with velocity gradient dU / dy ≠ 0 x
Free Stream U∞ y
(x)
B.L. u 
Velocity gradient due to shear stress, 

6. Kinetic energy of fluid
 is the boundary layer thickness defined as the value of y at which u = 0.99 U∞. At y =  dU/dy and τ are negligible.
The local friction coefficient is calculated from:
The surface frictional drag, FD, is calculated using Cf
Assuming a Newtonian fluid, τs is calculated from:
Where μ is the dynamic viscosity of the fluid, and du/dy is the velocity gradient perpendicular to the wall.
Wall shear
Kinetic energy of fluid
also

7. Thermal Boundary Layer
Thermal B.L. → Region with temperature gradient dT/dy
t is the thermal boundary layer thickness, defined as the value of y at which: x T∞ y t
t(x) U∞ Ts
Isothermal Surface

8. For a constant surface temperature,Ts
The local convective heat transfer coefficient may be calculated by first finding qs” from energy balance at the surface. Fourier Law (No fluid motion at the surface, conduction only)
For a constant surface temperature,Ts
Ts - T∞ = constant
Since t increases with x, ∂T/∂y must decrease with x
Therefore h and q” also decrease with x
also
Since
decreases with x

9. Concentration Boundary Layerx y c
c(x) U∞
C A,s
Concentration
Boundary Layer
C A, ∞
C A
Free Stream
Mixture
of A & B
Fick’s Law
Where DAB or Dv = mass diffusion coefficient, m/s2
Mass diffusivity or mass diffusion coefficient has the same units as thermal diffusivity, α (m/s2)
As with heat transfer, we can write
Heat flux (qx”)  Fourier’s Law
Molar or Mass flux  Fick’s Law

10. Three boundary layers:
SUMMARY:
Three boundary layers:
Velocity B.L. ()
Velocity distribution (V)
Wall friction ()
Thermal B.L. (t)
Temperature distribution (T)
Convection heat transfer (h, Nu)
Concentration B.L. (c)
Concentration distribution (C)
Convection mass transfer (hm, Sh)

11. Problem 6.1

12. Laminar & Turbulent Flow & Reynolds Number
Convection transfer rates and surface friction depend on whether the flow is laminar or turbulent. Laminar or turbulent flow is determined by the value of Re
For internal flow: Reynolds Number (Re)
where ReD,c is the critical Reynolds number
For external flow: Reynolds Number (Re)
where Rex,c is the critical Reynolds number

13. U∞ Laminar xc Transition Turbulent Turbulent Laminar
Ordered fluid motion with identifiable streamlines on which fluid particles move.
Turbulent
Irregular fluid motion with velocity fluctuations which increase energy, mass, and momentum transfer. Larger t with flatter velocity, temperature and concentration profiles.

14. Regions in turbulent Boundary Layer
Turbulent region – Transport mainly by
turbulent mixing
Buffer Zone – Transport by diffusion and
Viscous or Laminar Sub layer – Transport
by diffusion and linear velocity profile
h(x)
(x) x
h , δ
Laminar
Transition
Turbulent
Variation of Velocity B.L.
Thickness () and local heat
transfer coefficient (h) for flow
over an isothermal surface

15. Problem 6.14

16. Boundary Layer Equations
Consider a steady , 2-D flow of a viscous incompressible fluid in a Cartesian coordinate as shown below. We wish to obtain a set of differential equations that governs the velocity and temperature distributions in the fluid to solve V, T, C and  (force), q” (heat transfer), m” (mass transfer). T∞ U∞
Thermal B.L.
Concentration B.L.
Velocity B.L.
qs”
N A,s”
C A, ∞
C A y Ts t C
C A, s
Mixture of
A & B dx dy  x

17. The velocity Boundary Layer
The equations are based on application of conservation principles on a differential control volume of the fluid as demonstrated below for conservation of mass over a control volume:
The velocity Boundary Layer
Continuity equation
Conservation of mass over a control volume, in – out = 0 u v y x
= mass velocity (kg / m2 s)
For an incompressible fluid where ρ = constant:
See Appendix E for detailed development of conservation of mass, momentum, energy and chemical species

18. The following differential equations are therefore obtained for steady, 2-D flow for an incompressible fluid with constant properties (ρ, μ, cv, cp, k, etc.)
Continuity Equation: Conservation of Mass
Momentum Equation in x-direction: Velocity B.L.
Momentum Equation in y-direction: Velocity B.L.
Where X = Body
force in x direction
X=ρgx
Where Y = Body
force in y direction
Y=ρgy

19. Energy Equation: Thermal B.L.
Viscous Energy Dissipation Equation
Species Transfer or Continuity Equation: Species Concentration B.L.
Where μΦ = viscous
energy dissipation

20. Approximations and Special Conditions
A most common situation is one in which the 2-D boundary layer can be characterized as
Steady (time independent)
Incompressible (ρ is constant)
Having constant properties (ρ, μ, k, etc.) with temperature
Having negligible body forces ( X = Y = 0)
Non-reacting (no chemical reaction) without internal heat generation
In addition, since  is very small, the following B.L. approximations apply:
Velocity B.L.

21. This means that normal stresses in x-direction are negligible, and the wall shear stress reduces to:
These simplification reduce the B.L. equations to the following equations.
Continuity Equation:
Thermal B.L.
Concentration B.L.

22. Momentum equation – in x-direction, and y-direction
The continuity and momentum equations can be used to solve for the spatial variations of u and v.
Energy Equation:
Species Continuity Equation ,
This can be used to
solve for the temperature or
This can be used to solve for concentration variation

23. Boundary Layer Similarity and Normalized Convection Equations
Examination of the simplified convection equations repeated below shows a strong similarity between the momentum equation and energy equation:
Similarly, there is a strong similarity between mass transfer and momentum and energy equations as indicated below:
Momentum equation
in x-direction for
velocities u and v
Advection
Diffusion
Energy equation for
temperature T
Mass Transfer
Equation
Advection
Diffusion

24. Similarity Parameters for Boundary Layers
Dimensionless independent variables defined as:
Dependent dimensionless variables are:
The dimensionless independent and dependent variables may be substituted in to the momentum and energy equations to obtain dimensionless forms of conservation equations (see table 6.1)
Similarity parameters make it possible to apply results obtained for a surface experiencing one set of conditions to geometrically similar surfaces experiencing entirely different conditions.
Where L = characteristic
length for the surface
and
Where
V = Upstream velocity , ,
CA = mass concentration
of species A ,

26. Dimensionless groups = Inertia Forces Reynolds Number (Re)
Viscous Forces
Reynolds Number (Re)
μ Is the viscosity (kg / s m)
ν is the kinematic viscosity (m2/s)
Prandtl Number (Pr)
3. Nusselt Number (Nu) = The dimensionless temperature gradient at the surface.
4. Stanton Number (St)
= Momentum diffusivity
Thermal diffusivity
α is the thermal diffusivity (m2/s)
(A modified
Nusselt Number)

27. Schmidt Number (Sc)
Sherwood Number (Sh) = Dimensionless concentration gradient at the surface
In terms of dimensionless groups, the complete set of dimensionless convection equations are therefore:
= Momentum diffusivity
Mass diffusivity
DAB = mass diffusivity (m2/s)
Velocity (Continuity equation)
Velocity
(Momentum
equation)

28. Thermal
(Energy Equation)
Solution Forms:
Mass Concentration
(Mass Diffusion
Equation)

29. ,
Nusselt Number:
Convection mass transfer: , ,
Where hm = mass transfer coefficient, m/s (convection)

30. Physical Significance of the Dimensionless Parameters
Sherwood Number:
Physical Significance of the Dimensionless Parameters
Reynolds Number
Inertia forces therefore dominate for large values of Re and viscous forces dominate for small Re values. Viscous forces dominate in laminar flow but become progressively less important than inertia forces as Re increases.
= Inertia Forces
Viscous Forces

31. = Momentum diffusivity Thermal diffusivity Prandtl Number, Pr
Large differences in Pr are associated with large variations in the fluid viscosity, μ. The spectrum of Prandtl Numbers of fluids is given below.
Normally
δ is the velocity B.L. thickness and δt is the thermal B.L. thickness.
10-2
10-1
100 10
102
103
Liquid metals
Gases
Water
Light Organics
Oils
Where n is a positive integer

32. For gases,
For liquid metals (very high k, low Pr)
For oils (very high μ, high Pr)
Mass transfer and Lewis Number, Le
Table 6.2 (text) lists several dimensionless parameters that are generally relevant in heat and mass transfer

36. Boundary Layer Analogies
Heat and mass transfer analogy
Heat and mass transfer are analogous
Heat and mass transfer relations are therefore interchangeable for a particular geometry
For example, it the form of relationship for a convection heat transfer problem involving x*, Re, and Pr, that is f4(x*, Re, Pr) has been obtained for a particular surface geometry, the results may be used for convection mass transfer for the same geometry simply by replacing Nu with Sh and Pr with Sc
The analogy can also be used to relate two convection coefficients. By analogy:

38. Problem 6.46

39. Modified Nusselt Number
Reynolds Analogy
When dP*/dx*= 0 and Pr = Sc = 1, the boundary layer equations (momentum, energy and mass concentration) are exactly of the same form.
If dP*/dx*= 0 then U∞ = V = upstream fluid velocity, and the boundary conditions are also of the same form.
Solutions for U*,T*, CA* must also be of the same form.
It follows that:
(1)
Modified Nusselt Number
Mass Transfer Stanton Number

40. From (1) we have: Reynolds Analogy,
The Reynolds analogy can be used, provided these restrictions are satisfied
dP*/dx* ≈ 0 , Pr ≈ 1 , Sc ≈ 1
In general application, the modified Reynolds analogy is:
Where jH, and jm are the Colburn j factor for heat and mass transfer
Equations (2) and (3) are approximate for laminar flow, but more accurate or valid for turbulent flow.
If one parameter is known, it can be
used to obtain others.
For 0.6 < Pr < 60
(2)
For 0.6 < Sc < 3000
(3)

41. Evaporative Cooling: Simultaneous Heat & Mass Transfer
As shown in the figure below, the Evaporative cooling process of heat and mass transfer occurs when a gas flows over a liquid surface.
There are numerous industrial and environmental applications of evaporative cooling process.
In the evaporative cooling process the energy required for evaporation of the liquid comes from the internal energy of the liquid. As a consequence, the temperature of the liquid reduces or “cooling effect” occurs
Liquid Layer (species A)
Gas Layer
(species B)
q”evap
q”add
q”conv
Latent and sensible heat exchange at a gas liquid interface

42. If there is no heat addition we have
Under steady state operation, the latent energy lost by the liquid must be replenished by energy transfer to the liquid from its surroundings or by energy addition by other means (e.g. electrical heating of the fluid
By energy conservation on a control surface on the liquid surface, we have:
Where
qevap” = evaporative cooling load
qconv” = convection sensible heat transfer from the gas to the liquid
qadd” = heat addition to the liquid
If there is no heat addition we have
Where ρA,sat is the saturated vapor density at the surface temperature, Ts.

43. expressed as:
Since by heat and mass transfer analogy, and for exponent n=1/3
With mA << mB, the evaporative cooling effect may be
approximately calculated from or
R = universal gas
constant
Where MA and MB
are the molecular
weight of the species
A and B, respectively or

44. Solution Methods in Internal and External Convection Transfer
The primary objective in internal and external convection transfer is to obtain convection coefficients for different flow geometries. The convection coefficients are subsequently used to obtain heat or mass transfer rates.
Solution methods in obtaining convection coefficients include:
Experimental or Empirical Method
Theoretical Method
Non-dimensionalized approach is generally followed, and the local and average convection coefficients are generally correlated by the following form of equations:
Heat Transfer
Mass Transfer

45. Experimental or Empirical Method
The empirical method involves heat and mass transfer measurements and data correlation using appropriate dimensionless parameters
For fixed Prandtl numbers for a given fluid, log-log plots of Re versus Nu generally are straight lines as illustrated below (left).
The straight line plots may be represented by an equation of the form:
Where C, m, n are independent
constants

46. Log-log plot of Re versus the ratio Nu / Prn combine into a single straight line for all the Pr as illustrated above (right).
The corresponding correlation equation for mass transfer is of the form:
Where the independent constants C, m, and n are the same as obtained for heat transfer for similar geometry

47. Theoretical method
Special case of exact solution to convection transfer equations (example 6.4 in text)
Parallel flow or Couette flow is one of the situations where exact solutions can be obtained for convection transfer equations. A special case of parallel flow involves stationary and moving plates of infinite extent separated by a distance L with the intervening space filled by an incompressible fluid as shown below:
With the assumption of steady state conditions, incompressible fluid with constant properties, no body forces (I.e. X=0, and Y=0), and no internal energy generation the convection transfer equations reduce to the following:
Engine oil
Moving Plate
Stationary Plate L TL T0
x,u
y,v U

48. Continuity Equation
For incompressible fluid (ρ = constant) and parallel flow
(v = 0) the continuity equation reduces to:
The x velocity component is therefore independent of x, and the velocity field is said to be fully developed.
2. Momentum Equation
With ∂u / ∂v = 0, v = 0 and X = 0, the momentum equation reduces to:
In parallel flow the pressure gradient ∂P / ∂x = 0, and the x-momentum equation reduces to:

49. Finally, the velocity distribution is given by:
3. Energy equation
For the 2-D, S.S. condition with y=0, energy generation q=0 and ∂u/∂x = 0 for fully developed temperature field, the energy equation reduces to:
Integrating twice and solving, the temperature distribution is given by:
B.C.’s
T(0) = To
T(L) = TL
Tb = bearing Temp.

50. Surface Heat Fluxes
The surface heat fluxes are obtained from the temperature distribution by applying the Fourier’s Law:
At the bottom and top surfaces, the heat fluxes are respectively given be:
Location of Maximum Temperature
The location of the maximum temperature may be found from the requirement:

51. The maximum temperature occurs in the fluid and there is heat transfer to the hot and cold plates. The temperature distribution is a string function of the velocity of the moving plate.
6. Viscous Energy Dissipation
The viscous energy dissipation was defined as:
With v = 0, and du/dx = 0, the viscous energy dissipation equation reduces to:
For U = 0 there is no viscous dissipation, and the temperature distribution is linear.