CONVECTIVE HEAT TRANSFER

Mohammad GoharkhahDepartment of Mechanical Engineering, Sahand Unversity of Technology,

Tabriz, Iran

LAMINAR BOUNDARYLAYER FLOW

CHAPTER 3- PART2

Boundary Layer Equations- Exact Solutions

CONVECTIVE HEAT TRANSFER- CHAPTER3By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

Exact solutions to the boundary layer problem

SIMILARITY SOLUTIONS

1-flow problem Blasius

2-heat transferproblem Pohlhausen

Classic problem of flow over a semi-infinite flat plate

CONVECTIVE HEAT TRANSFER- CHAPTER3By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

SIMILARITY SOLUTIONS- Flow ProblemGoverning equations and boundary conditions

Boundary layer momentum equation contain three unknowns: u, v, and P .pressure in boundary layer problems is independently obtained from theinviscid flow solution outside the boundary layer.The inviscid region can be modeled as uniform inviscid flow over the thinboundary layer δ. Thus, the inviscid problem can be assumed to be auniform flow over a flat plate of zero thickness.Since the fluid is assumed inviscid, the plate does not disturb the flow andthe velocity remains uniform. Therefore, the solution to the inviscid flowoutside the boundary layer is:

CONVECTIVE HEAT TRANSFER- CHAPTER3By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

Flow Problem- Blasius solution

The boundary layer momentum and continuity equation for this problem becomes:

Blasius used similarity transformation to combinethe two independent variables x and y into a singlevariable (x, y) and postulated that u/V depends ononly ŋ

CONVECTIVE HEAT TRANSFER- CHAPTER3By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

the similarity variable η is proportional to y and the proportionality factor depends on x

Based on the scaling laws , η must be proportional to y/δ(x), andδ ∼ x Re - ½ . We assume,therefore, that f’ accounts for the shape of the master profile

Imagine that the two profiles u1(y) and u2(y) were drawn by an artist whousing the master profile; like the elastic metal band of a wristwatch, this masterprofile can be stretched appropriately at x1 and x2 so as to fit the actualvelocity profiles. Mathematically, the stretching of a master profile means:

The basic idea isthe observationthat from onelocation x toanother, the u andT profiles looksimilar.

CONVECTIVE HEAT TRANSFER- CHAPTER3By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

Flow Problem- Blasius solution

Integration by parts

1, 2, 3, 4, 5

1 2

3

4

5

The governing partial differential equations are transformed into an ordinary differential equation

Boundary condition

CONVECTIVE HEAT TRANSFER- CHAPTER3By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

Flow Problem- Blasius solution

Although the mathematical problem is reducedto solving a third order ordinary differentialequation, the difficulty is that this equation isnonlinear.Blasius obtained a power series solution.The tabulated values for f and its derivatives areavailable for the determination of u and v.

Blasius solution gives the boundary layer thicknessand the wall shearing stress

CONVECTIVE HEAT TRANSFER- CHAPTER3By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

Heat transfer Problem- Pohlhausen solution

1

2 3

4 5 6

1, 2, 3, 4, 5, 6

The governing partial differential equation is successfully transformed into an ordinary differential equation

7

CONVECTIVE HEAT TRANSFER- CHAPTER3By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

Heat transfer Problem- Pohlhausen solution

The Prandtl number Pr is the single parameter characterizing the equation. The function f represents the effect of fluid motion on temperature distribution. It is obtained from Blasius solution.

Eq. (7) can be integrated, keeping in mind that f(η) is a known function available in tabular form . Via separation of variables:

CONVECTIVE HEAT TRANSFER- CHAPTER3By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

Heat transfer Problem- Pohlhausen solution

Now we can obtain the thermal boundary layerthickness, heat transfer coefficient, and Nusselt number

CONVECTIVE HEAT TRANSFER- CHAPTER3By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

Heat transfer Problem- Pohlhausen solution

h and Nu depend on the temperature gradient at the surface. This key factor depends on the Pr.

CONVECTIVE HEAT TRANSFER- CHAPTER3By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

Heat transfer Problem- Pohlhausen solution

The overall Nusselt number formulasAnother overall Nusselt number expressionthat covers the entire Prandtl number range was recommended by Churchill and Ozoe

The use of Pohlhausen’s solution to determine heat transfercharacteristics requires the determination of fluid properties such askinematic viscosity, thermal conductivity, and Prandtl number.

All fluid properties in Pohlhausen’s solution are assumed constant. Infact they are temperature dependent. When carrying out computationsusing Pohlhausen’s solution, properties are evaluated at the filmtemperature , defined as

CONVECTIVE HEAT TRANSFER- CHAPTER3By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

Example1- Blasius Solution

Assumptions. (1) Continuum(2) Newtonian fluid(3) steady state(4) constant properties(5) two-dimensional(6) laminar flow (7) viscous boundary layer flow (Rex > 100)(8) uniform upstream velocity(9) flat plate(10) negligible changes in kinetic and potential energy (11) no buoyancy

CONVECTIVE HEAT TRANSFER- CHAPTER3By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

Example1- Blasius Solution

The flow is laminar

boundary layer thickness

At location 0 where x = 150 mm and y = 2 mm

From the table:

CONVECTIVE HEAT TRANSFER- CHAPTER3By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

Example2- Laminar Convection over a Flat PlateAssumptions. (1) Continuum(2) Newtonian fluid(3)two-dimensional process(4) negligible changes in

kinetic and potential energy,

(5) constant properties, (6) boundary layer flow, (7) steady state(8) laminar flow(9) no dissipation, (10) no gravity,(11) no energy generation, (12) flat plate,(13) negligible plate thickness, (14) uniform upstream vel. (15) uniform upstream temp.,(16) uniform surface temp. (17) no radiation.

CONVECTIVE HEAT TRANSFER- CHAPTER3By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

Properties of water are evaluated at the film temperature

boundary layer approximations can be made and the flow is laminar at x = 7.5 m

At the edge of the thermal boundary layer

CONVECTIVE HEAT TRANSFER- CHAPTER3By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

Heat transfer coefficient.

Heat transfer rate.

Doubling the length of plate doubles the corresponding Reynolds number at the trailing end. There is a possibility that transition to turbulent flow may take place. For a plate of length 2L, the Reynolds number isthe flow at the trailing end is turbulent and consequently Pohlhausen's solution is not applicable.

CONVECTIVE HEAT TRANSFER- CHAPTER3By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

Comparison between the exact solution and Scaling Estimate of Heat Transfer Rate

Scaling of h for Pr >>1 gives

CONVECTIVE HEAT TRANSFER- CHAPTER3By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

Questions?