Convection: Internal Flow 8'18'6

1
Chapter 8

• Convection Internal Flow (8.1-8.6)

2
Introduction

• In this chapter we will obtain convection
  coefficients and heat transfer rates for
  geometries involving internal flow, such as flow
  in tubes
• Recall Newtons law of cooling
• For flow inside a tube we cannot define T?
• Must know how temperature evolves inside the pipe
  and find alternative expressions for calculating
  heat flux due to convection.

3
Flow Conditions for Internal Flow

• Onset of turbulent flow at

• Hydrodynamic entry length
• Laminar flow
• Turbulent flow

4
Mean Velocity

• Velocity inside a tube varies over the cross
  section. For every differential area dAc

• Overall rate of mass transfer through a tube with
  cross section Ac

(8.1)
where um is the mean (average) velocity
(8.2)

• Can determine average velocity at any axial
  location (along the x-direction), from knowledge
  of the velocity profile

5
Velocity Profile in a pipe

• Recall from fluid methanics that for laminar flow
  of an incompressible, constant property fluid in
  the fully developed region of a circular tube
  (pipe)

(8.3a)
(8.3b)
(8.3c)
6
Thermal Considerations Mean Temperature

• We can write Newtons law of cooling inside a
  tube, by considering a mean temperature, instead
  of T?

(8.4)
where Tm is the mean (average) temperature

• For constant r and cp, Tm is defined

(8.5)
7
Fully Developed Conditions

• For internal flows, the temperature, T(r), as
  well as the mean temperature, Tm generally vary
  in the x-direction, i.e.

8
Fully Developed Conditions

• Although T(r) changes with x, the relative shape
  of the temperature profile remains the same Flow
  is thermally fully developed.

• A fully developed thermally region is possible,
  if one of two possible surface conditions exist
• Uniform wall temperature (Tsconstant)
• Uniform heat flux (qxconst)
• Thermal Entry Length

9
Fully Developed Conditions

• It can be proven that for fully developed
  conditions, the local convection coefficient is a
  constant, independent of x

10
Mean temperature variation along a tube

• We are still left with the problem of knowing
  how the mean temperature Tm(x), varies as a
  function of distance, so that we can use it in
  Newtons law of cooling to estimate convection
  heat transfer.

Recall from Chapter 1, page 10 that by
simplifying the energy balance for flow inside a
control volume
For flow inside a pipe
(8.6)
where Tm,i and Tm,o are the mean temperatures of
the inlet and outlet respectively
11
Mean temperature variation along a tube
Psurface perimeter
For a differential control volume
where Psurface perimeter p D for circular
tube, width for flat plate
(8.7)

• Integration of this equation will result in an
  expression for the variation of Tm as a function
  of x.

12
Case 1 Constant Heat Flux

• Integrating equation (8.7)

(8.8)
where Psurface perimeter pD for circular tube,
width for flat plate
13
Example (Problem 8.15)

• A flat-plate solar collector is used to heat
  atmospheric air flowing through a rectangular
  channel. The bottom surface of the channel is
  well insulated, while the top surface is
  subjected to a uniform heat flux, which is due to
  the net effect of solar radiation absorption and
  heat exchange between the absorber and cover
  plates.
• For inlet conditions of mass flow rate0.1 kg/s
  and Tm,i40C, what is the air outlet
  temperature, if L3 m, w1 m and the heat flux is
  700 W/m2? The specific heat of air is cp1008
  J/kg.K

14
Case 2 Constant Surface Temperature,Tsconstant

• From eq.(8.7)

with Ts-TmDT
Integrating for the entire length of the tube
(8.9)
(8.10)
where
(8.11)
As is the tube surface area, AsP.LpDL, DTlm is
the log-mean temperature difference
15
Case 3 Uniform External Temperature

• Replace Ts by and by (the
  overall heat transfer coefficient, which includes
  contributions due to convection at the tube inner
  and outer surfaces, and due to conduction across
  the tube wall). Equations (8.9) and (8.10) become

(8.11)
(8.12)
16
Reminder from Chapter 3, p. 19 lecture notes
17
Example (Problem 8.55)

• Water at a flow rate of 0.215 kg/s is cooled
  from 70C to 30C by passing it through a
  thin-walled tube of diameter D50 mm and
  maintaining a coolant at 15C in cross flow over
  the tube. What is the required tube length if the
  coolant is air and its velocity is V20 m/s? The
  heat transfer coefficients are hi680 W/m2.K for
  flow of water inside the tube and ho83.5 W/m2.K
  for a cylinder in air cross flow of 20 m/s

18
Summary (8.1-8.3)

• We discussed fully developed flow conditions for
  cases involving internal flows, and we defined
  mean velocities and temperatures
• We wrote Newtons law of cooling using the mean
  temperature, instead of
• Based on an overall energy balance, we obtained
  an alternative expression to calculate convection
  heat transfer as a function of mean temperatures
  at inlet and outlet.
• We obtained relations to express the variation of
  Tm with length, for cases involving constant heat
  flux and constant wall temperature

(8.4)
(8.6)
(8.9)
(8.8)
19
Summary (8.1-8.3)

• We used these definitions, to obtain appropriate
  versions of Newtons law of cooling, for internal
  flows, for cases involving constant wall
  temperature and constant surrounding fluid
  temperature

(8.10-8.12)

• In the rest of the chapter we will focus on
  obtaining values of the heat transfer coefficient
  h, needed to solve the above equations