Deduction of Fundamental Laws for Heat Exchangers

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Deduction of Fundamental Laws for Heat Exchangers

• P M V Subbarao
• Professor
• Mechanical Engineering Department
• I I T Delhi

Modification of Basic Laws for Design of General
Templates for HXs?!?!?!
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Thermodynamic Vision of Science Engineering
Accept Theory
Test Theory
Particular Relations
INDUCTION
DEDUCTION
Propose Theory
Engineering Relations
Secondary concepts
Industrial Processes
Primitive concepts
Impact on Society
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Increase or Decrease of Temperature Fluid in A
Container

• Heating of a control mass

1Q2 U2 U1
Constant Volume Heating

• Consider a homogeneous phase of a substance with
  constant composition.
• Define Specific Heat The amount of heat required
  per unit mass/mole to raise the temperature by
  one degree.
• No change in other forms of energy, except
  internal energy.

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Increase or Decrease of Temperature Flowing
Fluid
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Thermodynamic Perspective of HX.

• The rate of enthalpy gained by a cold fluid

• The rate of enthalpy lost by hot fluid

• Thermal Energy Balance

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Heat Transfer Perspective of HX.

• Estimation and Creation of primary driving
  force.
• To the hot fluid loose thermal energy?
• To help cold fluid gain thermal energy?
• Provision of thermal infrastructure to satisfy
  law of conservation of energy.

• How to model this mutual interaction using
  principles of Heat Transfer ?

Understanding of precise role of thermodynamic
Parameters
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Incompleteness in Basic Laws of Heat Teat Transfer

• A Simple adiabatic Heat Exchanger model.

• For Heat communication between cold and hot.

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Fouriers law of heat conduction
A Constitutive Relation
This is called as Fourier Law of Conduction
Global heat transfer rate
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Use of Fourier Law of Conduction for HXs
Local Heat flux in a slab
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Mathematical Description

• Temperature is a scalar quantity.
• Heat flux is defined with direction and Magnitude
  A Vector.
• Mathematically it is possible to have

Using the principles of vector calculus
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Further Physical Description

• Will k be constant from one end of HX to the
  other end?
• Will k be same in all directions?
• Why k cannot be different each direction?
• Why k cannot be a vector variable?

Will mathematics approve this ?
What is the most general acceptable behavior of
k, approved by both physics and mathematics?
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Most General form of Fourier Law of Conduction
Local Heat flux in a slab along x-direction
Local Heat flux vector
We are at cross roads !!!!!
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Physical-mathematical Feasible Model

• Taking both physics and mathematics into
  consideration, the most feasible model for
  Fouriers Law of conduction is

Thermal conductivity of a general material is a
tensor.
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Surprising Results !!!
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Newtons Law of Convection Cooling

• Convection involves the transfer of heat between
  a surface at a given temperature (Ts) and fluid
  at a bulk temperature (Tb).
• Newtons law of cooling suggests a basic
  relationship for heat transfer by convection

h is called as Convection Heat Transfer
Coefficient, W/m2K
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Realization of Newtons Law Cooling

• A general heat transfer surface may not be
  isothermal !?!
• Fluid temperature will vary from inlet to exit
  !?!?!
• The local velocity of flow will also vary from
  inlet to exit ?!?!
• How to use Newtons Law in a Real life?

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Local Convection Heat Transfer
Consider convection heat transfer as a fluid
passes over a surface of arbitrary shape Apply
Newtons law cooling to a local differential
element with length dx.
h is called as Local Convection Heat Transfer
Coefficient, W/m2K
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Radiation from a Thermodynamic System
The total energy emitted by a real system,
regardless of the wavelengths, is given by

• where esys is the emissivity of the system,
• Asys-surface is the surface area,
• Tsys is the temperature, and
• s is the Stefan-Boltzmann constant, equal to
  5.6710-8 W/m2K4.
• Emissivity is a material property, ranging from 0
  to 1, which measures how much energy a surface
  can emit with respect to an ideal emitter (e 1)
  at the same temperature

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Radiative Heat Transfer between System and
Surroundings
Consider the heat transfer between system surface
with surroundings, as shown in Figure. What is
the rate of heat transfer into system surface ?
To find this, we will first look at the emission
from surroundings to system. Surrounding Surface
emits radiation as described in
This radiation is emitted in all directions, and
only a fraction of it will actually strike system
surface. This fraction is called the shape
factor, F.
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The amount of radiation striking system surface
is therefore
The only portion of the incident radiation
contributing to heating the system surface is
the absorbed portion, given by the absorptivity
aB
Above equation is the amount of radiation gained
by System from Surroundings. To find the net
heat transfer rate for system, we must now
subtract the amount of radiation emitted by
system
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The net radiative heat transfer (gain) rate at
system surface is
Similarly, the net radiative heat transfer (loss)
rate at surroundings surface is
What is the relation between Qsys and Qsur ?
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Wall Surfaces with Convection
Boundary conditions
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A Simple Heat Exchanger
Annular Flow
Tubular Flow
Overall heat transfer coefficient of a used HX,
based on outside area
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Overall heat transfer coefficient of a
new/cleaned HX, based on outside area
Thermal resistance of any annular solid structure
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Mean Temperature Difference
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Simple Counter Flow Heat Exchangers C gt1
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Simple Counter Flow Heat Exchangers C lt 1
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Simple Parallel Flow Heat Exchangers
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Thermodynamics of An Infinite HX

• All properties of thermal structure remain
  unchanged in all directions.
• All properties of thermal structure are
  independent of temperature .
• An unique surface area of heat communication is
  well defined.

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Thermal Resistance of infinitesimal Heat Exchanger
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Variation of Local temperature difference for
heat communication
Heat Transfer in an infinitesimal HX
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Synergism between HT TD
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For A finite HX
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For A finite HX
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A representative temperature difference for heat
communication
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Discussion on LMTD

• LMTD can be easily calculated, when the fluid
  inlet temperatures are know and the outlet
  temperatures are specified.
• Lower the value of LMTD, higher the value of
  overall value of UA.
• For given end conditions, counter flow gives
  higher value of LMTD when compared to co flow.
• Counter flow generates more temperature driving
  force with same entropy generation.
• This nearly equal to mean of many local values
  of DT.