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?!?!?!

Thermodynamic Vision of Science & Engineering Primitive concepts Accept Theory Secondary concepts Propose Theory Test Theory Impact on Society Industrial Processes Engineering Relations Particular Relations INDUCTION DEDUCTION

Increase or Decrease of Temperature : Fluid in A Container Heating of a control mass: Constant Volume Heating 1 Q 2 = U 2 – U 1 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.

Increase or Decrease of Temperature : Flowing Fluid

Thermodynamic Perspective of HX. The rate of enthalpy gained by a cold fluid The rate of enthalpy lost by hot fluid Thermal Energy Balance:

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…

Incompleteness in Basic Laws of Heat Teat Transfer For Heat communication between cold and hot. A Simple adiabatic Heat Exchanger model.

Fourier’s law of heat conduction This is called as Fourier Law of Conduction A Constitutive Relation Global heat transfer rate:

Use of Fourier Law of Conduction for HXs Local Heat flux in a slab:

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:

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?

Most General form of Fourier Law of Conduction We are at cross roads !!!!! Local Heat flux in a slab along x-direction : Local Heat flux vector :

Physical-mathematical Feasible Model Taking both physics and mathematics into consideration, the most feasible model for Fourier’s Law of conduction is: Thermal conductivity of a general material is a tensor.

Surprising Results !!!

Newton’s Law of Convection Cooling Convection involves the transfer of heat between a surface at a given temperature (T s ) and fluid at a bulk temperature (T b ). Newton’s law of cooling suggests a basic relationship for heat transfer by convection: h is called as Convection Heat Transfer Coefficient, W/m 2 K

Realization of Newton’s 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 Newton’s Law in a Real life?

Local Convection Heat Transfer Consider convection heat transfer as a fluid passes over a surface of arbitrary shape: Apply Newton’s law cooling to a local differential element with length dx. h is called as Local Convection Heat Transfer Coefficient, W/m 2 K

The total energy emitted by a real system, regardless of the wavelengths, is given by: where ε sys is the emissivity of the system, A sys-surface is the surface area, T sys is the temperature, and σ is the Stefan-Boltzmann constant, equal to 5.67×10 -8 W/m 2 K 4. 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 (ε = 1) at the same temperature Radiation from a Thermodynamic System

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 ? 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. To find this, we will first look at the emission from surroundings to system. Surrounding Surface emits radiation as described in

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 α B : 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:

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 Q sys and Q sur ?

Wall Surfaces with Convection Boundary conditions: R conv,1 R cond R conv,2 T1T1 T2T2

didi dodo DiDi Tubular Flow Annular Flow Overall heat transfer coefficient of a used HX, based on outside area: A Simple Heat Exchanger

Overall heat transfer coefficient of a new/cleaned HX, based on outside area: Thermal resistance of any annular solid structure:

Mean Temperature Difference

Simple Counter Flow Heat Exchangers: C >1

Simple Counter Flow Heat Exchangers: C < 1

Simple Parallel Flow Heat Exchangers

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.

Thermal Resistance of infinitesimal Heat Exchanger Thermal resistance of an Infinitesimal adiabatic Heat Exchanger ThTh TcTc

Variation of Local temperature difference for heat communication: Heat Transfer in an infinitesimal HX

Synergism between HT & TD:

For A finite HX:

A representative temperature difference for heat communication:

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  T.