Transport Phenomena L4

Equations of Change in Nonisothermal Systems

Topic Objectives:Topic Objectives:

To understand the concept of Energy Conservation To understand the concept of Energy Conservation

To obtain expression describing the temperature To obtain expression describing the temperature profile in the nonisothermal flow

To use the Equations of Change to solve steady state problemsproblems

Conservation of Energy

The law of conservation of energy is an extension of the classical 1st Law of Thermodynamics, which concerns the diff i i l i f ilib i f difference in internal energies of two equilibrium states of a closed system because of the heat added to the system and the work done on the system (U = Q + W)

The changes in Kinetic Energy and Internal Energy of a system are related to convective heat transport as well as conductive heat transport due to molecular transport.

Conservation of Energy

(i) Kinetic Energy: Energy due to observable motion of the fluid per unit volume(i) Kinetic Energy: Energy due to observable motion of the fluid per unit volume

(v is the fluid velocity vector)

(ii) I l E ki i i i l l d ib i (ii) Internal Energy: kinetic energies in molecules + energy due to vibrations & rotational motions in molecules. Assumption: U = f(, T)

Energy Equations

The rate of increase in kinetic and internal energy:Note: = the internal energy per unit mass

The rate of energy enters and leaves the faces of volume element The rate of energy enters and leaves the faces of volume element:

where vector e (energy flux) includes convective transport of kinetic and internal energy, the heat conduction, and work associated with molecular processes.

The rate at which work is done on the fluid by the external force is the dot yproduct of the fluid velocity v and the force acting (gravitational) on the fluid (xyz)g :

Energy Equations

e vgAssignment 1a

Special form of Energy Equations

Equation of Change for Internal Energy (U)q g gy ( )

Note: Obtained through the Equation of Mechanical Energy and Energy Equationgy q

Assignment 1b

The term (v), with double dot multiplication of two tensors appears from Equation of Mechanical Energy. gy

This (v) term describes the degradation of mechanical energy into thermal energy gy gy(viscous dissipation heating)

Energy Equations

Assignment 1 (expected NOT more than 5 pages)g ( p p g )

a) For energy flux expression:

Explain the terms [ v] and q by giving the definitions of the symbols used Explain the terms [v] and q, by giving the definitions of the symbols used. Give brief description of [v] derivation

Refer to text book by Bird, Steward & Lightfoot: Chapter 9 on Energy Transport, S9 7: Convective Transport of Energy S9.7: Convective Transport of Energy

S9.8: Work associated with molecular motions

b) For viscous heating term (v) in the equation of mechanical energy:Explain the (v) term and give brief description of its derivationsRefer to text book by Bird, Steward & Lightfoot: Chapter 1 on Momentum Transport -Refer to text book by Bird, Steward & Lightfoot: Chapter 1 on Momentum Transport

S1.2: Newtons Law of Viscosity & Appendix A


Equation of Change for Temperature (T)q g p ( )

This is the general form for equation of change for temperature, in terms of substantial time derivative, heat flux vector q and the viscous momentum flux tensor the viscous momentum flux tensor .

Note: P is the pressure and is the fluid density


Equation of Change for Temperature:q g p

If Fouriers Law is applied for (q) term, then:q = k2T

**Provided the thermal conductivity k is assumed constant



If Newtons Law of Viscosity is applied, the viscous heating term can expressed by:



Using Equation of Using Equation of Continuity, / T = -( v)Ideal gas LawP = RT/Mr

d R C C For an ideal gas ( In / In T)p = 1, thus: and R = Cp- Cv

or

Note: viscous heating term is normally ignored unless the flow has enormous Note: viscous heating term is normally ignored unless the flow has enormous velocity gradient



For a fluid flowing in a constant pressure system, DP/Dt = 0,thus:



For a fluid with constant density ( In / In T)p = 0, thus:



For a stationary solid, v is zero and ( In / In T)p = 0, thus:


Equation of Change for Temperature

The last five equations are the most frequently encountered in te tbooks and research p blications textbooks and research publications.

When the needs for more accuracy arise, the less restrictive y ,expressions can be developed from the general form of equations of change.

The terms involving chemical, electrical and nuclear sources can be added to the general form of equation of change.


In general, to describe the nonisothermal flow of a Newtonian fluid requires:

the Equation of Continuity (Mass Balance) the Equation of Motion (Moment m Balance) the Equation of Motion (Momentum Balance) the Equation of Energy (Energy Balance)

Additionally: the related thermal equations: [ p = p(,T); Cp = Cp(,T) ] expressions for the density and temperature dependence of

the viscosity, dilatational viscosity, and thermal conductivity the boundary and initial conditions (for numerical solution) the boundary and initial conditions (for numerical solution)


In principle, the entire set of equations (Continuity, Motion and Energy p p q ( y gyEquations) can then be solved to get the pressure, density, velocity, and temperature profiles, as functions of position and time. (steady state: differential time derivative term = 0)( y )

Generally, numerical methods have to be used.

Standard assumptions normally used to simplify the calculations:

C t t h i l ti bt i l ti l l ti Constant physical properties may obtain analytical solution

Assume zero fluxes (momentum flux and heat flux) for certain cases

Temperature Profile in Nonisothermal Systems

Example: Heating of a fluid through a circular tube. Find expression for temperature profile for the fully developed laminar flows

1) Continuity: Rate Increase = In - Out 0 = 0

2) Motion :

or

( = P + gh )( P + gh )


3) Energy:) gy

Dividing the energy balance equation by 2rz gives:


In the limit as r and z approaching zero, and dividing by r:

Now,


Viscous heating term

Corresponds to heat conduction in axial direction

z-component of Equation of Motion

Through assumptions & simplifications, finally:

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Transport Phenomena L4