We have so far considered convection in situations where an external force has played a role.We will now turn our attention to cases where an external force is absent.

Free ConvectionThe most important physical phenomenon in free convection is buoyancy.

T1T2r1r2Gravity We will focus on gravity driven buoyancy effects but it is important to note that there are other driving forces such as centrifugal forces in spinning machinery or Coriolis forces

Free ConvectionWhat happens when the temperature difference is reversed ?

T1T2r1r2Gravity The buoyancy forces balance the gravitational forces and the system is stablebuoyancy

Heat transfer modes

Stable system heat transfer occurs by CONDUCTIONUNSTABLE system heat transfer occurs by FREE CONVECTION

Flow regimes

Buoyant jet into quiescent gas Tate, J 2002 PhD thesis UNSWWe will focus on buoyant flow in quiescent gas bounded by a surface

Ts > TT , r

gRatio of inertia to viscous forces Volumetric thermal expansion coefficientA measure by which a fluid density changes as a function of temperature at constant pressureRatio of buoyancy forces to viscous forces

GrL/Rel2>>1 free convection GrL/Rel2

Volumetric thermal expansion coefficient for an ideal gasFor an ideal gas For an ideal gas

Volumetric thermal expansion coefficientBoussinesq Approximation-when used in conjunction with the Conservation of mass momentum and energy equations leads aset of equations which can be solved to characterise laminar flow in regimes of free convection

Governing equations Conservation of mass Conservation of momentum Conservation of Energy

Laminar Flow Laminar

Empirical correlationsFor Laminar free convection only past a vertical plate Ra < 109 it is claimed that the equation below is slightly more accurateFor Laminar free convection past a vertical plate Ra < 109 and for turbulent free convection past a vertical plate Ra > 109

It has been shown that

Inclined plates

Vertical PlateTs > TVertical PlateTs < TInclined PlateTs > TVertical PlateTs < T

Inclined platesSubstitute gcosq instead of g

Horizontal platesUpper surface of hot plate or lower of cold plate

Horizontal platesLower surface of hot plate or upper of cold plate

Cylinder and Sphere CylinderSphere

Heat Exchangers

Heat exchangers

Shell and tube heat exchangers

200MW Heat exchanger. For a pulp mill in Kalanti, Finland

Heat Exchangers

Concentric tube Parallel flow Concentric tube counter flow

Cross flow Heat Exchangers

Cross flow mixed Cross flow unmixed

Shell and tube heat exchanger

T1,h1T2,h2 x x=LTs2Ts1Ts3Ts4KAKBKCq

Where

The heat transfer rate 1/RwFor clean unfinned surfaces

For Heat exchangers- fouled and finned Overall Fin surface efficiency Fouling factor

Heat transfer from the fin Heat transfer from the body if the entire fin was at the base temperatureFin efficiencyWhere Fin thickness

wtLwtL

Fouling factors

For unfinned tubular heat exchanger i-inner o-outer surfaces

Log mean temperature analysisIf : Heat transfer between the fluid and the surroundings is negligible and there is negligible change in the specific heat of the fluids then we can say that q the total heat transfer rate between the hot and cold fluid is given by

qMass flow rate of hot fluid Enthalpy difference of hot fluid Mass flow rate of cold fluid Enthalpy difference of cold fluid

To predict the performance of heat exchangers We need to relate the performance of the heat exchangers to the inlet and outlet temperatures- this is what we will do now

Heat exchangers The equation can be re-written as

Another useful expression relies on the difference between the meat hot and cold temperatures

Parallel flow HE dqChCcThTh+dThTc+dTcTcThe heat transfer rates are Local and varies with location Hence the derivativeDT1DT212ThiThoTciTcoDT

Integrating from the inlet (1) to outlet (2) gives

Substituting the integral of the equality dq=UDTdA and rearranging givesWhich is After some manipulation the above equation becomes

Which can be rewritten as Substituting the integral of the equality dq=UDTdA and rearranging givesWhich is We know Ch = DTh/q andCc = DTc/q Hence the above equation becomes

Then Since for a parallel flow HE DT1 is And DT2 isThen Which is Where

DT1DT21ThiThoTciTcoDT2DT1DT21ThiThoTciTcoDT2

The above equations apply to both parallel flow and counterflow Heat Exchangers with the difference that

For Parallel flow DT1=Th1-Tc1=Thi-TciDT2=Th2-Tc2=Tho-TcoFor Parallel flow DT1=Th1-Tc1=Thi-TcoDT2=Th2-Tc2=Tho-Tci

Let us first define the maximum possible heat transfer rate qmax Consider the case when Cc < Ch ie | dTc | > | dTh|Which means Now define effectiveness e asWhich means NTU effectiveness method

LMTD method can be used when the inlet and outlet temps are known but what if theyre not ? What if only inlet temps are known ?then here we use the effectivenesss NTU method

Which means that q can be written as For any heat exchanger it has been shown that Where NTU is the number of transfer units given by

LMTD method can be used when the inlet and outlet temps are known but what if theyre not ? What if only inlet temps are known ?then here we use the effectivenesss NTU method

Such as boilers independent of the arrangementExact solution for e =1 but valid for 0

In design calculations it is more convenient to work with equations in the form NTU =f(e, ..)

Procedure for DesignIn heat exchanger design problem you know The inlet temps and flow rates as well as a required outlet (hot or cold) temperature. Then you have to find the type and the size i.e. surface Area A This is done with custom-designed products

To predict the performance of heat exchangers We need to relate the performance of the heat exchangers to the inlet and outlet temperatures- this is what we will do nowLMTD method can be used when the inlet and outlet temps are known but what if theyre not ? What if only inlet temps are known ?then here we use the effectivenesss NTU method LMTD method can be used when the inlet and outlet temps are known but what if theyre not ? What if only inlet temps are known ?then here we use the effectivenesss NTU method