Heat Exchanger AnalysisLog Mean Temperature Difference (ΔTm)ln Method

1. Log Mean Temperature Difference (ΔTm)ln

Method• To design or to predict the performance of a heat exchanger, it is essential to

relate the total heat transfer rate to quantities such as the inlet and outlet fluidtemperatures, the overall heat transfer coefficient, and the total surface area forheat transfer.

• Two such relations may readily be obtained by applying overall energy balancesto the hot and cold fluids, as shown in Figure.

Log Mean Temperature Difference (ΔTm)ln

Method• In particular, if q is the total rate of heat transfer between the hot andcold fluids and there is negligible heat transfer between theexchanger and its surroundings, as well as negligible potential andkinetic energy changes, application of the steadyflow energyequation, Equation, gives;

Log Mean Temperature Difference (ΔTm)ln

Method• where i is the fluid enthalpy. The subscripts h and c refer to the hotand cold fluids, whereas the subscripts i and o designate the fluid inletand outlet conditions. If the fluids are not undergoing a phase changeand constant specific heats are assumed, these expressions reduce to;

• where the temperatures appearing in the expressions refer to themean fluid temperatures at the designated locations.

Log Mean Temperature Difference (ΔTm)ln

Method• Note that these equations are independent of the flow arrangement

and heat exchanger type.

• Another useful expression may be obtained by relating the total heat transfer rate q to the temperature difference ΔT between the hot and cold fluids, where;

Log Mean Temperature Difference (ΔTm)ln

Method• Such an expression would be an extension of Newton's law of cooling,

with the overall heat transfer coefficient U used in place of the singleconvection coefficient h. However, since Δ T varies with position in theheat exchanger, it is necessary to work with a rate equation of theform;

• where Tm is an appropriate mean temperature difference.

The Parallel-Flow Heat Exchanger

• The hot and cold mean fuidtemperature distributionsassociated with a parallel flow heatexchanger are shown in Figure.

• The temperature difference ΔT isinitially large but decays withincreasing x, approaching zeroasymptotically. It is important tonote that, for such an exchanger, theoutlet temperature of the cold fluidnever exceeds that of the hot fluid.

The Parallel-Flow Heat Exchanger

• The energy balances and thesubsequent analysis are subject to thefollowing assumptions.

• The heat exchanger is insulated fromits surroundings, in which case theonly heat exchange is between the hotand cold fluids.

• Axial conduction along the tubes isnegligible.

• Potential and kinetic energy changesare negligible.

• The fluid specific heats are constant.

• The overall heat transfer coefficient isconstant.

The Parallel-Flow Heat Exchanger

• The specific heats may of coursechange as a result of temperaturevariations, and the overall heattransfer coefficient may changebecause of variations in fluidproperties and flow conditions.

• However, in many applicationssuch variations are not significant,and it is reasonable to work withaverage values of cp,c, cp,h, and Ufor the heat exchanger.

The Parallel-Flow Heat Exchanger

• Recognizing that, for the parallel flow heat exchanger

The Counterflow Heat Exchanger

• The hot and cold fluid temperaturedistributions associated with a counter flowheat exchanger are shown in Figure.

• In contrast to the parallel flow exchanger, thisconfiguration provides for heat transferbetween the hotter portions of the two fluidsat one end, as well as between the colderportions at the other.

• For this reason, the change in the temperaturedifference, ΔT =Th-Tc, with respect to x isnowhere as large as it is for the inlet region ofthe parallel flow exchanger.

• Note that the outlet temperature of the coldfluid may now exceed the outlet temperatureof the hot fluid.

The Counterflow Heat Exchanger

• Equations and apply to any heatexchanger and hence may be used forthe counterflow arrangement.

The Counterflow Heat Exchanger

• Note that, for the same inlet and outlet temperatures, the log meantemperature difference for counterflow exceeds that for parallel flow,ΔTlm,CF>ΔT lm,PF.

• Hence the surface area required to effect a prescribed heat transferrate q is smaller for the counter flow than for the parallel-flowarrangement, assuming the same value of U.

• Also note that Tc,o can exceed Th,o for counter flow but not for parallelflow.

Overall Heat Transfer Coefficient

• An essential, and often the most uncertain, part of any heat exchangeranalysis is determination of the overall heat transfer coefficient due theparameters that it includes such as geometry, convection andconduction.

Overall Heat Transfer Coefficient

• A heat exchanger typically involves two flowingfluids separated by a solid wall.

• Heat is first transferred from the hot fluid to thewall by convection,

• Through the wall by conduction,• From the wall to the cold fluid again by

convection.• Any radiation effects are usually included in theconvection heat transfer coefficients.

• The thermal resistance network associated withthis heat transfer process involves twoconvection and one conduction resistances, asshown in Figure.

Overall Heat Transfer Coefficient

• For a double-pipe heat exchanger, we have Ai=DiL and Ao=DoL, and thethermal resistance of the tube wall in this case is;

• where k is the thermal conductivity of the wall material and L is thelength of the tube. Then the total thermal resistance becomes;

Overall Heat Transfer Coefficient

• The Ai is the area of the inner surface ofthe wall that separates the two fluids,

• The Ao is the area of the outer surface ofthe wall.

• In other words, Ai and Ao are surface areasof the separating wall wetted by the innerand the outer fluids, respectively.

Overall Heat Transfer Coefficient

• For thin tubes, Di=Do and thus Ai=Ao, thickness of the inner pipe is ignored,so thermal resistance of pipe is extractred from equation. In such thiscase Do and Ao are used for outer tube.

• In the analysis of heat exchangers, it is convenient to combine all thethermal resistances in the path of heat flow from the hot fluid to the coldone into a single resistance R, and to express the rate of heat transferbetween the two fluids as;

• where U is the overall heat transfer coefficient, whose unit is W/m2°C,which is identical to the unit of the ordinary convection coefficient h.

Overall Heat Transfer Coefficient

• When the wall thickness of the tube is small and the thermalconductivity of the tube material is high, as is usually the case, thethermal resistance of the tube is negligible (Rwall =0) and the inner andouter surfaces of the tube are almost identical (Ai≈Ao≈As).

• Then for the overall heat transfer coficient simplifies to;

Overall Heat Transfer Coefficient

• The overall heat transfer coefficient U is dominated by the smallerconvection coefficient, since the inverse of a large number is small.

• When one of the convection coefficients is much smaller than theother (say, hi << ho), we have 1/hi >>1/ho, and thus U≈hi.

• Therefore, the smaller heat transfer coefficient creates a bottleneckon the path of heat flow and seriously impedes heat transfer.

• This situation arises frequently when one of the fluids is a gas and theother is a liquid. In such cases, fins are commonly used on the gasside to enhance the product UAs and thus the heat transfer on thatside.

Overall Heat Transfer Coefficient

• The overall heat transfer coefficient relation given above is valid forclean surfaces and needs to be modified to account for the effects offouling on both the inner and the outer surfaces of the tube. Shell-and-tube heat exchanger, it can be expressed as;

• where Ai =πDiL and Ao=πDoL are the areas of inner and outer surfaces,and Rf,i and Rf,oare the fouling factors at those surfaces.

Special Operating Conditions

• It is useful to note certain special conditions under which heatexchangers may be operated.

Example:

Example:

Example:

Example:

Example:

Example:

Example:

Example: