Countercurrent Double-Pipe Heat Exchanger - Maple Help
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Countercurrent Double-Pipe

Heat Exchanger


This application models the temperature dynamics of a countercurrent double-pipe heat exchanger. Three partial differential equations describe the following:



The heat balances across the tube- and shell-side liquids


A heat balance across the tube-wall (taking into account the heat flow from the shell- and tube-side liquids, and conduction along the length of the tube)


The equations are solved numerically, and the temperature profiles are plotted.  The heat exchanger is assumed to be perfectly insulated. Densities, specific heat capacities, heat transfer coefficients, and thermal conductivities are assumed to be constant.



Specific heat capacities for the tube-side fluid, shell-side fluid, and the tube wall:




Flow rates:


Heat transfer coefficients:


Thermal conductivity of tube-wall:


Length of heat exchanger:


Internal and external diameter of inner tube, and internal diameter of outer tube:


Partial Differential Equations Derived from Energy Balances

Tube wall heat balance:

pde1π4Do2Di2CpwrhowtTwx,t=Ut π Di Ttx,tTwx,tUs π DoTwx,tTsx,t+kw π4Do2Di22x2Twx,t:

Tube-side heat balance:

pde2rhot Cpt π4Di2tTtx,t=CptFtxTtx,tπDiUtTtx,tTwx,t:

Shell-side heat balance:

pde3rhos Cps π4Dis2Do2tTsx,t=Cps Fs xTsx,t+πDoUsTwx,tTsx,t:

Initial and boundary conditions:


Solution and Results

solpdsolvepde1,pde2,pde3,ibc,numeric,time=t,range=0.. L:

p1sol:-plotTt,x= 0.5 L,t=0..5,axes=boxed,color=black,legend=Tube-side liquid:p2sol:-plotTs,x=0.5 L,t=0..5,axes=boxed,color=blue,legend=Shell-side liquid:p3sol:-plotTw,x=0.5 L,t=0..5,axes=boxed,color=red,legend=Tube wall: plotsdisplayp1,p2,p3,labels=Time (s),Temperature (C),title=Temperature Halfway along Heat Exchanger,titlefont=Helvetica, 18,Bold

p1sol:-plotTt,t=5,x=0..L,axes=boxed,color=black,legend=Tube-side liquid:p2sol:-plotTs,t=5,x=0..L,axes=boxed,color=blue,legend=Shell-side liquid:p3sol:-plotTw,t=5,x=0..L,axes=boxed,color=red,legend=Tube wall:plotsdisplayp1,p2,p3,labels=Distance (m),Temperature (C),title=Temperature Profile along Heat Exchanger,titlefont=Helvetica, 18,Bold