<Text-field style="Heading 1" layout="Heading 1">Restart</Text-field>

<Text-field style="Heading 1" layout="Heading 1">Application of Pi Theorem</Text-field>

To use the PiTheorem module, you must first define Maple variables as the physical parameters of a physical system. Maple knows the dimensions for numerous physical parameters. Here are its known physical dimensions.

GetDimensions();

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

The AddDimension command lets you add new dimensions to the system. For example, basis weight in paper manufacture is defined as the grams per square meter of surface area. It is defined to the system in terms of previously defined dimensions as follows using AddDimension:

AddDimension(basis_weight=mass/length^2);

Overview:

Define the physical parameter variables with their unit types.

Create a module instance for performing the analysis via a call to GeneratePiTheorem. It creates the module with the desired name such as Case: GeneratePiTheorem(Case):

Case:-Exclude is available for very special cases when one or more fundamental units must be dropped from the analysis. The software will advise when it is necessary to use exclude.

Initialize Case with its parameters: Case:-Initialize(........)

Check for valid data: Case:-Check()

Find the \316\240 groups: Case:-FindPiGroups

Print the \316\240 groups: Case:-PrintPiGroups

Several examples are provided to show its usage.

<Text-field style="Heading 2" layout="Heading 2">Traditional Pendulum Example</Text-field>

The period of a pendulum is related to the pendulum length, the acceleration of gravity and the release angle. First, enter the physical parameter variables:

t:=time: # period (time for a complete cycle in movement) l:=length: # length of the pendulum arm g:=acceleration: # acceleration of gravity theta:=1: # release angle: it is unitless for this case

Create a PiTheorem module instance called Case1 for this example.

GeneratePiTheorem(Case1):

Initialize the module instance with the physical parameter data as follows. Note that parameter t is placed first in the list since it is the main variable of concern. The method will create LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5Mjg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy groups with t found only in 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The variable names are entered to the initialization as a 'name'=name. 'name' delays the evaluation so that the actual name gets passed to the routine.

Case1:-Initialize(['t'=t,'l'=l,'g'=g,'theta'=theta]);

Perform the initial calculations and check for valid solutions,

Case1:-Check();

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

Find the \316\240 groups,

Case1:-FindPiGroups();

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

Print the \316\240 groups from the rows of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEpUGlDb2VmZnNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn:

Case1:-PrintPiGroups();

NyQvJkkjUElHNiI2IyIiIkkmdGhldGFHRiYvJkYlNiMiIiMqKEkidEdGJkYoKUkibEdGJiNGKEYtISIiKUkiZ0dGJkYyRig=

Then, the resulting mathematical relationship for the pendulum is in the form of 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Therefore, 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.

What if you attempted to add the mass of the pendulum as a physical parameter to the model?

m:=mass; # new variable m for the pendulum mass GeneratePiTheorem(Case1a): # new module instance Case1a:-Initialize(['t'=t,'l'=l,'g'=g,'theta'=theta,'m'=m]); Case1a:-Check();

SSVtYXNzRzYi

Fundamental unit ( mass ) has only one entry within the groups. You must drop the parameter containing it or add another physical parameter with that fundamental unit.

Error, (in Check) there is a row with only one entry for the fundamental unit

The check step now fails. One of the rules for the Pi Theorem is that each fundamental unit within the proposed physical parameters must occur in at least two physical parameters. Otherwise, it is not possible to form a dimensionless group with the physical parameter. If the pendulum mass is actually required to model the physical system, then there must be another physical parameter added that includes mass in its dimensions.

<Text-field style="Heading 2" layout="Heading 2">Terminal Raindrop Velocity</Text-field>

Falling raindrops reach a maximum velocity as they fail to earth. It is proposed that the terminal velocity is related to the radius of the raindrop, the density of the air, the viscosity of the air and the earth's acceleration of gravity. The Pi Theorem is applied as follows:

v:=speed: # terminal raindrop velocity rho:=mass_density: # air density r:=length: # radius of raindrop mu:=dynamic_viscosity: # air viscosity g:=acceleration: # earth's gravity

Create a PiTheorem module instance for this example,

GeneratePiTheorem(Case2):

Initialize the module instance with the physical parameter data as follows. Note that parameter v is placed first in the list since it is the main variable of concern.

Case2:-Initialize(['v'=v,'rho'=rho,'r'=r,'mu'=mu,'g'=g]);

Perform the initial calculations and check for valid solutions,

Case2:-Check();

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Case2:-FindPiGroups();

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Print the \316\240 groups,

Case2:-PrintPiGroups();

NyQvJkkjUElHNiI2IyIiIiooSSJ2R0YmRigpSSJyR0YmI0YoIiIjISIiKUkiZ0dGJkYtRi8vJkYlNiNGLioqSSRyaG9HRiZGKClGLCMiIiRGLkYoSSNtdUdGJkYvRjBGKA==

Note that there are two LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5Mjg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy groups and that the first two physical parameters in the entry list were selected to be in only one of the two LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5Mjg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy groups and to be raised to the first power. By placing the terminal velocity as the first physical parameter entry, it gets to be only within the first LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5Mjg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy group. That is a good feature for our model of terminal raindrop velocity to have.

You may change the second physical parameter in the list and get another set of Pi groups:

GeneratePiTheorem(Case2a):

Case2a:-Initialize(['v'=v,'r'=r,'rho'=rho,'mu'=mu,'g'=g]);

Case2a:-Check();

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mnspanGQ"16"-I$mtdG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6(-I%mrowG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6$-I#moG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6-Q*&uminus0;6"/%,mathvariantGQ'normal6"/%&fenceGQ&false6"/%*separatorGQ&false6"/%)stretchyGQ&false6"/%*symmetricGQ&false6"/%(largeopGQ&false6"/%.movablelimitsGQ&false6"/%'accentGQ&false6"/%'lspaceGQ,0.2222222em6"/%'rspaceGQ,0.2222222em6"-I#mnG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6$Q"26"/%,mathvariantGQ'normal6"/%)rowalignGQ!6"/%,columnalignGQ!6"/%+groupalignGQ!6"/%(rowspanGQ"16"/%+columnspanGQ"16"/%)rowalignGQ!6"/%,columnalignGQ!6"/%+groupalignGQ!6"/%&alignGQ%axis6"/%)rowalignGQ)baseline6"/%,columnalignGQ&right6"/%+groupalignGQ'|grfrleft|grhr6"/%/alignmentscopeGQ%true6"/%,columnwidthGQ%auto6"/%&widthGQ%auto6"/%+rowspacingGQ&1.0ex6"/%.columnspacingGQ&0.8em6"/%)rowlinesGQ%none6"/%,columnlinesGQ%none6"/%&frameGQ%none6"/%-framespacingGQ,0.4em|ir0.5ex6"/%*equalrowsGQ&false6"/%-equalcolumnsGQ&false6"/%-displaystyleGQ&false6"/%%sideGQ&right6"/%0minlabelspacingGQ&0.8em6"/%,mathvariantGQ'normal6"/%%openGQ"[6"/%&closeGQ"]6"F'-I#moGF$6-Q",F'/%,mathvariantGQ'normalF'/%&fenceGQ&falseF'/%*separatorGQ%trueF'/%)stretchyGF8/%*symmetricGF8/%(largeopGF8/%.movablelimitsGF8/%'accentGF8/%'lspaceGQ&0.0emF'/%'rspaceGQ,0.3333333emF'-F,6#QL/%-FundamentalsG7#7%%'lengthG%%massG%%timeGF'F/-F,6#QO/%+ParametersG7#7'Q"v6"Q"r6"Q$rho6"Q#mu6"Q"g6"F'F/-F,6#Qcq/%3UnitlessParametersG-I#mnG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6$Q"06"/%,mathvariantGQ'normal6"F'

Case2a:-FindPiGroups();

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

The new \316\240 groups are:

Case2a:-PrintPiGroups();

NyQvJkkjUElHNiI2IyIiIioqSSJ2R0YmRigpSSRyaG9HRiYjRigiIiRGKClJI211R0YmRi0hIiIpSSJnR0YmRi1GMS8mRiU2IyIiIyoqSSJyR0YmRigpRiwjRjdGLkYoKUYwRjtGMUYyRig=

The analysis does not work for all cases. If \317\201 and \316\274 were entered first and second in the Initialize list, then the analysis tries to find a solution where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEnJiM5Mjg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnRjhGOC8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjg= has \317\201 to the first power without \316\274 present and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEnJiM5Mjg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnRjhGOC8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjg= has \316\274 to the first power without \317\201 present. It is not feasible. The module responds with an error and recommends changing the order of the physical parameters.

GeneratePiTheorem(Case2b): Case2b:-Initialize(['rho'=rho,'mu'=mu,'v'=v,'r'=r,'g'=g]); Case2b:-Check(); Case2b:-FindPiGroups();

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Error, (in FindPiGroups) The determinant of the right square A matrix of UnitMatrix is zero. Shift the order of the physical paramters in Initialize to get a non-zero determinant.

<Text-field style="Heading 2" layout="Heading 2">Heat Transfer in a Pipe</Text-field>

The rate of heat transfer of a fluid in a pipe is given by 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEiLkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDRi8=

Heat Flow = heat transfer coefficient * area * temperature difference between the fluid and the pipe wall

The heat transfer coefficient is related to six other parameters. The parameter list is:

h:=heat_transfer_coefficient: # heat transfer coefficient rho:=mass_density: # density of the fluid in the pipe Cp:=specific_heat_capacity: # fluid heat capacity k:=thermal_conductivity: # fluid thermal conductivity mu:=dynamic_viscosity: # fluid viscosity v:=speed: # fluid velocity Dia:=length: # pipe inside diamter

GeneratePiTheorem(Case3): # new module instance Case3:-Initialize(['h'=h,'rho'=rho,'Cp'=Cp,'k'=k,'mu'=mu,'v'=v,'Dia'=Dia]); Case3:-Check();

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Case3:-FindPiGroups();

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The resulting \316\240 groups are:

Case3:-PrintPiGroups();

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The groups are known as the Nusselt number, Reynolds number and Prandtl number.

<Text-field style="Heading 2" layout="Heading 2">Find All Pi Groups Feasible for Heat Transfer in a Pipe</Text-field>

The seven physical parameters for heat transfer in a pipe are:

The heat transfer coefficient is kept as the first parameter and the other six parameters are tested in all permutations in order to find all of the feasible groups from the solution technique.

i:=0: # enter the other parameters (not the prime parameter h) in the following special form Params:=['''rho'=rho'','''Cp'=Cp'','''k'=k'', '''mu'=mu'','''v'=v'','''Dia'=Dia'']: StorePiGroups:-Initialize(): # use it to store all of the returned Pi groups # enter the prime parameter h as follows: StorePiGroups:-EnterParameters(["h",seq(convert(rhs(Params[j]), `string`),j=1..nops(Params))]): for Item in combinat[choose](Params,3) do Params2:=[]: for j from 1 to nops(Params) do if not member(eval(Params[j]),Item) then Params2:=[op(Params2),j] fi end do; j:='j'; for j from 1 to nops(Params) do if member(eval(Params[j]),Item) then Params2:=[op(Params2),j] fi end do; Params3:=[''h'=h',seq(eval(Params[Params2[k]],1),k=1..nops(Params2))]: i:=i+1: try GeneratePiTheorem(Case2_||i): # new module instance Case2_||i:-Initialize(Params3): Case2_||i:-Check(): val:=Case2_||i:-FindPiGroups(): StorePiGroups:-StoreCoefficients(rhs(val[1]),rhs(val[2])): catch: end try; end do:

GroupData:=StorePiGroups:-PiGroupSummary();

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, 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 and 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 correspond to the groups found in the previous section. There are six more \316\240 groups that are feasible. The \316\240 theorem requires that the three selected \316\240 groups are independent.

Suppose that you select the 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 and 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 groups. Which of the remaining seven Pi groups are independent of these two Pi groups. The module provides a routine to find them.

GroupDefs:=convert(rhs(GroupData[2]),Matrix): # enter the two selected group numbers # **** Review the results from (3.4.1) to find the two desired groups. # **** The numbers change between different executions of the commands. SelectedGroups:=[1,5]: i:='i': DeleteList:=[]: for i from 1 to LinearAlgebra[RowDimension](GroupDefs) do if not member(i,SelectedGroups) then DeleteList:=[i,op(DeleteList)] fi end do; GroupDefs:=LinearAlgebra[DeleteRow](GroupDefs,DeleteList): IndGroups:=StorePiGroups:-FindIndependentGroups(GroupDefs);

NiQ8KCIiIyIiJCIiJyIiKCIiKSIiKi9JKVBpR3JvdXBzRzYiNygvJkkjUElHRiw2I0YkKipJImhHRiwiIiJJJHJob0dGLCEiIkkjQ3BHRixGNkkidkdGLEY2LyZGMDYjRiUqLEY1RjZGN0Y2SSJrR0YsRjRGOEY2SSREaWFHRixGNi8mRjA2I0YmKihGN0Y0Rj1GNkkjbXVHRixGNC8mRjA2I0YnKixGNUY0RjdGNEY9RjZGOEY0Rj5GNC8mRjA2I0YoKipGM0Y0RjdGNkZDRjZGPkY0LyZGMDYjRikqKEY3RjZGPUY0RkNGNg==

6 groups are independent from the two selected groups.

Now try again with a selection of Pi groups 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 and 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.

GroupDefs:=convert(rhs(GroupData[2]),Matrix): # enter the two selected group numbers # **** Review the results from (3.4.1) to find the two desired groups. # **** The numbers change between different executions of the commands. SelectedGroups:=[1,6]: i:='i': DeleteList:=[]: for i from 1 to LinearAlgebra[RowDimension](GroupDefs) do if not member(i,SelectedGroups) then DeleteList:=[i,op(DeleteList)] fi end do; GroupDefs:=LinearAlgebra[DeleteRow](GroupDefs,DeleteList): IndGroups:=StorePiGroups:-FindIndependentGroups(GroupDefs);

NiQ8JSIiIyIiJiIiKS9JKVBpR3JvdXBzRzYiNyUvJkkjUElHRik2I0YkKipJImhHRikiIiJJJHJob0dGKSEiIkkjQ3BHRilGM0kidkdGKUYzLyZGLTYjRiUqKEYwRjFJImtHRilGM0kkRGlhR0YpRjEvJkYtNiNGJioqRjBGMUY0RjNJI211R0YpRjNGO0Yx

The three LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5Mjg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy groups that contain the heat transfer coefficient are then feasible as independent groups to match with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JJW1zdWJHRiQ2JS1GLDYlUSUmUGk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYkRitGOi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEiPUYnRjovJSZmZW5jZUdGOS8lKnNlcGFyYXRvckdGOS8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZWLUkmbWZyYWNHRiQ2KC1GIzYmLUYsNiVRI0NwRicvRjhRJXRydWVGJy9GO1EnaXRhbGljRictRkM2LVExJkludmlzaWJsZVRpbWVzO0YnRjpGRkZIRkpGTEZORlBGUi9GVVEmMC4wZW1GJy9GWEZjby1GLDYlUSUmbXU7RidGN0Y6RjotRiM2JC1GLDYlUSJrRidGW29GXW9GOi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGYnAvJSliZXZlbGxlZEdGOUY6RitGOg== and 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as the final LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5Mjg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIn5GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSUYygroup.

<Text-field style="Heading 2" layout="Heading 2">Coffee Warmer</Text-field>

Reference: Dr. Thomas Szirtes, "Applied Dimensional Analysis and Modeling," McGraw-Hill, (1998), ISBN 0-07-062811-4, pp. 356-358.

A cup of coffee is being heated on a hot plate as follows:

heat, 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 added from the hot plate to the coffee through the bottom flat surface of the mug

heat, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSIsRidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0ZJLyUqc3ltbWV0cmljR0ZJLyUobGFyZ2VvcEdGSS8lLm1vdmFibGVsaW1pdHNHRkkvJSdhY2NlbnRHRkkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJ0Y+is lost from the coffee to the side wall of the mug

the mug has a lid that is assumed to perfectly insulate the top of the mug

the heat transfer from the mug wall is influenced by the heat transfer coefficient, the area and the temperature difference between the coffee and room air

The physical system is proposed to be given by five quantities:

Delta_t:=thermodynamic_temperature: # delta temp between coffee and air h:=heat_transfer_coefficient: # heat transfer coef from cup to air AddDimension(heat_flux=energy/time): Q:=heat_flux: # heat flux to coffee from hot plate b:=length: # height of coffee in mug Dia:=length: # diameter of mug

GeneratePiTheorem(Case4): # new module instance #Case4:-Exclude([time]): Case4:-Initialize(['Delta_t'=Delta_t,'b'=b,'Q'=Q,'h'=h,'Dia'=Dia]); Case4:-Check();

-I%mrowG6#/I+modulenameG6"I,TypesettingGI(_syslibGF'6)-I*mverbatimGF$6#Qjddl/%+UnitMatrixG-I(mfencedG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6&-I%mrowG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6#-I'mtableG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"68-I$mtrG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6*-I$mtdG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6(-I#mnG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6$Q"06"/%,mathvariantGQ'normal6"/%)rowalignGQ!6"/%,columnalignGQ!6"/%+groupalignGQ!6"/%(rowspanGQ"16"/%+columnspanGQ"16"-I$mtdG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6(-I#mnG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6$Q"16"/%,mathvariantGQ'normal6"/%)rowalignGQ!6"/%,columnalignGQ!6"/%+groupalignGQ!6"/%(rowspanGQ"16"/%+columnspanGQ"16"-I$mtdG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6(-I#mnG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6$Q"26"/%,mathvariantGQ'normal6"/%)rowalignGQ!6"/%,columnalignGQ!6"/%+groupalignGQ!6"/%(rowspanGQ"16"/%+columnspanGQ"16"-I$mtdG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6(-I#mnG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6$Q"06"/%,mathvariantGQ'normal6"/%)rowalignGQ!6"/%,columnalignGQ!6"/%+groupalignGQ!6"/%(rowspanGQ"16"/%+columnspanGQ"16"-I$mtdG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6(-I#mnG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6$Q"16"/%,mathvariantGQ'normal6"/%)rowalignGQ!6"/%,columnalignGQ!6"/%+groupalignGQ!6"/%(rowspanGQ"16"/%+columnspanGQ"16"/%)rowalignGQ!6"/%,columnalignGQ!6"/%+groupalignGQ!6"-I$mtrG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6*-I$mtdG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6(-I#mnG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6$Q"06"/%,mathvariantGQ'normal6"/%)rowalignGQ!6"/%,columnalignGQ!6"/%+groupalignGQ!6"/%(rowspanGQ"16"/%+columnspanGQ"16"-I$mtdG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6(-I#mnG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6$Q"06"/%,mathvariantGQ'normal6"/%)rowalignGQ!6"/%,columnalignGQ!6"/%+groupalignGQ!6"/%(rowspanGQ"16"/%+columnspanGQ"16"-I$mtdG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6(-I#mnG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6$Q"16"/%,mathvariantGQ'normal6"/%)rowalignGQ!6"/%,columnalignGQ!6"/%+groupalignGQ!6"/%(rowspanGQ"16"/%+columnspanGQ"16"-I$mtdG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6(-I#mnG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6$Q"16"/%,mathvariantGQ'normal6"/%)rowalignGQ!6"/%,columnalignGQ!6"/%+groupalignGQ!6"/%(rowspanGQ"16"/%+columnspanGQ"16"-I$mtdG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6(-I#mnG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6$Q"06"/%,mathvariantGQ'normal6"/%)rowalignGQ!6"/%,columnalignGQ!6"/%+groupalignGQ!6"/%(rowspanGQ"16"/%+columnspanGQ"16"/%)rowalignGQ!6"/%,columnalignGQ!6"/%+groupalignGQ!6"-I$mtrG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6*-I$mtdG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6(-I#mnG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6$Q"16"/%,mathvariantGQ'normal6"/%)rowalignGQ!6"/%,columnalignGQ!6"/%+groupalignGQ!6"/%(rowspanGQ"16"/%+columnspanGQ"16"-I$mtdG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6(-I#mnG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6$Q"06"/%,mathvariantGQ'normal6"/%)rowalignGQ!6"/%,columnalignGQ!6"/%+groupalignGQ!6"/%(rowspanGQ"16"/%+columnspanGQ"16"-I$mtdG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6(-I#mnG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6$Q"06"/%,mathvariantGQ'normal6"/%)rowalignGQ!6"/%,columnalignGQ!6"/%+groupalignGQ!6"/%(rowspanGQ"16"/%+columnspanGQ"16"-I$mtdG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6(-I%mrowG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6$-I#moG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6-Q*&uminus0;6"/%,mathvariantGQ'normal6"/%&fenceGQ&false6"/%*separatorGQ&false6"/%)stretchyGQ&false6"/%*symmetricGQ&false6"/%(largeopGQ&false6"/%.movablelimitsGQ&false6"/%'accentGQ&false6"/%'lspaceGQ,0.2222222em6"/%'rspaceGQ,0.2222222em6"-I#mnG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6$Q"16"/%,mathvariantGQ'normal6"/%)rowalignGQ!6"/%,columnalignGQ!6"/%+groupalignGQ!6"/%(rowspanGQ"16"/%+columnspanGQ"16"-I$mtdG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6(-I#mnG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6$Q"06"/%,mathvariantGQ'normal6"/%)rowalignGQ!6"/%,columnalignGQ!6"/%+groupalignGQ!6"/%(rowspanGQ"16"/%+columnspanGQ"16"/%)rowalignGQ!6"/%,columnalignGQ!6"/%+groupalignGQ!6"-I$mtrG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6*-I$mtdG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6(-I#mnG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6$Q"06"/%,mathvariantGQ'normal6"/%)rowalignGQ!6"/%,columnalignGQ!6"/%+groupalignGQ!6"/%(rowspanGQ"16"/%+columnspanGQ"16"-I$mtdG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6(-I#mnG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6$Q"06"/%,mathvariantGQ'normal6"/%)rowalignGQ!6"/%,columnalignGQ!6"/%+groupalignGQ!6"/%(rowspanGQ"16"/%+columnspanGQ"16"-I$mtdG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6(-I%mrowG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6$-I#moG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6-Q*&uminus0;6"/%,mathvariantGQ'normal6"/%&fenceGQ&false6"/%*separatorGQ&false6"/%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Case4:-FindPiGroups();

Error, (in FindPiGroups) There are too many accepted fundamental units: , [length, mass, thermodynamic_temperature, time], . Use Exclude to remove , 1, of them prior to the Initialize statement.

# Repeating the calculations excluding time dimension from the analysis, GeneratePiTheorem(Case4): # new module instance Case4:-Exclude([time]): Case4:-Initialize(['Delta_t'=Delta_t,'b'=b,'Q'=Q,'h'=h,'Dia'=Dia]); Case4:-Check();

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Case4:-FindPiGroups();

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Case4:-PrintPiGroups();

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