.MCAD 304020000 1 79 611 0
.CMD FORMAT  rd=d ct=10 im=i et=3 zt=15 pr=3 mass length time charge temperature tr=0 vm=0
.CMD SET ORIGIN 1
.CMD SET TOL 0.001000000000000
.CMD SET PRNCOLWIDTH 8
.CMD SET PRNPRECISION 4
.CMD PRINT_SETUP 1.000000 1.000000 0.750000 0.500000 1
.CMD HEADER_FOOTER 1 1 *empty* Properties^of^Pipe^Flow *empty* 0 1 |F |P |D^|T
.CMD HEADER_FOOTER_FONT fontID=14 family=Arial points=10 bold=0 italic=0 underline=0 colrid=-1
.CMD HEADER_FOOTER_FONT fontID=15 family=Arial points=10 bold=0 italic=0 underline=0 colrid=-1
.CMD DEFAULT_TEXT_PARPROPS 0 0 2
.CMD DEFINE_FONTSTYLE_NAME fontID=0 name=Variables
.CMD DEFINE_FONTSTYLE_NAME fontID=1 name=Constants
.CMD DEFINE_FONTSTYLE_NAME fontID=2 name=Text
.CMD DEFINE_FONTSTYLE_NAME fontID=4 name=User^1
.CMD DEFINE_FONTSTYLE_NAME fontID=5 name=User^2
.CMD DEFINE_FONTSTYLE_NAME fontID=6 name=User^3
.CMD DEFINE_FONTSTYLE_NAME fontID=7 name=User^4
.CMD DEFINE_FONTSTYLE_NAME fontID=8 name=User^5
.CMD DEFINE_FONTSTYLE_NAME fontID=9 name=User^6
.CMD DEFINE_FONTSTYLE_NAME fontID=10 name=User^7
.CMD DEFINE_FONTSTYLE fontID=0 family=Times^New^Roman points=10 bold=0 italic=0 underline=0 colrid=-1
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.CMD DEFINE_FONTSTYLE fontID=4 family=Math^Ext points=10 bold=0 italic=0 underline=0 colrid=-1
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.CMD UNITS U=1
.CMD DIMENSIONS_ANALYSIS 0 0
.CMD COLORTAB_ENTRY 0 0 0
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{\rtf\ansi \deff0{\colortbl;\red0\green128\blue128;\red0\green0\blue128;}{
\fonttbl{\f0\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard {\cf2 
Darcy-Weisbach equation for frictional head loss.}}
.EQN 4 0 189 0 0
{0:h.f}NAME÷{1:ƒ}NAME*({0:L}NAME)/({0:D}NAME)*(({0:v}NAME)^(2))/(2*{0:g}NAME)
.EQN 0 19 217 0 0
{0:h.f}NAME÷8*{1:ƒ}NAME*({0:L}NAME*({0:W}NAME)^(2))/(({0:D}NAME)^(5)*({0:\p}NAME)^(2)*{0:g}NAME)
.EQN 7 -19 256 0 0
{0:h.f_dw}NAME({1:ƒ}NAME,{0:L}NAME,{0:D}NAME,{0:W}NAME):8*{1:ƒ}NAME*({0:L}NAME*({0:W}NAME)^(2))/(({0:D}NAME)^(5)*({0:\p}NAME)^(2)*{0:g}NAME)
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{\rtf\ansi \deff0{\colortbl;\red0\green128\blue128;\red0\green0\blue128;}{
\fonttbl{\f0\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard {\cf2 ; 
Darcy-Weisbach equation solved for head loss (}{\cf2\i h}{\cf2\i\dn f}{
\cf2 ) as a function of friction factor (}{\object{\*\objclass \eqn}
\rsltpict{\*\objdata 
.EQN 17 46 466 0 0
{1:ƒ}NAME
}}{\cf2 ), pipe length (}{\cf2\i L}{\cf2 ), pipe diameter (}{\cf2\i D}{
\cf2 ) and volumetric flow rate (}{\cf2\i W}{\cf2 ).  }}
.EQN 8 -28 255 0 0
{0:D.dw}NAME({1:ƒ}NAME,{0:L}NAME,{0:h.f}NAME,{0:W}NAME):(5){80}(8*{1:ƒ}NAME*({0:L}NAME)/({0:h.f}NAME)*(({0:W}NAME)^(2))/(({0:\p}NAME)^(2)*{0:g}NAME))
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{\rtf\ansi \deff0{\colortbl;\red0\green128\blue128;\red0\green0\blue128;}{
\fonttbl{\f0\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard {\cf2 ; 
Darcy-Weisbach equation solved for }{\cf2\i D}{\cf2  as a function of }{
\object{\*\objclass \eqn}\rsltpict{\*\objdata 
.EQN 23 66 467 0 0
{1:ƒ}NAME
}}{\cf2 , }{\cf2\i L}{\cf2 , }{\cf2\i h}{\cf2\i\dn f}{\cf2 , and}{\cf2
\i  }{\cf2\i W}{\cf2 .  }}
.EQN 8 -28 254 0 0
{0:L.dw}NAME({1:ƒ}NAME,{0:D}NAME,{0:h.f}NAME,{0:W}NAME):({0:h.f}NAME*({0:D}NAME)^(5))/(8*{1:ƒ}NAME)*(({0:\p}NAME)^(2)*{0:g}NAME)/(({0:W}NAME)^(2))
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{\rtf\ansi \deff0{\colortbl;\red0\green128\blue128;\red0\green0\blue128;}{
\fonttbl{\f0\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard {\cf2 ; 
Darcy-Weisbach equation solved for }{\cf2\i L}{\cf2  as a function of }{
\object{\*\objclass \eqn}\rsltpict{\*\objdata 
.EQN 31 57 468 0 0
{1:ƒ}NAME
}}{\cf2 , }{\cf2\i D}{\cf2 , }{\cf2\i h}{\cf2\i\dn f}{\cf2 , and}{\cf2
\i  }{\cf2\i W}{\cf2\i .}{\cf2   }}
.EQN 8 -28 253 0 0
{0:W.dw}NAME({1:ƒ}NAME,{0:L}NAME,{0:D}NAME,{0:h.f}NAME):\(({0:h.f}NAME)/(8*{1:ƒ}NAME)*(({0:D}NAME)^(5))/({0:L}NAME)*({0:\p}NAME)^(2)*{0:g}NAME)
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{\rtf\ansi \deff0{\colortbl;\red0\green128\blue128;\red0\green0\blue128;}{
\fonttbl{\f0\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard {\cf2 ; 
Darcy-Weisbach equation solved for }{\cf2\i W}{\cf2  as a function of }{
\object{\*\objclass \eqn}\rsltpict{\*\objdata 
.EQN 39 66 469 0 0
{1:ƒ}NAME
}}{\cf2 , }{\cf2\i L, }{\cf2\i D}{\cf2 , and}{\cf2\i  h}{\cf2\i\dn f}{
\cf2 .  }}
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C x1,1,0,0
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{\rtf\ansi \deff0{\colortbl;\red0\green128\blue128;\red0\green0\blue128;}{
\fonttbl{\f0\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard {\cf2 
Colebrook equation for friction factor }using Mathcad's solve block 
function. (MathCad implements the {\b\ul\link1 Levenberg-Marquard} for 
to solve for several constraints simultaneously){\b .}\par }
.ATT
.ATT_END
.ATT
.LINK http://www.3-cities.com/~tyroneb/mathcad/SolveBlocks.mcd
.ATT_END
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{1:ƒ}NAME:.002
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{\rtf\ansi \deff0{\colortbl;\red0\green128\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard ; initial guess (seed)}
.EQN 4 -8 93 0 0
{0:Given}NAME
.EQN 6 3 94 0 0
(1)/(\({1:ƒ}NAME))÷-0.86*{0:ln}NAME(({5:M}NAME)/(3.7*{0:D}NAME)+(2.51)/({4:Á}NAME*\({1:ƒ}NAME)))
.EQN 6 -3 336 0 0
{1:ƒ.t}NAME({5:M}NAME,{0:D}NAME,{4:Á}NAME):{0:find}NAME({1:ƒ}NAME)
.EQN 5 0 342 0 0
{1:ƒ.l}NAME({4:Á}NAME):(64)/({4:Á}NAME)
.EQN 5 0 505 0 0
{1:ƒ.cb}NAME({5:M}NAME,{0:D}NAME,{4:Á}NAME):{0:if}NAME({4:Á}NAMEò3000,{1:ƒ.t}NAME({5:M}NAME,{0:D}NAME,{4:Á}NAME),{1:ƒ.l}NAME({4:Á}NAME))
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{\rtf\ansi \deff0{\colortbl;\red0\green128\blue128;\red128\green0\blue0;}{
\fonttbl{\f0\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard {\cf2 
Velocity in Tubes}}
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\pard ; velocity in the pipe (tubes) as a function of mass flow rate ({
\i w}), diameter ({\i D}), density ({\f1\i r}) and the number of tubes 
carrying the flow ({\i N}).}
.EQN 1 -23 497 0 0
{0:v.t}NAME({0:w}NAME,{0:D}NAME,{0:\r}NAME,{0:N}NAME):({0:w}NAME)/({0:N}NAME*{0:\r}NAME*({0:\p}NAME)/(4)*({0:D}NAME)^(2))
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{\rtf\ansi \deff0{\colortbl;\red0\green128\blue128;\red128\green0\blue0;}{
\fonttbl{\f0\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard {\cf2 
Reynolds Number}}
.EQN 6 3 496 0 0
{0:Re.v}NAME({0:v}NAME,{0:D}NAME,{0:\r}NAME,{0:\m}NAME):({0:v}NAME*{0:D}NAME*{0:\r}NAME)/({0:\m}NAME)
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\cf1\fs20 \pard {\cf2 ; Reynolds number (}{\object{\*\objclass \eqn}
\rsltpict{\*\objdata 
.EQN 113 41 307 0 0
{4:Á}NAME
}}{\cf2 )as}{\cf2 , }{\cf2 velocity (}{\cf2\i v}{\cf2 ), diameter (}{
\cf2\i D}{\cf2 ), density (}{\cf2\f1 r}) and viscosity ({\f1 m})}
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{0:Re.W}NAME({0:W}NAME,{0:D}NAME,{0:\r}NAME,{0:\m}NAME):(4*{0:W}NAME)/({0:\p}NAME*{0:D}NAME)*({0:\r}NAME)/({0:\m}NAME)
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\cf1\fs20 \pard {\cf2 ; Reynolds number (}{\object{\*\objclass \eqn}
\rsltpict{\*\objdata 
.EQN 120 41 307 0 0
{4:Á}NAME
}}{\cf2 )as a function of volumetric flow rate (W}{\cf2 ), diameter (}{
\cf2\i D}{\cf2 ), density (}{\cf2\f1\i r}{\cf2 ) and viscosity (}{\cf2
\f1\i m}{\cf2 ).}}
.EQN 8 -23 508 0 0
{0:Re.W\n}NAME({0:W}NAME,{0:D}NAME,{0:\n}NAME):(4*{0:W}NAME)/({0:\p}NAME*{0:D}NAME*{0:\n}NAME)
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{\rtf\ansi \deff0{\colortbl;\red0\green128\blue128;\red128\green0\blue0;}{
\fonttbl{\f0\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard {\cf2 
Prandtl Number}}
.EQN 5 3 494 0 0
{0:Pr}NAME({0:c.p}NAME,{0:\m}NAME,{0:k.w}NAME):({0:c.p}NAME*{0:\m}NAME)/({0:k.w}NAME)
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{\rtf\ansi \deff0{\colortbl;\red0\green128\blue128;\red0\green0\blue128;}{
\fonttbl{\f0\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard {\cf2 Swamee 
and Jain equation for flow rate (}{\cf2\i W}{\cf2 ) based on Colebrook 
equation for friction factor and Darcy-Weisbach formula for head loss.}}
.EQN 9 1 489 0 0
{0:W.sj}NAME({5:M}NAME,{0:\n}NAME,{0:D}NAME,{0:L}NAME,{0:h.f}NAME):-0.955*({0:D}NAME)^(2)*\(({0:g}NAME*{0:D}NAME*{0:h.f}NAME)/({0:L}NAME))*{0:ln}NAME(({5:M}NAME)/(3.7*{0:D}NAME)+(1.775*{0:\n}NAME)/({0:D}NAME*\(({0:g}NAME*{0:D}NAME*{0:h.f}NAME)/({0:L}NAME)
)))
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{0:D}NAME:30*{0:in}NAME
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{\rtf\ansi \deff0{\colortbl;\red0\green128\blue128;}{\fonttbl{\f0
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.EQN 4 -10 579 0 0
{0:Given}NAME
.EQN 8 0 580 0 0
{0:W}NAME÷-0.955*({0:D}NAME)^(2)*\(({0:g}NAME*{0:D}NAME*{0:h.f}NAME)/({0:L}NAME))*{0:ln}NAME(({5:M}NAME)/(3.7*{0:D}NAME)+(1.775*{0:\n}NAME)/({0:D}NAME*\(({0:g}NAME*{0:D}NAME*{0:h.f}NAME)/({0:L}NAME))))
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{0:D.sj}NAME({5:M}NAME,{0:\n}NAME,{0:L}NAME,{0:h.f}NAME,{0:W}NAME):{0:Find}NAME({0:D}NAME)
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{\rtf\ansi \deff0{\colortbl;\red0\green128\blue128;\red0\green0\blue128;}{
\fonttbl{\f0\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard {\cf2  
Darcy-Weisbach equation solved for }{\cf2\i W}{\cf2  as a function of }{
\object{\*\objclass \eqn}\rsltpict{\*\objdata 
.EQN 203 42 469 0 0
{1:ƒ}NAME
}}{\cf2 , }{\cf2\i L, }{\cf2\i D}{\cf2 , and}{\cf2\i  h}{\cf2\i\dn f}{
\cf2 .\par \par                                                 }{
\object{\*\objclass \eqn}\rsltpict{\*\objdata 
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{0:W}NAME÷\(({0:h.f}NAME)/(8*{1:ƒ}NAME)*(({0:D}NAME)^(5))/({0:L}NAME)*({0:\p}NAME)^(2)*{0:g}NAME)
}}{\cf2  }{\cf2 \par \par }{\cf2 Note that for an installed design the 
diameter, length, and friction factor should remain constant (actually 
diameter, effective length and friction factor can all change due to 
degradation/plugging of the piping).  Thus flow can be expressed as:
\par \par                                                }{\object{
\*\objclass \eqn}\rsltpict{\*\objdata 
.EQN 224 25 586 0 0
{0:W}NAME({0:K}NAME,{0:h.f}NAME):{0:K}NAME*\({0:h.f}NAME)
}}{\cf2  \par \par where K is calculated from actual plant operating 
data and/or from the design data:\par \par                                    
        }{\object{\*\objclass \eqn}\rsltpict{\*\objdata 
.EQN 236 23 607 0 0
{0:K.dw}NAME({1:ƒ}NAME,{0:L}NAME,{0:D}NAME):\((({0:D}NAME)^(5)*({0:\p}NAME)^(2)*{0:g}NAME)/(8*{1:ƒ}NAME*{0:L}NAME))
}}{\cf2 \par \par                                                         
or }{\cf2 \par \par                                              }{
\object{\*\objclass \eqn}\rsltpict{\*\objdata 
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{0:K.pr}NAME({0:W}NAME,{0:h.f}NAME):({0:W}NAME)/(\({0:h.f}NAME))
}}{\cf2 \par \par                                                  }}