Example 9.4-1
Prediction of the Thermal Conductivity of a Liquid
The density of
liquid carbon tetrachloride at 20 C and 1 atm is 1.595 g/cm^3, and its
isothermal compressibility (1/rho)(drho/dp) is 90.7*10^-6 atm^-1. What
is its thermal conductivity?
This problem can be readily solved through the use of MAPLE.
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restart;
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k:=1/3*rho*C[v]*U*lambda=rho*C[v]*Uy*a;
This is the equation stated in the terms of Equation 9.4-1. This is based on a reinterpretation of the thermal conductivity for a monatomic gas via the rigid-sphere gas theory.
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k:=2.80*(N[n]/V[n])^(2/3)*kappa*v[S];
This is the modified version of Bridgman's equation as stated in Equation 9.4-3.
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v[S]:=sqrt(C[p]/C[v]*diff(p(rho),rho)[T]);
This is the equation for the velocity of low-frequency sound as stated in Equation 9.4-4.
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comp:=rho->inv(rho)*diff(rho(p),p[T]);
This helps set the isothermal compressibility as is given in the problem.
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eqn:=diff(p(rho),rho)[T]=1/(rho*comp);
This is the setup for the isothermal conductivity.
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rho:=1.595*g/cm^3;comp:=90.7e-6/atm;assume(cm>0);assume(s>0);atm:=1.0133e6*g/cm/s^2;
This are the given values stated in the problem.
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eqn;
This is the value for the (dp/drho) term in terms of cm^2/s^2. BS&L provide 7.00 * 10^9 cm^2/s^2 as the answer.
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C[p]:=1;C[v]:=1;
These help set Cp/Cv to unity, since carbon tetrachloride is not near its critical point in this problem.
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v[S];
This gives the velocity of low-frequency sound for carbon tetrachloride under the given conditions. BS&L provide 8.37 * 10^4 cm/s as the answer.
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M:=153.84*g/mol;V[n]:=M/rho;N[n]:=6.023e23/mol;kappa:=1.3805e-16*erg/K;
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k;
This gives the thermal conductivity in terms of erg/cm/K/s. BS&L provide 1.10 * 10^4 erg/cm/K/s as the answer.
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erg:=0.001*W*s;cm:=0.01*m;
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simplify(k);
This gives the thermal conductivity in terms of W/cm/K/. It would equal 0.1097022822 W/m/K. BS&L provide 0.110 W/m/K as the answer.
This concludes the work done for Example 9.4-1.
This was taken from P-280 of the second edition of TRANSPORT PHENOMENA
by Bird, Stewart, and Lightfoot (2002).