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.


> restart;

> k:=1/3*rho*C[v]*U*lambda=rho*C[v]*Uy*a;

[Maple Math]

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.

> k:=2.80*(N[n]/V[n])^(2/3)*kappa*v[S];

[Maple Math]

This is the modified version of Bridgman's equation as stated in Equation 9.4-3.

> v[S]:=sqrt(C[p]/C[v]*diff(p(rho),rho)[T]);

[Maple Math]

This is the equation for the velocity of low-frequency sound as stated in Equation 9.4-4.

> comp:=rho->inv(rho)*diff(rho(p),p[T]);

[Maple Math]

This helps set the isothermal compressibility as is given in the problem.

> eqn:=diff(p(rho),rho)[T]=1/(rho*comp);

[Maple Math]

This is the setup for the isothermal conductivity.

> rho:=1.595*g/cm^3;comp:=90.7e-6/atm;assume(cm>0);assume(s>0);atm:=1.0133e6*g/cm/s^2;

[Maple Math]

[Maple Math]

[Maple Math]

This are the given values stated in the problem.

> eqn;

[Maple Math]

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.

> C[p]:=1;C[v]:=1;

[Maple Math]

[Maple Math]

These help set Cp/Cv to unity, since carbon tetrachloride is not near its critical point in this problem.

> v[S];

[Maple Math]

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.

> M:=153.84*g/mol;V[n]:=M/rho;N[n]:=6.023e23/mol;kappa:=1.3805e-16*erg/K;

[Maple Math]

[Maple Math]

[Maple Math]

[Maple Math]

> k;

[Maple Math]

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.

> erg:=0.001*W*s;cm:=0.01*m;

[Maple Math]

[Maple Math]

> simplify(k);

[Maple Math]

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).