Homework 6
- Your homework solution has to be handed in as a group solution via Moodle.
- Clearly state name and matriculation number of each student
1 Stationary convection-diffusion
Let’s assume we have given a heat equation of constant advection speed \(-v\):
\[ \begin{aligned} \underbrace{\frac{\partial}{\partial t} T}_{\left(\mathrm{I}\right)} - \underbrace{v \frac{\partial}{\partial x} T}_{\left(\mathrm{II}\right)} = \underbrace{\lambda \frac{\partial^2}{\partial x^2} T}_{\left(\mathrm{III}\right)} \end{aligned} \tag{1}\]
on a domain \(x=\left[0,\infty\right)\) with boundary conditions
\[ \begin{aligned} \left. T \right|_{x=0} = T_\mathrm{wall}, \quad \left. T \right|_{x \rightarrow \infty} = T_\mathrm{inf} = \text{const.}, \quad T_\mathrm{wall} > T_\mathrm{inf}. \end{aligned} \tag{2}\]
1.1 Tasks
Task 1
Explain the physical meaning of the terms (I), (II), and (III) of the heat equation.
Task 2
Give the SI base units of \(t\), \(x\), \(v\), and \(\lambda\).
Task 3
We introduce the dimensionless variables
\[ \begin{aligned} t:=t_0\tilde{t}, \, x:=x_0\tilde{x}, \, v:=v_0\tilde{v}, \text{and } T:=\underbrace{\left(T_\mathrm{wall} - T_\mathrm{inf}\right)}_{T_0} \tilde{T} + T_\mathrm{inf}. \end{aligned} \]
Write both equation Equation 1 and the boundary conditions Equation 2 in dimensionless variables. Identify the Peclet number \(\mathrm{Pe}=\frac{v_0 x_0}{\lambda}\).
Now consider a physical regime characterized by \(t_0 \gg \frac{x_0^2}{\lambda}\).
Task 4
How does the equation read in this physical regime?
Task 5
Solve for the temperature profile and sketch the solution.
Task 6
How does the profile change as the thermal diffusivity decreases? Indicate it in the sketch!
Task 7
If \(\lambda\) doubles, how does the velocity need to change to retain the same profile?
2 Stefan problem: Solidification of a composite liquid
The standard Stefan problem has been discussed in class. Once the liquid is no longer pure, the situation gets more complicated as the solid phase might have a different composition than the fluid phase. Think of the solidification of salt water into ice. Salt is typically rejected and needs to be transported away from the interface. Let’s go through it step by step:
The coupled system reads
\[ \begin{aligned} & \text{heat conduction: } & \partial_t T = \alpha \partial_x^2 T, \\ & \text{salt mass diffusion: } & \partial_t C = D \partial_x^2 C, \end{aligned} \tag{3}\]
where \(\alpha\) is the thermal diffusivity and \(D\) is the mass diffusivity.
Cooling from the left will induce a phase interface to propagate from left to right through the domain. It’s position is described by \(X(t)\) (see lecture). The boundary and interface conditions are given by:
Temperature \(T\):
- \(T(t,0) = T_\mathrm{B}\)
- \(\lim_{x \rightarrow \infty} T(t,x) = T_\infty\)
- \(T(t, X(t)) = T_\mathrm{i}\)
- \(\rho L \dot{X}(t) = k \partial_x \left. T \right|_{x=X(t)^-} - k \partial_x \left. T \right|_{x=X(t)^+}\)
Concentration \(C\):
- \(\lim_{x \rightarrow \infty} C(t,x) = C_\infty\)
- \(C(t, X(t)) = C_\mathrm{i}\)
- \(T_i=T_m-mC_i\), with a constant parameter \(m\)
- \(\left(C_i-C_\mathrm{S}\right) \dot{X}(t) = - D \partial_x \left. C \right|_{x=X(t)^+}\), where \(C_\mathrm{S}\) is the concentration in the solid (here: \(C_\mathrm{S} = 0\)).
2.1 Tasks
Task 1
Interpret the physical meaning of each of these conditions.
Task 2
Introduce the similarity variable \(\eta := \frac{x}{2\sqrt{Dt}}\) and write both temperature and mass diffusion (Equation 3), in terms of the similarity variable.
Task 3
The resulting ODE system can be solved in terms of the (unknown) interface position \(X_\mathrm{m}=X(t)\) and written in \(x\) and \(t\) again. Derive the solution (get inspired by the lecture).
Task 4
Discuss which of the relations/boundary conditions 1. – 8. are fulfilled versus what is still unknown.
Task 5
Make the Ansatz: \(X_\mathrm{m}(t) = 2\lambda\sqrt{Dt}\) (similar to lecture, but with \(D\) instead of \(\alpha\)). Substitute the derived solutions for \(T\) and \(C\) into the Stefan condition and salt rejection and derive an algebraic system that allows us to determine \(C_\mathrm{i}\), \(T_\mathrm{i}\) and \(\lambda\).
Task 6
Combine the derived equations into one equation for \(\lambda\). Determine \(\lambda\) for freezing sea water with \(T_\mathrm{B} = -10^\circ C\), \(T_\infty=2^\circ C\), \(T_\mathrm{m}=0^\circ C\), \(C_\infty=3.5\%\), \(D=1.33 \times 10^{-9}m^2s^{-1}\) (Na-Ions in Water), \(\alpha_\mathrm{ice} = 1.203 \times 10^{9}m^2s^{-1}\), \(\alpha_\mathrm{water} = 0.14 \times 10^{9}m^2s^{-1}\), \(L=334 \times 10^{3} J\), \(c_p=2.05 \times 10^{3} J kg^{-1} K^{-1}\) and \(m\approx 1 K \%^{-1}\) (i.e., for each percent of salt the melting temperature is reduced by 1K; valid until about 20 % of salts). Plot the solution and discuss what you see.