Thermodynamics of a basaltic dike (dyke)

[hokulea]

Hello,
Im newly registered to this forum, though I've been here from time to time, and have always enjoyed the level of guidance and general willingness of the forum members to help others. Now that I have an actual question, I figured it was time to sign up :)

Here is my problem, technically it's not homework or coursework, its a research question I am trying to work out, but I have no formal training in thermodynamics and could use some advice.

Homework Statement

Basically, the question is this. I would like to determine the time it takes for a 'chilled margin' to form on the edge of a basaltic dike. The 'chilled margin' is a glassy portion on the outer part of the dike that comes into contact with the cooler country rock. As the country rock is no doubt much cooler than the intruding dike, a glassy layer forms. Now I am still working on determining the actual temperature values under which the glass can form (it is a range) and depends on the particular composition of the basalt I am studying. But to generalize let's just say that the glass forms between 1150 and 1100 C.

Now I have field measurements of these margins that range from 1cm to ~10cm, with the dike generally being about 1 m in total width. However, I am only considering the 1/2 width as there is another chilled margin on the other side of the dike.

So it seems to me that I can use the thermal diffusivity and conductivity, to describe the range of times that this margin took to form.

I have calculated the thermal diffusivity;

$$\alpha$$= k/($$\rho$$*K)
where k= thermal conductivity of basalt, 1.8 W/mK; $$\rho$$= the mean density of basaltic glass, 2.772 kg/m^3; and K=degrees Kelvin

In this case the 'country rock' the dike has intruded is actually the volcano itself, so the composition and density are the same, and for these purposes I think I should ignore other heat transfer modes such as radiation, and convection. So basically, steady-state.

Please let me know if there is any relevant information that I have left out, and I really appreciate any advice given.

 

[hokulea]

Ok, here is my stab at starting to solve this problem:

Problem (re-stated specifically for the following solution):
How long would it take to form a 'glassy or chilled margin' on a basaltic dike? The dike is initially molten at 1140 C and is solid at 880 C. Assume steady-state, 1 dimension.

Parameters:

T_l=temp. of liquidus=1140 C
T_s=temp. of solidus=880 Cp
ρ=density of basalt=2.772 g/cm^3
k=thermal conductivity=1.8 W/mC
Cp=specific heat= 1.05 J/gC

alpha=thermal diffusivity= k/(ρ*Cp)=6.1*10^-3 cm^2/s

Initial Condition:

T(x,0)= T_s for x₀ < x < ∞


PDE:

∂T/∂t=alpha(∂^2 T)/(∂x^2 ) for x₀ < x < ∞


Boundary Condition 1:

T(x₀,t)= T_l for t>0


Boundary Condition 2:

T(x,t) ---> T_s as x ---> ∞ and t>0

Solution for T(x,t) is...

T(x,t)= T_s + (T_l - T_s) erfc[(x-x₀)/(2√αt)]

where erfc(gamma)=1-erf(gamma)

(Im having trouble with typing equations here...)

Basically, I solved erf(gamma) for the temp. I was interested in 880 C, which came to gamma=3.2

T=880 + (1140-880)*(1-0.99999)=880

and then solved for x=10cm

x= gamma* 2(aplha*t)^1/2 => solving for t = x^2/(4*gamma^2*alpha) = ~400s

I hope this isn't too confusing, and I could really use some feedback. I think that 400s is pretty reasonable for bringing the temp. down from 1140 to 880, but I think I might need to set the problem up slightly differently as the rock surrounding the dike is probably much, much cooler, and also gaining heat as the dike is giving off heat. This would no doubt slow down the rate at which the margin would form.