[RESOLVED] modelling problem

hi

im stuck on an additional homework problem, any ideas on how to solve appreciated

A sphere of ice i melting so its volume decreases whilst it maintains the same shape (but not size).

(a) Use volume balancing to derive differential equations describing the rate of change of the radius of the sphere , r, if it is melting:

i) with a constant rate of volume loss;
ii) at a rate of volume loss proportional to the surface area of the sphere;
iii) at a rate of volume loss proportional to the volume itself.

-constants of proportionality are k1, k2, and k3

-solve each of the equations to find an expression for r as a function of time t, assuming that the initial radius of the sphere is ro.

-ro = 2x10^-3, k1= 10^-6 m^3 s^-1
k2 and k3 should be chosen to make the initial rate of melting the same in all 3 cases.

thanks,

Donna

Quote:

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Originally Posted by hani1987

hi

im stuck on an additional homework problem, any ideas on how to solve appreciated

A sphere of ice i melting so its volume decreases whilst it maintains the same shape (but not size).

(a) Use volume balancing to derive differential equations describing the rate of change of the radius of the sphere , r, if it is melting:

i) with a constant rate of volume loss;
ii) at a rate of volume loss proportional to the surface area of the sphere;
iii) at a rate of volume loss proportional to the volume itself.

-constants of proportionality are k1, k2, and k3

-solve each of the equations to find an expression for r as a function of time t, assuming that the initial radius of the sphere is ro.

-ro = 2x10^-3, k1= 10^-6 m^3 s^-1
k2 and k3 should be chosen to make the initial rate of melting the same in all 3 cases.

thanks,

Donna

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Hi, welcome to the forums.

The volume is V = 4*Pi*r³/3 so the rate of change is:

dV/dt = d(4*Pi*r³/3)/dt = 4*Pi*r²dr/dt

This is our basic equation and I'll call it (E0).

Part (a) Constant rate of volume loss. This means dV/dt = constant, i.e. dV/dt = -k₁
where I have assumed k₁ is positive, so I use a negative sign as the volume decreases, i.e. the rate is negative.

From (E0):

4*Pi*r²dr/dt = -k₁ or 4*Pi*r²dr = -k₁dt

Integrating this you get,

4*Pi*r³/3 = -k₁ + C

where C is the integration constant to be determined. If the initial radius of the sphere (when t=0) is r₀ then substituting you get C = 4*Pi*r₀³/3

Putting C into the equation and rearranging you finally obtain:

r = [r₀³ - 3k₁t/(4*Pi)]^1/3

Parts (b) and (c) are similar, all you have to do is write the adequate expressions for dV/dt, i.e.

(b) dV/dt = -k₂*4*Pi*r²
and
(c) dV/dt = -k₃*V = -4*k₃*Pi*r³/3

and substitute in (E0).

thanks!

Quote:

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Originally Posted by hani1987

thanks!

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You're welcome. Post back if you need any more help.

Hey thanks for helping out, there is only one more thing i need to know before its all solved:

for part c, my final equation is ln r = -1/3k3t + C

so to find c you put t= 0, which gives ln r0 = c?

and taking exponentials, gives r = e^(-1/3k3t) + r0?

should i work out the constant before or after integrating?

thanks!

Your final equation should be

dr/r = -k₃dt/3 and after integrating,

ln(r) = -k₃t/3 + C

C = ln(r₀) and all you have to do is substitute and operate:

ln(r) = -k₃t/3 + ln(r₀)
ln(r) - ln(r₀) = -k₃t/3
ln(r/r₀) = -k₃t/3
r/r₀ = e^-k₃t/3
r = r₀e^-k₃t/3