Growth of gas bubble in superheated liquid: Explain "explosive growth in which..."

[yli]

Growth of gas bubble in superheated liquid: Explain "explosive growth in which..."

I'm having trouble understanding this question and how I would go about explaining the meaning of what I am assuming is a rate of change.

---

2. In his 2008 textbook Fundamentals of Multiphase Flows, the author Christopher Brennan describes the expansion of a gas bubble in a superheated liquid. Under certain conditions -- for example, in the absence of thermal effects -- it may be determined that:

. . . . .\(\displaystyle \dfrac{dR}{dt}\, \sim\, \sqrt{\dfrac{\strut 2\, (p_V\, -\, p_{\infty}^{*})}{3\rho_L}\,}\)

where R is the radius of the bubble, p_V is a constant describing vapor pressure within the bubble, \(\displaystyle p_{\infty}^{*}\) is a constant describing pressure from the superheated liquid, and \(\displaystyle \rho_L\) is a constant describing the density of the superheated liquid. Brennan writes that, under these conditions, the bubble experiences "explosive growth ... in which the volume displacement is increasing like \(\displaystyle t^3\)". Explain in three or four sentences what this means.

---

I have been left confused by this problem and would appreciate someone explaining it to me. Thanks in advance for any help.

 

[mmm4444bot]

I'm having trouble understanding this question and how I would go about explaining the meaning of what I am assuming is a rate of change.

---

2. In his 2008 textbook Fundamentals of Multiphase Flows, the author Christopher Brennan describes the expansion of a gas bubble in a superheated liquid. Under certain conditions -- for example, in the absence of thermal effects -- it may be determined that:

. . . . .\(\displaystyle \dfrac{dR}{dt}\, \sim\, \sqrt{\dfrac{\strut 2\, (p_V\, -\, p_{\infty}^{*}}{3\rho_L}\,}\)

where R is the radius of the bubble, p_V is a constant describing vapor pressure within the bubble, \(\displaystyle p_{\infty}^{*}\) is a constant describing pressure from the superheated liquid, and \(\displaystyle \rho_L\) is a constant describing the density of the superheated liquid. Brennan writes that, under these conditions, the bubble experiences "explosive growth ... in which the volume displacement is increasing like \(\displaystyle t^3\)". Explain in three or four sentences what this means.

The rate of change for the sphere's radius is given (it's dR/dt), but you're not asked to explain the meaning of that rate.

You're asked to explain why a different rate -- the rate at which the sphere's volume changes -- is similar to t^3. Therefore, you need to think about expressing the volume as a function of time.

Did you notice that the expression for dR/dt does not contain a variable? In other words, dR/dt is a constant.

Perhaps, it will help you to simplify the notation. Instead of the given expression for dR/dt, how about we use symbol A?

A = sqrt(2·[p_v - p_∞*]/[3·p_L])

Now we can think of the sphere's radius (R) as a function of time (t), using symbol A for the rate at which it's changing.

R(t) = A·t

Next, what is V(t)? That is, what function can you write, to see how the sphere's volume (V) changes over time?

Hopefully, this line of thought will give you some ideas about the explanation they want.

If I wrote anything that you don't understand, please ask. :cool:

 

[yli]

Thanks for giving me a place to start. Would V(t) =V*t ? I'm still a bit confused about the volume relates to time.

 

[stapel]

Thanks for helping me get started. Would V(t) = V*t? Where is volume represented in this problem? I'm still a bit confused about how to relate time back to volume.

How does "radius" relate to "volume" for a sphere? How does the change in the radius relate to the change in the volume for a sphere?

 

[mmm4444bot]

I'm still a bit confused about [how] the volume relates to time.

The volume of a sphere depends upon the sphere's radius.

If this radius changes over time, the volume also changes.

At any given instant, radius = A·t

What happens if you use the expression A·t for the radius (in the formula for a sphere's volume), instead of using symbol r?