How Long Does It Take for a Falling Water Drop to Freeze at -20°C?

[pulkit675]

Hi all,

I have a heat transfer problem. In this problem, a water drop (assume sphere) is falling from a very high building. The outside temperature is below zero say -20 degree centigrade. How much time will it take to freeze?

I am not able to understand how to model this problem. Which equations to use because it analyze this problem from the reference frame of drop then the flow is unsteady accelerating with an acceleration of 'g'. I shall be very thankful if anybody can help me out solving this problem.

Pulkit

 

[jbsteele]

To model this problem, you should first consider the laws of thermodynamics. Specifically, the heat transfer equation and the energy equation. The equation for heat transfer is: q = -kA(T2-T1)/L where q is the rate of heat transfer, k is the thermal conductivity, A is the surface area, T2 and T1 are the temperatures at each end of the conduction path, and L is the length. The energy equation is: dQ/dt = mCpdT/dt where dQ/dt is the rate of change of energy, m is the mass of the water drop, Cp is the specific heat, and dT/dt is the rate of change of temperature. Using these equations and assuming a constant temperature of the environment, you can calculate the rate at which the drop loses energy due to conduction and the rate at which it gains energy due to its fall. You can then use these values to calculate the time it takes for the drop to freeze.

 

[thomate1]

Hello Pulkit,

This is an interesting problem to tackle. To solve this, we will need to consider a few factors such as the temperature difference between the water drop and the outside air, the shape and size of the drop, and the air resistance it experiences during its fall.

Firstly, we can use the heat transfer equation to determine the rate of heat transfer from the drop to the surrounding air. This equation takes into account the temperature difference, surface area, and thermal conductivity. We can also consider the latent heat of fusion, which is the amount of heat required to change the state of water from liquid to solid.

Next, we will need to consider the shape and size of the drop. A smaller drop will have a larger surface area to volume ratio, leading to a faster rate of heat transfer and a quicker freezing time. The shape of the drop can also affect its aerodynamics and the air resistance it experiences during its fall, which can impact its acceleration and therefore the time it takes to freeze.

Finally, we can use the equations of motion to analyze the drop's trajectory and determine the time it takes to reach the ground. Combining this with the heat transfer equation, we can estimate the time it takes for the drop to freeze.

I hope this helps you in solving your problem. Good luck!