Chapter 9: Problem 1

Draw the temperature profile that is assumed in the derivation of the d2 -law for evaporation inside and outside of a droplet as a function of the radius.

Short Answer

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Question: Describe the temperature profile inside and outside of a droplet as a function of radius, according to the d2-law for evaporation. Answer: According to the d2-law for evaporation, the temperature profile inside the droplet is assumed to be constant (T_i) and independent of the radial distance (r) from the center. In contrast, the temperature profile outside the droplet is assumed to be a linear function of distance from the droplet surface, given by T(r) = T_i + K(r-R) for r > R, where K is a constant determining the temperature gradient and R is the droplet radius.

Step by step solution

Understanding the d2-law and its assumptions

The d2-law is a simplified model for the evaporation rate of a droplet. It assumes that the temperature profile inside the droplet remains uniform, and the temperature outside the droplet is a linear function of the distance from the droplet surface. Additionally, the d2-law considers that the mass transfer is governed by diffusion.

Temperature inside the droplet

As per the d2-law, the temperature profile inside the droplet is assumed to be constant. This means that the temperature is the same at any point within the droplet, regardless of distance from the center. We can represent this as: T(r) = T_i, for 0 ≤ r ≤ R, where T(r) is the temperature at a radius 'r', T_i is the constant temperature inside the droplet, and R is the droplet radius.

Temperature outside the droplet

The d2-law assumes the temperature profile outside the droplet to be a linear function of the distance from the droplet surface. We can represent this linear relationship as: T(r) = T_i + K(r-R), for r > R, where K is a constant determining the temperature gradient, and r is the distance from the center of the droplet.

Plotting the temperature profile

To plot the temperature profile, follow these steps: 1. On the horizontal axis, represent the radial distance (r) from the center of the droplet, ranging from 0 to the desired maximum distance. 2. On the vertical axis, represent the temperature (T). 3. Draw a horizontal line at T_i for the region 0 ≤ r ≤ R, representing the constant temperature inside the droplet. 4. Draw a straight line with a slope K, starting from the point (R, T_i) and extending further for r > R, representing the linear temperature gradient outside the droplet. The resulting plot represents the assumed temperature profile in the d2-law evaporation model, inside and outside of the droplet as a function of the radius r.

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Draw the temperature profile that is assumed in the derivation of the d2 -law for evaporation inside and outside of a droplet as a function of the radius.

Short Answer

Step by step solution

Understanding the d2-law and its assumptions

Temperature inside the droplet

Temperature outside the droplet

Plotting the temperature profile

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