A new lotion is developed at a well-known cosmetic company, and the engineers at this company...

  

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A new lotion is developed at a well-known cosmetic company, and the engineers at this company are test- ing to ensure that this lotion can absorb and diffuse into the skin within a reasonable time. The principal task is to develop a scheme to measure the diffusiv- ity of the lotion as it courses through the skin. Here is the description of the problem. (1) The skin is very thin so you can ignore curva- ture effects and the thickness of the skin is taken to be L. (ii) The initial thickness of a very thin film of lotion above the skin is &n = 8(0). (iii) The lotion dissolves into the skin with a con- stant solubility, Co, and the concentration units can be taken to be moles of lotion per unit vol- ume of the skin. (iv) The diffusion of the lotion within the skin depends on the concentration and the flux equa- tion takes the form J- DdC/dx. Here you can assume that the diffusivity is a lincar function of concentration, D=D (1+ aC) (v) You may assume that the lotion concentration at the bottom of the skin layer is negligible, and the total layer of skin is initially void of lotion. (a) Show that the mass balance equation to describe the distribution of lotion in the skin is (you may assume a quasi-steady state for the transport of lotion in the skin): d Do(1 dx dC «C) dx A new lotion is developed at a well-known cosmetic company, and the engineers at this company are test- ing to ensure that this lotion can absorb and diffuse into the skin within a reasonable time. The principal task is to develop a scheme to measure the diffusiv- ity of the lotion as it courses through the skin. Here is the description of the problem. (1) The skin is very thin so you can ignore curva- ture effects and the thickness of the skin is taken to be L. (ii) The initial thickness of a very thin film of lotion above the skin is &n = 8(0). (iii) The lotion dissolves into the skin with a con- stant solubility, Co, and the concentration units can be taken to be moles of lotion per unit vol- ume of the skin. (iv) The diffusion of the lotion within the skin depends on the concentration and the flux equa- tion takes the form J- DdC/dx. Here you can assume that the diffusivity is a lincar function of concentration, D=D (1+ aC) (v) You may assume that the lotion concentration at the bottom of the skin layer is negligible, and the total layer of skin is initially void of lotion. (a) Show that the mass balance equation to describe the distribution of lotion in the skin is (you may assume a quasi-steady state for the transport of lotion in the skin): d Do(1 dx dC «C) dx