4 In modelling the digestion process in insects it is believ

4. In modelling the digestion process in insects, it is believed that digestion yield rate Y , in mass per time, is related to the concentration C of the limiting nutrient, the residence time T in the gut, the gut volume V , and the rate of nutrient breakdown r, given in mass per time per volume. Show that for fixed T, r, C, the yield is positively related to the gut volume.

4. In modelling the digestion process in insects, it is believed that digestion yield rate Y, in mass per time, is related to the concentration C of the limiting nutrient, the residence time T in the gut, the gut volume V, and the rate of nutrient breakdown r, given in mass per time per volume. Show that for fixed T, T, C, the yield is positively related to the gut volume.

Solution

Given the dimension of digestion yield rate, Y = mass per time = [M¹L⁰T^-1]

and Y = f (C, T, V, r)

The dimension of rate of nutrient breakdown, r = mass per time per volume = [M¹L^-3T^-1]

The dimension of residence time T in the gut = time = [M⁰L⁰T¹]

The dimension of gut volume V = volume = [M³L⁰T⁰]

The dimension of concentration C of the limiting nutrient  = [M⁰L⁰T⁰] , as concentration is dimensionless

Thus from the above dimensions,

Y = f (C, T, V, r)

or, [M¹L⁰T^-1] = f([M⁰L⁰T⁰], [M⁰L⁰T¹], [M³L⁰T⁰], [M¹L^-3T^-1])

or, [M¹L⁰T^-1] is directly proportional to [M⁰L⁰T¹] * [M³L⁰T⁰] * [M¹L^-3T^-1]

or, Y is directly proportional to T*V*r

if T & r are constant then

Y is directly proportional (positively related) V.