Sustainability: Humanity Perspective - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 768

For a uniform stand of trees on a lot with uniform soil moisture and nutrients and constant
atmospheric CO 2 , the specific growth rate can be represented by

n

X
X N

m G ¼ m max 1 (15.8)

where specific biomass growth rate m G is that normalized by the standing biomass density or

the amount of standing biomass over the area of the stand X; m max is the maximum specific

biomass growth rate; and X is the maximum amount of standing biomass achievable or the

N

carrying capacity. Both m max and X are functions of growth conditions and their relations to

N

the biomass species. When n 1, the growth rate is reduced to the logistic rate. The growth is

proportional to the amount of live standing biomass when the tree is “young”. As the tree
ages or biomass increases, the specific growth rate decreases. When the saturation biomass
level or carrying capacity is reached, the specific growth rate is zero. Therefore, the biomass
growth is dependent on both the amount of biomass and the population distribution.
Mass balance over a stand leads to
d d t ¼ r X ¼ m G X

¼

(15.9)

which describes the relationship of the standing biomass as a function of the growth time.
The Maximum amount of sustainable biomass available would be the maximum woody
biomass production on site. The biomass production rate for a cycle time t is given by

X X 0
t

P X ¼ (15.10)

where X 0 is the standing biomass of seedlings (or at time t ¼ 0). At maximum sustainable rate

t max ðX max X 0 Þ
t 2 max

d d t

t max ¼ t max

d P X
d t

0 ¼ (15.11)

which leads to

X max X 0
t max

P X max ¼ r X j t¼t max ¼ (15.12)

Here t max is the recycle time for maximum biomass production P Xmax , and X max is the
standing biomass at time t max .
Therefore, how the harvest is conducted is directly affecting the maximum amount of
biomass available at a sustainably renewable level. The optimum and sustainable production
rate of a biomass stand is the one that with final incremental growth rate at harvest equal to
the average overall growth yield.

Example 15-5. Pinch line and optimum sustainable biomass production.
For a biomass growth curve shown in Fig. 15.12 , what is the optimum recycle time,
biomass yield, and production rate?
Solution. Since the growth curve is given as biomass changing with time, graphical
method can be applied to solve the problem. Eqn (15.12) is the principle behind a simple
graphical solution scheme.