Creatinine is a waste product of creatine, a compound used to supply energy to muscles in the human body. Creatinine is cleared from the body entirely through the kidneys and the urinary tract. We can model the production of creatinine in the muscle as an input to a two-compartment model, one piece containing muscle mass and the other containing blood plasma. A diagram of the system is provided at right. Each compartment (m for muscle, p for plasma) has a fixed and constant volume, and at any point contains a number of moles of creatinine n. After a workout, the muscles produce a creatinine at a constant rate ǹ, which has units [mol/time]. Vp ĥin k₁ Vm nm Пр k₂ K3 The transfer of creatinine between compartments is governed by a first-order rate law based on transfer coefficients k. The mass balances on each compartment, treated as well-mixed control volumes, are: dnp(t) = dt dnm(t) dt = - kmnm (t) (k₁ + k3)np(t) - k₁np(t) k₂nm(t) + nin(t) 2.1. Assuming the initial conditions for all states and inputs are zero and that the concentration of creatinine in a given control volume i can be calculated as c₁(t) = i(t), derive the following first-order transfer functions G₁(S), G₂(s) and G3(s): vi G₁(s) = Cp(s) Cm(s) Cm(s) Cm(s) G₂(s) = G3(S) = Cp(s) Nin(s) 1 Note: G₁(s) is not the same as 2.2. Draw a block diagram of this system, labeling the blocks as G₁(s), G₂(s) and G¸(s). √293 dnm (t) = K₁np(t) - k₂ ^m(t) + "(+) G3(5)= Cm(s) dnplt) = K(+)-(k₁+k₂) np (t) 'm ↓223 KmNm(s) - (K1+k3) Np (s) dt SNpcs) = ] = Ν km Nm (s) Npcs)= km Nm (s) Npls) [s+k+k] G₁ (5) = Cp(s) Cm(s) = Np (5) Vm Km km Nm(s) Vp 5+ky+43 Vp 62(5) = Çm(s) Nm (5) Vp = s+k₂+3 Vp km = Nin (5) Npcs) Vm Nmcs). I Vm Nin (s) = 1 vm Vm 5+k₂-kkm S+kytk3 5Nm(s) = K₁ Np(s)- K₂ Nm (s) + Nin(s) Cm937 Nin (s). ← G3 (5) G₁(5) - Nm (5) [ 3+k₂ = K₂ km ] = Nm (5) Cpcs)
Creatinine is a waste product of creatine, a compound used to supply energy to muscles in the human body. Creatinine is cleared from the body entirely through the kidneys and the urinary tract. We can model the production of creatinine in the muscle as an input to a two-compartment model, one piece containing muscle mass and the other containing blood plasma. A diagram of the system is provided at right. Each compartment (m for muscle, p for plasma) has a fixed and constant volume, and at any point contains a number of moles of creatinine n. After a workout, the muscles produce a creatinine at a constant rate ǹ, which has units [mol/time]. Vp ĥin k₁ Vm nm Пр k₂ K3 The transfer of creatinine between compartments is governed by a first-order rate law based on transfer coefficients k. The mass balances on each compartment, treated as well-mixed control volumes, are: dnp(t) = dt dnm(t) dt = - kmnm (t) (k₁ + k3)np(t) - k₁np(t) k₂nm(t) + nin(t) 2.1. Assuming the initial conditions for all states and inputs are zero and that the concentration of creatinine in a given control volume i can be calculated as c₁(t) = i(t), derive the following first-order transfer functions G₁(S), G₂(s) and G3(s): vi G₁(s) = Cp(s) Cm(s) Cm(s) Cm(s) G₂(s) = G3(S) = Cp(s) Nin(s) 1 Note: G₁(s) is not the same as 2.2. Draw a block diagram of this system, labeling the blocks as G₁(s), G₂(s) and G¸(s). √293 dnm (t) = K₁np(t) - k₂ ^m(t) + "(+) G3(5)= Cm(s) dnplt) = K(+)-(k₁+k₂) np (t) 'm ↓223 KmNm(s) - (K1+k3) Np (s) dt SNpcs) = ] = Ν km Nm (s) Npcs)= km Nm (s) Npls) [s+k+k] G₁ (5) = Cp(s) Cm(s) = Np (5) Vm Km km Nm(s) Vp 5+ky+43 Vp 62(5) = Çm(s) Nm (5) Vp = s+k₂+3 Vp km = Nin (5) Npcs) Vm Nmcs). I Vm Nin (s) = 1 vm Vm 5+k₂-kkm S+kytk3 5Nm(s) = K₁ Np(s)- K₂ Nm (s) + Nin(s) Cm937 Nin (s). ← G3 (5) G₁(5) - Nm (5) [ 3+k₂ = K₂ km ] = Nm (5) Cpcs)
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Please confirm that my solution is correct, especially the block diagram. Please DRAW (not type) what the block diagram would look like if it's incorrect.
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![Creatinine is a waste product of creatine, a compound used to
supply energy to muscles in the human body. Creatinine is cleared
from the body entirely through the kidneys and the urinary tract.
We can model the production of creatinine in the muscle as an
input to a two-compartment model, one piece containing muscle
mass and the other containing blood plasma. A diagram of the
system is provided at right.
Each compartment (m for muscle, p for plasma) has a fixed and
constant volume, and at any point contains a number of moles of
creatinine n. After a workout, the muscles produce a creatinine at a
constant rate ǹ, which has units [mol/time].
Vp
ĥin
k₁
Vm
nm
Пр
k₂
K3
The transfer of creatinine between compartments is governed by a first-order rate law based on transfer coefficients
k. The mass balances on each compartment, treated as well-mixed control volumes, are:
dnp(t)
=
dt
dnm(t)
dt
=
-
kmnm (t) (k₁ + k3)np(t)
-
k₁np(t) k₂nm(t) + nin(t)
2.1. Assuming the initial conditions for all states and inputs are zero and that the concentration of creatinine in
a given control volume i can be calculated as c₁(t) = i(t), derive the following first-order transfer functions
G₁(S), G₂(s) and G3(s):
vi
G₁(s) =
Cp(s)
Cm(s)
Cm(s)
Cm(s)
G₂(s) =
G3(S) =
Cp(s)
Nin(s)
1
Note: G₁(s) is not the same as
2.2. Draw a block diagram of this system, labeling the blocks as G₁(s), G₂(s) and G¸(s).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fca6b5de9-d666-4be4-bec5-372f49facd74%2Ff48d2979-75a8-44e0-9e16-2ea73a4e90ca%2Fg8bersk_processed.png&w=3840&q=75)
Transcribed Image Text:Creatinine is a waste product of creatine, a compound used to
supply energy to muscles in the human body. Creatinine is cleared
from the body entirely through the kidneys and the urinary tract.
We can model the production of creatinine in the muscle as an
input to a two-compartment model, one piece containing muscle
mass and the other containing blood plasma. A diagram of the
system is provided at right.
Each compartment (m for muscle, p for plasma) has a fixed and
constant volume, and at any point contains a number of moles of
creatinine n. After a workout, the muscles produce a creatinine at a
constant rate ǹ, which has units [mol/time].
Vp
ĥin
k₁
Vm
nm
Пр
k₂
K3
The transfer of creatinine between compartments is governed by a first-order rate law based on transfer coefficients
k. The mass balances on each compartment, treated as well-mixed control volumes, are:
dnp(t)
=
dt
dnm(t)
dt
=
-
kmnm (t) (k₁ + k3)np(t)
-
k₁np(t) k₂nm(t) + nin(t)
2.1. Assuming the initial conditions for all states and inputs are zero and that the concentration of creatinine in
a given control volume i can be calculated as c₁(t) = i(t), derive the following first-order transfer functions
G₁(S), G₂(s) and G3(s):
vi
G₁(s) =
Cp(s)
Cm(s)
Cm(s)
Cm(s)
G₂(s) =
G3(S) =
Cp(s)
Nin(s)
1
Note: G₁(s) is not the same as
2.2. Draw a block diagram of this system, labeling the blocks as G₁(s), G₂(s) and G¸(s).
![√293
dnm (t) = K₁np(t) - k₂ ^m(t) + "(+)
G3(5)= Cm(s)
dnplt) = K(+)-(k₁+k₂) np (t)
'm
↓223
KmNm(s) - (K1+k3) Np (s)
dt
SNpcs)
=
] =
Ν
km Nm (s)
Npcs)=
km
Nm (s)
Npls) [s+k+k]
G₁ (5) = Cp(s)
Cm(s)
=
Np (5) Vm
Km
km
Nm(s) Vp
5+ky+43
Vp
62(5)
=
Çm(s)
Nm (5) Vp
=
s+k₂+3 Vp
km
=
Nin (5)
Npcs) Vm
Nmcs). I
Vm
Nin (s)
=
1
vm
Vm
5+k₂-kkm
S+kytk3
5Nm(s)
=
K₁ Np(s)- K₂ Nm (s) + Nin(s)
Cm937
Nin (s).
←
G3 (5)
G₁(5)
-
Nm (5) [ 3+k₂ = K₂ km ] = Nm (5)
Cpcs)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fca6b5de9-d666-4be4-bec5-372f49facd74%2Ff48d2979-75a8-44e0-9e16-2ea73a4e90ca%2Fh26ro2o_processed.png&w=3840&q=75)
Transcribed Image Text:√293
dnm (t) = K₁np(t) - k₂ ^m(t) + "(+)
G3(5)= Cm(s)
dnplt) = K(+)-(k₁+k₂) np (t)
'm
↓223
KmNm(s) - (K1+k3) Np (s)
dt
SNpcs)
=
] =
Ν
km Nm (s)
Npcs)=
km
Nm (s)
Npls) [s+k+k]
G₁ (5) = Cp(s)
Cm(s)
=
Np (5) Vm
Km
km
Nm(s) Vp
5+ky+43
Vp
62(5)
=
Çm(s)
Nm (5) Vp
=
s+k₂+3 Vp
km
=
Nin (5)
Npcs) Vm
Nmcs). I
Vm
Nin (s)
=
1
vm
Vm
5+k₂-kkm
S+kytk3
5Nm(s)
=
K₁ Np(s)- K₂ Nm (s) + Nin(s)
Cm937
Nin (s).
←
G3 (5)
G₁(5)
-
Nm (5) [ 3+k₂ = K₂ km ] = Nm (5)
Cpcs)
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