A1. Derivatives of the Liquid Continuity Equations with Respect to the Parameters
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(1) ≡
Tdb are as follows:
where:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(2) ≡
Tdp are as follows:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(3) ≡
Tw,in are as follows:
where:
and where
a2,ρ = –0.26847207;
a3,ρ = –1.8113691 × 10
–3;
a4,ρ = –1.7041217 × 10
–6.
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(4) ≡
Patm are as follows:
where:
Note: The term on the right hand side of Equation (A25) stems from the following relation:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(5) ≡
wtsa are as follows:
where:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(6) ≡
ksum are as follows:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(7) ≡
μ are as follows:
where:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(8) ≡
υ are as follows:
where:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(9) ≡
kair are as follows:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(10) ≡
fht are as follows:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(11) ≡
fmt are as follows:
where:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(12) ≡
f are as follows:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(13) ≡
a0 are as follows:
where:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(14) ≡
a1 are as follows:
where:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(15) ≡
a0,cpa are as follows:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(16) ≡
a1,cpa are as follows:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(17) ≡
a2,cpa are as follows:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(18) ≡
a0,dav are as follows:
where
was defined previously in Equation (A16), and:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(19) ≡
a1,dav are as follows:
where
was defined previously in Equation (A16), and:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(20) ≡
a2,dav are as follows:
where
was defined previously in Equation (A16), and
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(21) ≡
a3,dav are as follows:
where
was defined previously in Equation (A16), and
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(22) ≡
a0f are as follows:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(23) ≡
a1f are as follows:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(24) ≡
a0g are as follows:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(25) ≡
a1g are as follows:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(26) ≡
a0,Nu are as follows:
where
was defined previously in Equation (A23), and
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(27) ≡
a1,Nu are as follows:
where
was defined previously in Equation (A23), and:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(28) ≡
a2,Nu are as follows:
where
was defined previously in Equation (A23), and:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(29) ≡
a3,Nu are as follows:
where
was defined previously in Equation (A23), and:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(30) ≡
Wdkx are as follows:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(31) ≡
Wdky are as follows:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(32) ≡ Δ
zdk are as follows:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(33) ≡ Δ
zfan are as follows:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(34) ≡
Dfan are as follows:
where
and
were defined previously in Equations (A23) and (A24), respectively, and:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(35) ≡ Δ
zfill are as follows:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(36) ≡ Δ
zrain are as follows:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(37) ≡ Δ
zbs are as follows:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(38) ≡ Δ
zde are as follows:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(39) ≡
Dh are as follows:
where:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(40) ≡
Afill are as follows:
where:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(41) ≡
Asurf are as follows:
where:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(42) ≡ Pr are as follows:
where:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(43) ≡
Vw are as follows:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(44) ≡
Vexit are as follows:
where
and
were defined previously in Equations (A23) and (A24), respectively, and
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(45) ≡
mw,in are as follows:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(46) ≡
Ta,in are as follows:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(47) ≡
ma are as follows:
where
and
were defined previously in Equations (A23) and (A24), respectively.
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(48) ≡
ωin are as follows:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(49) ≡ Re
d are as follows:
where
was defined in Equation (A23), and:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(50) ≡
Sc are as follows:
where:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(51) ≡
Sh are as follows:
where:
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter
α(52) ≡
Nu are as follows:
where
was defined in Equation (A23).
A2. Derivatives of the Liquid Energy Balance Equations with Respect to the Parameters
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(1) ≡
Tdb are as follows:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(2) ≡
Tdp are as follows:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(3) ≡
Tw,in are as follows:
where
was defined in Equation (A20), and:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(4) ≡
Patm are as follows:
where
and
were defined in Equations (A24) and (A25), respectively, and:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(5) ≡
wtas are as follows:
where:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(6) ≡
ksum are as follows:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(7) ≡
μ are as follows:
where:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(8) ≡
υ are as follows:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(9) ≡
kair are as follows:
where:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(10) ≡
fht are as follows:
where:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(11) ≡
fmt are as follows:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(12) ≡
f are as follows:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(13) ≡
a0 are as follows:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(14) ≡
a1 are as follows:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(15) ≡
a0,cpa are as follows:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(16) ≡
a1,cpa are as follows:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(17) ≡
a2,cpa are as follows:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(18) ≡
a0,dav are as follows:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(19) ≡
a1,dav are as follows:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(20) ≡
a2,dav are as follows:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(21) ≡
a3,dav are as follows:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(22) ≡
a0f are as follows:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(23) ≡
a1f are as follows:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(24) ≡
a0g are as follows:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(25) ≡
a1g are as follows:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(26) ≡
a0,Nu are as follows:
where
was defined in Equation (A105) and
was defined in Equation (A59).
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(27) ≡
a1,Nu are as follows:
where
was defined in Equation (A105) and
was defined in Equation (A61).
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(28) ≡
a2,Nu are as follows:
where
was defined in Equation (A105) and
was defined in Equation (A63).
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(29) ≡
a3,Nu are as follows:
where
was defined in Equation (A105) and
was defined in Equation (A65).
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(30) ≡
Wdkx are as follows:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(31) ≡
Wdky are as follows:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(32) ≡ Δ
zdk are as follows:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(33) ≡ Δ
zfan are as follows:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(34) ≡
Dfan are as follows:
where
was defined in Equation (A105), while
and
were defined in Equations (A24) and (A71), respectively.
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(35) ≡ Δ
zfill are as follows:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(36) ≡ Δ
zrain are as follows:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(37) ≡ Δ
zbs are as follows:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(38) ≡ Δ
zde are as follows:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(39) ≡
Dh are as follows:
where
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(40) ≡
Afill are as follows:
where:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(41) ≡
Asurf are as follows:
where:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(42) ≡ Pr are as follows:
where:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(33) ≡
Vw are as follows:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(44) ≡
Vexit are as follows:
where
was defined in Equation (A105), while
and
and were defined previously in Equations (A24) and (A86), respectively.
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(45) ≡
mw,in are as follows:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(46) ≡
Ta,in are as follows:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(47) ≡
ma are as follows:
where
and
were defined previously in Equations (A24) and (A105), respectively.
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(48) ≡
ωin are as follows:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(49) ≡ Re
d are as follows:
where
was defined in Equation (A105), and
was defined in Equation (A93).
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(50) ≡
Sc are as follows:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(51) ≡
Sh are as follows:
The derivatives of the liquid energy balance equations [cf. Equations (A4)–(A6)] with respect to the parameter
α(52) ≡
Nu are as follows:
where
was defined in Equation (A105).
A3. Derivatives of the Water Vapor Continuity Equations with Respect to the Parameters
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(1) ≡
Tdb are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(2) ≡
Tdp are as follows:
where:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(3) ≡
Tw,in are as follows:
where
was defined in Equation (A20).
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(4) ≡
Patm are as follows:
where
was defined in Equation (A25) and:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(5) ≡
wtsa are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(6) ≡
ksum are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(7) ≡
μ are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(8) ≡
υ are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(9) ≡
kair are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(10) ≡
fht are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(11) ≡
fmt are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(12) ≡
f are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(13) ≡
a0 are as follows:
where:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(14) ≡
a1 are as follows:
where:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(15) ≡
a0,cpa are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(16) ≡
a1,cpa are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(17) ≡
a2,cpa are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(18) ≡
a0,dav are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(19) ≡
a1,dav are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(20) ≡
a2,dav are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(21) ≡
a3,dav are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(22) ≡
a0f are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(23) ≡
a1f are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(24) ≡
a0g are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(25) ≡
a1g are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(26) ≡
a0,Nu are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(27) ≡
a1,Nu are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(28) ≡
a2,Nu are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(29) ≡
a3,Nu are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(30) ≡
Wdkx are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(31) ≡
Wdky are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(32) ≡ Δ
zdk are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(33) ≡ Δ
zfan are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(34) ≡
Dfan are as follows:
where
was defined in Equation (A71).
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(35) ≡ Δ
zfill are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(36) ≡ Δ
zrain are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(37) ≡ Δ
zbs are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(38) ≡ Δ
zde are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(39) ≡
Dh are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(40) ≡
Afill are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(41) ≡
Asurf are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(42) ≡ Pr are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(43) ≡
Vw are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(44) ≡
Vexit are as follows:
where
was defined in Equation (A86).
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(45) ≡
mw,in are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(46) ≡
Ta,in are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(47) ≡
ma are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(48) ≡
ωin are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(49) ≡ Re
d are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter α
(50) ≡ Sc are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(51) ≡
Sh are as follows:
The derivatives of the water vapor continuity equations [cf. Equations (A7)–(A9)] with respect to the parameter
α(52) ≡
Nu are as follows:
A4. Derivatives of the Air and Water Vapor Energy Balance Equations with Respect to the Parameters
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(1) ≡
Tdb are as follows:
Note: The value of the inlet air temperature is set equal to dry-bulb temperature, although these quantities are treated as two different parameters in the model. The dry-bulb temperature is used in mass diffusivity calculations. The relation between the two parameters, i.e.,
Ta,in =
Tdb, needs to be accounted for when computing the respective derivatives: the derivative of Equation (A12) with respect to the dry-bulb temperature must be the same as the derivative of Equation (A12) with respect to the inlet air temperature.
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(2) ≡
Tdp are as follows:
where
was defined in Equation (A166).
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(3) ≡
Tw,in are as follows:
where
was defined in Equation (A20), and
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(4) ≡
Patm are as follows:
where
and
were defined in Equations (A24) and (A105), respectively, while
and
were defined in Equations (A25) and (A171), respectively.
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(5) ≡
wtsa are as follows:
where
was defined in Equation (A107).
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(6) ≡
ksum are as follows:
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(7) ≡
μ are as follows:
where
was defined in Equation (A110).
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(8) ≡
υ are as follows:
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(9) ≡
kair are as follows:
where
was defined in Equation (A113).
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(10) ≡
fht are as follows:
where
was defined in Equation (A115).
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(11) ≡
fmt are as follows:
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(12) ≡
f are as follows:
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(13) ≡
a0 are as follows:
where
was defined in Equation (A182).
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(14) ≡
a1 are as follows:
where
was defined in Equation (A185).
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(15) ≡
a0,cpa are as follows:
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(16) ≡
a1,cpa are as follows:
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(17) ≡
a2,cpa are as follows:
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(18) ≡
a0,dav are as follows:
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(19) ≡
a1,dav are as follows:
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(20) ≡
a2,dav are as follows:
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(21) ≡
a3,dav are as follows:
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(22) ≡
a0f are as follows:
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(23) ≡
a1f are as follows:
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(24) ≡
a0g are as follows:
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(25) ≡
a1g are as follows:
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(26) ≡
a0,Nu are as follows:
where
and
were defined previously in Equations (A59) and (A105), respectively.
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(27) ≡
a1,Nu are as follows:
where
and
were defined previously in Equations (A61) and (A105), respectively.
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(28) ≡
a2,Nu are as follows:
where
and
were defined previously in Equations (A63) and (A105), respectively.
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(29) ≡
a3,Nu are as follows:
where
and
were defined previously in Equations (A65) and (A105), respectively.
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(30) ≡
Wdkx are as follows:
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(31) ≡
Wdky are as follows:
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(32) ≡ Δ
zdk are as follows:
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(33) ≡ Δ
zfan are as follows:
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(34) ≡
Dfan are as follows:
where
and
were defined previously in Equations (A105) and (A24), respectively, while
was defined in Equation (A71).
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(35) ≡ Δ
zfill are as follows:
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(36) ≡ Δ
zrain are as follows:
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(37) ≡ Δ
zbs are as follows:
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(38) ≡ Δ
zde are as follows:
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(39) ≡
Dh are as follows:
where
was defined in Equation (A145).
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(40) ≡
Afill are as follows:
where
was defined in Equation (A147).
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(41) ≡
Asurf are as follows:
where
was defined in Equation (A149).
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(42) ≡ Pr are as follows:
where
was defined in Equation (A151).
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(43) ≡
Vw are as follows:
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(44) ≡
Vexit are as follows:
where
and
were defined previously in Equations (A105) and (A24), respectively, while
was defined in Equation (A86).
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(45) ≡
mw,in are as follows:
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(46) ≡
Ta,in are as follows:
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(47) ≡
ma are as follows:
where
and
were defined previously in Equations (A24) and (A105), respectively.
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(48) ≡
ωin are as follows:
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(49) ≡ Re
d are as follows:
where
was defined in Equation (A105) and
was defined in Equation (A93).
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(50) ≡
Sc are as follows:
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(51) ≡
Sh are as follows:
The derivatives of the air/water vapor energy balance equations [cf. Equations (A10)–(A12)] with respect to the parameter
α(52) ≡
Nu are as follows:
where
was defined in Equation (A105).