Appendix A
Merkel and Poppe Equation Derivation Merkel equation derivation The Merkel equation can be derived as follows. Recall Eqns. A.1 and A.2, ma dω + dmw = 0,
(A.1)
ma dima − mw diw − iw dmw = 0.
(A.2)
Substitute Eqn. A.1 in Eqn. A.2 and re-arrange to get Eqn. A.3: # " ma dima − Tw dω . dTw = mw Cpw
(A.3)
By neglecting the change in water mass flow rate, dω is removed from the equation so it can be written as Eqn. A.4 with respect to vertical contact area. This yields a simplified energy balance where the change in water enthalpy is equal to the change in air enthalpy. This can be re-arranged to Eqn. A.5,
dTw ma 1 dima = , dA mw Cpw dA
(A.4)
dima mw = Cpw , dTw ma
(A.5)
180
APPENDIX A. MERKEL AND POPPE EQUATION DERIVATION 181 so
dima dTw
is a constant. This is needed to solve the final Merkel equation. Now
recall Eqn. A.6 and Eqn. A.7. ma dima = iv dmw + h(Tw − Ta )dA,
(A.6)
′′ dmw = hm [ω(T − ω] · dA. w)
(A.7)
Substitute Eqn. A.7 into Eqn. A.6 to give, # " hm h dima ′′ = (Tw − Ta ) (ω(Tw ) − ω)iv + dA ma hm
(A.8)
Now take the difference (i′′(Tw ) − ima ), where i′′(Tw ) and ima are given by, ima = Cpa Ta + ω · [if gwo + Cpv Ta ],
(A.9)
′′ i′′(Tw ) = Cpa Tw + ω(T · [if gwo + Cpv Tw ] w)
= Cpa Tw + ωiv + (ω ′′ − ω)iv ,
(A.10)
recalling that iv evaluated at the water temperature is given by, iv = [if gwo + Cpv Tw ].
(A.11)
If small differences in specific heats which are evaluated at different temperatures are ignored [1], then the result of the difference (i′′(Tw ) − ima ) can be
given as,
(Tw − Ta ) =
′′ − ω)iv ] [i′′(Tw ) − ima − (ω(T w)
(Cpa + ωCpv )
.
(A.12)
Substituting Eqn. A.12 into Eqn. A.8 and re-arranging gives the following: ## " " ′′ ′′ [i(Tw ) − ima − (ω(T − ω)i ] v hm h dima w) ′′ (A.13) = (ω(T − ω)iv + w) dA ma hm (Cpa + ωCpv ) The Lewis factor relates the heat and mass transfer coefficients and is given in Eqn. A.14, Lef =
h , hm Cpm
(A.14)
APPENDIX A. MERKEL AND POPPE EQUATION DERIVATION 182 where Cpm is the specific heat of the air water vapour mixture and is given by Eqn. A.15, Cpm = Cpa + Cpv ω.
(A.15)
Substituting for Cpm and the Lewis factor gives the following relationship: " # dima hm ′′ = Lef (i′′ma(Tw ) − ima ) + [1 − Lef ](ω(T − ω)iv ] w) dA ma
(A.16)
Now if the Lewis factor is taken such that Lef = 1 then Eqn. A.16 simplifies to,
hm ′′ dima = (i − ima ). dA ma ma(Tw )
(A.17)
The driving force force for heat and mass transfer has been reduced down to the enthalpy difference between the water surface and the air stream. The Merkel number is finally found by combining Eqn. A.17 and Eqn. A.4: Me =
hm A = mw
Z
T wi
T wo
Cpw dTw ′′ (ima(Tw ) − ima )
(A.18)
Poppe equation derivation The Poppe equations can be derived as follows. Take Eqn. A.8. In the Merkel derivation, this relationship was simplified with a substitution of (A.12). This step was also taken in the original Poppe derivation [10] but will be omitted here as the final form of the equations does not require this substitution. Now substitute Eqns. A.8 and A.7 into Eqn. A.2 and rearrange to find Eqn. A.19: " # diw hm h ′′ ′′ = · (ω(T − ω)iv · + (Tw − Ta ) − Cpw Tw (ω(T − ω) (A.19) w) w) dA mw hm The Poppe equations are in the form (dω/dTw ) and (dha /dTw ) and can be found from the above results using Eqn. A.20, dω dA dω dA dω = = Cpw , dTw dA dTw dA diw
(A.20)
APPENDIX A. MERKEL AND POPPE EQUATION DERIVATION 183 and rearranging Eqn. 2.1 and Eqn. 2.2 we get Eqn. A.21: ′′ hm [ω(T − ω] dω w) = dA ma
(A.21)
So substituting Eqn. A.21 and Eqn. A.19 into Eqn. A.20, and substituting h hm
= Lef Cpma , from the Lewis factor definition (Eqn. A.14), gives the first
of the Poppe equations, Eqn. A.22: # " ′′ Cpw (mw /ma ) · (ω(T − ω) dω w) = ′′ ′′ dTw iv · (ω(T − ω) + Lef Cpma (Tw − Ta ) − Cpw Tw (ω(T − ω) w) w) (A.22) Now find (dima /dTw ) by substitution Eqn. A.8 and Eqn. A.19 into Eqn. A.23 below:
dima dia dA dω dA = = Cpw dTw dA diw dA diw
(A.23)
The result of this substitution is Eqn. A.24, the second of the Poppe Equations. dima dTw
" mw ′′ ′′ = Cpw 1 + Cpw Tw (ω(Tw ) − ω) iv · (ω(T − ω) + w) ma # ′′ (A.24) Lef Cpma (Tw − T a) − Cpw Tw (ω(T − ω) w)
The Merkel number for the Poppe equations can be derived as follows. Combine Eqn. A.7 and Eqn. A.1 and re-arrange to get, hm dA =
ma dω . ′′ (ω(Tw ) − ω)
(A.25)
Divide through by mw and dTw /dTw and then integrating gives, Z
hm dA = mw
Z
ma dω/dTw · dTw . ′′ mw (ω(T − ω) w)
(A.26)
The Merkel number for the Poppe method, M ep , can then be given as, hm A = M ep = mw
Z
ma dω/dTw · dTw . ′′ mw (ω(T − ω) w)
(A.27)
Substituting Eqn. A.22 into Eqn. A.27 and re-arranging gives last of the
