This section outlines a step-by-step procedure for the thermal calculations involved in a rating problem. Steps 4–9 are required for each fluid stream.
- Surface geometry parameters: Given β, Af /A, and Dh, calculate γ and σ for each stream
- Using the given heat exchanger size, calculate the frontal mass velocity (Gfr) for each stream, then calculate Gc (= Gfr /σ).
- Estimate the heat exchanger thermal effectiveness (ε) to allow calculation of the average fluid temperature for each stream. This may be based on an outright guess, or a preliminary calculation following steps 4–12.
- Evaluate fluid properties (pm, ηm, cpm) at the estimated average fluid temperature.
- Calculate Re = Dh Gc /ηm.
- Determine f and j (or Nusselt number Nu) from f and j versus Re plots for the surface, or from tabled laminar flow solutions, where appropriate.
- Calculate the heat transfer coefficient, α = j Gc cpm Pr ^(2/3 ) or α = Nu k /Dh.
- Calculate fin efficiency. With m l = (2α /k δ)^(1/2 ) l calculated, the fin efficiency (ηf) is determined from the appropriate fin efficiency chart or equation.
- Calculate surface efficiency, η0 = 1 – (1 – ηf ) Af /A.
- Calculate heat transfer surface areas and determine U A. A simpler approach is to use the given heat exchanger volume (V) and calculate U A from
where subscripts 1 and 2 refer to stream 1 and 2, respectively, and V = heat exchanger volume, γ1 = A1 /V, ηs = fin surface efficiency, λ = tube thermal conductivity, A1 = Surface area on side 1, and A2 denotes surface are on side 2, and Rw = tube wall thermal resistance - Calculate Cmin /Cmax and NTU = U A /Cmin.
- Using the parameters in step 11, determine ε from the ε-NTU-Cmin /Cmax chart (or equation) for the given heat exchanger flow arrangement. The ε-NTU chart or equation can be found in most heat transfer textbooks.
- Compare the calculated ε with estimated ε. Repeat steps 4–12 as necessary to obtain desired convergence of ε.
Nomenclature
Ac exchanger minimum free flow area
Af exchanger total fin area on one side
Afr exchanger total frontal area
b plate spacing (or rectangular fin height)
C flow stream capacity rate (ṁ cp); Cc (cold fluid), Ch (hot fluid), Cmin (minimum), Cmax (maximum)
cp specific heat at constant pressure
Dh hydraulic diameter of any internal passage (Dh = 4rh = 4Ac L /A)
f mean friction factor, defined on the basis of mean surface shear stress
Gc flow stream mass velocity based on minimum flow area, ṁ cp /Ac
Gfr flow stream mass velocity based on flow frontal area, ṁ cp /Afr
L total heat exchanger flow length; also, flow length of uninterrupted fin
m fin effectiveness parameter (2α /k δ)^(1/2)
ṁ mass flow rate
NTU number of heat transfer units of an exchanger (= U A /Cmin)
Nu Nusselt number (α Dh /k)
R flow stream capacity rate ratio (= Cmin /Cmax)
rh hydraulic radius (Ac L /A)
U unit overall thermal conductance
V volume
α convection heat transfer coefficient
β ratio of total heat transfer area on one side of a plate-fin heat exchanger to the volume between the plates on that side
γ ratio of total transfer area on one side of the exchanger to total volume for the exchanger
δ fin thickness
ε heat exchanger thermal effectiveness, dimensionless
η dynamic viscosity
ηf fin efficiency, dimensionless
ηs fin surface efficiency, dimensionless
λ thermal conductivity
σ ratio of free-flow area to frontal area, Ac /Afr, dimensionless
Subscripts
m mean conditions, defined as used
max maximum
min minimum
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