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Calculation Procedure for a Rating Problem
DOI 10.1615/hedhme.a.000304
3.9.9 Calculation procedure for a rating problem
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 Pr2/3 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.
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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
\[ \frac{1}{UA}=\left(\frac{1}{\eta_{s}\alpha A}\right)_{\!1}+R_{w}+\left(\frac{1}{\eta_{s}\alpha A}\right)_{\!2}=\frac{1}{V}\left[\left(\frac{1}{\eta_{s}\alpha\lambda}\right)_{\!1}+\frac{a}{2k_{w}}(b_{1}+b_{2}+2_{a})+\left(\frac{1}{\eta_{s}\alpha\lambda}\right)_{\!2}\right] \]
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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