Heat processes. Heat exchange

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Rodney Kelley Bell
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1 Heat processes Heat exchange Heat energy transported across a surface from higher temperature side to lower temperature side; it is a macroscopic measure of transported energies of molecular motions Temperature a macroscopic measure of kinetic energies of molecules Mechanisms of heat transport i) conduction transport mediated by individual molecules ii) convection in fluids the transport of heat may occur so that large assemblies of molecules transport their energy as a whole a) free convection due to density difference b) forced convection due to pumping iii) radiation mediated by electromagnetic waves Next we express heat flow in two basic situations: - heat flow in medium at rest - heat flow in a flowing medium near interface - heat flow radiated from hot body - conduction of heat applies here Fourier law (empirical observation) dq = λ dt dq dt da or q = = λ dx da dx Heat flow in non-moving environment λ thermal conductivity dt dx temperature gradient (= driving force) q heat flux λ 0 2 thermal insulators λ 0 2 thermal conductors (e.g. metals) Conduction in a plate: Q = λ dt dx A at steady state Q = const dt dx = t x = t t 0 = t t 0 x x 0 δ

2 thus Q = λ t t 0 A δ or Q = driving force t 0 t A δ λ resistance multiple layers: Q = t t n n A i= R i where R i = δ i λ i resistance in the i-th layer Conduction in a cylinder at steady state Q = const Q = λ dt dt A = λ dr dr 2πrL L length of the tube by integration r t dr = 2πLλ dt r 0 r Q t 0 ln r = λ 2πL (t r t 0 ); ( r = d ) 0 Q r 0 d 0 or Q = 2πλL (t 0 t ) ln d = π d 0 driving force (t 0 t ) d 0 resistance 2λ ln d L

3 multiple layers Q = π (t 0 t ) ln d L i 2λ i d i Heat transfer from a flowing fluid to an interface with a solid wall Heat transfer Newton law of cooling (empirical) dq = α(t t w )da or q = dq da = α(t t w) α [ W m 2 K ] (t t w ) t t w heat transfer coefficient driving force average temperature in the fluid temperature at the wall Theoretically: q = δ λ l (t t w ) but the thickness of the laminar sublayer λ l is difficult to evaluate Instead: α is calculated by using dimensionless numbers and dimensionless relation (=correlation) obtained from experiments. Remark: dq = (t t w ) α α Dimensionless numbers used is resistance against heat transfer Nu = αl λ Pe = vl a Re = vl = vlρ ν η Ar = gl3 ρ ν 2 ρ Gr = gl3 ν 2 β t Pe = ν a Nusselt number (l is characteristic length) Péclet number (a = Reynolds number Archimedes number Grasshof number λ ρc p thermal diffusivity) Prandtl number (material properties)

4 α is calculated from Nu which in turn is calculated from a dimensionless correlation Nu = Nu(Re, Pe, Ar, Gr, Pr, ) Examples of correlations: a) free convection (e.g. in pipes with large diameter) Nu = C(GrPr) n ; C, n are empirical values b) forced convection Nu = 0.023Re 0.8 Pr 0.4 ; Re > 0000 c) heat transfer in boiling liquids or condensing vapors correlations are complex but qualitative explanation is as follows (i) boiling bubble boiling = bubbles arise at the wall and rise film boiling = bubbles at the wall merge into a film of vapor which has a low conductivity, therefore α drops (ii) condensation Presence of non-condensing inert in the vapor (air) makes α smaller because it prevents vapor from flowing toward wall

5 Table comparing typical values of α transfer to/from gas ~ 0 W m -2 K - transfer to/from liquid ~ 0 2 W m -2 K - bubble boiling ~ 0 3 W m -2 K - condensation ~ 0 4 W m -2 K - α Heat transfer in a cylinder (=pipe) dq = α(t w t)da = α(t w t)πddl = π (t w t) ; αd resistance αd Overall heat transfer This is a combination of heat transfer from fluid A to a wall, heat flow through the wall and heat transfer from the wall to fluid B. By combining heat flow expressions and summing up all resistances we get: dq = (t A t B ) α + n δ i i= + da = K(t A t B )da; K overall heat transfer coefficient (planar geom.) A α B λ i [K] = W m 2 K For pipes (cylinders)

6 π(t A t B ) dq = + ln d i + dl = K L (t A t B )da; K L overall heat transfer coef. (pipes) α A d A 2λ i d i α B d [K L ] = W m K Used for: - heating - cooling - boiling - condensation Notation: Heat exchangers A hotter fluid B cooler fluid c pa = c pa c pb = c pb t Bi, t Ai t Be, t Ae average thermal capacity of A average thermal capacity of B temperature at inflow temperature at outflow Most frequent arrangement is counter-current as shown in the figure, also possible is co-current or cross-flow. Geometry: - planar: plate heat exchangers - cylindric: o tube-within-a-tube o tubular exchangers with a bundle of tubes in a shell Temperature profiles

7 - counter current heating of B + cooling of A - condensation of A + heating of B - condensation + boiling - co-current heating of B+ cooling of A t driving force normally varies along the exchanger, constant for case C Enthalpy balance of counter-current heat exchanger H Ai = H Ae + Q H Bi + Q = H Be H Ai + H Bi = H Ae + H Be H Ai = m Ac pa (t Ai t ref ); H Ae = m Ac pa (t Ae t ref ) Q = m Ac pa (t Ai t Ae ) = m Bc pb (t Be t Bi ) c pa evaluated at t A = t Ai + t Ae 2 ; c pb evaluated at t B = t Bi + t Be 2

8 Once Q is calculated from the enthalpy balance, the rate equation for heat transfer can be used to calculate the area A or the length L of exchanger. However, because the driving force t varies along the exchanger, a differential enthalpy balance must be used in the first place + combined with the heat transfer equation: or dq = K (t A t B ) da = m Ac pa dt A = m Bc pb dt B t K overall heat transfer coefficient for a plate exchanger dq = K L (t A t B )dl = m Ac pa dt A = m Bc pb dt B K L overall heat transfer coefficient for a tubular exchanger By integration we obtain three useful formulae: (i) the first one: Q = K t l.m. A or Q = K L t l.m. L where the logarithmic mean temperature t l.m. is t l.m. = t t 2 (t Ai t Be ) (t Ae t Bi ) ln t = ln t Ai t Be t 2 t Ae t Bi The differences t and t 2 are driving forces at both ends of the exchanger. They are called approaches. (ii) another useful formula is A = m Ac pa t Ai t Ae m Bc pb t Be t Bi = K t l.m. K t l.m. A A N A A B N B for plate heat exchangers A A, A B are areas of transfer units N A, N B are numbers of transfer units L = m Ac pa t Ai t Ae = K L t l.m. N A L A m Bc pb K L L B t Be t Bi t l.m. N B for tubular heat exchangers L A, L B are lengths of transfer units (iii) final formula is

9 ζ AB η A η A = exp(n A ( ζ AB )) where ζ AB = m Ac pa m Bc pb thermal capacity ratio η A = t Ai t Ae temperature difference in phase A = effectiveness t Ai t Bi maximal temperature difference This formula is useful, when the area A or length L are known but both output temperatures t Ae, t Be are unknown. Then η A can be evaluated from the formula iii) and from the η A then t Ae. The second unknown t Be is then calculated from the enthalpy balance Remark: m Ac pa (t Ai t Ae ) = m Bc pb (t Be t Bi ) If condensation or boiling occurs, formulae i) and ii) still apply, but the enthalpy balance is for example for condensation of A: Q = m Ah cond = m Bc pb (t Be t Bi ) Evaporation - removal of a liquid solvent from solution of a nonvolatile component by boiling and removal of the vapor - typically boiling is caused by heating similar to a heat exchanger For more effective operation, vapor could be compressed and used for heating of the same unit or subsequent units. For example: co-current two-stage evaporator

10 Description of a single unit Mass balance of the evaporator - total: m R0 = m R + m B - non-volatile component: w 0 m R0 = w m R Enthalpy balance of the evaporator Enthalpy balance of the reboiler Q + m R0h R0 = m Rh R + m Bh B Q = m P(h P h ) = m P h cond ; h cond condensation enthalpy per unit mass Enthalpy balance of the condenser m P(h P h B ) = m C(h C h C ) = Q condenser The enthalpy h R is found at temperature t = t s 0 + t e t s 0 t e boiling point of the solvent temperature elevation due to dissolved component

11 Area of the heat exchange in the reboiler is calculated from the heat transfer equation: Q = K(t P t )A t P t K Q temperature of the steam in the reboiler temperature in the evaporator overall heat transfer coefficient heat flow is calculated from the enthalpy balance

Mechanical Engineering. Postal Correspondence Course HEAT TRANSFER. GATE, IES & PSUs

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