3.2 Cryogenic Convection Heat Transfer

„ Involves process of heat transfer between q solid material and adjacent cryogenic fluid f = TT mean „ Classic heat transfer problem (Newton’s law) 2 m& f q(kW/m ) = h (Ts –Tf) „ Configurations of interest

„ = TT Internal forced flow (single phase, f mean ) q „ Free convection (single phase, f = TT ∞ ) „ Internal two phase flow „ Pool boiling (two phase) „ Understanding is primarily empirical leading to correlations based on dimensionless numbers

„ Issue is relevant to the design of: Liquid T „ Heat exchangers f Ts „ Cryogenic fluid storage „ Superconducting magnets „ Low temperature instrumentation

USPAS Short Course Boston, MA 6/14 to 6/18/2010 1 Single phase internal flow heat transfer

Forced T D ,Tm f Convection s & Q Classical fluid correlations

„ The heat transfer coefficient in a classical fluid system is generally correlated in the form where the Nusselt number, hD where, NuD ≡ and D is the characteristic length k f „ For laminar flow, NuD = constant ~ 4 (depending on b.c.) „ For turbulent flow (Re > 2000) D μ C mn Pr ≡ pf D = fNu D = C D PrRePr),(Re and (Prandtl number) k f „ Dittus-Boelter Correlation for classical fluids (+/- 15%) 4 2 5 5 NuD = PrRe023.0 Note that fluid properties should be +TT fs computed at T (the “film temperature”): T ≡ f f 2 USPAS Short Course Boston, MA 6/14 to 6/18/2010 2

Johannes Correlation (1972)

„ Improved correlation specifically for helium (+/- 8.3%) − 716.0 4 2 ⎛ T ⎞ Nu = 5 PrRe0259.0 5 ⎜ s ⎟ D D ⎜ ⎟ ⎝ Tf ⎠

„ Last factor takes care of temperature dependent properties „ Note that one often does not know Tf, so iteration may be necessary.

Example

USPAS Short Course Boston, MA 6/14 to 6/18/2010 3 Application: Cryogenic heat exchangers

„ Common types of heat exchangers used

in cryogenic systems P1 , T1

P , T „ Forced flow single phase fluid-fluid 2 2

„ E.g. counterflow heat exchanger in refrigerator/liquefier

„ Forced single phase flow - boiling liquid (Tube in shell HX)

„ E.g. LN precooler in a cooling circuit 2 m

„ Static boiling liquid-liquid

„ E.g. Liquid subcooler in a magnet system

USPAS Short Course Boston, MA 6/14 to 6/18/2010 4 Simple 1-D heat exchanger

Ts D & ,Tm f

„ Differential equation describing the temperature of the fluid in the tube:

dTf & Cm ()TThP Sf =−+ 0 dx

„ For constant Ts, the solution of this equation is an exponential

Ts -Tf . ⎛ hP ⎞ Increasing m −=− TTTT fsfs 0 exp)( ⎜− x⎟ ⎝ & Cm ⎠

x

USPAS Short Course Boston, MA 6/14 to 6/18/2010 5 He (Ti = 300 K) He (Tf = 80 K) Liquid nitrogen precooler

Ts D & ,Tm f

„ Assumptions & givens

„ Ts is a constant @ 77 K (NBP of LN2) LN2 (Ts = 77 K) -6 3 „ Helium gas (Cp = 5.2 kJ/kg K; μ = 15 x 10 Pa s; ρ = 0.3 kg/m , k = 0.1 W/m K) „ Allowed pressure drop, Δp= 10 kPa Properties are average values „ Helium mass flow rate = 1 g/s between 300 K and 80 K „ Find the length and diameter of the HX (copper tubing) „ Total heat transfer: = 1 g/s x 5.2 J/g K x 220 K Q = m& C p (Tin −Tout ) = 1144 W „ Δ Log mean T: ΔT (x = 0)− ΔT (x = L) f f (300-77) –(80-77) K ΔTlm = = ⎛ΔTf ()x = 0 ⎞ ln⎜ ⎟ ln [(300-77)/(80-77)] ⎝ ΔTf ()x = L ⎠ = 51 K USPAS Short Course Boston, MA 6/14 to 6/18/2010 6 Liquid nitrogen precooler (continued)

Q πDLhUA == = 1144 W/51 K = 22.4 W/K ΔTlm

„ The heat transfer coefficient is a function of ReD and Pr = 0.67 „ Assuming the flow is turbulent and fully developed, use the Dittus Boelter correlation

hD 3.08.0 4m& NuD == D PrRe023.0 and ReD = k f Dμπ

Substituting the ReD and solving for h 8.0 0.0247 x 0.1 W/m K x (10-3 kg/s)0.8 k f ⎛ 4m& ⎞ 3.0 h = 023.0 ⎜ ⎟ Pr = D ⎝ Dμπ⎠ (15 x 10-6 Pa s)0.8 x D1.8 = 0.07/D(m)1.8

hπDL = 0.224 x (L/D0.8) = 22.4 W/K 0.8 0.2 or L/D = 100 m 1 equation for two unknowns

USPAS Short Course Boston, MA 6/14 to 6/18/2010 7 Liquid nitrogen precooler (continued)

„ Pressure drop equation provides the other equation for L & D 2 L ⎛ m& ⎞ =Δ fp ⎜ ⎟ ; A = πD2/4 and f ~ 0.02 (guess) 2ρD ⎜ A ⎟ flow ⎝ flow ⎠ 4m Re = & D Dμπ 4 x 0.001 kg/s -6 Substituting for ReD and f π x 0.0062 m x 15 x 10 Pa s 2 ~ 13,700 Lm -3 2 L p ≈Δ 016.0 & 0.016 x (10 kg/s) ⎛ ⎞ 5 = ⎜ 5 ⎟ ρ D 0.3 kg/m3 ⎝ D ⎠ ⎛ L ⎞ 2nd equation for Δp= 5.33 x 10-8 ⎜ ⎟ Substitute: ⎝ D5 ⎠ two unknowns Eq. 1: L = 100 D0.8 Δp= 5.33 x 10-6/D4.2 with Δp = 10,000 Pa

D = [5.33 x 10-6/Δp]1/4.2 = 6.2 mm and L = 100 x (0.0062 m)0.8 = 1.7 m

USPAS Short Course Boston, MA 6/14 to 6/18/2010 8 Single phase free convection heat transfer g g Q L or Q Q

D

„ Compressible fluid effect: Heat transfer warms the fluid near the heated surface, reducing density and generating convective flow. „ Free convection heat transfer is correlated in terms of the Rayleigh number, 3 n βΔTLg L = ()GrfNu ~Pr, CRaL where L GrRa Pr =≡ Dthν where g is the acceleration of gravity, β is the bulk expansivity, ν is the kinematic viscosity (μ/ρ) and D is the thermal diffusivity (k/ρC). th L (or D) are the scale length of the problem (in the direction of g)

USPAS Short Course Boston, MA 6/14 to 6/18/2010 9 Free convection correlations

Free convection correlation for low „ For very low Rayleigh temperature helium number, Nu = 1 corresponding to pure conduction heat transfer „ For Ra < 109, the boundary layer flow is laminar (conv. fluids) 25.0 NuL ≈ 59.0 RaL

„ For Ra > 109, the boundary layer is turbulent 33.0 L ≈ 1.0 RaNu L

Example

USPAS Short Course Boston, MA 6/14 to 6/18/2010 10 Pool Boiling Heat Transfer (e.g. Helium)

Factors affecting heat transfer curve „ Surface condition (roughness, insulators, oxidation) „ Orientation „ Channels (circulation) „ Time to develop steady state (transient heating) Liquid Heated Note: Other cryogenic fluids have basically Surface the same behavior, although the numerical Q values of q and ΔT are different. USPAS Short Course Boston, MA 6/14 to 6/18/2010 11 Nucleate Boiling Heat Transfer

„ Nucleate boiling is the principal heat transfer mechanism for static liquids below the peak heat flux (q* ~ 10 kW/m2 for helium) „ Requirements for nucleate boiling „ Must have a thermal boundary layer of superheated liquid near the surface ΔTk δ = f ~ 1 to 10 μm for helium th q

„ Must have surface imperfections that act as nucleation sites for formation of vapor bubbles.

USPAS Short Course Boston, MA 6/14 to 6/18/2010 12 Critical Radius of Vapor Bubble

r 2σ Pv PL PP Lv += σ is the surface tension r

„ Critical radius: For a given T, p, the bubble radius that determines whether the bubble grows of collapses

„ r > rc and the bubble will grow

„ r < rc and the bubble will collapse „ Estimate the critical radius of a bubble using thermodynamics „ Clausius Clapeyron relation defines the slope of the vapor pressure line in terms of fundamental properties

dp ⎞ Δs hfg fg ph ⎟ = = ≈ 2 If the gas can be approximated as ideal dT ⎠sat Δv ()− vvT Lv RT and vv >> vL

USPAS Short Course Boston, MA 6/14 to 6/18/2010 13 Critical radius calculation

„ Integrating the Clausius Clapeyron relation between

ps and ps + 2σ/r

2 2 −1 2σ fg Δ RTTh 2σRTs rc = (e 1) ≈− ps sfg ΔTph

„ Example: helium at 4.2 K (NBP) „ Empirical evidence indicates that ΔT ~ 0.3 K

„ This corresponds to rc ~ 17 nm „ Number of helium molecules in bubble ~ 10,000 „ Bubble has sufficient number of molecules to be treated as a thermodynamic system „ Actual nucleate boiling heat transfer involves heterogeneous nucleation of bubbles on a surface. This is more efficient than homogeneous nucleation and occurs for smaller ΔT.

USPAS Short Course Boston, MA 6/14 to 6/18/2010 14 Nucleate Boiling Heat Transfer He I

2.5 Note that hnb is not constant because Q ~ ΔT

USPAS Short Course Boston, MA 6/14 to 6/18/2010 15 Nucleate Boiling Heat Transfer Correlations

„ The mechanism for bubble formation and detachment is very complex and difficult to model „ Engineering correlations are used for analysis

„ Kutateladse correlation

6.0 125.0 2/1 ⎡ 2/1 ⎤ ⎡ 2 2/3 ⎤ 7.0 h ⎛ σ ⎞ qC ρ lpl ⎛ σ ⎞ ⎛ ρ ⎞ ⎛ σ ⎞ ⎛ p ⎞ ⎜ ⎟ ×= 1025.3 −4 ⎢ ⎜ ⎟ ⎥ ⎢ ⎜ l ⎟ ⎜ ⎟ ⎥ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ × g 2/1 gk ρ ⎢ ρ gkh ρ ⎥ ⎜ μ ⎟ ⎜ gρ ⎟ ⎜ ()gρσ ⎟ ⎝ ll ⎠ ⎣ ⎝ llvfg ⎠ ⎦ ⎣⎢ ⎝ l ⎠ ⎝ l ⎠ ⎦⎥ ⎝ l ⎠

„ Rearranging into a somewhat simpler form,

2 3125.0 5.1 ⎡ ⎛ ρ ⎞ ⎤ pχ 75.1 ⎛ ρ ⎞ 5.1 −9 l 3 ⎛ ⎞ l ⎛ C p ⎞ ⎛ k ⎞ 5.2 q ×= 1090.1 ⎢g⎜ ⎟ χ ⎥ ⎜ ⎟ ⎜ ⎟ ×⎜ ⎟ ⎜ l ⎟()−TT ⎢ μ ⎥ ⎝ σ ⎠ ρ ⎜ ⎟ ⎜ ⎟ bs ⎣ ⎝ l ⎠ ⎦ ⎝ v ⎠ ⎝ h fg ⎠ ⎝ χ ⎠ 2/1 ⎛ σ ⎞ 2 5.2 where χ = ⎜ ⎟ cmWq 8.5)/( Δ= T For helium at 4.2 K ⎝ gρl ⎠

USPAS Short Course Boston, MA 6/14 to 6/18/2010 16 Peak Heat Flux (theory)

„ Understanding the peak nucleate boiling heat flux is based on empirical arguments due to instability in the vapor/liquid flow

Instability in the vapor-liquid boundary v vv v v vL vL v vL v

c L vv λ

„ Instability due to balance between surface energy and kinetic energy

2 2πσ ρ ρvL 2 c = − 2 ()− vv Lv ()+ ρρλvL ()+ ρρvL

Transition to unstable condition when c2 = 0

USPAS Short Course Boston, MA 6/14 to 6/18/2010 17 Peak Heat Flux Correlations

„ Zuber correlation:

1 1 4 2 * ⎡ ()− ρρσvl g ⎤ ⎡ ρl ⎤ ≈ Khq ρvfg ⎢ 2 ⎥ ⎢ ⎥ ⎣ ρv ⎦ ⎣ + ρρvl ⎦

„ Empirical based on Zuber Correlation

1 1 * 2 4 ≈ 16.0 vfg []ghq ()− ρρσρvl

~ 8.5 kW/m2 for He I at 4.2 K

„ Limits:

„ T Tc; q* 0 since hfg 0 and σ 0 1/2 „ T 0; q* ~ ρv (decreases)

„ q*max near 3.6 K for LHe

USPAS Short Course Boston, MA 6/14 to 6/18/2010 18 Film Boiling

„ Film boiling is the stable condition when the surface is blanketed by a layer of vapor „ Film boiling heat transfer coefficient is generally much less that that in nucleate boiling

„ Minimum film boiling heat flux, qmfb is related to the stability of the less dense vapor film under the more dense liquid

liquid

vapor ~ 10 μm

Q

“Taylor Instability” governs the collapse of the vapor layer

USPAS Short Course Boston, MA 6/14 to 6/18/2010 19 Film Boiling Heat Transfer Correlations

„ Factors affecting the process

„ Fluid properties: Cp, hfg, σ, ρl, ρv „ Fluid state: saturated or pressurized (subcooled) „ Heater geometry (flat plate, cylinder, etc.) „ Breen-Westwater correlation

8/1 4/1 ⎛ σ ⎞ ⎛ μ ()−TT ⎞ 2/1 h ⎜ ⎟ ⎜ bsv ⎟ ⎛ σ ⎞ fb 3 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ += 28.037.0 2 ⎝ g()− ρρvl ⎠ ⎝ k ()− vlvv gλρρρ' ⎠ ⎜ ⎟ ⎝ gD ()− ρρvl ⎠

Where, [ + 34.0 ( −TTCh )]2 λ'= fg bspv hfg „ Simplified form for large diameter

4/1 8/1 3 ⎛ g()− ρρvl ⎞ ⎛ k ()− vlvv gλρρρ' ⎞ 3 h = 37.0 ⎜ ⎟ ⎜ ⎟ 4 fb ⎜ ⎟ ( −≈ TTq fs ) ⎝ σ ⎠ ⎝ μ ()−TT bsv ⎠

USPAS Short Course Boston, MA 6/14 to 6/18/2010 20 Minimum film boiling heat flux

2 LHe @ 4.2 K „ Minimum film boiling heat Q/S, W/cm flux is less than the peak heat flux ≈ 0.9 „ Recovery to nucleate Film boiling

boiling state is associated Nucleate boiling with Taylor Instability. Transition boiling ≈ 0.3 1 4 ⎡ ⎤ qmfb g ()− ρρσvl mfb = 16.0 hq ρvfg ⎢ 2 ⎥ ⎣ ()+ ρρvl ⎦ „ Dimensionless ratio: ≈ 1 ≈ 4 1 (Ts-Tb), K 2 qmfb ⎡ ρv ⎤ ~ 0.36 for LHe @ 4.2 K = * ⎢ ⎥ ~ 0.1 @ 2.2 K q ⎣ + ρρvl ⎦ ~ 1 @ Tc = 5.2 K 2 ~ 0.09 for LN2 @ 80 K where q* ~ 200 kW/m

USPAS Short Course Boston, MA 6/14 to 6/18/2010 21 Prediction of Nucleate/Film Boiling for Helium

USPAS Short Course Boston, MA 6/14 to 6/18/2010 22 Experimental Heat Transfer (Helium)

USPAS Short Course Boston, MA 6/14 to 6/18/2010 23 Prediction of Nucleate/Film Boiling for Nitrogen

USPAS Short Course Boston, MA 6/14 to 6/18/2010 24 Experimental Heat Transfer (Nitrogen)

USPAS Short Course Boston, MA 6/14 to 6/18/2010 25 Internal two phase flow

„ Heat transfer depends on

various factors & 2 φ , Tm f Ts D „ Mass flow rate Q „ Orientation w/r/t gravity & 2 φ , Tm f „ Flow regime „ Quality (χ) g „ Void fraction (α) „ Total heat transfer rate D = + QQQ bfcT Ts where: Q is convective and Q is fc b Q gravity enhanced boiling.

Depending on factors above, either contribution may dominate

USPAS Short Course Boston, MA 6/14 to 6/18/2010 26 Horizontal flow two phase heat transfer

& 2 φ , Tm f Ts D Q „ Consider the case where gravitational effects are negligible „ Horizontal flow at moderate Re so that inertial forces dominate „ Correlation based on enhanced Nusselt number 9.0 1.0 5.0 Nu2φ ⎛1− x ⎞ ⎛ μ ⎞ ⎛ ρ ⎞ = f ()χ where χ = ⎜ ⎟ ⎜ L ⎟ ⎜ v ⎟ tt tt x ⎜ μ ⎟ ⎜ ρ ⎟ NuL ⎝ ⎠ ⎝ v ⎠ ⎝ L ⎠ (1− xm ) 4.08.0 Re = & and NuL = PrRe023.0 LL ; L μ AflowL Typical correlation (de la Harpe): Nu 2φ ≈ Aχ −m tt 1 for χtt large NuL with m ~ 0.385 and A ~ 5.4

USPAS Short Course Boston, MA 6/14 to 6/18/2010 27 Vertical channel heat transfer

& 2 φ , Tm f „ Main difference between this problem and pool boiling is that the fluid is confined within channel g D „ At low mass flow rate and self driven flows (natural Ts circulation) the heat Q transfer is governed by buoyancy effects „ Process is correlated against classical boiling heat transfer models „ In the limit of large D the correlation is similar to pool boiling heat transfer (vertical surface)

USPAS Short Course Boston, MA 6/14 to 6/18/2010 28 Vertical channel maximum heat flux

Critical flow in an evaporator 1 Empirical observations 2 * w h ρvfg ⎡ gχ ⎛ []()βχ−+ 11ln ⎞⎤ ρ q = ⎢ c ⎜1− c ⎟⎥ β = l „ q* ~ w for small w β −12 ⎜ ()βχ−1 ⎟ z ⎣ ⎝ c ⎠⎦ ρv „ q* ~ z-1/2 for w/z < 0.1 „ q* ~ constant for w/z m& v ⎞ and χc = ⎟ “Critical quality” >> 1 m& ⎠ max ~ 0.3 for helium

USPAS Short Course Boston, MA 6/14 to 6/18/2010 29 Example: Cryogenic Stability of Composite Superconductors (LTS in LHe @ 4.2 K)

„ Used in large magnets where flux jumping and other small disturbances are possible and must be arrested „ General idea: in steady state ensure that cooling rate exceeds heat generation rate (Q > G) „ Achieved by manufacturing conductor with large copper (or aluminum) fraction and cooling surface

„ Lower overall current density G/S I2R/S „ Potentially high AC loss (eddy currents)

Current Fully Sharing normal Composite Conductor

T (K) T (K) T (K) T (K) Copper/Aluminum stabilizer b cs c Insulating spacer (G-10)

NbTi/Nb3Sn Icu Isc

V=IcuR cu

USPAS Short Course Boston, MA 6/14 to 6/18/2010 30 Cryogenic Stability (LHe @ 4.2 K)

Case 1: Unconditional stability, recovery to fully superconducting state occurs uniformly over length of normal zone Case 2: Cold End recovery (Equal area criterion): Excess cooling capacity (area A) > Excess heat generation (area B)

Q/S is the LHe boiling heat Q/S transfer curve for bath cooling or A normalized per surface area G/S 2 B 1 G/S is the two part heat generation Curve for a composite superconductor Tcs is the temperature at which Top = Tc

Tb (K) Tcs (K) Tc (K) Tx (K)

USPAS Short Course Boston, MA 6/14 to 6/18/2010 31 Transient Heat Transfer

„ Heat transfer processes that occur on time scale short compared to boundary layer thermal diffusion. Why is this important in cryogenics? -4 2 2 (Dth (copper) ~ 10 m /s @ 300 K; ~ 1 m /s @ 4 K) „ Normal liquid helium has a low thermal conductivity and large heat capacity

k f D = ~ 3 x 10-8 m2/s (LHe @ 4.2 K) f ρC q T(x) π 1 2 -4 1/2 δth = ( f tD ) ~ 3 x 10 t [m] for LHe @ 4.2 K 2 ~ 1.5 t1/2 [m] for copper @ 4.2 K δth hL „ Lumped capacitance condition: Bi <<= 1 ~ 10L [m] k

„ Note that this subject is particularly relevant to cooling superconducting magnets, with associated transient thermal processes „ Important parameters to determine * „ ΔE = q Δt (critical energy) „ T – surface temperature during heat transfer USPAS Shorts Course Boston, MA 6/14 to 6/18/2010 32 Transient Heat Transfer to LHe @ 4.2 K

Time evolution of the temperature difference following a step heat input: Steady state is reached after ~ 0.1 s

USPAS Short Course Boston, MA 6/14 to 6/18/2010 33 Critical Energy for Transition to Film Boiling

Hypothesis: The “critical energy” is determined by the amount of energy that must be applied to vaporize a layer of liquid adjacent

to the heated surface. „ Energy required Δ = ρ hE δ thfgl q* LHe „ Layer thickness determined by diffusion π 1 2 δth = ( f tD ) 2 δth „ Critical flux based on heat diffusion:

1 π ⎛ k ⎞ 2 * = ρ hq ⎜ l ⎟ fgl ⎜ * ⎟ 2 ⎝ ρl ΔtC ⎠ ~ 0.09 Δt-1/2 [W/cm2] @ 4.2 K

USPAS Short Course Boston, MA 6/14 to 6/18/2010 34 Surface Temperature (Transient HT)

„ During transient heat transfer, the surface temperature will be higher than the surrounding fluid due to two contributions: „ Fluid layer diffusion: transient conduction in the fluid layer will result in a finite temperature difference „ Kapitza conductance: At low temperatures, there can be a significant temperature difference, ΔTk, due to thermal impedance mismatch (more on this subject later). This process is dominant at very low temperatures, but is small above ~ 4 K, so is normally only important for helium systems. „ Fluid layer diffusion equation: 2 Δ∂ Tf 1 Δ∂ Tf k 2 = ; D f = ∂ Dx f ∂t ρC p

„ Boundary conditions:

„ ΔTf (x,0) = 0; initial condition Solution is a standard second order differential equation with two „ ΔTf (infinity, t) = 0; isothermal bath spatial and one time boundary „ q = -k dΔT /dt) ; heat flux condition f f x=0 condition. „ Solid is isothermal (Bi = hL/k << 1)

USPAS Short Course Boston, MA 6/14 to 6/18/2010 35 Transient diffusion solution

ΔTf „ Integrating the diffusion equation:

21 q ⎡ ⎛ tD ⎞ ⎛ x2 ⎞ ⎛ x ⎞⎤ q T =Δ ⎢2⎜ f ⎟ exp⎜− ⎟ − xerfc⎜ ⎟⎥ f k ⎜ π ⎟ ⎜ 4 tD ⎟ ⎜ 2()tD 21 ⎟ f ⎣⎢ ⎝ ⎠ ⎝ f ⎠ ⎝ f ⎠⎦⎥

„ Evaluating at x = 0 (surface of heater) x 21 2 ⎛ tq ⎞ ()xT 0 ==Δ ⎜ ⎟ f ⎜ ⎟ π ⎝ ρ kC fp ⎠

„ The transient heat transfer coefficient can then be defined 21 q π ⎛ ρ kC fp ⎞ h = = ⎜ ⎟ ΔTf 2 ⎝ t ⎠ 1.0 h ≈ ; kW/m2K for helium near 4 K t

2 2 At t = 10 μs; h ~ 30 kW/m K and for q = 10 kW/m ; ΔTf ~ 0.3 K Note: this value of h >> h (nucleate boiling HT coefficient) USPAS Short Course Bostonnb, MA 6/14 to 6/18/2010 36 Summary Cryogenic Heat Transfer

„ Single phase heat transfer correlations for classical fluids are generally suitable for cryogenic fluids „ Free convection „ Forced convection „ Two phase heat transfer also based on classical correlations „ Nucleate boiling „ Peak heat flux „ Film boiling „ Transient heat transfer is governed by diffusive process for ΔT and onset of film boiling

USPAS Short Course Boston, MA 6/14 to 6/18/2010 37